Chemical modelling: applications and theory

Specialist Periodical Reports Chemical Modelling: Applications and Theory comprises critical literature reviews of all ...

Author: Alan Hinchliffe

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Specialist Periodical Reports

Chemical Modelling: Applications and Theory comprises critical literature reviews of all aspects of molecular modelling. Molecular modelling in this context refers to modelling the structure, properties and reactions of atoms, molecules and materials. Each chapter provides a selective review of recent literature, incorporating sufficient historical perspective for the non-specialist to gain an understanding. With chemical modelling covering such a wide range of subjects, this Specialist Periodical Report serves as the first port of call to any chemist, biochemist, materials scientist or molecular physicist needing to acquaint themselves with major developments in the area.

Editor A Hinchliffe

Chemical Modelling: Applications and Theory Volume 5

ISBN 978-0-85404-248-7

www.rsc.org/spr

Hinchliffe

9 780854 042487

Chemical Modelling: Applications and Theory Volume 5

Specialist Periodical Reports Specialist Periodical Reports provide systematic and detailed review coverage in major areas of chemical research. Compiled by teams of leading experts in their specialist fields, this series is designed to help the chemistry community keep current with the latest developments in their field. Each volume in the series is published either annually or biennially and is a superb reference point for researchers.

Specialist Periodical Reports

Chemical Modelling Applications and Theory

Volume 5

A Specialist Periodical Report

Chemical Modelling Applications and Theory Volume 5 A Review of the Literature Published between June 2005 and May 2007 Editor A. Hinchliffe, School of Chemistry, The University of Manchester, Manchester, UK Authors D. J. Evans, Australian National University, Canberra, Australia P. B. Karadakov, University of York, York, UK J. R. Kitchin, Carnegie Mellon University and National Energy Technology Laboratory, Pittsburgh, PA, USA R. A. Lewis, Novartis Institutes for Biomedical Research, Basel, Switzerland S. D. Miller, Carnegie Mellon University, Pittsburgh, PA, USA E. A. Moore, The Open University, Milton Keynes, UK A. J. Mulholland, University of Bristol, Bristol, UK A. H. Pakiari, Shiraz University, Shiraz, Iran D. Pugh, University of Strathclyde, Glasgow, UK D. J. Searles, Griffith University, Brisbane, Australia D. S. Sholl, Carnegie Mellon University and National Energy Technology Laboratory, Pittsburgh, PA, USA T. E. Simos, University of Peloponnese, Tripolis, Greece M. Springborg, University of Saarland, Saarbru¨cken, Germany S. Wilson, University of Oxford, Oxford, UK and Comenius University, Bratislava, Slovakia C. J. Woods, University of Bristol, Bristol, UK

If you buy this title on standing order, you will be given FREE access to the chapters online. Please contact [email protected] with proof of purchase to arrange access to be set up. Thankyou.

ISBN 978-0-85404-248-7 ISSN 1472-0965 A catalogue record for this book is available from the British Library r The Royal Society of Chemistry 2008 All rights reserved Apart from any fair dealing for the purpose of research or private study for non-commercial purposes, or criticism or review as permitted under the terms of the UK Copyright, Designs and Patents Act, 1988 and the Copyright and Related Rights Regulations 2003, this publication may not be reproduced, stored or transmitted, in any form or by any means, without the prior permission in writing of The Royal Society of Chemistry, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to The Royal Society of Chemistry at the address printed on this page. Published by The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 0WF, UK Registered Charity Number 207890 For further information see our web site at www.rsc.org Typeset by Macmillan India Ltd, Bangalore, India Printed by Henry Ling Ltd, Dorchester, Dorset, UK

Preface DOI: 10.1039/b801788n Welcome to Volume 5 of the ‘Chemical Modelling’ SPR. Naturally, I want to start by thanking my team of Reporters for the hard work they have put into making this the best and most comprehensive volume so far. We hope you will derive benefit and perhaps even pleasure from our efforts. It seems a long time since I wrote the following in my Preface to Volume 1 (1999) . . . ‘Starting a new SPR is never easy, and there was the problem of where the contributors should start their accounts; since time began? five years ago? An SPR should be the first port of call for an up-to-the-minute account of trends in a specialist subject rather than a dull collection of references. My solution was to ask contributors to include enough historical perspective to bring a non-specialist up to speed, but to include all pertinent references through May 1999. Volume 2 will cover the literature from June 1999 to May 2001 and so on. In subsequent Volumes, I shall ask those Contributors dealing with the topics from Volume 1 to start from there. New topics will be given the same generous historical perspective opportunity as Volume 1 but will have to cover the literature to 2001 + n where n = 0, 2, 4, . . . . This process will continue until equilibrium is reached.’ Equilibrium was reached a couple of Volumes ago and some mature topics don’t need cover every Volume. My final sentence for Volume 1 was ‘I am always willing to listen to convincing ideas for new topics’ as indeed I am. Alan Hinchliffe Manchester 2008 alan.hinchliff[email protected]

RSC Specialist Periodical Report Chemical Modelling: Applications and Theory (Volume 5) Chapter 1 2 3 4 5

Corresponding author Dr Adrian Mulholland Dr Richard Lewis Prof Michael Springborg Dr Elaine A Moore Prof David Sholl

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Prof Debra Searles

7

Dr Steven Wilson

8

Dr David Pugh

9

Prof Ali Pakiari

10 11

Dr Peter Karadakov Prof Theodore Simos

Topic Multiscale modelling of biological systems Computer-aided drug design 2005–2007 Solvation effects The solid state DFT studies of alloys in heterogeneous catalysis Fluctional relations, free energy calculations and irreversibility Many body perturbation theory and its application to the molecular structure problem Experiment and theory in the determination of molecular hyperpolarizabilities in solution The Floating Spherical Gaussian Orbital (FSGO) method Advances in valence bond theory Numerical methods in chemistry

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The Royal Society of Chemistry 2008

CONTENTS Cover The icosahedral ’golden fullerene’ WAu12 reproduced by permission of Pekka Pyykko¨, Chemistry Department, University of Helsinki, Finland.

Preface Alan Hinchliffe

7

Multiscale modelling of biological systems

13

Christopher J. Woods and Adrian J. Mulholland Introduction Interfacing QM with MM models Interfacing atomistic with coarse grain models Interfacing particle with continuum models Beyond continuum models Conclusion

13 15 26 41 45 46

Computer-aided drug design 2005–2007 Richard A. Lewis Introduction QSAR and ADMET Structure-based drug design Virtual screening De novo structure generation Fragment-based screening

51 51 51 54 56 60 61


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Target fishing Library design Conclusions

62 63 64

Solvation effects

67

Michael Springborg Introduction Fundamental methods Recent studies Conclusions

67 68 75 114

The solid state E. A. Moore Introduction Interatomic potential methods Ab initio methods QM/MM Molecular dynamics and related methods Properties Applications Minerals Conclusions

119

Density functional theory studies of alloys in heterogeneous catalysis

150

John R. Kitchin, Spencer D. Miller and David S. Sholl Introduction Segregation Adsorption properties on alloy surfaces Reaction kinetics Miscellaneous Electrocatalysis Conclusions

150 153 158 166 170 172 176

Fluctuation relations, free energy calculations and irreversibility

182

Debra J. Searles and Denis J. Evans Introduction Fluctuation relations Free energy relations

182 184 191

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119 121 124 130 131 133 134 141 142

Fluctuation relations and irreversibility Comment on the interpretation of the fluctuation relation Conclusions

201 201 202

Many-body perturbation theory and its application to the molecular structure problem

208

S. Wilson Introduction An overview of previous reports Applications An overview of applications of second order theory Application area 1: periodic systems Application area 2: DNA bases and amino acids Application area 3: DFT benchmarking Application area 4: basis set extrapolation and the calibration of general energy models Summary and future directions

208

Experiment and theory in the determination of molecular hyperpolarizabilities in solution; pNA and MNA in dioxane David Pugh Introduction General theory of the response to frequency-dependent electric fields The sum over states method General theory of the EFISH experiment Ab initio and DFT calculations of the pNA b tensor Gas phase measurement Solution EFISH studies of pNA and MNA in dioxane Theoretical approaches to the calculation of the EFISH nonlinearity of pNA in solution Recent work on pNA/MNA Appendix I. Conversion of units

249

The floating spherical Gaussian orbital (FSGO) method A. H. Pakiari Introduction Part I Part II, Development of original theory Part III, Application of FSGO method

279

208 209 211 230 237 238 241 242 243

249 251 253 254 257 260 261 271 275 276

279 280 282 296


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Part IV, Using the FSGO concept in other methods Appendix

298 307

Advances in valence bond theory

312

Peter B. Karadakov Introduction Comparison of the MO and VB approaches Developments in VB methodology Applications of VB theory Concluding remarks

312 313 316 328 346

Numerical methods in chemistry T. E. Simos Newton–Cotes formulae for the numerical integration of the Schro¨dinger equation Stabilization of a multistep exponentially-fitted methods and their application to the Schro¨dinger equation Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G Appendix H Appendix I


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350 350 380

399 408 421 427 439 444 453 464

Multiscale modelling of biological systems Christopher J. Woods and Adrian J. Mulholland DOI: 10.1039/b608778g

1. Introduction At what point does a collection of molecules become a biomolecular system? At what length scale does biology begin, and chemistry end? Biological phenomena involve the flow of information across a range of length and timescales. For example, a cell may be placed under physical stress at the macroscopic level, which causes an increase in pressure within its protective membrane. This pressure has the effect of opening1 or closing2 mechanosensitive ion channels, thereby changing the flow of individual ions into the cell. This changes the ionic concentration within the cell, which then acts as the trigger for a signal sent via a protein signalling pathway. A chemist would look at this as a molecular system that was capable of converting mechanical forces into electrical signals. A biologist would however look at this as the mechanism a cell uses to adapt to stress, and thereby stay alive. Biology is full of such examples. Every thought we have involves the passage of signals between neurons, which itself requires the conversion of electrical signals into flows of ions. These ions trigger the release of neurotransmitter molecules, which cross the synaptic gap between neurons, and bind to individual receptor proteins at the synapse. This causes a change in protein conformation, which open nearby ion channels, causing ions to rush in or out of the neuron, thereby continuing the signal. Information is constantly flowing between the macroscopic world and the atomic, chemical world. Indeed it is this interplay between the chemical and macroscopic worlds that is a real beauty of biology, and it is the recent advances made by the science of biochemistry that has revealed the elegance of the chemicals of life to all. However, while it is possible to use a microscope to watch how an individual cell responds to external stimuli, it is not possible to ‘zoom in’ further and observe what is occurring at the chemical level. Experiments can infer what is happening, and can provide supporting evidence for a particular hypothesis, but there is no experimental technique or microscope that allows us to watch a chemical reaction within an enzyme active site. Until such techniques are developed, the most appealing route that currently exists is to use computers to create models of the biochemical world. Computational scientists can create virtual enzymes, and models of cell membranes, and then use these to provide a window through which the interactions of biomolecules can be observed. If the models are constructed on the firm foundations of physics and chemistry, and if their predictions are carefully compared and validated against experiment, then simulations using these models can provide the valuable insight necessary to link the chemical and biological worlds. Computational scientists have developed many tools for modelling molecules. Computer models are not perfect recreations of reality. Instead, approximations and assumptions have to made, and the model compromised for the sake of computational efficiency. As the size of the system gets larger, and so the size and number of molecules increases, so to does the computational expense of the calculation. This means that the larger the system, the more compromises and approximations must be made. This act of compromise has led computational scientists to develop four main levels of biomolecular modelling: 1. Quantum mechanics (QM). Quantum chemical calculations model the fine detail of the electrons in the molecule. They achieve this by modelling the electrons as a quantum mechanics wavefunction that interacts with the electrostatic potential Centre for Computational Chemistry, School of Chemistry, University of Bristol, Bristol, UK BS8 1TS


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field generated by the atomic nuclei. Quantum chemical calculations provide the most physically realistic and accurate models of molecules, but this accuracy comes at a cost. While methods have been developed that allow QM calculations on complete proteins,3 in general the high computational expense of QM methods limits their application to small molecular systems. 2. Molecular mechanics (MM). Atomistic molecular mechanics calculations apply the assumption that the fine detail information about the behaviour of the electrons can be ignored, and instead they are approximated by representing their effects using simple descriptors such as atomic partial charges or polarisabilities. By modelling the electrons implicitly, MM methods are much less expensive than QM methods, and so they are able to model significantly larger systems. By including atomic detail, MM models are still limited to the molecular level, and even today’s largest applications can only achieve the modelling of hundreds of thousands of atoms over hundreds of nanoseconds. 3. Coarse grain (CG). Coarse grain (or coarse grained) calculations apply the assumption that the fine detail information about the position of each atom in the molecule can be ignored, and instead groups of atoms are approximated by smearing them out into single ‘beads’. So, for example, rather than modelling each atom in a protein, a CG representation would portray each residue as a single bead. This approximation allows CG simulations to achieve length and timescales that are far beyond those possible using atomistic MM models. 4. Continuum. Continuum models apply the assumption that the fine detail information about the location of any particles or groups can be ignored, and instead systems are modelled as continuum regions. For example, implicit solvent models ignore the location of each individual solvent molecule, but instead represent the complete solvent as a fuzzy dielectric continuum. Equally, continuum models of a cell membrane ignore the individual locations of each lipid molecule, and instead model the membrane as a homogenous elastic sheet. By ignoring particles, and instead modelling biological systems as continuous fields or homogenous assemblies, continuum models are able to simulate the largest length scales and longest timescales of any of the four levels. These four levels of biomolecular modelling are each well-suited to modelling phenomena at the length and timescales for which they were designed. However, what makes biology work, and what makes it scientifically interesting, is the interplay and flow of information across the different length and timescales. It is not possible for simulations at any one of these biomolecular modelling levels to represent these complex, multiscale biological phenomena on their own, and so methods that allow the combination of different levels of biomolecular model together must therefore be sought. Multiscale modelling, in which calculations at multiple length and/or timescales are combined together into a single simulation, is now becoming popular, and its development is now the focus of significant research effort. Multiscale modelling is not new, for example combined QM/MM methods, and MM/continuum implicit solvent methods have been used for over 30 years, and multiscale methods have a rich heritage of applications in the fields of materials modelling and nanomaterials,4 and modelling fluid and gas flow.5,6 Recently, there has been a huge increase in the development and application of multiscale methods for biomolecular modelling. This review focuses on these developments, in particular the application of multiscale methods to biomolecules covering the period from 2005 to 2007. Coveney7 has produced a review of biological multiscale modelling that covers the period up to 2005. Before starting this review, there first needs to be a definition of what is meant by a multiscale method. There are several different definitions that vary depending on the type of coupling between the different modelling levels. This review will adopt perhaps the most broad definition of a multiscale method, namely that it is any method that involves a flow of information from one modelling level to another. By definition, if there is a flow of information from one level to another, then there must 14 | Chem. Modell., 2008, 5, 13–50 This journal is

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be an interface between the levels through which this information will flow. Throughout this review it will become clear that there are four main classes of interface; 1. One-way, bottom-up interfaces. These involve a one-way, often one-time transfer of information from a lower level of modelling to a higher level. Examples include using a QM calculation to parameterize an MM forcefield, or using an MM simulation to parameterize a CG potential. 2. One-way, top–down interfaces. These involve a one-way transfer of information from a higher level of modelling to a lower level. Examples include using a CG model to reconstruct an atomistic model of a protein, or using a continuum model to provide the boundary conditions for an atomistic simulation. 3. Two-way parallel interfaces. These involve a two-way dynamic transfer of information between two simulations running in parallel at two different modelling levels. An example includes running both an MM and CG simulation of a system and using replica exchange8–10 moves to exchange coordinates between the two levels. 4. Two-way embedded interfaces. These involve embedding a low modelling level region within a simulation at a higher level, e.g. embedding a QM model of a substrate and active site within an MM model of the enzyme, or embedding an MM model of an ion channel within a CG model of a membrane. This review is therefore organised according to the different interfaces between levels (QM/MM, atomistic/CG, particle/continuum), and then by the different classes of interface that are used between these levels.

2. Interfacing QM with MM models The most accurate physical description of atoms and molecules is provided by quantum chemical calculations. Quantum chemical calculations are capable of correctly predicting the energetics and conformations of small molecules from first principles, using broadly applicable approximations (e.g. the Born-Oppenheimer approximation) and nothing more than fundamental physical constants as input.11 Quantum chemical calculations model electrons as a quantum mechanics (QM) wavefunction that interacts with the electrostatic potential field created by the atomic nuclei of the molecule. QM provides the most exact physical model of matter at the atomic scale, and QM calculations are capable of predicting chemical bonding and chemical reactivity. There are several recent reviews of quantum chemical methods,11–13 and QM methods may now be used across a length and timescale that ranges from modelling the femotosecond interactions of infra-red laser light with carbon monoxide,14 to modelling the sub-nanosecond dynamics of a complete protein.15 There are a range of QM methods available with varying degrees of approximation, with a range that includes fast semi-empirical Hamiltonians such as AM116 or PM3,17,18 and highly exact coupled cluster methods such as LCCSD(T).19 Because QM methods include an explicit representation of electrons, they are able to model chemical processes such as charge transfer, bond breaking and formation, and changes of molecular polarisation. However, the high computational expense of QM methods prevents their application to the large length and timescales that are required to understand complex biomolecular processes. MM methods provide a simpler representation of molecules, in which the fine detail of the electrons represented implicitly via partial charges and, is some cases, molecular polarisabilities.20,21 MM models represent molecules as a collection of atoms interacting through classical potentials. There are several MM models (or forcefields), and they differ in the functional forms of the interaction potential used between atoms, and in the means by which these interaction potentials are parameterized. Several good recent reviews of MM forcefields have been produced.22–26 Several MM forcefields have been developed for application to biomolecular systems. The most popular of these are the CHARMM,27 AMBER,28,29 Chem. Modell., 2008, 5, 13–50 | 15 This journal is

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GROMOS30–32 and OPLS33,34 forcefields. Each of these forcefields has evolved over time, with different versions produced periodically. However, despite the proliferation and evolution of MM forcefields for biomolecular modelling, their functional forms all remain broadly similar. Each atom is modelled as a single point in space. Pairs of atoms in separate molecules interact through a pairwise non-bonded potential, Enb, which depends only on the distance between the atoms, r. The electrostatic part of this non-bonded potential, Eelec, is modelled using Coulomb’s law, assigning a fixed partial charge to each atom in the molecule. The non-bonded potential must also model the van der Waals (vdW) forces between the molecules, which result from the combination of the Pauli repulsion that results from the inability of two electrons to occupy the same space with the same set of quantum numbers, and the attractive dispersion (London) forces whose physical basis lies in the ‘instantaneous dipoles’ that result from the wavefunctions of close atoms moving in phase. These vdW forces have their origin in the behaviour of electrons, which are not explicitly modelled in MM forcefields. These forces must therefore be approximated. The most common approximation used for biomolecular applications is the Lennard-Jones (LJ) potential. This approximates the vdW interactions using a 12–6 repulsive–attractive potential, ELJ, " 6 # sij 12 sij ELJ ðrij Þ ¼ 4eij ; ð1Þ rij rij where ELJ(rij) is the Lennard-Jones energy between atom i and atom j, rij is the distance between the pair of atoms, and sij and eij are parameters that are tuned to reproduce the strength of the vdW forces between this pair of atoms, often by fitting to macroscopic properties. Note that this is a pairwise potential that acts only between pairs of atoms. This is despite the fact that unlike the permanent electrostatic forces, vdW forces are not pairwise in nature. Indeed, while permanent electrostatic forces are pairwise, MM forcefields use Coulomb’s law and fixed atomic partial charges to model both the permanent electrostatics of the molecule and, implicitly, its polarisation. Charge polarisation is also not a pairwise phenomenon. The non-bonded potential energy between two molecules is given by the sum of the Coulomb and LJ energies between all pairs of atoms in the molecules. It is therefore an effective pair potential, as the derivation of the partial charges and LJ parameters must account for the errors implicit in only using a pairwise sum over atoms, and must therefore include 3-, 4- to n-body effects implicitly. Modelling the electronic detail of a molecule, as well as providing an explicit representation of polarisation and vdW forces, is also responsible for giving a correct representation of chemical bonding. As MM forcefields do not explicitly model electrons, they must include classical interaction potentials that mimic the effects of chemical bonding. MM forcefields include classical intramolecular interaction potentials, e.g. a harmonic bond potential, Ebond that acts between bonded atoms (called 1–2 atoms), a harmonic angle potential, Eangle that acts on the angle between a series of three bonded atoms (1–3 atoms), and a torsional potential, Etorsion, that acts about the dihedral formed by four bonded atoms (1–4 atoms). Atoms that are separated by more than three bonds (1–5+ atoms) are treated as being non-bonded, and so their interaction energy is calculated using the sum of their Coulomb and LJ interaction energies. The total intramolecular energy of a molecule is then given by the sum of the bond energy between all 1–2 atoms, the sum of the angle energy between all 1–3 atoms, the sum of torsion energy between all 1–4 atoms, and the sum of the non-bonded Coulomb and LJ energies between all pairs of 1–5+ atoms. The total energy of a system of molecules can then be calculated as the sum of the intramolecular energies of all of the molecules, together with the sum of the nonbonded potential energies between all pairs of molecules. By using classical potentials, MM models allow for a very rapid evaluation of the energy and forces acting on each atom within a large biomolecular system. This 16 | Chem. Modell., 2008, 5, 13–50 This journal is

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rapid evaluation allows these forces and energies to be used by statistical conformational sampling methods, such as molecular dynamics (MD)35,36 or Monte Carlo (MC),36,37 to generate large ensembles of configurations of the system, from which macroscopic (thermodynamic) properties may be evaluated. However, by not explicitly modelling electrons, MM models struggle to model many chemically important phenomena, such as chemical bond breaking and formation, electronic polarisation and charge transfer. There is therefore a strong motivation to combine MM models with quantum mechanics (QM) calculations within a multiscale modelling framework, and combined QM/MM biomolecular simulation methods have a rich history of application and evolution since their original inception in the early 1970s.38,39 Using a broad definition, QM/MM multiscale methods are those that involve a transfer of information across an interface between the QM and MM levels of modelling. There are several different types of interface in use, which fall into four categories: 1. One-way bottom-up methods. These involve a single transfer of information from a QM calculation to a classical simulation, e.g. by using QM calculations to parameterize the classical potentials used in MM force fields. 2. Two-way dynamic parameterisation methods. These involve a dynamic transfer of information between separate classical and quantum calculations, e.g. using successive QM calculations to dynamically re-parameterize the classical potentials the QM atoms of an MM forcefield during a live simulation. 3. Two-way embedded methods. These involve embedding of molecules or parts of molecules modelled using QM into a system of molecules modelled using MM, e.g. using QM to model a substrate, and MM to model the enzyme and solvent. 4. Two-way parallel methods. These involve the running in parallel of classical and quantum simulations, and dynamically sharing information between them at run time. Examples of each of these different types of interface, and recent developments in their methodology, will now be discussed in turn. 2.1 One-way bottom-up QM/MM interfacing methods As described in the last section, molecular mechanics (MM) forcefields use classical potentials to calculate the interaction energy between pairs of atoms. Several types of interaction potential are required to fully describe the MM energy of a set of molecules: 1. Non-bonded potentials. These typically take the form of a Coulomb potential between non-bonded pairs of atoms to describe polarisation and permanent electrostatics, and a Lennard-Jones (LJ) potential between non-bonded atom pairs to describe the vander Waals (vdW) interactions. 2. Bonded potentials. These typically involve harmonic terms that are applied between 1–2 bonded and 1–3 bonded pairs of atoms, and cosine terms between 1–4 bonded pairs. These potentials try to model the effect of chemical bonding. These classical interaction potentials must be parameterized, e.g. the magnitude of the partial charges on each atom in the molecule must be assigned, and the equilibrium bond length and size of the harmonic force constant must be attached to each bond. In the early biomolecular MM forcefields, these parameters were developed to produce molecular models that could reproduce known experimental properties of the bulk system. For example, several MM water models have been developed.26,40,41 One of the earliest successful models, TIP3P,42 was parameterized such that simulations of boxes of TIP3P molecules reproduced known thermodynamic properties of water, such as liquid density and heats of vaporisation. Such a parameterisation scheme is to be applauded, as it ties the molecular model closely to experiment. Indeed many of the common MM models of amino acids were developed by comparison to experiment, e.g. OPLS.33 Indeed it is such a good Chem. Modell., 2008, 5, 13–50 | 17 This journal is

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scheme that even some modern water models, like TIP5P,43 are still parameterized in this way. However, it was quickly realised that parameterisation against experiment required large amounts of physical data that was just not available for the novel molecules being conceived during rational drug design. The developers of biomolecular forcefields therefore created recipes that allowed for the parameters of new molecules to be derived from quantum chemical calculations. One popular example of such a recipe is GAFF (generalised AMBER forcefield),44 which is an MM forcefield for small drug-like ligands that is compatible with AMBER. GAFF uses generic atom types that are assigned to each atom in the drug-like molecule, e.g. aliphatic carbon or aromatic hydrogen. These atom types are used to assign LJ, bond, angle and dihedral parameters to the molecule from a large parameter library. The partial charges for the atoms are derived by performing an AM116 semiempirical QM calculation, and calculating BCC45,46 charges. In a very broad sense, this parameterisation is a multiscale method, as information (the charge distribution) from a QM calculation is transferred to an MM simulation via the parameterisation of the atomic partial charges. This multiscale parameterisation therefore represents a one-time, one-way flow of information up from the QM level to the MM level. A similar scheme is available for the OPLS forcefield,47 which uses CM1A charges48 that are also derived from semiempirical AM1 QM calculations. Wang and Sandberg49 used a more complex multiscale parameterisation to derive intramolecular CHARMM parameters for the interaction of DNA bonded to a gold surface. The parameters were calculated by fitting to density functional theory (DFT) QM calculations. 2.2 Two-way dynamic parameterisation methods QM calculations are now used routinely as the source of parameters for MM forcefields. Indeed this application is now so routine that most workers would not consider forcefield parameterisation to be an example of a multiscale method. However, there is a drawback with using QM calculations to provide MM forcefield parameters. The problem is that the information flow is only in one direction, from the QM to the MM level. This means that the QM-derived parameters for a molecule have to be very general, and be able to represent the molecule in a variety of conformations and environments. This is an unreasonable requirement, as it is clear that the charge distribution and polarisation of the molecule depends on both its conformation and its environment, e.g. whether it is in bulk solvent or whether it is bound to the active site of a protein. Because information only flows from the QM to MM level, there is no mechanism that allows information about the environment and conformation of the molecule experienced during the MM simulation to be fed back to the QM calculation. A solution is to modify multiscale parameterisation so that there is a two-way interface between the QM and MM levels of modelling. There have been two recent applications that have made such a modification: one targeted at creating a QM/MM multiscale docking method,50 and another targeted at QM/MM multiscale free energy calculations.47,51,52 Docking is one of the primary tools used during the process of rational drug design.53 The aim of docking is to predict the binding mode of a ligand with a protein. Because docking calculations are typically used to study how libraries of thousands of ligands bind to a protein, the calculations involved must be simple and efficient. This means that the interaction potentials used in docking tend to be based on molecular mechanics forcefields.53 MM forcefields struggle to model the changes in polarisation upon protein-ligand binding, an effect that is thought to account for as much as 10–40% of the binding affinity.54 Multiscale docking methods that attempt to use QM calculations to overcome this problem have therefore been developed.50,55 Cho et al.50 have developed a QM/MM docking method that uses multiscale parameterisation dynamically throughout a docking calculation. The electrostatic interaction energy between the ligand being docked and the protein is 18 | Chem. Modell., 2008, 5, 13–50 This journal is

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calculated using Coulomb’s law, with atomic partial charges placed on the protein and ligand atoms. Cho et al. first showed that the docking prediction is improved if the atomic partial charges on the ligand are derived from a QM calculation of the ligand in the bound geometry. They demonstrated this by selecting several test protein-ligand systems whose bound geometries were known via crystal structures available in the protein databank (PDB). They calculated the partial charges by performing a density functional theory (DFT) calculation of the ligand in the bound geometry. The wavefunction was polarised by embedding the partial charges from the protein within the Hamiltonian of the QM calculation. By using the bound geometry, and by embedding the partial charges of the protein, information about the environment of the ligand in the protein active site was made available to the QM calculation. The polarised wavefunction was used to obtain the molecular electrostatic potential (MEP) surface around the ligand. Partial charges were generated using an electrostatic potential fitting procedure to reproduce the QM MEP. By calculating these partial charges from a QM calculation that had information about the bound geometry, it could be argued that these atomistic partial charges were optimised for the bound geometry. Cho et al. demonstrated this50 by running redocking calculations where the ligands were docked using both optimised and generic partial charges. The results demonstrated that docking calculations using the optimised charges were significantly more likely to rediscover the known experimental binding modes. To turn this observation into an effective docking algorithm, Cho et al. created an iterative dynamic reparameterisation algorithm; 1. Dock the ligand using the default partial charges taken from the docking MM forcefield. 2. Perform a QM calculation on the predicted binding mode to obtain optimised partial charges for the ligand. 3. Dock the ligand again, this time using the optimised partial charges. 4. Keep iterating until the charges and predicted binding mode converge to within a set limit. By dynamically reparameterising the ligand throughout the docking calculation, Cho et al. allow information to flow both ways between the QM and MM levels. A similar idea has been developed by Jorgensen and co-workers to create a QM/MM multiscale method for free energy calculations.47,51,52 Jorgensen and co-workers developed a method to obtain atomic partial charges efficiently from a QM calculation that were compatible with the partial charges from the standard OPLS all-atom forcefield.47 The charges were calculated using the charge model 1 (CM1A)48 analysis of an AM1 semiempirical QM calculation. However, as CM1A charges were parameterized to reproduce gas-phase dipole moments,48 they had to be scaled by a factor of 1.2 so that an implicit account could be made for the extra polarising effects of polar solvents. Jorgensen and co-workers first used this method to perform QM/MM hydration free energy calculations. The solute molecule was modelled using QM (AM1), while the solvent molecules were modelled using MM (OPLS). The QM calculation was used to obtain the intramolecular energy of the solute. The QM calculation was also used to obtain the atomic partial charges on the solute using AM1/CM1A. These partial charges were used to calculate the electrostatic interaction energy between the solute and solvent via Coulomb’s law. The LJ equation was used to obtain the vdW interaction between the solute and solvent using pre-assigned OPLS e and s LJ parameters. The QM calculation in this method was used only to dynamically re-parameterize the MM atomic partial charges during the simulation. The only information flow from the MM to QM level was the change in conformation of the solute. The solvent environment around the solute was not passed explicitly, as it was not included within the QM calculation. The effect of the solvent was only felt implicitly in the QM calculation via the application of the scale factor. Despite the lack of explicit modelling of the solvent environment in the QM calculation, Jorgensen and co-workers have successfully used this method to study solution-phase Diels-Alder reactions,56 and to study the enzyme-catalysed Claisen Chem. Modell., 2008, 5, 13–50 | 19 This journal is

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rearrangement reaction of chorismate to prephenate.51 Jorgensen and co-workers have since adapted this method57–59 to use the PDDG/PM3 semiempirical QM Hamiltonian,60 using the CM361 method to extract charges, which are then scaled by a factor of 1.14,62 again to provide an implicit account for the extra polarising effects of the solvent. 2.3.

Two-way embedded methods

Periodic multiscale reparameterisation, i.e. periodically during a simulation, provides a two-way conduit for information flow between the QM and MM levels of modelling. However, the coupling between levels is not particularly strong. The interchange of information between levels occurs only periodically, which can lead to the MM level falling out of step with the QM. This problem was experienced during the development of the QM/MM docking method of Cho et al.50 If the ligand was initially docked in a poor configuration, then the partial charges derived for that conformation could bias the subsequent docking runs to rediscover the poor configuration in preference to the correct binding mode. Cho et al. developed a survival of the fittest algorithm50 that ran multiple docking runs in parallel, thereby preventing one poor result from biasing the rest of the calculation. A closer coupling between the QM and MM levels can be achieved using an embedded interace method. A QM region of the biomolecular system is embedded within a larger MM simulation. One of the primary application areas for embedded QM/MM methods is computational enzymology (the computational modelling of enzyme-catalysed reaction mechanisms), where typically a QM model of the substrate and part of the enzyme active site is embedded within an MM model of the rest of the enzyme and solvent.63 Embedding a QM model within an MM simulation creates a dynamic and permanent interface between the two levels, with information flow across that interface having to be managed for each configuration of the simulation. The ONIOM method, developed by Morokuma and co-workers,64 provides such an interface via the use of multilevel corrections. The ONIOM method partitions the system into multiple layers. For example, consider a two-layer system, where a low-level QM region, A, is embedded within a high-level MM region, B. The energy of both regions, A + B, is first calculated using only the MM Hamiltonian, giving EMM(A + B). This total energy is corrected by calculating the difference in energy between the QM and MM energies of the QM region, EQM(A) EMM(A). The total ONIOM energy of the system is therefore EMM(A + B) + EQM(A) EMM(A). The generalisation of this algorithm to multiple levels is straight-forward, e.g. a system can be divided into an ab initio QM region, A, which is embedded within a semiemprical QM region, B, which is itself embedded within an MM system, C. The ONIOM energy in this case would be EMM(A + B + C) + [Esemiemprical(A + B) EMM(A + B)] + [Eab initio(A) Esemiemprical(A)]. The ONIOM method allows the facile combination of QM and MM levels of modelling. However, the electrostatic interaction between the QM and MM regions in the original ONIOM implementation is handled at the MM level only, via EMM(A + B), based on partial atomic charges derived from the QM calculation. The use of this method, called classical or mechanical embedding,65 means that information about the electrostatics of the system flows only from the QM up to the MM level. There is no conduit by which the electrostatic environment of the MM atoms is able to flow down to the QM region, to polarise the QM wavefunction.65 An alternative method of interfacing QM and MM calculations, called electronic embedding, solves this problem. In electronic embedding, the partial charges of the MM atoms are embedded within the QM Hamiltonian. This allows the QM wavefunction to be polarised by the MM atoms, thereby providing a two-way conduit for electrostatic information between the QM and MM regions. The ONIOM method has since been extended to use electronic embedding,65 thereby overcoming one of the fundamental weaknesses of the algorithm. 20 | Chem. Modell., 2008, 5, 13–50 This journal is

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Electronic embedding has had a rich and long history of application in other QM/ MM schemes. The first application of electronic embedded QM/MM to a biomolecular system was in the ground-breaking work of Warshel and Levitt in 1976.39 They developed the method to study the reaction mechanism of hen egg-white lysozyme. Warshel has recently produced a detailed, clear and very interesting review63 of the details of the algorithm and the developments in QM/MM methodology since his pioneering work in the 1970s. Acceptance of this approach took a long time,63 but there is now a large body of QM/MM applications that use such methodology (see one of the very many reviews of QM/MM methods available in the literature66–69). The underlying theory of this method has been covered by many different authors68,69 so only a brief description will be given here. The biomolecular system is divided into a QM region, for example a substrate, and an MM region, e.g. an enzyme and surrounding solvent molecules. The total energy of the system is the sum of the energy of the MM region, evaluated using a standard MM forcefield, the energy of the QM region, and the QM/MM interaction energy between the two regions. The QM/MM interaction energy is split into two parts: an electrostatic part and a vdW part. The vdW part is calculated by assigning LJ parameters to all of the QM atoms and calculating the interaction between the QM and MM atoms using the LJ equation. The electrostatic part of the QM/MM interaction is calculated by bundling it together with the calculation of the QM energy of the QM region. This is achieved by embedding within the QM Hamiltonian the locations and partial charges of all of the MM atoms that are within a pre-determined cutoff distance of the QM atoms. Normally the MM atoms are represented as point charges in the QM calculation, but Gaussian charge distributions, with the width of the Gaussian that is similar to the covalent radii of the MM atoms may instead be used. 70 The MM partial charges act to polarise the QM wavefunction of the QM atoms, and therefore the evaluation of the energy of this wavefunction returns both the intramolecular energy of the QM atoms and the electrostatic interaction between the QM and MM atoms. The simple split of the QM/MM interaction into electrostatic and vdW parts is, however, only possible if the interface between regions lies between molecules, i.e. all molecules are either QM or MM, and there are no molecules that sit across the interface. It is desirable (e.g. within computational enzymology) for a single molecule to be able to bridge this interface, e.g. while the majority of an enzyme is modelled at the MM level, it is usually necessary to represent some (e.g. catalytic) active site residues at the QM level. The problem with having a single molecule straddle the interface is that the QM/MM interaction energy, as well as modelling electrostatic polarisation, must now also include terms that account for the chemical bonding between atoms modelled at the QM level and atoms modelled at the MM level. This is a particular problem with the QM side of the calculation, as when the boundary bisects a covalent bond, the electron density is terminated abruptly at the end of the QM region and electrons of the bonded atom are missing, potentially leading to unpaired electrons.66 The three most popular methods for solving this problem are the link atom method,66,71,72 the local selfconsistent field (LSCF) method73,74 and the generalized hybrid orbital (GHO) method.75,76 The ‘‘dummy junction atom’’ or link atom approach introduces so-called link atoms to satisfy the valence of the atoms on the QM side of the QM/MM interface.66,71,72 Usually this atom is a hydrogen, but other atom types have also been used, e.g. halogens such as fluorine or chlorine.77 The link atom method can be used with both the Warshel-type QM/MM methods and ONIOM methods.65,78 The link atom method has been criticised because it introduces extra unphysical atoms to the system, which come with associated extra degrees of freedom. Another problem is that a C–H bond is clearly not chemically exactly equivalent to a C–C bond. Despite these problems, the simplicity of the link atom method means that it is used widely in the QM/MM modelling of proteins and other biological molecules. 79


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Zhang et al.80,81 solved these problems with their pseudo bond method. If X and Y are the bonded QM and MM boundary atoms, respectively, then the link atom method effectively replaces the Y atom with a hydrogen, thereby changing X–Y into X–H. This has the problem that H is not necessarily chemically similar to Y, and also the X–H bond length may not be the same as the X–Y bond length. In the pseudobond method, Y is instead replaced by Yeff. This is a one-free valence boundary atom that has a parameterized effective core potential that mimics the strength of the X–Y bond. This method has been tested successfully80 with both Hartree Fock (HF), density functional theory (DFT) and MP282 QM/MM calculations. Antes and Thiel83 have also introduced a conceptually similar approach, which they call ‘‘adjusted connection atom’’, that works at the semiemprical QM level. Parameterisation of these effective link atoms is not straight-forward, as they must minimally perturb the electron density compared to a QM calculation of the entire molecule. Ro¨thlisberger and co-workers have developed a sophisticated scheme that derives parameters for these atoms via density functional perturbation theory.84 A second approach to define the bonding interface between the QM and MM regions is the local self-consistent field (LSCF) algorithm developed by Rivail and co-workers.73,74 In the LSCF method the bonds between the QM and MM atoms are represented by strictly localised bond orbitals, which are parameterized by separate QM calculations on small molecules.75 These localised orbitals are assumed to be transferable to the protein system, and are used, and kept constant, throughout the self-consistent field (SCF) QM calculation. An elegant feature of the LSCF method is that it does not require use of link atoms, and a comparison of the LSCF and link atom methods85 showed that both give equivalent energy results. However, the parameters for the localised bond orbitals have to be determined for each new system studied.75 Fonili et al.86 have recently shown that it is possible to use frozen core orbitals on the MM frontier atom within the LSCF scheme. This provides an explicit description of the core electrons of the atoms on the MM frontier atom, thereby improving the physical description of the interface, thus reducing the error on the calculation. While the LSCF method is elegant, and has been shown to work well, the parameters for the localised orbitals are not very portable. Gao et al. addressed this problem by developing generalised hybrid orbitals (GHOs).75,76 The QM boundary atom at the QM/MM interface has the standard valence s and p orbitals as all of the other non-hydrogen atoms in the QM region. These four sp orbitals are transformed into a set of orthogonal hybrid orbitals, which can be defined by the bound geometry of the atom.75 These hybrid orbitals are used, along with the atomic orbitals of the QM region, to determine the QM energy. However, only one hybrid orbital points along the bond between the QM and MM boundary atoms, and it is this active hybrid orbital that needs to be optimised. The complete set of this one active hybrid orbital, plus all of the atomic orbitals from the other QM atoms form the active set that is optimised during the SCF calculation. The remaining three hybrid orbitals, called auxiliary orbitals, act, together with the nucleus charge, to generate an effective core potential for the QM boundary atom. Gao et al. realised that these auxiliary orbitals may be parameterized to mimic the effective core potential for the active electrons from the MM region. Therefore rather than parameterising the charge density of the hybrid orbitals for each specific system, as is the case for the LSCF method, they instead optimise the semiemprical parameters for the boundary atom to reproduce the bonding properties of full QM systems. As a result, the parameters for this GHO method are expected to be general and transferable in the same way as all the semiemprical parameters. 75 The method has since been extended by Gao and co-workers87 to work with the selfconsistent charge density-functional tight binding (SCC-DFTB) method.88 The accuracy and efficiency of SCC-DFTB is making it a popular choice to model the QM region, and its use in computational enzymology has increased greatly in recent 22 | Chem. Modell., 2008, 5, 13–50 This journal is

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years.88 Gao and co-workers76,89 have also developed a GHO method that is suitable for ab initio Hartree Fock (HF)89 and density functional theory (DFT) calculations.76 Ko¨nig et al.90 recently compared several different algorithms to model chemical bonds between QM and MM atoms, and concluded that while none of the approaches were perfect, error cancellation meant that enzyme catalysed reaction free energies using an SCC-DFTB QM region were only marginally affected by the choice of algorithm, if total charge was conserved during the reaction. The conclusion contrasts with a comparison study in which we were involved,91 in which we compared the link atom and GHO boundary methods. While we found that both methods, when properly applied, can lead to similar behaviour, the inclusion of conformational sampling amplified the effects of the differences between the methods. This led to the two methods returning different free energy reaction profiles despite being applied to the same enzyme system. It is important to make clear that while the same enzyme system was used, the QM region was not the same for the two boundary methods. This was because of the different constraints on the partitioning of the two methods (the GHO method requires that the QM atom bound to an MM atom must be an sp3 carbon91). The origin of the observed difference may well be in the different QM and MM systems, rather than in the partitioning schemes themselves. Despite these questions over the fine detail of how QM/MM methods are applied, their development and application has now matured to a point where they can provide near-quantitive results for activation enthalpies and free energies of reaction.92 It is now possible to perform electronic structure calculations on large systems approaching chemical accuracy, thus allowing quantitative studies of reaction mechanisms in enzymes.92 2.4.

Two-way parallel QM/MM methods

Quantum mechanics calculations are computationally demanding. This has caused difficulties with their application to the calculation of thermodynamic properties, such as free energies. This is because thermodynamic properties are calculated as an average over a large ensemble of conformations of a system. Sampling methods, such as molecular dynamics (MD)35,36 or Monte Carlo (MC)36,37 must be used for the rigorous generation of such ensembles. The computational expense of QM calculations means that it is only practical to generate small ensembles via standard MD or MC, e.g. current methods are limited to picoseconds of molecular dynamics. Two-way parallel QM/MM interfaces have therefore been developed in an attempt to overcome this problem. We use the term parallel methods to mean those that use both a QM or QM/MM simulation running in parallel with a standard MM simulation, using only periodic exchange of information between the two simulation levels. Warshel and co-workers developed a successful QM/MM parallel method in a set of pioneering papers in the late 1990s.93,94 The aim of this method was to calculate the free energy difference between two systems, A and B. For example, system A could be a substrate bound to an enzyme, while system B could be the transition state. The free energy difference between these two corresponds to the activation free energy of the enzyme catalysed reaction. Warshel and co-workers calculated the relative free energy of A and B by first using a molecular mechanics type potential. Because an MM potential was used, molecular dynamics sampling was efficient, and therefore a large ensemble, and well-converged relative free energy were calculated. This relative free energy, DGMM(A - B), can only be as good as the MM potential used during the calculation. Ideally, this free energy should be calculated using a QM or QM/MM representation of A and B, giving DGQM(A - B). However, the computational expense of the QM calculation prevents the efficient generation of the large ensembles necessary to evaluate converged free energies. Warshel and coworkers solved this by rather than calculating DGQM(A - B) directly, they used the Chem. Modell., 2008, 5, 13–50 | 23 This journal is

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Fig. 1 The free energy cycle93,94 used to calculate the QM/MM free energy difference between systems A and B, DGQM/MM(A - B). The free energy difference between A and B is first estimated using an approximate potential (e.g. an MM potential), giving DGMM(A - B). This is then corrected to the QM/MM value by calculating the free energy necessary to perturb system A from MM to QM/MM (DGMM-QM/MM(A)) and the free energy to perturb system B from MM to QM/MM (DGMM-QM/MM(B)).

MM ensembles to calculate the difference in free energy between the QM and MM representations of A and B. In essence, Warshel and co-workers calculated the free energy error associated with using the MM forcefield. By calculating these errors, Warshel and co-workers were able to correct DGMM(A - B) so that it was formally equal to DGQM(A - B)93,94 (see Fig. 1). The correction free energies were calculated by generating ensembles for systems A and B using the MM model. The difference in energy between the QM and MM models was calculated for a subset of each ensemble, and the difference between these energies used as input to a single-step free energy perturbation (FEP)95,96 between the MM model (the FEP reference state) and the QM model (the FEP perturbed state). As long as the MM model is a good approximation of the QM model, i.e. the phase space overlap of the two models is good, then the average calculated via the FEP equation will converge to an accurate estimate of the correction free energy. The key advantage of this method is that all of the thermodynamic sampling is performed using only the MM model of the system. QM or QM/MM calculations are run in parallel with the MM sampling to estimate the correction free energies. The disadvantage of this method is the requirement of good overlap between the QM and MM models. Warshel and co-workers mitigate this disadvantage through the development of the empirical valence bond (EVB)93,94,97 forcefield, which has been designed to give energies that are in good agreement with experiment and QM calculations. In addition, the EVB potential has been developed so that it can be used to study chemical reactions, something that is not possible using most of the biological MM forcefields. The EVB potential and the Warshel parallel QM/MM method have been very successful, and have been used to study a variety of systems.98–102 Warshel and co-workers developed their method to avoid the problem of poor sampling of a QM or QM/MM Hamiltonian. MD methods are currently limited to picoseconds of QM/MM dynamics for typical biomolecular applications, even using relatively low levels of QM theory. Monte Carlo (MC) methods suffer from even greater problems. MC works by performing typically millions of small random moves of the biomolecular system, each of which are tested according to the change in energy associated with that move. MC sampling of a QM/MM Hamiltonian would potentially require millions of QM energy evaluations, which is impractical using current methods and computers. A second class of parallel QM/MM methods attempt to solve this problem. These methods use a novel Monte Carlo algorithm developed by Hastings in 1970.103 This is a multiscale sampling method that uses MC sampling at one modelling level to generate an ensemble which is formally correct for a different modelling level. The algorithm works by creating a new type of Monte Carlo move. The move starts at configuration i. The energy of this configuration is evaluated using both the fast, high-level model, giving Efast(i), and the slower, low-level forcefield, giving Eslow(i). A block of MC moves is then performed using the fast forcefield. This results in a new configuration, j. The energy of this configuration is evaluated using both forcefields, giving Eslow(j) and Efast(j). These 24 | Chem. Modell., 2008, 5, 13–50 This journal is

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Fig. 2 Application of the Metropolis-Hastings103 algorithm to accelerate sampling of a system represented using a QM/MM Hamiltonian.104,105 The Monte Carlo move starts at configuration i. The energy of this configuration is evaluated using the target QM/MM Hamiltonian (giving EQM/MM(i)) and on an approximate (MM) Hamiltonian (giving EMM(i)). Standard Metropolis Monte Carlo moves are then attempted from configuration i using only the approximate MM Hamiltonian, until after a set number of moves, the system is in configuration j. The energy of configuration j is evaluated using both the QM/MM and MM Hamiltonians (giving EQM/MM(j) and EMM(j)). Configuration j is then accepted into the QM/ MM ensemble according to the probability min{1,exp( DDE/kBT)} where DDE = (EQM/MM(j) EMM(j)) (EQM/MM(i) EMM(i)).

energies are used to test configuration j according to a new MC acceptance test. Configuration j is accepted into the ensemble if this test is passed (see Fig. 2). Otherwise the whole block of sampling is rejected and the simulation is reset to configuration i. The form of the MC test is such that even though the trial configurations are generated using the fast forcefield, they are accepted into the ensemble of the slow forcefield with the correct Boltzmann probability. This algorithm was popularised for applications to QM and QM/MM systems by Schofield and co-workers,104,106 who coined the phrase ‘‘molecular mechanics-based importance function’’ (MMBIF). One of the problems with this algorithm is that the acceptance ratio of the MC test will be low if there is poor overlap between the fast and slow forcefields. This is similar to the problem encountered in Warshel’s correction free energy method. Effort may therefore need to be spent optimising the fast forcefield such that it is a better match to the slow forcefield. We have developed a parallel QM/MM method105 that combines the advantages of both the Warshel and MMBIF algorithms. The method works by using the MMBIF method to generate QM/MM ensembles from which the Warshel correction free energies can be calculated in full. The correction free energy is calculated using thermodynamic integration (TI)107,108 over a fictional l scaling parameter that maps from the QM model to the MM model. This l parameter allows the QM model to be changed over a series of windows into the MM model. The MMBIF algorithm can then be used to generate ensembles of conformations of the system at different values of l, so that the gradient of the free energy with respect to l can be calculated at several points between the QM and MM models. These gradients can then be integrated across l to return the correction free energy. We overcome the problems associated with potentially poor overlap between the QM and MM models by using replica exchange8,9,109 moves across the l coordinate during the simulation. Replica exchange moves are additional Monte Carlo moves that lightly couple multiple trajectories together. All of the MC simulations at different l values are run in parallel. Neighbouring pairs of simulations are tested periodically according to a replica exchange MC acceptance test. If this test is passed, then the l values of the neighbouring pairs are swapped. This has the effect of allowing each MC simulation to sample multiple l values during the simulation. This enhances convergence of the free energy averages. We have used this method105 to calculate converged relative hydration free energies of water and methane, using an MP2 ab initio QM model of water and methane, solvated by a periodic box of MM waters. While this was a nonbiological application, we are currently using this method to calculate QM/MM relative binding free energies of protein-ligand systems (using a DFT QM model of the ligand and an MM model of the protein and explicit solvent), and are planning to use it to perform some computational enzymology calculations. Chem. Modell., 2008, 5, 13–50 | 25 This journal is

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3. Interfacing atomistic with coarse grain models In mid 2007, Leontiadou, Mark and Marrink110 produced a paper in which they used atomistic molecular dynamics simulations to model the effects of ionic concentration on the transport of ionic species across a pore in a lipid membrane. This work involved several large simulations, involving 128 dipalmitoylphosphatidylcholine (DPPC) lipids (each containing 130 atoms) and about 6000 water molecules. This work pushed the limits of what is achievable with current atomistic molecular dynamics, and, by using approximations such as modelling long-range electrostatics using a reaction field, and using bond constraints such that a 5 fs integration timestep could be used, they were able to run several simulations of between 50 ns to 100 ns in length. Despite the impressive size of these simulations, they are still limited to biologically small length and time scales. 128 lipids is merely an 8 8 membrane bilayer, which is too small to model effects such as membrane curvature or membrane waves.111,112 100 ns is also too short a time to capture events such as membrane protein aggregation or lipid raft formation within a membrane.111 Coarse grain models provide a route to longer time and length scales in biomolecular simulations. Coarse grain models are a class of mesoscale model that work by grouping several atoms together and modelling them as a single interaction site. In effect, groups of atoms are smeared together into beads. For example, a coarse grain model could be constructed that represents an amino acid residue as a single bead, and a protein as a string of beads. The use of coarse grain models reduces the computational expense of a simulation, as coarse graining reduces the number of interaction sites. In addition, CG models contain fewer degrees of freedom, and use forcefields that lead to smoother potential energy surfaces. The smoother potential energy surface reduces the problems associated with frustration or non-ergodic trapping, thereby leading to improved sampling and a lower correlation time. Also coarse graining tends to remove the stiffest degrees of freedom from the model (e.g. the C–H bond vibrational modes), thereby allowing a CG model to use a larger timestep. All of these effects mean that CG simulations provide a route to modelling length and time scales that are far beyond that which is practically achievable via atomistic molecular dynamics. Coarse grain modelling is currently undergoing a renaissance, and there is now significant international effort being spent developing and applying coarse grain methods to model biological systems. It is not the purpose of this review to cover all of these recent developments in depth, so the interested reader is directed to several excellent modern reviews of the development and application of CG methods.113–115 Coarse grain models allow simulators to routinely access length and time scales that are not practically possible using atomistic modelling methods. However, in smearing out the atomistic detail, CG models run the possibility of missing out important atomistic effects, much in the same way that molecular mechanics models, in smearing out all of the electronic detail, can fail to model important electronic effects such as polarisation. There is now significant interest in interfacing atomistic and coarse grain models within a multiscale framework, so that this problem may be overcome. Just as there is significant variation in the type and strength of interaction in the different methods developed to interface QM and MM models, so too is there significant variation in the type and strength of interaction of the different methods of interfacing atomistic and CG models. The type of interfaces broadly fall into four categories, which are similar in nature and definition to the interfaces that have been developed for QM/MM interfaces: 1. One-way, bottom-up methods. These involve a single transfer of information from atomistic simulations or calculations to the CG simulation, e.g. by using an atomistic simulation to parameterize a CG model. 2. One-way, top–down methods. These involve a transfer of information from the CG simulation to the atomistic simulation, e.g. using a CG model to enhance


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sampling, and then using an atomistic model to explore interesting conformations that are discovered. 3. Two-way parallel methods. These involve the running, in parallel, of both a CG and atomistic model, and dynamically exchanging information between them at run time. 4. Two-way embedded methods. These involve the running of simulations in which an atomistic region (e.g. a membrane protein) is embedded within a CG model (e.g. a lipid membrane). Examples of each of these different types of interface will now be presented and discussed in turn. 3.1 One-way, bottom-up interfacing methods Coarse grain models can be interfaced using a one-way, bottom-up scheme. In this scheme, an atomistic model of the system (be it MM or QM) is used to parameterize a coarse grain model of the system. This idea is not new, and was the parameterisation method used by Levitt and Warshel116,117 when they first introduced coarse grain models in their pioneering 1970s papers. Levitt and Warshel’s 1975 paper introduced the first coarse grain model of a globular protein. It used two CG particles per residues: one that represented the a-carbon of an amino acid (Ca), and one that represented the side-chain atoms. A torsion potential acted about Ca particles, while a Lennard-Jones type potential acted between pairs of side chain particles. This CG model was built to represent bovine pancreatic trypsin inhibitor (BPTI), and was very successful, being able to correctly refold the protein starting from a completely denatured configuration. This work is interesting not only because it represents the first coarse grain biomolecular model, but also because it represents the first multiscale MM/CG simulation. Levitt and Warshel generated the parameters for their coarse grain model by averaging over interaction energies calculated using an atomistic potential. The parameters for the interaction potential between side chain particles were calculated by assuming that the side chains had spherical symmetry. The effective potential between identical side chain particles was calculated at various distances apart by summing the interaction energies of all of the atoms in one sphere with all of the atoms in the other sphere using a molecular mechanics style potential. This interaction potential between like-particles was then used to parameterize a Lennard-Jones type function, and the parameters between different side chain particles were then obtained using a geometric combining rule. The torsion potential between Ca atoms was also calculated by fitting to atomistic calculations, with the torsion potential between a pair of residues based on a time average of the energy calculated from atomistic simulations of a set of dipeptides. 3.1.1. Coarse grain models of lipid membranes. Levitt and Warshel’s method of obtaining CG parameters from underlying atomistic calculations can now be recognised as an example of a multiscale parameterisation scheme, where there is a one-way, bottom-up, flow of information from the atomistic to the CG calculation. Levitt and Warshel were ahead of their time in using a multiscale parameterisation scheme for their CG model. Coarse grain models only became popular for biomolecular modelling in the 1990s, which saw the beginning of the development of CG representations of lipids. Because the description of the development of coarse grain lipid molecules is outside the scope of this review (excellent reviews of this subject113,115 exist already) only a brief history of CG lipid models will be presented, in particular to highlight how CG lipid models have recently moved to using multiscale parameterisation methods. The first applications of CG lipid models looked for qualitative, rather than quantitative predictions, and therefore did not use any information from atomisticlevel calculations in their parameterisation. One of the first models produced during this time was by Smit et al.118 This model was used to investigate the phases of an oil/ Chem. Modell., 2008, 5, 13–50 | 27 This journal is

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water/surfactant system. This was a qualitative model that used just two types of particle; an oil (o) particle and a water (w) particle. An oil molecule was represented using a single o particle, water using a single w, and a surfactant was modelled as a chain of two w particles followed by 5 o particles, all held together with harmonic springs. All particles interacted using a Lennard-Jones potential, with attractive o–o and w–w interactions, and a purely repulsive o–w interaction. This simple representation was qualitatively able to model micelle formation. Goetz et al.119 and Noguchi and Takasu120 also developed qualitative CG models of amphiphiles, and used them to investigate phenomena such as bilayer and vesicle formation. One of the problems with a CG model is that the form of the interaction potential between CG particles is not clear, and there are questions over whether simple potentials are sufficient. Noguchi and Takasu developed their qualitative model of amphiphiles using custom potentials; a pairwise exponential term to give a soft-core repulsive shape to the CG particles, and a many-body attractive potential that aimed to model the hydrophobic effect. Using these potential forms, and a small number of adjustable parameters, they were able to simulate the self-assembly of bilayer vesicles. Brannigan and Brown121 also developed a pure CG model with a complex form. They modelled lipids as soft spherocylinders that interacted via a short-range r 8 repulsive term and an isotropic attractive alignment potential, which was based on an r 2 attractive term that was mediated by an angular dependence. These terms gave the lipids shape and encouraged them to form an aligned lamellar phase. To encourage bilayer formation, they designated one end of the spherocylinder as the lipid tail, and added a r 6 Lennard-Jones type attractive term between tails. This model resulted in a very small number of parameters, which allowed its phase behaviour with respect to a parameter search to be investigated. With the right parameters, this model was observed to self-assemble into a bilayer. Michel and Cleaver122 were still investigating the correct functional form to use for coarse grain amphiphile molecules in 2007, in a paper that investigated the use of the GayBerne123 potential. The Gay-Berne potential is effectively an anisotropic version of the Lennard-Jones potential, where the r 12 repulsive and r 6 attractive terms are attenuated by the angle of interaction between two particles (using the dot product of the particles alignment vectors). This has the effect of elongating the LennardJones sphere along an alignment vector and turning it into an ellipsoidal rod. Michel and Cleaver performed a parameter space search for Gay-Berne particles, and were able to qualitatively observe several different liquid crystal phases. Despite Levitt and Warshel’s demonstration of the effectiveness of using quantitatively parameterized CG models, it took until 2001 before other workers began developing quantitative CG models for biomolecules. The Shelley model, developed by Shelley et al.124,125 was the first quantitative CG biomolecular representation that was used to model lipid membranes. Shelley et al. developed a CG model of the lipid dimyristoylphosphatidylcholine (DMPC). They represented the 46 atoms of DMPC with just 13 spherical interaction sites. Four sites were used for each hydrocarbon chain (three chain SM sites, and one terminating ST site), one site each for the esther link (E1), one site for the glycerol backbone (GL) one site for the choline (CH) and one site for the phosphate (PH) in the headgroup (see Fig. 3). In addition, they also used a spherical W particle, that represented a grouping of three water molecules. The interaction potentials between these spherical groups were based on LennardJones potentials, e.g. interactions between W particles used a LJ 6–4 potential (an r 6 repulsive term coupled with an r 4 attractive term). The parameters for these LJ potentials were obtained by reproducing thermodynamic properties, e.g. the sWW parameter was set to reproduce the experimental density of water at 303.15 K, while calculations of the vapour pressure of a box of W particles, where used to set eWW such that the experimental boiling point of water was reproduced. A Lennard-Jones 9–6 potential was used between non-bonded hydrocarbon particle sites (SM and ST), which the LJ parameters obtained by matching the densities and vapour pressures of boxes of nonane and dodecane. The Shelley CG model was mostly 28 | Chem. Modell., 2008, 5, 13–50 This journal is

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Fig. 3 A comparison of different coarse grain lipid models. The Shelley model124,125 of DMPC, and Marrink126 and Essex127 models of DPPC are compared to their atomistic equivalents (for ease of comparison, hydrogen atoms of the atomistic models are not shown). Solid lines represent harmonic bonds connecting CG particles, and the CG particle types for the Shelley and Marrink models are labelled (the labels are the same as those used in the main text). The point charges (represented by + and ) and point dipoles (represented by arrows) are shown for the Essex model (the charges and dipoles are located at the centre of their associated CG particle). The Shelley and Marrink models use LJ particles (represented by spheres), while the Essex model uses a combination of LJ particles (spheres) and Gay-Berne particles (ellipsoids). Finally, the ‘blob’ model proposed by Chao et al.128 is also shown for comparison. This model represents groups of atoms as rigid non-spherical ‘blobs’ that use interaction potentials based on multipole expansions.

parameterized from experimental and thermodynamic data, similarly to how the original atomistic molecular mechanics forcefields were parameterized from experimental and thermodynamic data. However, due to the importance of correctly modelling the interactions between head group particles, Shelley et al. did use a multiscale parameterisation method to generate the parameters for the head group particles from an underlying atomistic simulation. The interactions between head group particles were based on radial distribution functions (RDFs) between head group particles calculated from atomistic simulations. These RDFs were tabulated, and then used to create a potential of mean force (PMF) which was used as the interaction potential between the head group particles. Use of this PMF led to RDFs that disagreed with the atomistic simulation, so corrections to the interaction potential were calculated by comparing the atomistic and CG RDFs, and running an iterative scheme that updated the interaction potential until the differences Chem. Modell., 2008, 5, 13–50 | 29 This journal is

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between the atomistic and CG RDFs were minimised. The model semi-quatitatively reproduced the density profile of an aqueous DMPC bilayer, and could be used to simulate the self-assembly of the bilayer. Shelley et al. used this model to investigate diffusion of halothane through the membrane.125 Halothane was modelled using a LJ 6–4 potential, which was parameterized by matching the experimental density and boiling point of the molecule, but a lack of experimental data prevented the detailed parameterisation of the interactions between halothane and the particles in the lipids, and this led to a model which was shown to not be truly predictive. Lopez et al.129 used existing beads from the Shelley DMPC model to model a synthetic antimicrobial polymer, which was inspired by a natural antibiotic. They then qualitatively investigated the interaction between the antimicrobial model and a bilayer of Shelley DMPC molecules. Despite the quantitative foundations of the Shelley model, it has several limiting features which arise from the use of multiscale parameterization:126 the model is optimized for the bilayer phase of DMPC only (as it was an atomistic simulation of this phase that led directly to the head group parameters), the interaction potentials are more complicated, so short time steps must be used in the dynamics integrator, and it evaluates long-range interactions, so it is not possible to reduce the computational expense of the calculation by using a short-range cutoff. These disadvantages of multiscale parameterisation, the lack of portability and the complexity of the interaction potential, led Marrink and co-workers to develop a simple, experimentally parameterized CG lipid model.126 The Marrink model is semi-quantitative, and was developed to target four goals; speed, accuracy, applicability and versatility. Marrink et al. achieved speed by using only short range potentials, which were also very smooth, so that large integration timesteps may be used. Accuracy was maintained by deriving the parameters to match experimental results and by comparing results calculated using their model to results from equivalent atomistic simulations. The simplicity of the forcefield (it is composed as a set of generic particles), together with its flexibility to represent chemical groups achieved the aims of applicability and versatility. The coarse grain model is constructed as follows; on average four atoms are represented by a single interaction site. To keep the model simple, only four main types of interaction site are used; polar (P), nonpolar (N), apolar (C) and charged (Q). The nonpolar and charged groups are then futher divided into subtypes, which depend on whether or not hydrogen bonding is possible to these groups. Fig. 3 shows how a DPPC lipid is represented using this model. The coarse grain particles interact using a LennardJones function. However, only five levels of interaction are defined; attractive, where e = 5 kJ mol 1, semi-attactive, where e = 4.2 kJ mol 1, intermediate (e = 3.4 kJ mol 1), semirepulsive (e = 2.6 kJ mol 1) and repulsive (e = 1.8 kJ mol 1). The level of interaction between a particular pair of coarse grain particles is chosen from a table look-up based on the types of the interacting sites, e.g. a pair of P particles have an attractive level of interaction, while a P and a C have a repulsive level of interaction. The complete lookup table is provided in the original paper describing this model.126 These parameters were obtained by trial and error, running multiple oil and water CG simulations and modifying the small number of parameters to reproduce experimental properties (e.g. density of water and alkanes at room temperature, mutual solubility and diffusion rates). For all interaction levels, the same value of s was used (0.47 nm). To keep the interactions short-ranged, they were force shifted to zero between 0.9 nm and the user-defined parameter rcut using the standard Gromacs130,131 shift function. All coarse grain particles except for nearest neighbours interact through this Lennard-Jones potential. Nearest neighbours are connected via a weak harmonic spring, while next-nearest neighbours are connected via a harmonic angle potential. In addition to these interactions, and unusually for CG lipid models, charged (Q) particles were included, which interact via a Coulomb potential which used a relative dielectric constant of er = 20, to implicitly account for charge screening. This electrostatic interaction is also shifted using the standard 30 | Chem. Modell., 2008, 5, 13–50 This journal is

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GROMACS130,131 shift function between 0 nm and rcut. The Q particles were designed to represent groups that have a full charge, and the groups full charge should be used. The exception are small hydrated ions, that use a reduced charge, to take into account the effect of an implicit hydration shell. Energy, density, length, temperature and pressure all have the same meaning for in the Marrink CG model as they do for any atomistic MM model. Time is different though, as the CG interaction potential is very smooth, and therefore conformational sampling is faster as the system is not trapped in local minima. This means that CG time is probably 3–6 times faster than atomistic MD time. To achieve parity, Marrink et al. scale time by four. Part of the reason for the success of the model is that the smooth potential surface allows large timesteps to be used with the dynamics integrator—timesteps up to 50 fs can be used, though 40 fs gives more stable results. The Marrink model is implemented in Gromacs,130,131 and the input parameters for this model can be downloaded from http://md.chem.rug.ml/marrink/ coarsegrain.html. The Marrink model has been used in several applications110,132–134 and has been compared against a united atom atomistic force field (GROMOS)135–137 (a united atom MM forcefield is one in which non-polar hydrogen atoms are not modelled explicitly, but are instead combined into the representation of their bound heavy atom, e.g. ethane is represented as a pair of CH3 particles). This comparison included an estimate of the loss of configurational entropy incurred in moving from the atomistic to coarse grain model.135,136 These comparisons showed that while the CG model compared well, it failed to fully represent some thermodynamic properties, e.g. oil/oil interactions were too weak in the CG model, and water/oil repulsion appeared to be overestimated.137 In addition, the CG model showed less enthalpy/ entropy compensation than an atomistic model for the mixing of oil and water, leading the authors of the comparison to suggest that accurate solvation thermodynamics should be employed to help in the reparameterisation of CG models. 137 Marrink et al. have now performed this reparameterisation and have created the MARTINI forcefield.138 The MARTINI forcefield was parameterized in a consistent manner to reproduce the partitioning free energies between polar and apolar media of a large number of chemical compounds. The advantage of the Marrink model is that it provides a set of basic CG bead building blocks that may be used to create different types of molecules. Bond et al.139,140 used these building blocks to construct CG models of membrane proteins, so as to investigate membrane-protein insertion. They compared their results to fully atomistic simulations, and found qualitative agreement. Marrink and co-workers have also now parameterized CG beads to represent amino acid residues in proteins,134 and have used these parameters to perform a CG study of the selfassembly of G protein-coupled receptors in membrane bilayers.

3.1.2. Multiscale CG parameterisation. Despite the problems of multiscale parameterisation, e.g. the lack of portability of the parameters and the potentially complex form of the interaction potentials, and the success of the experimentally parameterized Marrink CG model, a trend over the last few years is that CG models of lipids have begun to move towards using parameters that are derived systematically from atomistic simulations. This is because multiscale parameterisation provides a route to systematically obtain CG potentials for mixed or complex systems, or for cases where experimental data is difficult to obtain. Parameterisation methods are appearing that use the type of multiscale one-way atomistic to CG information flows that were first used by Levitt and Warshel in the 1970s. This trend parallels the historical development of atomistic MM force fields, which also moved away from complex experiment-based parameterisation schemes towards using wellvalidated recipes (e.g. GAFF44) to get MM parameters from QM calculations. CG parameterisation schemes now seem to be following this trend, and systematic Chem. Modell., 2008, 5, 13–50 | 31 This journal is

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methods to obtain CG parameters from atomistic MM simulations are now becoming popular. Izvekov and Voth141,142 have developed a method that automates the generation of CG potentials and parameters from an atomistic MM simulation. Their method, which they term ‘‘multiscale coarse graining’’ (MS-CG), automatically generates pairwise interaction potentials for a defined CG model based on results from an atomistic simulation. They have developed a force matching method, which is an extension of the least-squares force matching approach developed by Ercolessi and Adams.143 The method is used to coarse-grain an underlying atomistic simulation trajectory. As in other coarse grain models, the atoms are combined together into beads, and the system as a whole is therefore represented as a set of interacting beads. The potential of mean force is estimated between each bead based on the average forces experienced during the atomistic trajectory. This is evaluated using the atomistic potential energy surface, and CG potentials are derived that minimise the difference between the CG force and the average atomistic force. An analytical derivation of the method has been published,144,145 and statistial mechanical justification of using pair potentials for the CG particles has been developed.146 If the CG potential depends linearly on the fitting parameters, then the fitting of the parameters can be written as a series of over-determined linear equations, and the least-squares solution to which can be found by orthogonal matrix triangulation (QR decomposition).141 The efficiency of the fitting can be enhanced by breaking the entire atomistic trajectory into small chunks and generating the CG parameters for each chunk independently. The final CG parameters are then found by averaging the parameters over each chunk. This method is very efficient, but it requires that the CG potential depends linearly on the CG parameters. This is achieved by using a series of splines, which are linear in the spline parameters, and combining this with a Coulomb potential for the electrostatics. The effect of this coarse-graining is to return a model of the forces between pairs of CG beads that accurately reflects the average forces experienced between the atoms contained within those beads during the course of the atomistic molecular dynamics trajectory. Izvekov et al. applied this method to generate a CG potential for a DMPC bilayer. They collapsed the forces over a 400 ps atomistic trajectory onto a bead representation of DMPC that was similar to the Shelley125 or Marrink126 models. These forces were input to the force matching equations to yield the CG potentials. Despite the fact that some of the CG beads were charged, they were able to omit the Coulomb term from the fitting, as it was found that due to charge screening, the effect of electrostatics could be handled sufficiently by the short range spline functions of the MS-CG forcefield. This method has since been used to derive a CG model of carbonaceous nanoparticles,147 monosaccharides in water148 and of ionic liquids.149 Because this method systematically derives CG parameters using only an atomistic simulation, it is much more straightforward to develop parameters for mixed systems, for which experimental data is either unavailable or difficult to obtain. Izvekov et al. demonstrated this advantage by deriving a CG model for a mixed DMPC/cholesterol bilayer.150 MS-CG effectively provides a CG model of a system that approximates the free energy surface. However, dynamics simulations using the MS-CG model cannot be related to the true dynamics of the atomistic model. Izvekov et al. have tried to overcome this problem by using the atomistic simulation to parameterize friction coefficients which can be used with a Brownian dynamics integrator.151 While this has only currently been applied to liquid methanol, they plan to apply this method to membrane systems. Lyubartsev152 has also developed a multiscale parameterisation method that has been used to systematically build a CG model of a DMPC bilayer. Lyubartsev uses an inverse Monte Carlo method153 to generate the CG parameters from an underlying atomistic simulation. The atomistic simulation trajectory is analysed to generate the radial distribution functions (RDFs) for the CG bead model. These RDFs can be converted into pairwise interaction potentials between the beads. The 32 | Chem. Modell., 2008, 5, 13–50 This journal is

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RDFs calculated from a CG simulation using these initial interaction potentials differ from those calculated from the atomistic simulation. An inverse Monte Carlo algorithm153 is therefore used to iteratively refine these interaction potentials by correcting them by the difference between the CG and atomistic RDFs. This is essentially the same method that was used by Shelley et al.125 to derive the parameters between the CG lipid head group particles, and is also similar to the Boltzmann inversion method,154 which also uses an iterative procedure that uses RDFs measured from atomistic simulations to derive CG interaction potentials. Lyubartsev has used this method to fully parameterize his own CG model of DMPC. Arkhipov et al. have taken multiscale coarse graining to an extreme in their recent work,155 in which they use atomistic models to parameterize CG models of a complete virus capsids. Each CG particle in the simulation represented about 200 atoms. Each CG particle interacted via a Lennard-Jones potential, which was parameterized to match the size of the domain the CG particle represented, as calculated from the radius of gyration of that domain measured from an atomistic model. The resulting CG model was able to simulate complete virus capsids (of dimensions 10 nm to 100 nm) over timescales of 1 ms to 10 ms. Chao et al.128 have developed a CG model that is based on the concept of soft rigid blobs. A blob is a grouping of atoms within a molecule that interacts with other blobs using a non-spherical, soft potential. A molecule is divided into blobs, e.g. see Fig. 3 for a proposed blob model of a phospholipid.128 Each blob represents a rigid grouping of atoms, and the interaction potential between a pair of blobs is calculated based on the underlying atomistic potential energy of interaction between the atoms within those blobs. However, unlike other CG models, which try to approximate this interaction potential using high symmetry beads, the rigid blob model expands this interaction potential as a Taylor series to produce interaction moment tensors between blobs. These are akin to the multipole tensors in a standard multipole expansion. The result is that the blob potential more accurately represents the three dimensional shape of the group of atoms that it represents. A further advantage is that the quality of the model can be increased by adding higher-order moments to the interaction potentials. 3.2 One-way top–down interfacing methods The multiscale atomistic/CG methods discussed so far all involve a single, one-way transfer of information from the atomistic model to the CG model. Essentially the atomistic models are used to parameterize a CG model, so that the CG model can then be used to explore the conformational space of the system more rapidly. However, in building the CG model, the atomistic fine detail of the system has been lost, as it has been smeared out into a set of beads. Recovering the atomistic fine detail would however be very useful, as this would allow the simulator to zoom back in and observe interesting configurations revealed during the CG trajectory at an atomistic level of detail. Knecht and Marrink133 realised the advantages of this approach, and have used it together with the Marrink CG lipid model to model vesicle formation. The CG model is used to self-assemble a vesicle, and this CG structure is then used as a template for the starting structure for an atomistic simulation. The conversion from the CG model to the atomistic model is straightforward. The coordinates from the CG simulation were scaled by a factor of 1.6. Atomistic models of the lipid, taken from an atomistic simulation of the planar bilayer, were fitted on top of their CG counterparts. The CG model had to be scaled to prevent steric clashes caused by the curvature of the CG vesicle bilayer. Molecular dynamics simulations were run on the atomistic model, allowing vesicle membrane fusion to be studied at the atomistic level. It is relatively straight-forward to move from a coarse grain model of a lipid to a fully atomistic model, as the main features of the lipid, namely head group and two tails, are still present in the CG model. It is less straight-forward to reconstruct the Chem. Modell., 2008, 5, 13–50 | 33 This journal is

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atomistic coordinates of a protein from a CG model. This is because many CG protein models represent only the backbone of the protein, and if the side chain is represented, it is done so using only one or two beads. There is a lot of atomistic structural information missing from a CG protein model, and reconstructing this fine detail is not trivial. McCammon and co-workers156 attempted in their multiscale exploration of ligand binding to HIV protease to map from a single residue per bead protein CG model back to an atomistic representation. The binding site of HIV protease is covered by two large flaps, which open and close on a timescale that is beyond that is accessible on the atomistic molecular dynamics timescale. McCammon and co-workers have, over a series of papers,157–162 developed a very successful CG model of HIV protease. This model uses a single bead per residue, and was parameterized based on a statistical analysis of the crystal structure of the protein. This CG model is capable of simulating opening and closing motion of the large flaps that gate the entrance to the active site.158,162 McCammon and co-workers used their CG model within a multiscale simulation to investigate the binding pathways of an HIV protease inhibitor. The CG simulation was used to investigate opening and closing of flaps which gate the binding site of the enzyme, to see how the presence of the inhibitor affected flap dynamics, and to test whether the inhibitor could bind with the flaps closed. Because the CG model lacks atomistic detail, McCammon and co-workers selected two configurations from the CG trajectory to act as starting points for fully atomistic, implicit solvent Brownian dynamics simulations of the protease plus inhibitor. While it was straight-forward to rebuild a fully atomistic model of the inhibitor from the CG model, it was not simple to rebuild an atomistic model of the enzyme. This is because the CG model is missing lots of important structural information, e.g. the orientation and location of the amino acid side chains. To avoid the difficulty of rebuilding the atomistic detail of the enzyme, they instead used crystal structure configurations of the protease that had backbone geometries that were similar to those observed in the two snapshots from the CG trajectory. McCammon and co-workers156 used the CG simulation to provide input to a pair of atomistic simulations, thereby performing a pair of one-way, top–down information transfers from the CG to atomistic level. They sidestepped the great challenge of top–down transfers, namely the difficulty of recovering fine atomistic detail, by only using the CG model to locate the binding geometry of the inhibitor, and as a means to select which crystal structures corresponded to that particular binding geometry. Other workers have directly tackled the problem of reconstructing atomistic detail from CG models. The RACOGS algorithm (Reconstruction Algorithm for Coarse Grain Structures)163 is an example of a recently developed algorithm that attempts to solve the reconstruction problem. The algorithm is designed to rebuild an atomistic protein model given only the position of the Ca atoms. The algorithm has several stages; 1. The backbone atoms (C, O, N) are located based on the input positions of the Ca atoms using the algorithm developed by Feig et al.,164 which is itself based on the work by Milik et al.165 The algorithm places the backbone atoms into the average positions based on a statistical analysis of the positions of backbone atoms taken from 4013 non-redundant protein structures from the PDB. 2. The next step is the positioning of side chain atoms. These are positioned using the algorithm developed by Xiang and Honig,166 which uses a rotamer library to initially place the side chain atoms. The algorithm starts by adding the side chain atoms to a residue based on selecting the rotamer for that residue that has the lowest energy of interaction with the backbone atoms of the other residues. The interaction energy is evaluated using the vdW and dihedral energy terms from the AMBER99 forcefield.167 In performing this step, the rotamer library is pruned of any rotamers that involve a steric clash with the backbone. This helps improve the efficiency of the algorithm by removing any structurally unrealistic conformations as quickly as possible. 34 | Chem. Modell., 2008, 5, 13–50 This journal is

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3. Once all of the side chains have been constructed, an iterative procedure is employed whereby each residue is considered in turn, and its energy of interaction of all possible rotamers of its side chain with all of the other residues’ side chains and backbone atoms is calculated. If another rotamer in the library has a lower energy, then it is selected in place of the existing rotamer. This iteration is repeated until an entire cycle over all of the residues in the protein is completed without a single residue being replaced. 4. The next step involves a minimisation of any high-energy side chains (defined by whether the energy of interaction of that side chain with the rest of the protein is greater than a specified cutoff). Each high energy side chain is, in turn, energy minimised while holding the rest of the protein fixed. The minimisation is performed using the bond, angle, dihedral and vdW terms from the AMBER99 forcefield. 5. Finally, hydrogen atoms are added using the LEAP module of AMBER 8, 168 and the entire protein is energy minimised using the conjugate gradient method, and the full generalized born/surface area (GB/SA)169 implicit solvent AMBER99 forcefield. Heath et al.163 were able to use this algorithm to hop between CG and atomistic models of the proteins src-SH3, S6wt and S6Alz. One of the problems of hopping from a CG to an atomistic model is that, because different forcefields are used, the free energy surfaces of the two models could be very different. Thus configurations that are sampled by the CG model may not be important for the atomistic model. Heath et al.163 investigated this for the CG model (the Das et al. model170) and atomistic model (AMBER99167) that they used. They found good agreement between the free energy surfaces, and were able to successfully use the CG model to investigate the folding and misfolding of these proteins, and to use their RACOGS algorithm to zoom in to obtain the atomistic detail of the misfolded structures. Ding et al.171 have also developed a method to reconstruct the atomistic fine detail of a protein structure from a CG model. They use a CG model that uses the positions of the Ca and Cb atoms to investigate domain swapping in seven proteins. The CG model enables the running of simulations that cover the timescale on which the domain swapping is seen to occur. A fully atomistic model of the protein is then reconstructed from the configurations generated from the CG simulation. The reconstruction algorithm is composed of the following stages; 1. The positions of the Ca and Cb atoms from the CG model are used to position the N and C backbone atoms so that the correct chirality is maintained (D-amino acids). To restrict the rotational freedom, a strong constraint was used, namely that neighbouring residues must have a planar peptide bond. Harmonic potentials were added between neighbouring backbone atoms, the equilibrium distances of which were taken from average distances calculated from the PDB structures. The Ca atoms were then immobilised by setting their masses to infinity, and a short molecular dynamics simulation was then run to relax the system. 2. Backbone hydrogens were added to create non-specific backbone hydrogen bonds. A CG potential was added between the Cb atoms, and a short molecular dynamics simulation was run to optimise the backbone hydrogen bond network. 3. The side chain and backbone O atoms were added using rotamers and a Monte Carlo simulated annealing algorithm. The scoring function for the simulated annealing used the vdW interactions from the Cedar forcefield,172 the EEF1 implicit solvation model173 and a statistical potential for hydrogen bonds, as proposed by Kortemme et al.174 This reconstruction algorithm was also used by Sharma, Ding and Dokholyan175 to reconstruct the atomistic detail from CG models of nucleosomes. This algorithms is very useful, as, like the RACOGS algorithm, it allows the CG model to be used to sample conformational change on timescale beyond that which is accessible to standard atomistic methods, yet it then allows the simulator to zoom in and reconstruct the atomistic data for interesting configurations.


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3.3 Two-way parallel interfacing methods Multiscale modelling across the atomistic/CG boundary is not limited to single, oneway transfers of information. Two-way interfacing methods, whereby the CG and atomistic levels exchange information dynamically throughout a simulation, are in active development and use. Two main schemes for linking CG and atomistic simulations have become popular; parallel methods, which use loosely-coupled atomistic and CG simulations running in parallel, and embedded methods, whereby atomistic regions are embedded within a CG model. 3.3.1. Parallel atomistic/CG methods. The replica exchange method, discussed in section 2.4 in terms of atomistic and QM calculations, has also been applied with CG models. For example, Nanias et al.176 have used replica exchange moves over temperature to enhance sampling within a CG protein simulation. Just as there has been interest in developing replica exchange moves that map between QM and MM models,105 so to has there been recent interest in developing replica exchange methodology that can map between atomistic and CG representations. Such replica exchange moves represent a parallel, two-way interface between the atomistic and CG levels, as atomistic and CG simulations are run in parallel, and information is only periodically exchanged between them. Lyman and Zuckerman177 have developed a atomistic/CG replica exchange method that they call ‘‘resolution exchange’’ (ResEx). The aim of this method is to run both an atomistic and CG simulation of a system in parallel, and to periodically attempt replica exchange moves between the simulations. The idea is that the enhanced sampling of the CG model will be shared with the atomistic model via the replica exchange moves. For this method to work, the coarse grain model must use a subset of the coordinates of the atomistic model. In this case, Lyman and Zuckerman achieved this by using a united atom model at the CG level (OPLS united atom forcefield33), and an all-atom model for the atomistic level (OPLS all atom forcefield34). While technically being a CG model, a united atom forcefield is more normally considered to be an atomistic representation. This is because a united atom forcefield does not use beads to represent residues or parts of residues, but instead merely removes aliphatic and aromatic hydrogens from the model by collapsing them into their bonded carbon. Despite this, Lyman and Zuckerman’s method can still be considered as a multiscale method, and this application points to some of the difficulties that more ambitious applications would have to overcome. One major difficulty is that the replica exchange moves between the united atom and all atom representations had a very low acceptance ratio. This is because the CG model uses only a subset of the atomistic coordinates, so replica swap moves can lead to unphysical conformations as this subset is moved out of step with the rest of the atomistic coordinates. To overcome this problem, Lwin and Luo,178 in a similar method, minimised the energy of the atomistic model after each swap, but before the Monte Carlo test that was used to accept the configuration. While this does successfully increase the acceptance ratio of the move, it violates detailed balance, so does not sample the required ensemble correctly. Lyman and Zuckerman’s solution to the problem is to use incremental coarsening. With this method, instead of switching the whole protein between the CG and atomistic models in one go, instead individual residues in the protein are switched. This leads to the problem that CG residues then need to interact with atomistic residues, so CG/atomistic crossterms need to be included in the Hamiltonian. Fortunately the OPLS united atom and all atom forcefields are compatible, and so can be combined in this way, at least for this application. Now multiple replicas of the system may be used, with each replica representing a different percentage of the protein using the CG forcefield. Lyman and Zuckerman tested their method by application to the pentapeptide metenkephalin. They used six replicas, the first replica using a fully atomistic model, then second replica using one CG residue and four atomistic, the second using two 36 | Chem. Modell., 2008, 5, 13–50 This journal is

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CG residues and three atomistic, and so on. In addition, they also increased the temperature of each replica, such that while the fully atomistic model was simulated at 298 K, the fully CG model was simulated at 700 K. However, despite the small size of the system, the small difference between the CG and atomistic models (OPLSUA vs. OPLSAA) and the use of incremental coarsening, the acceptance ratio of the swap moves was still very low, running between 2.5% and 5.8%. Applications of this method to larger systems, or using a greater difference between the atomistic and CG models therefore looks problematic. Christen and van Gunsteren10 have developed a novel multiscale method that they call ‘‘multigraining’’, which aims to use the CG model to enable both relaxation of large molecular systems and sampling of slow processes with concurrent atomic detail representation of the results. In this method, both an atomistic and a CG model of a molecule are used simultaneously. Each molecule in the simulation has both a set of atomistic (fine grain) coordinates, rFG, and a set of coarse grain coordinates, rCG. The coarse grain coordinates are constrained to map to the atomistic coordinates. The Hamiltonian of the system is composed of atomistic interaction terms, UFG(rFG), and CG interaction terms, UCG(rCG). Molecular dynamics is then performed using both the atomistic and CG potentials simultaneously. This is referred to as multigraining MD, and the algorithm is as follows; 1. First the coarse grain coordinates are determined by mapping them from the atomistic coordinates (e.g. by placing CG beads at the geometric centre of a residue). 2. Calculate the forces on the CG particles using the CG potential energy terms, and the forces on the atoms using the atomistic potential energy terms. 3. Distribute the forces from the CG particles back onto the atoms. 4. Propagate the coordinates of the atoms using the leap-frog integrator. By running both the CG and atomistic models in parallel, and by mapping the CG coordinates from the atomistic coordinates, this method avoids all of the problems associated with information loss when moving back and forth between the CG and atomistic representations. Also, the atomistic and CG potential energy terms are separate, therefore avoiding the problem of parameterising atomistic/CG crossterms. Also, by keeping the atomistic and CG potentials separate, Christen and van Gunsteren are able to use a l scaling parameter to switch between the atomistic and CG potentials, e.g. at l = 0, only the pure atomistic potential is used, at l = 1 only the pure CG potential is used, while at l = 0.5, a 50–50 mix of the CG and atomistic potentials are used. Christen and van Gunsteren then employ replica exchange moves over this l coordinate, thereby allowing several parallel trajectories to swap dynamically between CG and atomistic representations. They tested the method by using it to simulate liquid octance, but like Lyman and Zuckerman, they experienced problems with a low acceptance ratio for the replica exchange moves, and needed to use 24 replicas across the l coordinate. Using 24 replicas, they achieved an average acceptance ratio of just 24%. This figure masks the very poor acceptance ratio for replicas at l = 0, where the purely atomistic potential was used. Just 8% of the 300 replica exchange moves were accepted at l = 0, which is disappointing, given that it is the sampling at l = 0 which is important for the calculation of correct thermodynamic properties. However this method is useful as a means of performing a rapid equilibration or conformation search at the CG level, and then smoothly switching back to the atomistic model to explore interesting configurations.

3.3.2. Embedded atomistics/CG methods. Parallel atomistic/CG methods still use a simple interface between the CG and atomistic models, with information exchange occurring only during the replica exchange moves. This represents a loose coupling between the atomistic and CG models. An alternative to this loosely-coupled method is to develop a tightly coupled method whereby an atomistic model is linked directly to a CG model. This allows for an atomistic region, e.g. an atomistic model of an ion channel, to be embedded within a CG simulation, e.g. a CG model of a cell Chem. Modell., 2008, 5, 13–50 | 37 This journal is

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membrane. This requires that the atomistic and CG models are closely coupled together, with information exchange occurring every timestep via atomistic/CG cross-terms in the Hamiltonian. This is challenging, as it is not immediately obvious how to obtain the parameters that describe the atomistic/CG interaction, nor what the form of the interaction potential should take. In section 3.1.2 the multiscale coarse graining (MS-CG) method, developed by Izvekov and Voth, was presented as a method that could be used to systematically derive the functional form and parameters for MM-CG interactions. Shi,179 working with Izvekov and Voth, has used the MS-CG method to systematically derive the functional form and parameters for the atomistic/CG interaction of an atomistic gramicidin A polypeptide ion channel immersed in a CG membrane. The DMPC membrane and solvent water were modelled using the CG model developed by Izvekov and Voth,150 which was developed using the MS-CG method. This method derives the parameters for the CG interactions from the average forces experienced by the atoms during an atomistic dynamics simulation. These forces are then coarse-grained and a CG potential is fitted. Shi ran a short all-atom simulation of the whole membrane/ion channel system and used this to develop the CG/atomistic forcefield using the MSCG method. The forcefield was obtained by treating the atomistic and CG regions equally within the force matching algorithm. However, to reduce computational cost, some simplifications and approximations were made. First, the forcefield was split into three parts; atom–atom, atom-CG and CG-CG. The atom–atom forces were assumed to be the same as those from the wholly atomistic simulation, while the CG-CG forces were taken from an MS-CG application of a pure lipid bilayer embedded in water. This meant that only the atom-CG forces had to be derived from the fully atomistic simulation of the entire system. The atom-CG forces were obtained by subtracting the atom–atom and CG-CG forces from the reference forces from the fully atomistic simulation, and then fitting the residual forces using the MS-CG method. This method is not as rigorous as applying the full MS-CG method in a single step, but it is much more efficient, and Shi et al. found that this strategy worked quite well for this system. Essex and co-workers have developed a CG model that is designed from the outset to interact with an atomistic forcefield.127,180,181 They have developed a CG lipid model of DMPC127,180 that uses six Gay-Berne123 ellipsoids to represent the hydrocarbon tails, and two Lennard-Jones (LJ) particles to represent the head group. The glycerol region was modelled by 2 further Gay-Berne units, thereby allowing the entire DMPC model to be represented by just ten interaction sites (see Fig. 3). The key advance of the Essex model is that it was designed from the outset to be compatible with atomistic forcefields. The Gay-Berne potential is essentially an anisotropic form of the Lennard-Jones potential, which is the most common potential used to model van der waals (vdW) non-bonded forces between atoms. This means that it is possible to interface the Gay-Berne and LJ potentials so that the vdW interaction energies between Gay-Berne beads and Lennard-Jones (LJ) atoms can be calculated. However, to correctly handle the electrostatic interaction between the CG and atomistic levels, the CG model has to explicitly include a representation of the charge distribution of the lipid. The Essex model achieves this by including point charges in the two head group particles, and point dipoles in the two GayBerne particles that represent the glycerol region. Water was modelled explicitly using the soft sticky dipole model182 (SSD), which despite only using a single interaction site, is capable of accurately reproducing the structural, thermodynamic, dielectric and temperature-dependent properties of water.182–185 The lipid model was parameterized by trial-and-error molecular dynamics, whereby the parameter space was searched and molecular dynamics simulation run to reproduce the experimental structure of DMPC bilayers (to test whether the desired experimental properties were obtained). The point charges and dipoles of the model were chosen to reproduce the net electrostatics (charge and dipole) of each bead from the underlying atomistic level model. To parameterize the strength of interaction between atomistic 38 | Chem. Modell., 2008, 5, 13–50 This journal is

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and CG molecules, Essex and co-workers181 calculated solvation free energies of atomistic amino acid side chain analogues in boxes of CG octane, and compared this to experiment. The atomistic and CG interactions were found to be unbalanced, but Essex and co-workers found that simple scaling of the cross terms led to good agreement of comparable quality to atomistic simulations. To calibrate the scaling factor for the interaction of an atomistic inclusion with the CG water model, a similar set of calculations were made of the hydration free energy of atomistic amino acid side chain analogues in the SSD water model. The calculated hydration free energies were also compared with experiment. The calculation of these scaling factors removes the need for the atomistic forcefield to be reparameterized for use with the CG model. This is an interesting approach to the solution of the cross-term parameterisation problem of atomistic/CG multiscale modelling, and has the promise to be transferable across a range of solutes and proteins, thereby removing the need to reparameterize the model for each new system under study. The challenges in creating atomistic/CG multiscale methods parallel those of creating QM/MM methods. The definition of the QM/MM interface is relatively straightforward if the divide is between molecules. However the definition of the interface becomes more complicated if the divide is within a molecule, as then the Hamiltonian has to somehow include bonded terms between QM atoms and MM atoms. In much the same way, the definition of the atomistic/CG interface is more straightforward if it falls between molecules, rather then within a molecule, and that is why the majority of atomistic/CG multiscale simulations avoid this approach. The exceptions are the parallel method developed by Lyman and Zuckerman177 (the ResEx method discussed in the last section) and an embedded CG/atomistic method developed by Neri et al.186 Neri et al. have developed a method where a protein is modelled using a CG representation, but the active site is modelled using an atomistic representation. The aim is to allow efficient sampling of protein conformational change, due to the use of the CG model, but to still capture the fine-detail atomistic motions in the active site, which is the area of interest in protein-ligand binding or computational enzymology calculations. They tested the method by application to two proteins of active pharmaceutical interest: HIV protease, which is a target for anti-HIV medication, and human b-secretase (BASE) which plays a role in the onset of Alzheimer’s disease. The active site atoms were modelled using the GROMOS9630 atomistic forcefield, while the rest of the protein was modelled at the CG level by only considering Ca centroids. Solvent-protein interactions were modelled in terms of viscosity and the addition of random forces in the framework of a stochastic dynamics simulation. The interaction between the atomistic and CG regions was handled by the addition of an interface region. Several residues at the boundary of the atomistic and CG regions were modelled using both a CG and atomistic representation. The atomistic region residues interacted with the interface residues using the standard atomistic potential. To maintain backbone integrity between the interface and CG regions, harmonic bonds are added between the C a atoms of the interface residues with the Ca atoms of the neighbouring CG residues. In addition, an exponential-form non-bonded potential is added between CG and interface residues, and the parameters for the CG model and for the interface were chosen so as to reproduce the root mean square fluctuations (RMSF) of HIV protease as observed during fully atomistic and fully CG simulations. A key parameter is the thickness of the interface region, with the key concerns being that the interface region has to guarantee the correct geometry of the atomistic residues, it has to get the local electrostatics right, and it has to be able to transmit the modes of vibration experienced by the rest of the protein, and which are mimicked by the CG model. Neri et al. found that for both HIV protease and BASE they needed to use an interface region that included all residues that had at least one atom within 6 A˚ of any of the residues in the atomistic region. There is a clear separation in the methods presented so far between the atomistic and CG regions. One molecule, or part of one molecule (the protein), is treated at the Chem. Modell., 2008, 5, 13–50 | 39 This journal is

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atomistic level, while the remaining molecules (lipid membrane and solvent) are modelled using CG techniques. An alternative to defining the atomistic and CG levels by molecule is instead to define the levels by region, namely to divide the simulation space into an atomistic region and a coarse grain region. This creates a more dynamic split between the two models, as individual molecules are free to diffuse from one region to another, and therefore algorithms must be developed that account for the interconversion of CG and atomistic models. Praprotnik et al. have, over a series of insightful papers,187–191 developed just such algorithms. They have created a multiscale simulation method whereby the simulation space is divided into an atomistic region and a CG region, and the molecules in the system are free to diffuse between these two regions. The methods developed allow on-the-fly interchange between the molecules’ coarse grain and atomistic representations, enabling large length and time scales to be achieved, while still retaining the atomistic fine detail of parts of the system. The key development in this work is the methodology used to allow molecules to switch dynamically between CG and atomistic representations. They call these methods AdResS (adaptive resolution scheme). The original application of the method was to the multiscale dynamics of liquid methane.187 A CG model of methane was constructed based on the radial distribution function of methane obtained from a simulation of the reference atomistic model. The simulation box was split in half, creating an atomistic region and a CG region, and an identical number of methane molecules were placed in each half of the box. To ensure that the transition between the two regions was smooth, a ‘handshaking zone’ was created between the atomistic and CG regions. To achieve a smooth transition from the atomistic to CG forcefields, the forces are scaled in the handshaking zone. As the switching of the resolution can be considered as a first order phase transition, it is necessary for this scheme to be used in combination with a thermostat. Because the latent heat is generated at the interface region,187 it is important that a local thermostat is used that couples directly to local particle motion, e.g. Langevin or dissipative particle dynamics (DPD) thermostats. The other main challenge of an adaptive resolution method is how to handle the loss and reintroduction of atomic degrees of freedom at the atomistic/CG interface. It is important that the vibrational and rotational degrees of freedom at the atomistic model, which are missing in the CG model, are slowly reintroduced as a molecule diffuses across the interface between the CG and atomistic regions. To achieve this, when a molecule moves from the CG to atomistic region it is mapped onto an atomistic representation that has the same centre of mass and linear momentum as the CG representation. In addition, each atom is given rotational and vibrational velocities that come from a random molecule already in the atomistic region. Praprotnik used AdResS successfully to simulate a multiscale box of liquid methane.187 While the application worked well, there was a small problem with a pressure imbalance at the interface zone between the CG and atomistic regions. Praprotnik et al. solved this problem in a new version of their method,188 in which they also extended the scope of application to include spherical boundaries between the atomistic and CG regions. Since then, they have used the method to run multiresolution simulations of liquid water191 (using the CG parameterisation method of Lyubartsev152 to get the CG model of water) and have investigated the thermodynamic implications of running simulations that have a dynamic, and indeed fractional number of degrees of freedom.189 Ensing et al.192 have also developed a new multiscale method that allows for an atomistic region to be embedded within a CG simulation. Like AdResS, the method uses a spherical atomistic region that is interfaced to the CG simulation via a handshaking buffer zone. To ensure a smooth transition from the CG to atomistic representations, the potential energies, rather than the forces, are smoothly scaled from the CG to atomistic Hamiltonians as the molecule crosses the handshaking zone. Additional terms are also added to the Hamiltonian that account for the kinetic energy that is lost when the number of degrees of freedom is decreased when 40 | Chem. Modell., 2008, 5, 13–50 This journal is

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a molecule moves from the atomistic to the CG region. These additional terms ensure that the total energy of the system is conserved even as the number of degrees of freedom change. In addition to using a different method to switch between the CG and atomistic regions, Ensing et al. also use a different method to handle the change in dynamics between the CG and atomistic regions. Instead of destroying and recreating the atomistic degrees of freedom as the molecules move across the boundary, they instead use the RESPA193 multiple timestep molecular dynamics algorithm to freeze out the intramolecular degrees of freedom of molecules as they move into the CG region. RESPA is a molecular dynamics method whereby different timesteps can be used to sample different parts of the Hamiltonian. It has traditionally been used to separate the sampling of ‘‘fast’’ forces, such as bond potentials, from ‘‘slow’’ forces, such as those arising from the non-bonded potentials. Ensing et al. cleverly use RESPA to separate the sampling of the CG and atomistic Hamiltonians by region, and thereby effectively freeze the atomistic representation as it crosses into the CG region. This allows them to tune the molecular dynamics integrator optimally across the entire multiscale system. While this method has been applied only to a model system and liquid methane, it shows great promise as a method that could find widespread use in biomolecular modelling.

4. Interfacing particle with continuum models Continuum methods provide the largest time and space ranges considered in this review. While continuum models have been popular for the modelling of solids194 or of liquid or gas flow,195 they have only recently been applied to the study of biomolecular systems. This is because continuum methods tend to be best suited to periodic or smoothly varying systems, and biomolecular systems are distinctly inhomogenous, and exhibit physical behaviours that require the the modelling of fine-level atomistic detail. By far the most popular continuum models used in biomolecular modelling are those that replace atomistic waters with an implicit solvent model. Many implicit solvent models, which include the Poisson Boltzmann (PB)196,197 and Generalized Born (GB)198,199 methods, model the solvent as a dielectric continuum, and calculate the polarisation of that continuum caused by the charge distribution of the atomistic molecule model. This polarisation leads to an electrostatic reaction field with which the atomistic molecular model then interacts. Implicit solvent models have been used for many years in combination with QM,200,201 MM202,203 and CG204,205 calculations. These represent a large class of truly multiscale methods, and have been reviewed in detail many times.204,206–209 Implicit solvent models have been the dominant class of multiscale continuum methods over recent years. However exciting new classes of multiscale continuum models have recently been developed. These new methods fall into the following categories, which are of similar definition and type to the interfaces used between the QM/MM and atomistic/CG levels: 1. One-way bottom-up interfacing methods. These involve a single transfer of information from the atomistic or CG level to the continuum level, e.g. by using the atomistic level to provide parameters for the continuum model. 2. Two-way embedded interfacing methods. These involve embedding an atomistic or CG model within a continuum representation. Implicit solvent models fall into this category. New multiscale methods, which capture hydrodynamic and mechanical effects have now also been developed. 3. Two-way parallel interfacing methods. These involve the running of an atomistic or coarse grain level simulation in parallel with a continuum calculation, and interfacing the two using a lightly coupled interface, e.g. by using properties calculated from the continuum model to update the boundary conditions of the atomistic or CG simulation. Examples of each of these categories of multiscale continuum interfaces will now be discussed in turn. Chem. Modell., 2008, 5, 13–50 | 41 This journal is

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4.1 One-way bottom-up interfacing methods Continuum models can be interfaced with atomistic models using a one-way bottomup multiscale parameterisation scheme. In this scheme, an atomistic level calculation is used to provide the input parameters for the continuum model. This scheme is very simple, as it involves a single transfer of information from the atomistic level to the continuum level. Tang et al.210 have recently employed this scheme to get the parameters for a continuum model of a mechanosensitive ion channel embedded in a lipid membrane. The membrane was modelled as a homogenous elastic sheet of thickness 35 A˚, and area of 400 A˚ 400 A˚, using a finite element (FEM) model. The transmembrane helicies were represented by homogenous cylindrical elastic rods of diameter 5 A˚with spherical caps at each end. The helicies were embedded within the membrane, and were also modelled using a FEM model. What makes this a multiscale simulation is that the parameters for this model were calculated using information calculated using atomistic-level models. This represents a one-way, bottom-up information flow from the atomistic level to the continuum level. Elastic parameters for the model were calculated from the mechanical properties of the membrane and helicies measured by experiment and calculated from atomistic molecular dynamics simulations. The interactions between the helicies and the membrane were modelled using a Lennard-Jones style attractive/repulsive potential. The parameters for this potential were determined based on molecular mechanics calculations run using the CHARMM forcefield.27 The channel is comprised of ten helicies. The interaction potential between helicies was parameterized by fitting to CHARMM potential energy calculations. For each pair of helicies, the interaction energy in vacuum was calculated using the CHARMM19 forcefield, with the coordinates set to either X-ray or homology structures. This calculation was performed for different combinations of helix pairs, which effectively samples different orientations, and a range of centre of mass inter-helix separations that varied between 20 A˚ and 20 A˚. The interactions between each helix and the membrane were parameterized based on calculating the insertion energy profile of moving an atomistic model of the helix from an implicit solvent water model into an implicit lipid membrane model. In the implicit membrane models the membrane thickness was taken to be 23.5 A˚, which corresponds to the thickness of the hydrophobic part of the membrane. While this model was constructed as a proof of concept of FEM methods, it was able to reproduce the experimental gating tension of the channel, and the structural variations along the gating pathway as observed from biased atomistic molecular dynamics simulations. This application successfully demonstrated the potential of continuum models of biomolecular systems to reach length (sub-mm) and time (multi ms) scales that are not accessible to atomistic level calculations. 4.2 Two-way embedded interfacing methods Continuum models can be directly interfaced with atomistic or coarse grain models using a two-way embedded interface. In this scheme, the atomistic or CG model is embedded within a continuum model. Implicit solvent methods, in which an atomistic or CG model of a solute is embedded within a continuum model of the solvent, are popular and well-established examples of this type of interface. Implicit solvent models represent the solvent as a dielectric continuum, and allow the electrostatics of the atomistic or CG solute to polarise the continuum, which then results in an electrostatic reaction field that returns to interact with the solute. Implicit solvent models have been reviewed in detail many times before, 204,206–209 and enable the dynamic transfer of electrostatic information across the atomistic/ continuum or CG/continuum interfaces. Recently, new multiscale continuum methods have been developed that allow for the dynamic transfer of mechanical and hydrodynamic information across these interfaces. One example is the work by Villa 42 | Chem. Modell., 2008, 5, 13–50 This journal is

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et al.,211,212 who have developed a method that allows the transfer of mechanical information across the atomistic/continuum boundary. They have developed a multiscale model to investigate protein/DNA complexes. The method uses an atomistic model of the lac repressor protein and a continuum model of a large loop of DNA that is bound to the protein. The DNA loop is represented using a continuum elastic rod model, using elasticity theory. Protein MD simulations are used to position the DNA loop end points, and these end points then serve as crucial boundary conditions to elasticity theory to determine the shape of the loop. Elasticity theory then in turn provides the forces acting at the end points, which can be fed back into the protein molecular dynamics, completing the loop. Most atomistic/continuum embedded multiscale methods split the system into continuum and atomistic regions by molecule, e.g. the solvent is defined as being in the continuum, while the solute is atomistic. While being straight-forward to implement, such a split does not allow a complete representation of the physical detail of the system. One problem with continuum solvent models is that by representing all of the solvent as a uniform dielectric, they fail to capture atomic details, such as hydrogen bonding between the solute and solvent, which can be important factors to consider in protein-ligand binding or computational enzymology applications. It is therefore preferable to be able to spit the system spatially into atomistic and continuum regions, e.g. to model the solvent in the active site of the protein using an atomistic model, with the remainder modelled as a continuum. However, such a model must account for the diffusion of solvent between the atomistic and continuum regions. Schemes must be developed that allow for the conservation of momentum, mass and energy across the interface. While we are unaware of any such methods that have been developed for application to protein systems, Coveney et al. have, over a series of papers,6,213–219 developed an ambitious and highly promising multiscale method that allows the embedding of condensed phase atomistic simulations within continuum models for the purpose of studying hydrodynamics. The Coveney method works by dividing space into two main regions; an atomistic (particle) region, called P, and a continuum region, called C, modelled using computational fluid dynamics (CFD). An interface lies between these two regions. This interface is split into two parts; a C to P part, which sits on the continuum side of the interface, and a P to C part, which sits on the particle side. In the P region, the motion of the atomistic particles is solved using molecular dynamics. In the C region the motion of the continuum solvent is solved using standard continuum fluid dynamics (CFD). The hybrid scheme applied at the interface is designed to exchange fluxes of conserved quantities, namely energy, momentum and mass. In the C to P region, the fluxes from the continuum region are imposed on the particles. In the P to C region, the atomistic fluxes are coarse grained in time and space to form boundary conditions for the continuum domain.213 The details of how this is achieved is described in detail elsewhere.213,215,220 In essence, the stress induced in the P region by flow in the C region can be calculated and converted into a local momentum flux at the C to P interface. This flux can be converted into a force which is added to all of the particles in the C to P region. This pressure force from the continuum liquid is enough to stop the escape of particles from the C to P region, and, at a basic level, is responsible for the transfer of information from the continuum to the particle region. The P to C interface, on the other hand, is used to transfer information back to the continuum model. As the continuum model is a mesoscale representation of the solvent, the particle model must be averaged in both space and time. An averaging time of Dtavg is used, and the average momentum flux is calculated. This can then be used to establish the boundary condition of the continuum domain at the C to P interface. In addition to controlling the flow of momentum across the interface, the flow of mass must also be managed to ensure that the total mass in the whole system is conserved. This is achieved by calculating the mass flux at the interface and using that to determine whether particles need to be deleted Chem. Modell., 2008, 5, 13–50 | 43 This journal is

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(to represent flow of mass form the particle to continuum region), or whether particles need to be created (to represent flow back from the continuum region). Removing particles is simply a case of choosing those that are closest to the top of the molecular dynamics simulation and thus closest to the continuum region. Inserting particles is more difficult, as sufficient space has to be found for the new particle amongst the existing particles, and atomic fine detail (coordinates and momenta) have to be reconstructed. Coveney et al. have developed the USHER algorithm221 which efficiently finds a location and configuration of the new solvent molecule that releases an energy equal to the mean energy per molecule. The velocity of the new particle is obtained randomly from a Maxwell distribution with a zero mean velocity and temperature equal to the simulation temperature. One of the original applications of this method213 used a simple model system, with a model polymer attached to a wall, solvated by a single-particle model solvent. They were able to use this setup to study the dynamics of this polymer within a solvent which was undergoing shear flow. Coveney et al. refined their method214,219 to correctly couple hydrodynamics, using a fluctuating hydrodynamics (FH) continuum model,222 and were able to show that fluctuations, at both the atomistic and continuum level were thermodynamically consistent. They demonstrated their method by simulating sound waves passing through a continuum, then atomistic model of liquid water, and then being reflected by an atomistic model of a lipid membrane bilayer.214 The light coupling between the different types of simulation (MD and CFD or FH) makes it well-suited to application over a distributed computing cluster, and Coveney et al. have demonstrated the method’s successful deployment over computational GRIDs.6 4.3 Two-way parallel interfacing methods Creating a multiscale method that directly embeds an atomistic region within a continuum model introduces a physical interface between the atomistic and continuum models. This complicates the implementation, as techniques must be developed that allow for the transfer of information across this physical interface, e.g. the mass, momentum and energy flux across the interface must be controlled. Chang, Ayton and Voth223 have developed a multiscale atomistic/continuum method that avoids the creation of a physical interface between the two regions. They achieve this by running both a continuum and an atomistic model of the system in parallel, and only loosely coupling this pair of simulations by periodically transferring structural information between them. Chang et al. developed the method, called multiscale coupling (MSC),112,223,224 originally to study the effect of long range dynamics of a lipid bilayer membrane at the atomistic level. They loosely couple together an atomistic model of a bilayer with a continuum-based mesoscopic model. The continuum model is based on an elastic membrane (EM) implicit solvent model originally proposed by Ayton and Voth.225 This model was parameterized from mechanical properties calculated from atomistic MD simulations of a DMPC bilayer. To better capture the dynamics of a membrane, this model has been updated to include an explicit mesoscale model of water.223 The elastic membrane model was used to simulate a bilayer of dimension, 890 A˚ 890 A˚. To make this a multiscale simulation, a point on the elastic sheet representing the membrane is chosen, and an atomistic model of the membrane at that point is constructued. The atomistic model comprised 64 DMPC lipid molecules solvated by 1312 TIP3P waters. A real biological membrane is not subject to external constraints, and therefore it adopts a tensionless configuration.226 However simulations using the canonical (NVT) ensemble use a fixed number of particles and fixed membrane area. This results in a simulation of the membrane with a non-zero interfacial tension.227 This observation has resulted in the development of a new ensemble, whereby the membrane tension, g, is kept constant.226 This new ensemble, NPgT, allows the membrane tension to become part of the definition of the 44 | Chem. Modell., 2008, 5, 13–50 This journal is

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thermodynamic state of the system, as g has become a thermodynamic parameter. Chang et al. use this fact to couple their continuum-based mesoscopic membrane model to the atomistic model. They run the atomistic model using the NPgT ensemble, and update the value of g every 1 ps in response to the changes in surface tension measured at the corresponding location in the continuum-based mesoscopic model. This allows a dynamic, one-way transfer of information from the continuum model to the atomistic model throughout the simulation. The advantage of this approach is that the atomistic level simulation does not directly interface with the continuum level simulation, so standard approaches can be used, e.g. periodic boundaries and the use of the particle mesh Ewald sum to model long range electrostatics. Also, while this application only allows the dynamic flow of information from the continuum to atomistic level, in theory, mechanical properties calculated from the atomistic simulation could then be calculated and used to reparameterize the continuum model, thereby completing the cycle. Ayton and Voth228 have employed this multiscale coupling (MSC) method to study the effect of long range membrane motion on the dynamics of an ion-channel. An atomistic ion channel was embedded within an atomistic simulation of the solvated bilayer, and this was coupled to the continuum-based EM model using the MSC method. Because formally an ensemble of atomistic simulations should be connected to the continuum model,228 they ran eight atomistic simulations in parallel, and connected each of these to the same point in the continuum model. They compared the dynamics of the ion channel using both the MSC boundary conditions and standard NPT boundary conditions and found that the orientation of histidine residues in the channel appeared more disordered when the atomistic level simulation was coupled to the continuum. This suggested that the fluctuations in stress caused by the long range dynamics of the membrane modelled at the continuum level, could indeed affect the dynamics of the ion channel at the atomistic level.

5. Beyond continuum models Continuum models allow biological simulators to access extremely large length and timescales, e.g. there are continuum models of a complete human heart.229 There is, however, a level of modelling that addresses even larger length and time scales. At this level, biological network models are used to represent cells, 230,231 organs232–234 or creatures.235,236 Detailed reviews230,237,238 of how to build models at this level are available. In brief, the methods work by constructing differential equations that describe the flow of material through a biological system, e.g. using a differential equation to represent an ionic current.237 Several equations can be coupled together to form a model, e.g. of the electric current flow across the human heart.232 The whole network of equations can be run together to produce an output signal that describes the biological property that the simulator is interested in modelling, e.g. modelling the oscillations in concentration of cellular metabolites.238 Multiple levels of biological network models can be combined together. The Physiome project239 is an attempt to define standard interfaces between these levels so that they can be combined together easily. To make these models fully multiscale, they could also be interfaced with atomistic, CG, or continuum representations. The conduits for information flow are the kinetic parameters that calibrate the differential equations that make up the network.238 For example, QM/MM computational enzymology calculations can be used to obtain the rate constants of enzyme catalysed reactions, or MM/CG calculations can be used to obtain the diffusion constants of ions through membrane pores. In this way, biological multiscale modelling will allow the coupling of simulations of electrons all the way up to models of a human heart. Chem. Modell., 2008, 5, 13–50 | 45 This journal is

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6. Conclusion This review has outlined the techniques and methods involved in building computer models of biological systems that bridge between different length and time scales. Biomolecular multiscale modelling is not new: Levitt and Warshel pioneered much of the development of the foundations of the QM/MM and MM/CG interfaces in the 1970s. However, there has been an explosion of interest and activity in this area over the last few years that has been prompted both by the increases in available computer power, and the maturing of methodology at each of the different biomolecular modelling levels. As each level has matured, and become predictive, simulators are now rightly asking how can these methods be combined to answer the bigger questions of biology such as how does a cell maintain membrane cohesion, how do small changes in protein configuration initiate cell signalling pathways, and what is it about this collection of molecules that makes them a living system? While multiscale modelling methods are not yet capable of providing answers to these questions, the huge progress made over the last few years is now providing an unprecedented insight into the interplay between the biological and chemical worlds. Multiscale modelling methods will certainly be an exciting growth area of increasing importance in biology.

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Computer-aided drug design 2005–2007 Richard A. Lewis DOI: 10.1039/b609116b

1. Introduction At the 2005 Gordon Research Conference on computer-aided drug design, the delegates were polled for a series of big questions facing our field. These covered the fields of QSAR, structure-based design, ligand-based design, cheminformatics, library design and virtual screening. These questions are dispersed among the individual topics, but all focus on extending the scope and rigour of our methods. In the last two years, there has been a burst of activity around the fundamental science, reflecting the growing influence of modelling. It is no longer good enough to work only in the traditional areas of strength. Modelling is expected to make an impact now in all aspects of the drug discovery process.

2. QSAR and ADMET For the field of QSAR, the issues were: How global can good QSAR models be? How can we do simultaneous modelling against multiple pharmacological endpoints? How can we close the feedback loop so QSAR models could be used directly for generation of new structures? How can we make good predictive models of ADMET properties from molecular structure, everything from solubility to bioavailability to idiosyncratic toxicity? The valedictory article by Yvonne Martin 1 provides a reminder that modellers have most impact when they concentrate on the specific problems faced by medicinal chemists. Thus, providing models that we know to be flawed will have more effect than a ‘perfect’ model that is irrelevant to the daily experience of the users, or is not applicable to their compounds. The examples described are substructural alerts for reactive functional groups, which the chemists can modify and hence take ownership of, and C log P. The latter is a commonly used descriptor used as a surrogate for bioavailability despite the fact the calculation is in error by more than 1 log unit about a third of the time. Despite this, there is no pressing demand to devise a better method. 2.1 Model quality Guha et al.2 explore the relationship between global QSAR models and local models. A local model is obtained using only the compounds in the neighbourhood of the query molecule. This would be most useful when the global trend is non-linear, as one is approximating the first derivative of the non-linear curve with the local model. This has analogies to the kNN method. Of course, one must be careful how the distance metric and parameter k are chosen. In tests, the local models gave very similar global error predictions, but explained outliers in the global model much better. Where a local model falls down is when there are activity cliffs, that is, larger changes in activity for only small changes in structure, or when the neighbourhood is sparse. The key advantage is that no model needs to be built a priori, so that the method is well suited for analysing large data sets. A new method for making classification models from diverse structure, called line-walking recursive partitioning, has been published, using cyp450 data as the training set. The authors claim 85% accuracy, similar to SVM methods, and only 9 descriptors are used. The method was also able to uncover errors in the biological data, implying that it is Computer-aided Drug Discovery, Novartis Institutes for Biomedical Research, Basel CH-4002, Switzerland. E-mail: [email protected]


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robust even for noisy data sets.3 Gedeck et al.4 provide another view of QSAR, by looking at the quality of predictions as a function of the data set. Using 1000 datasets and creating QSAR models in identical fashions, they showed that the quality of the predictions correlates with the size of the data set and the dynamic range of the activity values, and that no one descriptor set is the best for all data sets. By performing randomised picking experiments, they also saw a wide variation in q2 values, indicating the instability of many of the data sets. The fraction of good models (q2 4 0.5) was low for all descriptor sets, with 2D descriptors performing better than quasi-3D descriptors such as Almond.

2.2 Matched pairs analysis Matched pair analysis (or Free-Wilson analysis) has come back into vogue with a number of papers employing the technique. As one is looking for changes in activity with small changes in structure, there is some tension with the similarity principle, but it is a valid approach. The argument is that classical QSAR models are subject to issues with descriptor validity, extrapolation and so on. By analysing a dataset of matched pairs, one can derive fragment-based rules for activity, with the hope that the rules are independent of the larger context of the rest of the molecule. Leach et al.5 have built a tool for extracting all matched pairs from a dataset, and for evaluating the statistical significance of each rule. They exemplify using solubility, plasma-protein binding and oral exposure in rats. Thus adding Br decreases solubility by 1 log unit in 98% of the cases investigated. Conversely, analysis of outliers can give hints about why the usual trend is bucked: adding a orthosubstituent may cause some change in conformational preference, i.e. the wider environment is affected and the change of fragment is not independent of the neighbourhood. Haubertin and Bruneau6 also use matched pairs in a similar fashion. Sheridan et al.7 employ the idea of a molecular transformation, that is the difference between two molecules, as a substructure difference vector. The key is to look at molecules and their activities in pairs, rather than as a whole data set. This allows one to pick up small signals that might otherwise be masked by other larger signals and errors. The substructure vector is based on the atoms left after subtraction of the maximal common substructure. By analysing all pairwise transformations, clustering them by similarity, then adding in the change in activity most strongly associated with the cluster, one can quickly see if a transformation was beneficial or deleterious, even if the original absolute activity was weak. A similar strategy can be applied to find which (beneficial) transformations derived from the training set could be applied to a query molecule. In studies using D2 agonists, DHFR inhibitors, ACE inhibitors, they showed that, in comparison with some global QSAR methods, their local method could pick up local trends that the global models missed, and vice versa. In addition, as the transformations have a well-defined neighbourhood, it is not generally possible to apply them multiple times, thus avoiding the unreasonable optimisation issues caused by applying global models to morph query structures in a de novo design paradigm. To provide balance, one can also look for matched pairs that do not show changes in activity at a position which has been shown to be important to the SAR. If this is combined by an analysis of the X-ray complexes, one can assume that there is bioisosterism. The work by Kennewell et al.8 does just that starting with an alignment of the relevant binding sites. The ligands are broken down into fragments by splitting at single bonds and the fragment pairs identified by measuring the volume overlap with a reference molecule. The analysis was performed across several targets and bioisosteric pairs are found. However the authors are keen to stress that bioisosteres are particular to the target and not necessarily universal. 52 | Chem. Modell., 2008, 5, 51–66 This journal is

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2.3 Multiobjective optimisation Inverse QSAR and multiobjective optimisation have been combined into a single workflow by Brown, McKay and Gasteiger,9 using a GA-based compound generator, and two QSAR models for solubility and polarizability. The issue of model knowledge domain is handled by Hotelling’s T2 statistic, and the multiobjective optimisation by Pareto ranking. They were able to show that the evolved molecules were consistent with the training sets. 2.4 Predictive models of ADMET The focus on producing general models for ADME phenomena can produce an over-reliance on the many robust methods for deriving QSAR models, while ignoring the underlying structure of the data. Good and Hermsmeier10 have re-examined the problem of predicting druglikeness, and show that there is considerable bias in the data, which can make the models look better than they really are. They argue for training set selection based on target class and ensuring that the decoy set is matched to the true actives by molecular weight as a minimal requirement. Random selection can fall foul of analogue bias, whereby certain classes of molecule are overrepresented, leading to recognition rather than prediction. Mutagenicity is an observation that will often halt the progress of a compound through the clinic. Based on the assumption that the features that give rise to mutagenic effects are quite localised, Kazius et al.11 have created a set of toxicophores based on a dataset of 4337 structures with Ames data. 29 toxicophores could classify the actives and inactives with an error of 18%, around the interlaboratory reproducibility level for the AMES test. An interesting case study is the aromatic nitro group. While is it strongly associated with mutagenic effects, its influence can be mitigated by other substituents on the ring: this observation lead to more specific rules, all of which are published as supplementary information in SMARTS format. This could form part of any assessment for compound purchasing or library assessment. Although it is included in the ADMET section, the review by Kalgutkar et al.,12 listing the bioactivation pathways of organic functional groups, has much wider relevance. Although the authors are careful to stress that no functional group should be ‘black-listed’, this survey could be used to bias away from certain groups during compound selection or library design or de novo design. Two new models for volume of distribution in humans have been published. This involved a very careful assembly of data, coming as it did from different data sources. In the Lombardo model,13 384 compounds were used in the study, most with quite diverse structures. As expected, C log P was an important descriptor, but also terms involving the fraction of the molecule ionised. It is clear that a good predictor for pKa, especially around physiological pH, will be key to future success in the modelling of pharmacokinetics. A model that predicts the pKa of amines accurately is not that helpful, if all the predictions are 48.5, as the compounds will be fully ionised. A program that could distinguish between a pKa of 6 and 7 would be immensely valuable. The model for VDss is accurate to 2-fold, as accurate as equivalent models based on animal data, and the underlying, curated data has been made available. The Gleeson model14 also includes rat data and is quoted as having an error of 0.37 and a r2 of 0.53. A curated data set has been put together using drugs marketed in Sweden.15 The total set of 673 pharmaceutical agents was reduced to 24 diverse compounds using multivariate analysis. Experimental data for these compounds was measured in a consistent way, to allow appropriate use in models (pKa, log P, solubility and CACO-2 permeability). Data on VDss, half-life and clearance were obtained from the literature. As all the compounds are commercially available at reasonable prices, this data set could be expanded: even in its present form, it will be a good standard for ADME modelling. Chem. Modell., 2008, 5, 51–66 | 53 This journal is

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Herg blockade is mesmerising the pharmaceutical industry, not just due to the implications for drug discovery, but also because the difficulty is understanding the interplay between displacement of dofetilide, patch clamp assays, dog telemetry and the occurrence of toursade des pointes in humans. Several groups have developed pharmacophore models, including one for non-basic molecules.16 However, as Pearlstein has pointed out,17 Herg does not have an endogenous ligand, so that the Herg binding site is under-evolved and promiscuous, and might be better viewed as a host–guest complex. 2.5 Metabolism There have been continued studies into the energetics of hydrogen abstraction as a key factor in both regioselectivity and relative rates of metabolism by Cytochrome P450s. Olsen et al.18 performed studies with B3LYP and AM1 Hamiltonians using 24 substrates covering most commonly found atom types, and a haem surrogate. The surrogates ranged from a full iron-porphyrine to a methoxy radical. By performing full comparisons at the different levels of theory, they showed that the error in a simple AM1 approach using PhOdwas under 4 kJ/mol, a quite acceptable margin. Bond Dissociation Energy was found to be the best correlate for activation energy. All of the results could be summarized into 5 rules based on atom type and local environment. The results were applied to two steroids, and the authors could predict, with good accuracy, the preferred sites of metabolism in both, even without taking into account the accessibility of the atoms to the iron in the cyp450 site. Sheridan et al.19 eschew the energetic approaches to a purely QSAR-based model of regioselectivity for metabolism by Cyp450’s. The best models use only three substructure descriptors and two property descriptors. In comparison with other methods, specifically Metasite,20 the simple QSARs seemed to do as well or better, although the authors acknowledge this could be an artefact, and true comparison of performance is not easy. The authors also not that the models only predict where a molecule might be oxidised, and not how fast or extensively, something the abstraction energies do address.

3. Structure-based drug design The key issues for structure-based drug design are: adequate sampling of protein conformational space; high quality but practical potentials for predicting protein structures and protein/ligand interactions; generation of homology models good enough for hit identification and lead optimization, and to what extent can we refine a homology model based on knowledge of the ligands known to bind? 3.1 Protein flexibility Protein plasticity is a common phenomenon, but not often explicitly addressed except via MD, and even then we cannot be sure that the key conformations have been sampled. The zero-order approximation is to assume rigidity, followed by the use of several rigid snapshots, or side chain flexibility (e.g. GOLD), or even some degree of induced fit is allowed (see below). What is lacking are methods that are fast and cover all the key conformations. To obtain this, we need to understand more about the important motions of protein structures. Kinases are a good test case, as they are plastic, but we have X-ray structures of many of the active and inactive states of the proteins, plus ligands, so we can try to assess the interplay between protein conformation and ligand structure. Subramanian et al.21 have used CDK kinases as a test case: the proteins were aligned to a standard, using all atoms. The size of the binding site was defined using 4 residues common to all the CDKs; this variable was correlated against standard descriptors generated from the ligand structures. The degree of stabilisation of active and inactive forms was found to 54 | Chem. Modell., 2008, 5, 51–66 This journal is

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depend strongly on the local hydrophobic character of the ligands, in addition to the usual size and shape constraints. Using the QSAR equations, one can predict which protein conformation is best suited for docking studies with a molecule. Another way to tackle this is to look at the formation of transient pockets. Eyrisch and Helms22 have done this specifically for protein–protein interactions. Starting with a crystal structure and running 10 ns of dynamics, they created multiple snapshots from the trajectories, each of which was examined using PASS, to detect cavities. The cavities were clustered to determine the location of more substantial pockets. The validity of the pockets (and the method) was assessed by superimposing the inhibitor structure: in all cases the pockets opened up in the regions one would expect ligand atoms to bind. The pockets also seemed to be the most non-polar. In docking studies, using the transient pockets gave better results compared to using the original apo-form from the X-ray. Apart from the formation of pockets, one also should consider if they are energetically favourable for binding. Landon et al.23 using computational solvent-mapping with small organic probes to identify energetic hotspots. Using renin as an example, they identified a pocket that is used by aliskiren, but not by other peptidomimetics. Using the Fesik24 training set of druggable and non-druggable proteins, they could shown that targets with a low ranked consensus site (among the probes used) were more likely to be the nondruggable ones. The exceptions to this were small confined sites such as PTB1B. Another way of looking at localised plasticity is to consider the structure ensembles generated by NMR methods.25 X-ray structures of complexes may have higher accuracy, but the protein is collapsed around the ligand, leading to the crossdocking problem, that one will not find other inhibitors of the same target, because the binding site is too closely modelled on one ligand. Using HIV protease as the exemplar, multiple pharmacophore models were built using benzene, ethane and methanol probes within an MD simulation, followed by clustering to identify hotspots of interaction. These models were used to query a conformer database. The effect of using multiple models is to increase the uncertainty (radius) of each feature, compared to using a model from an average structure. This is important when the pharmacophore contains more elements. The confusion matrices for identifying known hits from a database for presumed inactives (renin inhibitors were also found) were very good, with 90% true positive rate, 10% false negative. The study also compared the use of NMR structures to models derived from multiple X-ray structures. The recommendation is that NMR structures give better models, as the X-ray structures contain too little variation. The same group have also used conformations from MD sampling26 with similar findings. 3.2 Prediction of affinity The prediction of free energy, or at least consistent ranking of ligands according to free energy, has attracted much attention over the last few years, partly as a result of better hardware, but mainly due to the availability of more X-ray structures. The usual questions about sampling, the effect of solvent, force field accuracy have all been examined. Michel, Verdonk and Essex27 compared implicit and explicit solvent models for a series of inhibitors of neuraminidase, COX2 and CDK2. While they found that implicit solvent models are as good as explicit models in ranking ligands, while being much quicker to run, they urge caution, as changes in the conformation of the binding site can cause significant errors, so the methods should always be allied to X-rays or reliable docking predictions. Weis et al.28 used a MM/PBSA method to examine the effect of the force field and the starting geometries affected the calculated binding energies. While the force field didn’t matter much overall, mixing the force field for geometry and energy calculations did. They concluded, in contrast with Essex, that for their system explicit water should be used. They also conclude that simulations should be at least 450 ps long and that entropy is the most poorly handled term. Wiseman et al.29 describe their grand canonical Monte Carlo Chem. Modell., 2008, 5, 51–66 | 55 This journal is

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method for computing relative affinity. The ensemble is annealed with respect to free energy and the protein and target are treated as rigid entities, and conformational freedom is introduced by means of snapshots. This is less of an issue for fragments or small molecules. For the test case of T4-lysozyme complexes, the method was comparable with much more complex simulations. The goal of this method is to dock fragments, which can then be assembled into larger entities (see sec. 6). Finally, the empirical scoring function used in Glide has been upgraded with new terms, 30 for desolvation, for hydrophobic collapse, and for h-bonding. The force field was parameterised on 198 complexes and 15 screening sets. They claim an accuracy of 2–3 kcal/mol, depending on how well-behaved the system is (that is, can the ligand be docked into the target without substantial reorganisation). 3.3 Selectivity Methods for assessing cross-reactivity a priori are of interest for several classes of targets with similar binding sites, most notable the ATP-binding sites of kinases. Using their dehydron approach, Fernadez and Maddipati31 generate a descriptor for the ATP site that correlates very strongly with the pharmacological activity vector across a panel of 32 kinases. A dehydron is defined by the degree of non-polarity around an h-bonding group. Excessive non-polarity promoted dehydration as a means to enhance the electrostatic contribution of an h-bond to a ligand. A matrix of dehydron patterns can be built by assigning a score if two residues share a dehydron, 0 otherwise. This matrix can be compared between kinases using standard metrics. The descriptor can be sharpened by only looking at dehydrons that are known to interact with inhibitors. To extend the approach, the authors use the fact that there is a strong relation between dehydron wrapping and the disorder score,32 allowing the a priori prediction of cross-reactivity. There is a continuing debate about the ability of F bound to C as an h-bonding group, or rather the extent to which this phenomenon can be observed, and its contribution to affinity. There is also the other effect of F on pKa which can affect affinity. Razgulin and Mecozzi33 have performed a comprehensive analysis using high level ab initio methods of the energetics of C–F H interactions. They find that fluoroaromatics can h-bond to water with an energy of 2.6 kcal/mol; this is weaker than h-bonds involving O or N, but in the context of structure-based drug design, displacement on water could therefore be much more energetically favourable than was first thought.

4. Virtual screening Virtual screening covers both structure- and ligand-based design, in that the output from these fields is used to search for new ligands. The key issues are around scoring, or enriching actives over inactives in the final ranking. 4.1 Docking A significant advance in docking has been made by Sherman et al.34 by iteratively combining the usual rigid protein/flexible ligand approach (in this case GLIDE) with a program for protein structure prediction (PRIME). This allows the protein to be induced to fit a given ligand. Thus one can start with an inappropriate protein conformation (for example the apo-form) and mould the site around the ligand to produce a better model for docking. In self-docking experiments, the average RMSD could be decreased from 5.5 A˚ to under 1.8 A˚ for the 21 cases considered (chosen as they were challenges for the usual paradigm). The first docking starts with a softened steric term, and the induced fit dockings add in an extra term for the conformational strain of the protein. The induced fit method in the paper is restricted to side chain motions; there are obviously cases where there is much more extensive motion of 56 | Chem. Modell., 2008, 5, 51–66 This journal is

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loops and secondary structures. This is not an insuperable difficulty, as loop prediction can be added into the protocol. 4.2 Scoring The well-known bias of scoring functions with size is investigated in detail by Jacobsson and Karlen.35 Using a training set of 945 actives and 10k decoys, with 8 different scoring functions, three methods for correcting or improving the ranking of the actives were tried. MASC36 did not generally make the enrichments better. Normalisation by the square root of molecular weight or number of heavy atoms sometimes improved the enrichment rate. As the authors point out, using structural knowledge about the target is generally far more powerful than relying on the score. The other point to note that this study is concerned with improving the enrichment rates of a docking run, rather than improving the prediction of affinity. Cho et al.37,38 have performed an important study into the effect of ligand atomic charges on the accuracy of docking runs. In most protocols, it is not feasible to recalculate charges on the fly, so that standard force field charges are set once and maintained. Apart from the questionable accuracy for some atom types, this approximation neglects the effects of polarisation. In this study, a mixed QM/MM simulation was used to recalculate the ligand charges during several cycles of docking and minimisation. In the test set of redocking, using induced QM charges to seed the next docking, did not cause any harm, and in most cases gave improvements. However, in subsequent studies, if a local minimum is encountered, using QM charges can reinforce the ‘incorrect’ binding mode. This is countered by running the docking in a stochastic parallel mode to increase the sampling. 4.3 Validation The Astex test set for docking studies was updated in 2007.39 This valuable resource has been refined to just 85 structures of high quality, representing an unbiased sampling of available structures, to compensate for the large fraction of kinase and protease structures in the PDB. The Astex set also requires deposited structure factors, so that ligand positioning can be independently regenerated and assessed. Issues arising from steric clashes or symmetry have been addressed directly. The set was also subject to extensive manual curation around tautomer and protonation states. In a experiment using GOLD, it was shown that performance was improved, not due to better docking and scoring, more that the test set was freer from experimental artefacts that would have caused docking programs to fail. Interestingly, using this high quality test set can also reveal other, more subtle, effects that can impact performance, for example the method used to generate the starting conformation. Using Corina rather than the native conformation caused a degradation of 5% in redocking experiments. Huang, Shoichet and Irwin40 have provided another benchmarking set of ligands and decoys for 40 different targets. Given the correlation of scoring functions with size of lipophilicity, the decoy set has been chosen to resemble the active ligands physically while have distinct chemotypes. Each ligand has its own set of decoys. They showed that using uncorrected decoy sets, such as the MDDR, gave enrichments 0.5 log units better, which is clear evidence of bias. One of the key tools in evaluating virtual screening performance is some measure of enrichment, whether by ROC curves, enrichment factor and so on. What one wants to know is whether a screening method identifies the actives early in the overall prioritised list of compounds. Truchon and Bayley41 have performed a theoretical analysis to show that ROC curves may not be ideal for this purpose, despite their widespread use (and advocacy). A method can get a good ROC score, yet still fail to identify any actives in the top 20%. They propose a new metric based on an enhanced form of the ROC curve called BEDROC, which, via a parameter Chem. Modell., 2008, 5, 51–66 | 57 This journal is

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alpha, can be used to emphasis, or weight, the influence of the early part of the curve. Other effects, such as statistical error and saturation effects (the variation of a metric with the number of actives), are also discussed, leading to some practical recommendations for the conditions needed to perform a good evaluation of a new method. 4.4 Pharmacophores The old trick of using pharmacophores to pre-screen before performing expensive docking studies42 is enjoying a renaissance. Lu et al. employed this strategy to find leads against the MDM2-p53 interface.43 The pharmacophore constraints screened out all but 2599 structures from a starting set of 11k. Docking gave further enrichment of 10 confirmed hits out of 67 compounds assayed. As a pharmacophore, if properly constructed, will give very few false negatives (false negatives might arise out of different binding modes or poor feature definition), the scoring of docked hits becomes more useful. The strategy was also validated using known inhibitors, all of which passed the filters, and scored well. Baroni et al.44 combine 4-point pharmacophore fingerprints with a shape-based description of the binding site in the FLAP program. Molecular interaction fields (MIFs) are used the locate maxima and minima interaction regions within the site, which describe the pharmacophore points (which are hence independent of substructure). The same procedure can be applied in reverse to small molecules, leading to a fast fingerprint-based method for prefiltering large databases. The FLAP fingerprints can also be used for alignment-free comparison of binding sites to highlight regions of similarity and selectivity. Protein flexibility is handled at the side-chain level during the generation of the MIFs, and ligand flexibility by conformer sampling. FLAP shows good performance when evaluated against other docking programs (even though that it is not its primary purpose). A new methodology for combining pharmacophore detection and 3D QSAR, PHASE, has been described by Dixon et al.45 The workflow is similar to other procedures in the generation of conformers followed by mapping of pharmacophore features. These features can be the standard h-bond donors, acceptors, hydrophobes, charged groups plus other custom features. For hydrogen bonds, either vectors or projected binding points can be used, obviating to some extent the need to have the heavy atoms of the ligand feature superimposed, a limitation in many packages. Detection of a pharmacophore is also made more likely by reducing the restriction that all active molecules in the training set must display the same common pharmacophore. Normally several candidate pharmacophores can be proposed, so the next step is to score the models, which can be done using the fit of the actives to the model, which actives are fitted, conformational strain and volume overlap. Inactives are also used to penalise a model, but as a separate step. The selected model can, as in other cases, be used to define the alignment for a 3D QSAR model. PHASE uses an occupancy grid (that is, is the grid cube occupied by a feature from the ligand) rather than a probe energy grid as the descriptor for the QSAR; this leads to volume exclusion maps in a very facile manner. In a comparison with CATALYST (and with the caveat that this paper was written by the vendors of PHASE), PHASE seems to do better on the test set used by Patel et al.46 in their comparative study of pharmacophore generation methods. Another new tool for pharmacophore alignment, GALAHAD, has been described.47 This works by iterative construction of hypermolecules. A hypermolecule is generated by aligning successive pairs of molecules; it retains common features and individual atoms. It is therefore also dependent on the order in which the alignment pairs are processed, so the strategy is to align the most similar pairs first. The alignment is scored by the overlap of the pharmacophore features rather than just the atoms. Again, on the test set of Patel et al.,46 this method appears to find better pharmacophore models, but with the caveat that frozen conformations were used. 58 | Chem. Modell., 2008, 5, 51–66 This journal is

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Rules-based conformer generators often have fairly basic torsional parameters. Sadowski and Bostrom48 have developed a method based on MIMUMBA, to generate all possible for-atom torsional fragments, to search the Cambridge Crystallographic database with these fragments and to use the results to define torsional energy parameters. These more extensive rules improve the ability of OMEGA to find conformations closer to X-ray conformations. Interestingly, if the conformers are subsequently refined using an MMFF-like force field, the improved performance is removed. Agrafiotis et al.49 have performed a comparative study of conformational coverage provided by several common packages as well as their own stochastic proximity embedding method (SPE). They conclude that SPE, along with a poling method such as catCONF, provides better sampling, whereas the others produce more extended or compact geometries. Coverage was measured using pharmacophore triplets and radius of gyration. As the authors point out, the measure affects the relative performance, so that their conclusions are not a universal endorsement of a particular method. Stochastic methods are not perfect, and still produce horrors such as bonds penetrating through the centres of rings. 4.5 Scaffold hopping Scaffold hopping is the technique of identifying molecules that are structurally different but which might form similar interactions with a receptor. In its own way, scaffold hopping is a form of similarity search, but with more abstract, less connectivity-dependent descriptors. Gillet et al.50 describe the use of reduced graphs, which combine essential pharmacophore information within a graph; these graphs can then be compared using classical means. The methods seem to marginally outperform classical fingerprint similarity in enrichment experiments, as they pick up both the structurally similar and dissimilar actives. Stiefl et al.51 also use a reducedgraph formulation in their ErG program. ErG handles h-bond acceptor/donor groups as two distinct states, allowing for exact matches with structures that do not have a matching donor/acceptor, rather just, say, a donor. A new descriptor is generated for each state, leading to 2N descriptors, where N is the number of donor/ acceptor groups. The paper is also noteworthy for its analysis of why different levels of performance are seen, compared to classical 2D fingerprints, for enrichment experiments. Methods that are strongly dependent on pharmacophore features do well in cases where there is a clear pharmacophore exposed, and less well when the pattern is hidden but the class is structurally homogenous. The Cresset field point method52 uses molecular field extrema to create a representation of molecules. These field patterns can be compared to produce lists of similar molecules. This algorithm is effective at finding scaffold hops, particularly for ring systems. The field points convey the hydrophobic and electrostatic features in the neighbourhood of a molecule in one conformation. The overlay is performed using a rigid-body simplex optimisation, to optimise the fit of the extrema of one molecule onto the full field of another (and vice versa). The starting point is derived by matching the extrema patterns using coloured graphs, very much as in the reduced graph approach. One of the harder aspects is when molecules could be formally charged, as this will drastically affect the fields. The power of the method is illustrated using thrombin inhibitors from different classes and trying to regenerate the X-ray overlay without using any information from the X-ray complexes, which they were able to do. Maass et al.53 offer an update on the CAVEAT methodology with the RECORE program. As before, a database of compounds is shredded to create a fragment database. Fragments are created using the notion of cut points, that is, places at which the removal of a bond splits the structure into two. The cut points are further limited to preserve conformational restraints. Fragments are filtered by size and indexed by the vectors created by two or more cuts, as well as by pharmacophore features. Scaffold hopping is therefore performed by searching the database for Chem. Modell., 2008, 5, 51–66 | 59 This journal is

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defined vector geometries plus addition features. In three test cases, viable, unstrained replacements for query structures could be obtained, even for multiple vector queries, where one expect the density of suitable fragments to be much sparser. Bergmann et al.54 use a GRID-based method to combine the Cresset and CAVEAT strategies. The anchor points (as for CAVEAT) are provided by the user, and then a database is searched for similar scaffolds, which match the geometry of the anchor points and the molecular interaction field profile. Using protease test cases, the central core could be used as a query, and known active alternative chemotypes found. The author suggest that the other highly ranked scaffolds could well be bioisosteres. 4.6 Ligand preparation Successful virtual screening relies on an appropriate and correct description of the molecules being screened; two particular sources of error are protonation state and tautomeric form. Oellien et al.55 have developed an exhaustive tautomer enumeration approach based on a set of 21 predefined transforms. The group do not attempt to say which tautomer is favoured, only that every form is represented. Databases built using all tautomers show better performance especially when screened with pharmacophore queries, as the bias introduced by most compound registration tools as to the ‘correct’ tautomer is removed.

5. De novo structure generation Fink and Reymond56 have continued their exhaustive exploration of chemical space for molecules with up to 11 atoms, using the elements C, N, O, F. They could enumerate 26.4 million structures, in comparison with the 63k structures with similar constraints described in the literature, including 538 novel ring structures. The universe has been somewhat pruned by rules around ring strain and obvious instability. The druglikeness of the structures was assessed using Bayesian models to predict for GPCR, kinase and ion channel activity, and a surprising number of novel virtual hits were found. Fechner and Schneider57 have also published a de novo design program. It is based on the assembly of molecular building blocks derived from a dataset using a RECAP-like set of disconnections.58 The starting molecule is also disconnected using the same rules, and rebuilt. A population of structures can exchange fragments, leading to a GA implementation with cross-overs and mutations. The fitness function was defined as the similarity, using topological torsions, to a query template structure. There are also various filters, based on upper and lower bounds on defined properties or descriptors, for example, molecular weight. Some of the queries could be generated, but compounds that could not be well handled by the disconnection rules were not. Lameijer et al.59,60 used de novo methods to generate scaffolds that were subsequently hand-modified by chemists. The protocol is to generate molecules from atoms and some fragments, then quickly stop and seek input from the medicinal chemist, who selects and edits the suggestions. This cycle can be iterated. In the paper, 8 examples of evolved molecules were made, and 4 showed some activity against bioamine targets. The generation process was highly constrained to produce molecules with more leadlike properties. While the input from a chemist resolves questions about synthetic feasibility, it raises others around bias and prior knowledge of pharmacophoric requirements. However, the authors argue that it is easier to evaluate a suggestion, than to invent something new, hence the close coupling of the computer program with the human end-user. Boda and Johnson61,62 take a different tack, suggesting that complexity (as a surrogate for synthetic accessibility) can be measured by precedent. A database of fragments was created using standard procedures from the MDDR set. Query 60 | Chem. Modell., 2008, 5, 51–66 This journal is

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molecules from de novo generation can be fragmented in the same way. The frequency of occurrence of the fragments (together with terms for stereocentres and rotatable bonds) can be combined as penalty scores to give an index of complexity. Unusual fragments, or unknown fragments, will incur high penalties (for example pentasubstituted phenyl rings) whereas common fragments (3-substituted indoles) will not. Naturally, this does depend on the dataset and the fragmentation rules used to define the chemical precedent. Some well known drugs do have high complexity scores, reflecting this issue.

6. Fragment-based screening Fragment-based screening and drug design is intended to bridge the gap between the size of the druggable chemical space, and the relatively small numbers of compounds we have for screening. The argument is that good ligands can be built up from relatively small numbers of fragments, so by performing more sensitive screens with very small numbers (1000’s) of fragments, we can identify the key interactions, and then assemble the fragments back into more traditional ligands. One unresolved question is how to construct an optimal screening set of fragments, looking at diversity and screening set size as the parameters. Makara63 has conducted a retrospective study of 6 datasets, comparing an HTS deck, and virtual amide library and fragment libraries to look at these questions. There are other limiting factors to screening set size, based on the ability to follow up on any hits. The theoretical conclusion was that a few thousand (20k) compounds are required to represent the chemical space of commercially available amide space, and the hit rate from this library was comparable to screening the whole HTS deck. Mauser and Stahl64 also have looked at fragment selection using a backward approach, in contrast to the forward approach of Makara. A set of fragments was derived from bioactive compounds using RECAP rules,58 and filtered using some general leadlike criteria (ring size, rotatable bonds etc.). The fragments were clustered on the basis of maximal common substructure. Using different subsets of fragments of size 2000–4000 members, the chemical space that could be covered by assembling the fragments was also assessed using the Ftrees descriptors65 in comparison to the original database of bioactives. Each subset could represent about 20% of the original database of 85k compounds. Rings have long fascinated cheminformaticians, from determining the smallest set of rings in one structure, to determining the frequency of rings used in drug discovery,66 to try to understand which rings are acceptable, which are not. Much of this argumentation is, like the rings themselves, circular; rings may be favoured not for their biological properties but instead their synthetic accessibility, both historically and today in the cost of goods for synthesis of a drug in clinical development. Ertl et al.67 analysed the World Drug Index (bioactives) and compared this to compounds for sale, and a database of natural products. The diversity of simple aromatic rings was remarkably low, with only 780 scaffolds found from 150k compounds, including 216 singletons. Using exhaustive enumeration on a small number of frameworks, 580k possible scaffolds were created, and described using simple terms like C log P and PSA, as well as parameters derived from QM calculations. Using a Kohenen network as a classifier between the bioactive scaffolds and the virtual ones, it was observed that the bioactive scaffolds did indeed cluster into discrete regions: the descriptors most important to the classifications seemed to be heteroatom count and HOMO/LUMO energies. The authors conclude that we are only scratching the surface of structural chemistry space, even though bioactivity may be limited to small regions of that space. Mapping of hot spots of interaction has been a key concept in fragment-based design. Most scoring functions (for example GRID68) are based on reproducing crystal structures. In a paper by Ciulli et al.,69 the energetic contributions of fragments are measured experimentally using isothermal calorimetry and NMR. Chem. Modell., 2008, 5, 51–66 | 61 This journal is

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Using ketopantoate reductase/NAPHH as the model system, it was shown that the change in affinity per heavy atom added was below the figure proposed by Kuntz for drug-like molecules.70 Looking at individual components, adding ribose did not improve average ligand efficiency, but adding phosphate groups did. Thus two hot spots in the ligand-receptor complex could be identified, and targeted for a fragmentbased strategy. Hajduk71 returns to the question of lead-likeness and the minimal potency required for a fragment to be useful. By examining 18 highly optimised inhibitors, and deconstructing them stepwise back to fragments, he claims a linear relationship between weight and potency, i.e. the binding efficiency is almost constant. As a rule of thumb, one should expect 1 log unit of potency for every 64 Da added, if one is to proceed along a successful optimisation. A drop of Binding Energy Index of more than 10% is probably indicative that a plateau has been reached of the modification is not ideal. Working this through, a starting fragment of 250 Da and an affinity of 1 mM would need to grow to 750 Da to achieve a potency of 1 nM. To stay under 500 Da, the starting fragment would need to be less than 200 Da. Finally, Babaoglu and Shoichet72 have performed experiments to assess the fundamental principle of fragment-based design. If we assume that we can build a ligand from fragments, what happens if we deconstruct a ligand into fragments? Will the individual binding modes of the fragments be maintained? Unfortunately, the experiments suggests that this is not always the case, and that fragments can be too small to bind unambiguously and/or correctly. For the original binding mode to be favoured, a larger fraction of the original inhibitor needs to be in place.

7. Target fishing There is considerable interest in the new field of target fishing. This is basically the identification of the target to which a ligand binds. This can be at the level of finding an active compound in a whole cell assay, then driving down to the molecular target, or having a collection of structures, or combinatorial library, and estimating in silico which proteins are these compounds most likely to bind to, before conducting the in vitro assay. Rognan et al.73 have performed the latter, proposing five targets for a set of triazepandiones, and finding significant activity in one screen, sPLA2. The first step is to prepare an annotated library of binding sites. 5 representatives from the screening library were docked into the 2148 binding sites, and the sites that got the highest score were retained. The target proteins from which the sites were derived were assayed experimentally with members of the screening set, to give micromolar inhibitors. It can be argued that the control experiment, of assaying against all the proteins, was not done, but this was impractical. Also, the most potent compounds in the assay were not the one predicted by docking, again highlighting the errors in the virtual screen. However, it is now possible to deorphanise compounds in silico relatively quickly (docking is slow, but it can be performed in parallel), and this result will have major impact in library design and compound purchasing for enrichment of corporate compound archives. Langer et al.74 tackle the issue using pharmacophore models instead. They have built up a library of models against a variety of targets. By screening a set of compounds against these models, they try to predict the biological profile of the compounds, which they managed for 90 out of a set of 100. The models can be generated either from a set of ligands, or from the protein structure itself, as described elsewhere in this review. Using more extensive conformational analysis leads to more false positives and true positives. The quality of the model was critical (how many features, tolerances etc.), as was the stickiness of the site (how many models could be derived), as to the confidence for which a target could be linked with the query structure. Jenkins et al.75,76 have used ligand activity data from the well-curated WOMBAT database to build a series of 2D models to predict the most likely protein targets for 62 | Chem. Modell., 2008, 5, 51–66 This journal is

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the compounds in the MDDR database, as well as using a 3D pharmacophore-like descriptor for the same purpose. On average, the correct association of ligand to target could be found 90% of the time by one of the 2D methods, and some of the broader activity categories (for example, ‘‘kinase inhibitor’’) in the MDDR could be disambiguated to more precise targets. The 2D models could also be used for compound selection and library design, with the caveat that, as they are based on 2D fingerprints, they might bias against novel chemotypes (where 3D methods come more into their own). Indeed, the authors ascribe the very high success rate to the congeneric nature of the data. Cleves and Jain77 have used ligand-based methods to look at the link between similarity, pharmacological profile and target similarity for about 1000 therapeutics. The goal of the study was to rationalize, or even predict, off-target effects of compounds. The first stage was to curate the data, so that pharmacological target was named consistently. Their first result was to show that the similarity principle was generally true for the dataset. When ligand-based models were built for 22 targets, recovery rates of true ligands were between 60–70% with only 1–5% of false positives. By looking at the non-cognate ligands returned by the models, one could hop to a new target family, to hint at possible off-target pharmacology; the example given being the relationship between b-lactam antibiotics and beta2-adrenoreceptors. There is crossover between this topic and library design. Increasingly, groups are trying to introduce a target-family bias into new designs, or to select between designs on this basis. The SIFt methodology of Singh et al.78 encodes the pharmacophoric information within a site as a fingerprint that can be used for fast screening of large libraries. The first step is to dock a training library, followed by R-group analysis. Using the properties of the R-group as descriptors, and a SIFt analysis of the docking modes, more refined fingerprint patterns could be defined (describing which R groups make which interactions) which are used to build predictive QSAR models. These models were shown to be more sensitive to detecting compounds that could bind favourably than just using docking score alone. A related study by Fukunishi79 used dockings between known binders and inactive compounds, and set of pockets derived from protein active sites. Using PCA, they could show that the actives clustered together, and that using proximity to the active cluster for a given protein pocket was a good method for finding actives in a larger test set. Also they could cluster proteins with similar pockets together. This does require some fairly extensive calculations, and a large compound/target activity matrix, but only needs to be performed infrequently to generate the scoring model.

8. Library design Industry groups are now starting to publish their in-house tools for library design. Most of the innovation has occurred within Pharma research, as off-the-shelf packages often do not fit well into local workflows. Mosley et al.80 from Merck start us off with a tool for selecting reagents. The user selects the synthon, for example primary amine, and then refines the search using physicochemical properties. An important component is that the initial search is from a set of well-curated and readily available building blocks. One can also specify undesired functional groups which might interfere with the reaction scheme of require extra protection/ deprotection steps. Automatic exclusions are also triggered (i.e. choosing a secondary amine excludes the more reactive primary amine). The property refinement is performed with the true synthon, to remove effects of capping groups e.g. tosylates. The user can select the final reagents after a clustering process based on size and features. Truchon and Bayly81 propose a new algorithm, GLARE, for library subsetting, claiming very fast performance on libraries with up to 1012 members. As one of the slow steps has traditionally been the enumeration and scoring of the full product matrix, they use reagent-based approximate scores, that are 90% correlated with the Chem. Modell., 2008, 5, 51–66 | 63 This journal is

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product scores. This is well within the accuracy of most scoring metrics anyway, so the approach seems justified. At each iteration of the subsetting process (dimension reduction), the reagent fitness score is incremented by 1 if the product is scored is as good. This is basically reagent frequency filtering, and the reagents with the lower scores are pruned out, and the cycle continues until the product subset has achieved a ‘goodness’ threshold, as defined by the user. The number of reagents to prune at each stage is the novel twist. In methods like PLUMS82 it is one by one. Here, using a formula based on the approach to the final set, many reagents can be pruned at each iteration without having false negatives. Also reagents coming from a larger pool (e.g. amines) are pruned faster than those coming from a small pool (boronic acids). However, the real speed-up comes from the sampling of the whole library space by smaller partitions. The assumption is that the fitness of the reagents should have the same distribution in the small partitions as for the whole library. The ability to handle very large library at interactive speed makes this very attractive for webbased tools intended for use by chemists. The concept of privileged substructures is frequently invoked in library design, with the assumption that addition of such a motif to a structure will confer some specific activity or selectivity. Schnur et al.83 challenge this notion. Substructural analysis was used to generate potential privileged substructures, and the relative frequency of occurrence of these fragments was examined over a number of target sets. In most cases, no selectivity is observed, leading to the conclusion that the motifs are generally conducive to binding. This may in turn lead to undesired sideeffects or even promiscuity. Promiscuity in HTS screening decks is a perceived issue, in that years of experience looking at HTS results tells us that there are some compounds that are ‘old favourites’, but it is not clear how to quantify the extent, or how to assign guilt to scaffold or fragments, except by anecdote. Pearce et al.84 have developed two methods for mapping promiscuity based on linking 180 substructural queries with HTS data. The absolute number of promiscuous compounds (defined as active in 7 or more assays) is relatively small, but their representation in hit lists is significant. Also the polyhalophenol fragment is over represented in the promiscuous compound set. In contrast, property-based filters showed no association with promiscuity, whereas several substructural queries did. The trend is that promiscuous compounds tend to be smaller and more soluble. Using two measures of promiscuity, compounds can be ranked by consensus for potential promiscuity. In practise, this can reduce the number of compounds considered for triage by 12%.

9. Conclusions The impetus to generate models in the field of ADMET has been sustained, as more high quality data becomes available. There is a price to pay, and now more care is being taken to ensure that these models, which can be widely deployed to unsuspecting users, are robust and give confidence estimates with the predictions. As methods for virtual screening are also becoming more robust, there are increasing numbers of applications for target fishing—rather than use the target to find the ligands, structures are being used to deorphanise targets; this is a field of increasing importance as we explore the effects of compounds in whole cell assays. There have been significant advances in docking, scoring, and the handling of protein flexibility. Many of the big questions posed in 2005 remain unresolved, but the good news is that the amount of new methodological research suggests that we think that one day, we will have the answers.

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Solvation effects Michael Springborg* DOI: 10.1039/b608783n

I.

Introduction

The goal of electronic-structure calculations is to provide information on real materials. The field of electronic-structure calculations is therefore placed somewhere between experiment and theory. I.e., by carrying through such calculations on specific systems one obtains results that cannot be manipulated but instead have to be interpreted. This resembles the situation of experiment. On the other hand, by modifying the systems one is able to follow the changes in the properties also to idealized situations that lie outside the range of experiment, which is a typical approach within theory. However, electronic-structure calculations are only then useful, if they can provide results that are in qualitative and quantitative agreement with experimental results, whenever such ones are available. This means that not only idealized situations shall be accessible with such calculations, but also realistic situations that are as close as possible to those found in experiment. Nevertheless, the fact that electronic-structure calculations have to rely on a number of approximations (see, e.g., ref. 1) often puts serious limitations on the systems and situations that can be treated with such calculations. These limitations exist first of all concerning the size of the systems that can be treated, i.e., the number of symmetrically inequivalent atoms and/or the number of structural degrees of freedom. The reason for their occurrence is twofold: At first, the computational requirements of a typical electronic-structure calculation scale as the size of the system to some power that typically is 3 or larger. And second, if the system has no well-known symmetry, the number of independent local total-energy-minimum structures grows faster than any power in the size of the system (see, e.g., refs. 1 and 2). This means that only systems of not too large size can be treated computationally, although, due to the continuous developments in computer technology and computer programs, this limit is continuously been pushed towards larger systems. Nevertheless, there are finite limits for what is computationally possible. Therefore, finite systems without any interactions with any surroundings are the ideal systems for theoretical studies. On the other hand, experimental studies are often carried through under quite different conditions. This includes that the systems of interest may interact with a solvent or with one or more surfaces (for instance of a catalyst). As we shall see below, the presence of such extra media can modify the properties of the system of interest. Therefore, in order to obtain a proper connection to the experimental studies, it is important that the presence of these media are included in the calculations ‘somehow’. In the present contribution we shall discuss how the effects of a solvent can be included. The challenges are, however, huge. A solvent is a macroscopic system containing an enormous number of atoms and most often having essentially no symmetry. Therefore, a standard electronic-structure calculation on the complete system would require more computational resources than ever will be available. Instead, one has to apply special-purpose methods that have been developed with exactly the goal of studying the properties of a single molecule that is moving under the influence of the solvent molecules. Physical and Theoretical Chemistry, University of Saarland, D–66123, Saarbru¨cken, Germany. E-mail: [email protected]


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There are more different ways one can model this situation, depending on the precise type of interaction between the solvated molecule and the solvent. In many cases, these interactions are (partly or purely) of electrostatic nature, meaning that one may model the effects of the solvent as that of creating an external, electrostatic field in which the molecule of interest exists. But, there is also a feed-back between the solute and the solvent. I.e., the solute may in return influence the charge distribution of the solvent, thus ultimately leading to a modified effect of the influence of the solvent. Furthermore, in some cases (that are not at all atypical) hydrogen bonds between the solvent and the solute are important and should, therefore, also be included. It is the purpose of the present overview to discuss the various types of methods. In the next section we shall describe their theoretical foundations, and in the subsequent section we shall give some few selected examples of their applications. We add that the theoretical treatment of solvent effects is a currently very active research field, which can be seen both in the recent review by Tomasi et al.3 as well as from the fact that even special issues of certain journals have been devoted solely to this topic.4 Finally, we emphasize that we have not at all tried to be exhaustive in the choice of recent studies that shall be briefly mentioned below. Rather, they are meant as representing typical studies. Many others, not at all worse (or better), could have been selected.

II. Fundamental methods In this section we shall discuss the basic principles behind the different methods that have been developed for including solvation effects in electronic-structure calculations. Starting with the isolated molecules of interest, without the solvent, we shall subsequently see how the effects of the solvent can be included. We shall also briefly discuss the advantages and disadvantages of the various approaches. In the next section we shall then present different recent studies where the methods of the present section have been applied in calculating a number of different properties of different systems. A.

Without the solvent

In a typical electronic-structure calculation, the Born-Oppenheimer approximation is invoked. This means that for a system of M nuclei and N electrons one first seeks the electronic energy Ee by solving the electronic Schro¨dinger equation Hê Ce = EeCe

(1)

with ê ¼ 1 H 2

N X i¼1

r2! þ ri

N N X M N X X 1X 1 Zk ! þ Vext ðr i Þ: ! ! ! 2 i6¼j¼1 j r i r j j i¼1 k¼1 j !r R j i¼1 i k

ð2Þ

h 1. Moreover, ~ ri is the Here, we have used atomic units, me = |e| = 4pe0 = ~k that of the kth nucleus. Zk is the charge of the kth position of the ith electron, and R nucleus and, finally, we have assumed that the system is under the influence of an external, static field, Vext(~ r). Once Ee has been calculated, the total energy is obtained from Etot ¼ Ee þ

M M X ! 1 X Zk Zl þ Vext ðRk Þ: ! ! 2 k6¼l¼1 j R R j k¼1 k l

ð3Þ

~k) may actually also depend on the charge or mass of the kth nucleus (e.g., Vext(R when being of electrostatic or gravitational nature), but we shall not specify that dependence explicitly. 68 | Chem. Modell., 2008, 5, 67–118 This journal is

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In practical calculations, it is never possible to obtain the exact solution to eqn (1). Instead, approximate methods are used that are based on either the Hartree-Fock approximation or the density-functional formalism in the Kohn-Sham formulation. In the case of the Hartree-Fock approximation one may add correlation effects, which, however, is beyond the scope of the present discussion (for details, see, e.g., ref. 1). Then, in both cases the problem of calculating the best approximation to the ground-state electronic energy is transformed into that of solving a set of singleparticle equations, 1 ! ! ! ! ! r2 þ VC ðr Þ þ Vn ðr Þ þ V^ xc þVext ðr Þ ci ðr Þ ¼ ei ci ðr Þ: ð4Þ 2 Here, VC is the Coulomb potential from the electrons, Z ! rðr 0 Þ !0 ! dr VC ðr Þ ¼ ! ! jr 0 r j

ð5Þ

with the electron density being !

rðr Þ ¼

N X

jci ðr Þj2 : !

ð6Þ

i¼1

Moreover, Vn is the Coulomb potential from the nuclei !

Vn ðr Þ ¼

M X

Zk !

!

ð7Þ

k¼1 jr Rk j

and Vˆxc is either (in the Hartree-Fock approximation) the exchange operator that depends on all the occupied orbitals or (in the density-functional theory) the exchange-correlation potential that depends on the electron density. Independently of which approach is used, the (approximate) total energy of eqn (3) is a function of the nuclear coordinates, as is the (approximate) electronic energy, ~1, R ~2, . . . , R ~M), Etot = Etot(R

(8)

and the equilibrium structure is that for which Etot has its minimum. For isolated, finite molecules in the gas phase, the external potential Vext(~ r) = 0, and the computational requirements for solving eqn (4) scales as N to some power, typically 3, whereas the determination of the minimum of Etot as a function of the nuclear coordinates scales essentially exponentially with M. This means that exhaustive calculations are only for smaller systems possible. B. Supermolecule approaches When the molecule of interest is solvated in some solvent, interactions between the solvent and the solute have to be included. The most accurate approach would be to treat the complete system with the methods of the preceding subsection, which, however, due to the above mentioned scaling behaviours is computationally prohibitively demanding. One has, therefore, to restore to simpler approaches. In most cases, the solute consists of well-identified, neutral molecules that interact only weakly with each other. This means that reducing the solvent to consisting of just some few molecules around the solute may provide a reasonable approximation. In this case, the weak bonds (i.e., hydrogen bonds or van der Waals bonds) to more distant solvent molecules are broken, but it is expected that this will lead to only insignificant electronic redistributions. Ultimately, the finite system consisting of the solute and the smaller number of solvent molecules can be treated with the accurate methods of the preceding subsection. This is the supermolecule approach. Chem. Modell., 2008, 5, 67–118 | 69 This journal is

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The advantage of this approach is clearly that arbitrarily accurate methods can be applied. Thereby, also the weak interactions (first of all hydrogen bonds) can be treated accurately. On the other hand, when including more than those solvent molecules that are absolutely closest to the solute, the size of the system that shall be treated easily becomes so large that finite computational resources puts serious limits on the calculations. Thus, the computational requirements represent one disadvantage of this approach. Another is that longer-range interactions at first are not considered. For instance, if the solvent molecules possess a dipole moment, the electrostatic potential from more distant solvent molecules may be felt by the solute. Notice that since the number of dipoles scales essentially as r2 (with r being the distance to the solute) whereas the dipole potential scales as r2, long-range effects may be important although screening effects and averaging over different relative orientations lead to a faster convergence with respect to r. One way of introducing the interactions with more distant solvent molecules is to use a supercell approach (see, e.g., ref. 1). Then, a finite system including the solute and a smaller number of solvent molecules is repeated periodically in all three directions. Due to the periodicity, electronic-structure calculations can be carried through, but the drawback is that also the solute is repeated periodically. This means that if the repeated unit is too small, interactions between the solute molecules become non-negligible and may affect the results. C.

QM/MM approaches

A full quantum-mechanical treatment of the solute plus the solvent molecules may appear to be a too complicated approach. It would yield detailed information about the internal properties of the solvent molecules, that is not sought, and it would do so at the cost of large computational requirements. Only the properties of the solute are of interest as well as to which extent they are influenced by the presence of the solvent. Thus, only the effects of the solvent molecules outside their inner parts are of relevance. Therefore, simplified approaches that describe only those effects may be useful. This is the basic idea behind the QM/MM (quantum-mechanical/molecularmechanical) approaches. The solute is treated at the quantum-mechanical level, but the presence of the solvent is modeled through an external potential in which the solute moves. This external potential takes the role of Vext(~ r) discussed in section IIA and, therefore, the solute molecule can be treated using the same methods as for the isolated molecule in the gas phase, except that Vext(~ r) has to be included. The success of such an approach relies on the ability to devise realistic potentials Vext(~ r). One possibility is to distribute a set of solvent molecules at random and with random orientations in a finite, sufficiently large volume under a set of constraints. I.e., the density of the solvent molecules should be realistic, no two solvent molecules should come two close to each other, and no solvent molecule should come too close to the solute. Most often it is then assumed that the external potential Vext(~ r) is of electrostatic nature and can be modeled as a superposition of those of the solvent molecules. These can, e.g., in turn be approximated as a superposition of potentials from point charges placed at the positions of the nuclei. Improvements will allow the point charges to be placed at other positions, and also higher-order multipoles can be included. These potentials are devised so that selected properties of the solvent are reproduced accurately. Since the results may be biased by the randomly chosen (fixed) positions and orientations of the solvent molecules, one may repeat the calculation for more such arrangements and ultimately average the calculated properties of the solute. In refined models there may be a feedback between solute and solvent molecules. Thus, through the charge distribution of the solute, the solvent molecules feel the presence of the former. By calculating the forces acting on the solvent molecules, these may be allowed to move, ultimately leading to a molecular-dynamics simulation for the solvent. In turn, the shifted positions of those will modify the external 70 | Chem. Modell., 2008, 5, 67–118 This journal is

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potential acting on the solute, which, therefore, may respond with a changed charge distribution. Ultimately, the structure of the lowest total-energy for the complete system may be identified. In such simulations it is important that the interatomic interactions go beyond those of point charges (and, eventually, multipoles) in order to avoid that the atoms come too close. At least, short-ranged repulsions shall be included. Moreover, one may also include the internal degrees of freedom of the solvent molecules and, accordingly, not consider them solely as rigid bodies. Since the solvent molecules are included explicitly in the calculations, such approaches are called ‘explicit’. They may be quite accurate of the simple reason that, in many cases, there is a natural separation of the system into entities that interact only weakly (i.e., the solute and solvent molecules). Because the force fields that are used in the molecular-mechanics part are not unique, different models may differ in these force fields. Also differences in the quantum-mechanical approach for treating the solute may lead to different models. The advantage is that quite large systems can be treated. Disadvantages include that relatively stronger interactions between the molecules (first of all hydrogen bonds but also van der Waals interactions) are seldom described accurately. Moreover, there is no electrostatic feedback between solute and solvent molecules at this level of theory. I.e., the charge distribution of the individual solvent molecules remains largely unaffected by the solute. More recent developments aim to at improve this situation, as we shall discuss in the next subsection. D.

Polarizable-molecules approaches

Since the main parts of the solute–solvent interactions are of electrostatic nature, one may improve the models mentioned in the previous subsection by including the response of the solvent molecules to the electric field generated by the solvent and the solute. In the general case, the dipole moment of a (solvent) molecule changes under the influence of an electrostatic field, !

!

!

m ¼ m0 þa E DC þ ;

ð9Þ

~DC is the field vector, a is the where ~ m0 is the dipole moment in the field-free case, E polarizability tensor, and higher-order terms have been ignored. By knowing the polarizability (e.g., from experiment or from calculations), one can calculate the dipole moment of the solvent molecules when these are influenced by the electrostatic field generated by the other solvent molecules and by the solute. This leads to a modified dipole moment that in terms leads to a new electrostatic potential created by the solvent molecule, etc. Ultimately this leads to improved QM/MM models compared to those described in the preceding subsection. Although these approaches go back to the end of the 1970s,5–9 they are still widely used as we shall see in the next section. Finally, as was the case for the approaches of the preceding subsection, different descriptions of the interatomic interactions for the MM part lead to different models, and further differences in the results of such calculations may be due to differences in the approach used for the quantum-mechanical treatment of the solute. E. Continuum approaches The approach of the preceding section was improved over the one of section IIC by including an electrostatic response of the solvent to the presence of the solute (and of the other solvent molecules). It is thereby taken into account that the most important inter-molecular interactions in such a system most often are of electrostatic nature and an accurate description of those is essential. Since the approach considers the individual molecules, it is an explicit approach. Chem. Modell., 2008, 5, 67–118 | 71 This journal is

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A similar rationale is behind the polarizable-continuum approaches. Here, the solvent is modeled by a homogeneous continuum, i.e., the individual solvent molecules are not treated, and, therefore, this corresponds to a so called implicit approach. There are two central features of the model that can influence the outcomes of the calculations critically. At first, the solute is supposed to occupy some cavity in the solvent, whereby the precise definition of the size and shape of the cavity is not unique. Second, the continuum is able to respond to the charge distribution of the solute and, thereby, in turn by creating an electrostatic potential in which the solute exists to influence the properties of the solute. Again, the precise description of this response can differ among different approaches. We shall here briefly outline the main ideas behind this class of models. For more details, the interested reader is referred to the recent review by Tomasi et al.3 The size and shape of the cavity is one critical issue. The calculations become easier when simple shapes like spheres and ellipsoids are used, but in very many cases it is unrealistic to assume that the solute has such a simple shape. Instead, it is common practice to construct the cavity by superposing spheres placed at the positions of the atoms. Thus, any point that is inside at least one such sphere is considered a part of the cavity, and all other points are part of the polarizable continuum. Often, the radii of the spheres around each atom are chosen as van der Waals radii as defined, e.g., by Bondi10 or by Pauling,11 but other approaches exist, too (see, e.g., ref. 3). The surface of a cavity constructed as the overlap of atom-centered spheres has kinks where two spheres meet. Connolly12,13 has proposed a method that can lead to a more smooth surface and that very often is used nowadays. The idea is to let a smaller sphere ‘roll’ on the outer surface of the cavity. In most cases, this smaller sphere touches the cavity at exactly one point, but close to the regions where two (or more) spheres meet, the smaller sphere will touch all these spheres. In that case, that part of the smaller sphere that is closest to the cavity is used as a part of the cavity surface replacing the parts where the kink occurs. The presence of the continuum modifies the electrostatic potential. Formally, one may split this into one part from the solute and one part from the continuum, V(~ r) = Vm(~ r) + Vc(~ r),

(10)

but only for the total potential, one may set up the Poisson equation, r[e(~ r)rV(~ r)] = 4pr(~ r),

(11)

r(~ r) = rm(~ r) + rc(~ r),

(12)

where

is the total charge distribution from the molecule (rm) and from the continuum (rc). In the electronic-structure calculation for the solute, we replace in eqn (4) V(~ r) = VC(~ r) + Vn(~ r) + Vext(~ r).

(13)

In order to solve eqn (11) we also need the boundary conditions. First of all, very far from the system, r - N, we require that limr-N rV(~ r) and limr-N r2V(~ r) remain finite. More important is it that V(~ r) is continuous across the cavity/continuum interface, !

!

lim Vðr Þ limþ Vðr Þ ¼ 0;

!

!

!

r!rC

!

ð14Þ

r!rC

rC (~ rC+) is a point just inside (outside) the where ~ rC is a point on the interface, and ~ cavity next to ~ rC. Moreover, the derivative of the potential experiences a discontinuity !

!

@Vðr Þ @Vðr Þ e limþ ¼ 0: ! ! ! ! @n @n r!rC r!r lim

C


c


ð15Þ

Here, ~ n is an outward pointing normal to the interface. Moreover, we have assumed that inside the cavity the relative dielectric constant takes the value 1 (as in vacuum) and in the continuum the constant value e. The systems of eqns (11)–(15) is under-determined because we know neither the potential V, nor the charge density rc. However, ultimately we are only interested in the potential inside the cavity and can therefore choose V and rc arbitrarily as long as they satisfy all equations above. One suggestion is then to assume that rc vanishes except that we have a charge density on the cavity/continuum interface that is determined so that it satisfies the Poisson equation with boundary conditions. This is the strategy behind the polarizable continuum model that goes back to the beginning of the 1980s.14 Then e is given the value that is typical for the macroscopic solvent. A modified version, the conductor-like screening model (COSMO), was later proposed by Klamt and Schuü¨rmann,15 where the equations above were solved for the continuum being a conductor with e - N. Subsequently, the potential Vc was scaled in order to take the finite value of e into account. Since the shape of the cavity is that of overlapping spheres, an alternative, natural strategy could be to expand the potential V(~ r) in harmonics (i.e., potentials from monopoles, dipoles, quadropoles, . . . ). Also here various strategies have been proposed.16–19 F. Hybrid approaches None of the methods we have mentioned is new, and all of them are still the subject of improvements. Here we shall not discuss those, as we consider that beyond the scope of this presentation. However, as a single example of recent developments we mention the work of Li and Gordon.20 These authors presented a hybrid method that aims at merging the two approaches of sections IID and IIE into one hybrid method. G.

Frozen-density approaches

Alternatively, there have been other approaches earlier that, currently, seem to be less frequently used than those discussed above. Among those is the frozen-densityfunctional approach of Wesolowski and Warshel.21,22 This is an explicit approach where solvent molecules are directly included in the calculations. It is based on density-functional theory and a separation of the total electron density into that of the solute, r1, and that of the solvent, r2. According to the density-functional theory, we may write the total electronic energy as a functional of r(~ r) = r1(~ r) + r2(~ r),

(16)

Ee[r] = Ee[r1 + r2] = Ee[r1] + Ee[r2] + Ee,non-add[r1,r2],

(17)

i.e., as

where the last term is the non-additive part of the total electronic energy. Since we are mainly interested in the properties of the solute, we may choose to use less accurate approaches for describing those part of the total electronic energy that originate from the solvent. Doing so we end up with a method that scales much more favourable with system size than the supermolecule approaches of section IIB. H.

Recent developments

As may have become clear, methods for the quantum-mechanical treatment of molecules in solutions were originally developed more than 20 years ago. But as in any other active field, the methods have continuously been improved, partly in order to increase accuracy and computational speed and partly to make possible the Chem. Modell., 2008, 5, 67–118 | 73 This journal is

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calculation of further properties, for instance such that are experimentally accessible or relevant. Many of the developments are directly tied to specific scientific questions and will in some cases be discussed further in the next section where we focus on applications, but here we shall briefly discuss some few recent works where methodological aspects have been put at the foreground. Iozzi et al.23 considered systems where the solute was experiencing surroundings that had some kind of structure. When modifying the polarizable continuum model of section IIE, this means in effect that the continuum was heterogeneous, i.e., e takes different values in different parts of space. As examples the authors mention different biological systems like water-soluble proteins where some residue interact both with the water and with the bulk proteins. Furthermore, molecules in lipid bilayers interact both with the water solvent and with the lipid bilayer. In order to treat such situations the authors suggested an extension of the continuum method where different parts of the surroundings have different relative dielectric constants. This leads to some complications that, however, could be solved. Finally, the authors demonstrated the applicability of the approach by addressing two biological issues, i.e., the protonation energies of the histidine residue, and the calculation of the nitrogen hydrogen-fine-splitting of the tempo radical in biological membranes. Here, we do not consider it relevant to discuss these issues further. In another recent development, Mennucci24 presented an extension of the polarizable continuum method that allows for studying time-dependent phenomena. Such approaches are, e.g., relevant when studying fast processes like electronic excitations or transfers. Due to the changed charge distribution of the solute following electronic excitations or transfers, the interaction with the solvent will change which in turn may change the properties of the solute, as we have discussed in this section. Since these are dynamical processes, a theoretical study of the time evolution is highly relevant. In particular when the time scale of the processes in the solute is comparable with that of the response of the solvent, resonance phenomena beyond those of a static treatment may be important. Mennucci presented an approach for such studies. Subsequently, she applied it to two processes, a charge transfer from a donor to an acceptor in a smaller molecule, as well as to the calculation of the time dependence of a Stokes shift which is used spectroscopically to measure polar solvation dynamics. An interesting approach was proposed by Yamamoto and Kato.25 They observed that calculations for a solute in some solvent bear a close resemblance to a standard electronic-structure calculations within the Born-Oppenheimer approximation. In ~1, R ~2, . . . , R ~M) R ~ [together with the the latter case, the nuclear coordinates, (R ~ define some external parameters on which nuclear charges, (Z1, Z2, . . . , ZM) Z], the solution to the (electronic) Schro¨dinger depends. When including the solvent, one in essence introduces one further dependence of the solution to the electronic Schro¨dinger equation, i.e., on the potential generated by the solvent Vext(~ r). In their approach, Yamamoto and Kato25 let Vext(~ r) be determined through a non-polarizable model for the solvent so that Vext(~ r) can be written as a superposition of Coulomb potentials from point charges representing the solvent. They showed, subsequently, how electron-transfer processes could be treated within their approach, even in the case when quantum effects of light nuclei (most notably, protons) need to be included. Finally, they presented results for a specific electrontransfer process. Tan and Luo26 have presented an improved implicit model for solvation effects. In the discussion above, in section IIE, the continuum is considered being a medium with a certain charge density. This charge density may be vanishing in large parts of space, except in those parts where the presence of the solute is being felt. The solute is in most cases treated quantum-mechanically, which for larger (e.g., biologically relevant) systems may provide a computationally too large challenge. Therefore, simpler treatments of the solute could be useful. Instead of replacing the continuum 74 | Chem. Modell., 2008, 5, 67–118 This journal is

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description of the solvent with a force field approach that, then, could include the solute, too, Tan and Luo26 suggested to use a polarizable model for describing the properties of the solute. The interior of the solute is modeled through a dipolemomentum density that then replaces the quantum-mechanically treatment of the solute. A feed-back between the solvent and the solute adjusts the densities of both media. This approach is significantly simpler than that of section IIE and could, therefore, be relevant for more complex systems. Tan and Luo carried through a set of test calculations using this proposal. They found that the model was robust, i.e., model parameters could be transferred between different structures and surroundings of the solute. In some cases, the performance was even better than that of a force field, although the computational requirements of the latter are larger. Not only the development of theoretical approaches and accompanying computer programs has led to an increase in the accuracy that can be achieved and the sizes of the systems that can be treated, i.e., number of atoms in the system as well as the length of the time for which a molecular-dynamics simulation can be carried through. Also developments in the computer powers are important. This was recently demonstrated impressively by Saito and Okazaki.27 They carried through a molecular-dynamics simulation over 45 ns for hemoglobin in water using the ‘Earth Simulator’ computer that 2002–2004 was the worldwide fastest supercomputer. They considered a system consisting of the human adult hemoglobin molecule surrounded by water molecules so that the complete system occupied a sphere with a radius of 66 A˚. This system, consisting of 119 421 atoms, was treated with a force-field approach, and longer-range solvent effects were treated using a polarizable continuum approach. From the simulation the authors were able to study the dynamics of even the tertiary and quaternary structures of the hemoglobin molecule.

III.

Recent studies

In the preceding section we outlined briefly the main ideas behind the different methods/models that have been introduced for the theoretical treatment of solvated molecules. We emphasize that we concentrated on presenting the foundations and not on discussing the many, also recent, improvements that have been introduced in order to increase both the accuracy and the efficiency of the different approaches, which has led to a large variety of methods, each with its advantages and disadvantages. Thus, if the reader gets the impression that all currently applied methods were developed long time ago, this is only partly the case. However, the reader who is interested in the recent technical and methodological developments is referred to the original literature. Here, we shall instead present results of recent studies on various systems and in the discussion use the rough classification of the methods from the preceding section. We shall briefly discuss the results of various, subjectively chosen, recent studies, emphasizing that these were chosen not as an attempt of identifying the scientifically best studies, but more as representing typical studies at the time of writing. A.

Comparing the methods

None of the approaches that has been proposed is exact. Instead, each approach has its advantages and disadvantages, and each approach may be more accurate for certain systems and/or properties than for others. Thus, one of the first questions to address is, how well the various approaches perform. This has been the topic of some recent studies. Jacob et al.28 compared a frozen-density and a polarizable-molecules approach for the calculation of various molecular properties. In both approaches they used a density-functional method for the quantum-mechanical treatment of the solute. Chem. Modell., 2008, 5, 67–118 | 75 This journal is

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Table 1 Various properties for a water molecule in the gas phase and in an water solvent. The FD results were obtained with a frozen-density approximation, and the QM with a polarizablemolecules approximation. Listed are the size of the dipole moment, the three independent components of the quadrupole moment, excitation energies, static polarizability, and static hyperpolarizability. All results are from ref. 28 Property

Quantity

Dipole moment (D) Quadrupole moment (a.u.)

m 1.80 Qxx 1.79 Qyy 1.86 Qzz 0.07 7.76 11A1 - 11B1 11A1 - 11A2 9.61 11A1 - 21A1 9.72 a 9.40 g 0.91

Excitation energy (eV)

Static polarizability (a.u.) Static hyperpolarizability (a.u.)

Gas phase

Solution (FD) Solution (QM) 2.66 2.05 2.15 0.11 8.41 10.38 10.40 9.62 0.72

2.71 2.09 2.17 0.08 8.88 10.69 11.01 8.67 0.50

Their test system was a single water molecule embedded inside a solvent consisting of 127 further water molecules. For the latter a fixed structure was used. In Table 1 we summarize their main findings. For the frozen-density calculations they considered two different approaches, one where the solvent-molecule density was kept fixed and one where it was allowed to relax. In the table we have only shown the results for the latter, which according to the authors led to an improved accuracy. The table shows that the dipole and the quadrupole moments are very similar for both approaches, which is to a lesser extent the case for the excitation energies and the static (hyper)polarizabilities. The latter were calculated using timedependent density-functional theory. In order to understand this discrepancy the authors used also a supermolecule approach with just two solvent molecules. By comparing with results from calculations with the frozen-density and the polarizable-molecule approaches on the same system they concluded that the frozen-density approach was the more accurate one in calculating the responses to electromagnetic fields. Although we in this report mainly is concerned with the behaviour of a single solute molecule in some solvent, the study of mixed systems can give some insight into the accuracy of the approaches. Therefore, we shall next discuss the work of Chevrot et al.29 who reported a study of the interface between water and the ionic liquid [BMI][PF6] (1-butyl-3-methyl-imidazolium hexafluorophosphate). They used molecular-dynamics simulation with different force fields to describe the interactions between the atoms of the system. They constructed an interface by joining a box of 2119 water molecules with a box of 181 BMI+PF6 molecules. In addition, they studied demixing of the two systems by considering a system that initially consisted of a mixture of water and the ionic liquid. In Table 2 we summarize some of their findings. The table shows the amount of water in the ionic liquid, the amount of ionic liquid in the water, and the width of the interface separating the two phases at the end of the simulation for the different force fields. It is seen that the popular TIP3P model gives quite different results from those of the other force fields and in the simulation of the mixing process there is not even a well defined interface. The other two force fields give similar results. All simulations support that the ionic liquid is essentially water-free, as should be. In order to understand the differences between the results of the different models, the authors also calculated the interaction energies between water and the ions of the ionic liquid. The results are reproduced in Table 3. The table shows at first that the variation in the interaction energies for the different computational approaches is small. This demonstrates that even small inaccuracies can have significant effects on the output of a simulation. Moreover, the table shows also that the TIP5P model 76 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 2 Molar fraction of water in the ionic liquid and of the ionic liquid in water, as well as width of the interface as found in the molecular-dynamics simulations with different force fields. q(IL) describes the effective charge of the ions in the ionic liquid used in the force field for this system, and FF(H2O) describes the force field for the water. The results for the ‘mixing’ simulations were obtained by considering a system where the ionic liquid and the water initially were separated, whereas ‘demixing’ marks results where the two systems initially formed a mixture. All results are from ref. 29 Simulation

q(IL)

FF(H2O)

xH2O

xIL

Width (A˚)

Mixing

1 1 0.9 0.9 0.9 0.9 0.9 0.9

SPC TIP5P TIP3P SPC TIP5P TIP3P SPC TIP5P

0.65 0.45 0.66 0.23 0.27 0.71 0.29 0.25

0.023 0.004 0.008 0.004 0.001 0.009 0.002 0.001

— 14.3 18.9 10.9 9.3 22.1 10.6 9.4

Demixing

with scaled effective charges of the ionic liquid is the model that at best reproduces the results of the parameter-free calculations. In section II we focused on an accurate description of electrostatic interactions between solute and solvent. Although these interactions account for the largest parts of the free energy of solvent, other interactions may contribute as well. These interactions, due to dispersion and repulsion, were studied by Curutchet et al.30 Amoville and Mennucci31,32 had earlier developed a formalism for including these interactions in the polarizable continuum model and Curutchet et al. tested it on a larger set of solvent-solute systems for which they also calculated these interactions with force-field methods. Since these interactions only rarely are discussed, we list typical results from their work in Table 4. It is seen that fairly large basis sets are needed in order to obtain results that can be considered close to converged. Moreover, the density-functional B3LYP method yields larger absolute values than does the Hartree-Fock method. Finally, the repulsion contribution amounts to only some 10–20% of the dispersion contribution. Using the B3LYP method for reference they compared subsequently the performance of four force-field methods for reproducing the B3LYP results for the dispersion and repulsive contributions to the free energy of solvation for 22 neutral molecules in four solvents. The results, Table 5, show that in most cases the force fields do not perform very well with the MST being an exception. Aidas et al.33 calculated NMR parameters for formaldehyde and acetone solvated in water using different models. Their main findings are summarized in Table 6. In Table 3 The interaction energies (in kcal/mol) between a water molecule and an ion from the ionic liquid, as obtained from calculations with Hartree-Fock (HF), density-functional (DFT), or force-fields. All results are from ref. 29 Method

q

PF6q H2O

BMI+q H2O

HF DFT TIP3P SPC TIP5P TIP3P SPC TIP5P

1 1 1 1 1 0.9 0.9 0.9

9.1 9.5 11.5 11.4 10.5 10.2 10.1 9.3

9.7 9.5 10.2 10.0 10.3 8.8 9.1 9.3


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Table 4 The dispersion and the repulsion contributions (in kcal/mol) to the free energy of solvation for C2H4O solvated in water or chloroform. The table shows the dependence of the results on the basis set and on the quantum-mechanical method for treating the solute. All results are from ref. 30 Solvent

Method

Basis set

DGdis

DGrep

Water

HF

6-31+G(d) cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ 6-31+G(d) cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ

7.4 7.3 9.2 9.1 10.2 8.4 8.5 10.9 10.3 12.0 8.6 10.8 10.6 11.9 10.0 12.8 12.0 14.0

3.1 2.6 3.1 2.9 3.1 3.4 2.7 3.5 3.2 3.4 1.7 2.0 2.0 2.0 1.8 2.4 2.1 2.3

B3LYP

Chloroform

HF

B3LYP

their calculations they compared the polarizable continuum models with QM/MM models and, moreover, they examined the effects of including two water molecules in the quantum-mechanical calculations. The results of Table 6 show that both longand short-range interactions are important. Thus, a quantum-mechanical treatment of the finite C3H6O + 2H2O system is not capable of reproducing the experimental results, showing the importance of long-range interactions. But without the two water molecules, the polarizable continuum model does not yield satisfactory results, which indicates that also short-range interactions are important. In this context it is interesting to notice that including the two water molecules in the part that is treated quantum-mechanically does not seem to be very important for the results with the QM/MM simulations. Sun et al.34 studied the properties of thirteen smaller peptides in an aqueous solution using both explicit models with different force fields for the solvent Table 5 A comparison between quantum-mechanical and force-field treatment of dispersion and repulsive contributions to the free energy of solvation. The force field is given under method, and for each solvent the first line gives the root mean square deviation (rmsd, in kcal/ mol) and the second line gives the coefficient c that gives the best fit of y = cx with y being the force-field results and x being the quantum-mechanical results. All results are from ref. 30 Method

Property

Water

Octanol

Chloroform

Carbon tetrachloride

Claverie

Rmsd c Rmsd c Rmsd c Rmsd c Rmsd c

1.9 0.87 4.0 0.71 5.7 0.59 5.6 0.59 1.2 1.01

5.4 0.68 7.5 0.56 5.4 0.68 5.6 0.67 3.1 1.17

9.8 0.48 8.8 0.54 9.2 0.51 9.1 0.52 3.6 0.82

10.3 0.47 8.9 0.54 9.6 0.50 9.6 0.50 3.8 0.81

MM3 AMBER OPLS MST


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Table 6 Isotropic shielding constant (in ppm) sX for X being oxygen or carbon and the shift dX compared to the gas-phase value. QM denotes the system that is treated quantum mechanically, and the water part describes the approach for treating the solvent. For the latter, PCM1 and PCM2 denotes polarizable continuum models and TIP3P and SPC two different force fields. marks gas-phase calculations. All results are from ref. 33 QM part

Water part

sO

dO

sC

dC

C3H6O C3H6O C3H6O C3H6O + 2H2O C3H6O C3H6O + 2H2O C3H6O + 2H2O C3H6O + 2H2O C3H6O + 2H2O Experiment

— PCM1 SPC SPC TIP3P TIP3P — PCM1 PCM2

339.2 285.6 259.3 258.7 250.6 251.0 297.3 262.7 250.1

— 53.6 79.9 80.5 88.6 88.2 41.9 76.5 89.1 75.5

21.5 33.2 37.0 37.7 38.2 38.9 31.1 36.6 38.2

— 11.7 15.5 16.2 16.7 17.4 9.6 15.1 16.7 18.9

0.7 0.8 0.6 0.6 0.5 0.5 0.5

0.7 0.8 0.6 0.6 0.5 0.5 0.5

0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.1 0.1 0.1 0.1

molecules and implicit models. Their interest focused on the interactions of the peptide with a hydrophobic surface, and to this end they studied the so called potential of mean force (PMF) for the solvated peptides using the different solvation methods. They calculated the PMF by considering the interaction between a selfassembled monolayer of some molecules on some surface. The important point is that the molecules were terminated by CH3 groups, which then are the ones that the peptide will interact with. The interactions will, of course, depend on the distance of the peptide to the CH3 group, but also on the relative orientation of the two as well as on the internal properties of the peptide. Here, the latter depends on the changes induced by the solvent, i.e., on the model for the solvent. Sun et al. found significant differences between the implicit and the explicit solvent models. However, lacking further information they were not able to conclude which of the models gave more accurate results. In a recent study, Kongsted and Mennucci35 compared explicit and implicit solvation models for the calculation of various properties of aqueous solutions of pyrazine, pyrimidine, and pyridazine, i.e., three diazines that can be obtained from benzene upon replacing two CH groups by two N atoms (cf. Fig. 1). For the quantum-mechanical part they used a density-functional approach and, moreover, they also considered cases where 2 or 4 H2O molecules of the solvent were included in the QM treatment. Some of their results are summarized in Table 7. Some spread in the results depending on the model is observed, and in this case we also notice that the results that are obtained with the continuum approaches depend on the size of the spheres that are used in the construction of the cavity. Ultimately, however, the authors showed that when including 2 water molecules in the quantum-mechanical treatment of the system (thereby allowing for hydrogen bonds) and allowing the complete quantum-mechanical part of the system to relax to the structure of the lowest total energy, a good agreement with experimental values could be obtained with the unscaled polarized-continuum model (Table 8).

Fig. 1 Schematic representation of (from left to right) pyrazine, pyrimidine, and pyridazine. Open and closed circles represent CH groups and N atoms, respectively.


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Table 7 Lowest electronic excitation energy (in eV) and isotropic shielding constant (in ppm) sN. n gives the number of water molecules that are explicitly treated quantum theoretically. PCM denotes the standard polarized continuum method, whereas in PCM* the radii spheres surrounding each atom in the construction of the cavity have been scaled by 91.7%. Finally, MM marks results obtained with an explicit description of the solvent. All results are from ref. 35 Property

n

Model


0 0 0 2 2 0 2 4 0 0 0 2 2 0 2 4

Vacuum PCM PCM* Vacuum PCM MM MM MM Vacuum PCM PCM* Vacuum PCM MM MM MM

sN (ppm)

Pyrazine 3.95 4.00 4.00 3.95 4.00 4.05 4.03 113.3 101.8 97.9 110.4 99.7 98.9 100.3

Pyrimidine

0.01 0.01 0.01 0.01

0.8 0.3 0.8 0.8

4.28 4.45 4.49 4.30 0.01 4.48 0.01 4.52 0.01 4.49 0.01 4.48 0.01 72.1 59.9 55.7 67.8 0.5 57.4 0.4 56.4 0.5 58.4 0.5 58.8 0.5

Pyridazine 3.55 3.89 3.95 3.65 3.92 3.85 4.03 210.4 176.1 168.8 198.5 171.0 178.5 179.7

0.01 0.01 0.01 0.01

1.0 0.6 1.0 1.0

As the last example we mention the study of Takemura and Kitao.36 They studied different models for molecular-mechanics simulations on water. To this end, they studied the dynamics of a ubiquitin molecule solvated in water. They performed molecular-dynamics simulations for a system with periodic boundary conditions. At first they considered pure water without the solute and studied boxes with 360, 720, 1080, and 2160 water molecules. It turned out that even for these fairly large systems, finite-size effects could be recognized. Thus, the translational diffusion constant was found to depend linearly on V1/3, where V is the volume of the repeated unit. Subsequently, they studied a periodically repeated volume containing the ubiquitin molecule as well as 4908, 9133, 13 200, or 29 182 water molecules. Again, the linear dependence of the translational diffusion constant of the protein on V1/3 was observed, which is a confirmation of well-known finite-size effects.37–39 Correcting the results for the finite-size effects the authors obtained the results of Table 9. A clear scatter in the calculated values is observed, making it non-trivial to determine the correct model for molecular-dynamics simulations for aqueous solutions. B. Aqueous solutions of atoms The transport of ions in aqueous solutions is of fundamental importance in chemistry and biology. However, despite its apparent simplicity, the transport of Table 8 Change in lowest electronic excitation energy (in eV) and in isotropic shielding constant (in ppm) sN due to solvation for the molecules of Fig. 1. PCM denotes results with the polarized-continuum model and Exp marks experimental results. Two water molecules have been included in the quantum-mechanical treatment. All results are from ref. 35 Property

Model

Pyrazine

Pyrimidine

Pyridazine


PCM Exp. PCM Exp.

0.16 0.19 20.8 16.9

0.31 0.33 20.2 16.8

0.55 0.48 59.2 41.6

sN (ppm)


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Table 9 The self-diffusion constant for water Dtw and the ubiquitin molecule Dtp in aqueous solutions as calculated with different force fields for the water. In addition, t1 and t2 are rotational relaxation times for the ubiquitin molecule. Where possible, the results are compared with experimental values. All results are from ref. 36 Model

Dtw (105 cm2/s)

Dtp (105 cm2/s)

t1 (ns)

t2 (ns)

TIP3P SPC SPC/E SPC/Fw Exp.

6.35 4.96 3.06 3.28

36.9 30.6 17.1 14.2 14.3, 14.9

4.09 4.88 9.13 8.18

1.33 1.78 2.97 2.69 4.09, 4.1

ions in water is a non-trivial process, where not only the ions but also a certain cloud of water molecules surrounding the ions is being transported. The study of these water shells around the ions is, therefore, still being pursued. Here, we shall discuss two such recent studies.40,41 Using a molecular-mechanics force-field model in combination with moleculardynamics simulations, Glezakou et al.40 studied the structure of K+ ions solvated in water. The simulation considered the potassium ion as well as 550 water molecules and was carried through at different temperatures. Among the results is the (timeaveraged) pair distribution function. Due to the finite temperature, such a function does not contain sharp peaks but more broad features from which one may try to extract the number of nearest neighbours. Using two different approaches to this end, Glezakou et al.40 obtained two different values, cf. Table 10. However, it is clear that an increase in the temperature results in a slight decrease in the number of water molecules in the first shell and a small increase in the distance between the ion and the water molecules. Glezakou et al.40 also performed experiment at 300 K, resulting in the values given in the table, too. With a supermolecule approach, Bock et al.41 studied the first and second shell of water molecules surrounding various metal ions. They studied M+q(H2O)18 clusters using parameter-free density-functional calculations and trying to optimize the structures in largely unbiased calculations. Due to the size of the system (55 atoms, or 19 building blocks when assuming that the H2O units stay intact), this is a far from trivial endeavour (see, e.g., ref. 2). Their results are summarized in Table 11. For all systems, a structure with one shell containing 6 water molecules and another shell with 12 water molecules is found, but for some cases (Be, Li, and Na) another structure with three shells of water with, respectively, 4, 8, and 6 molecules is also stable. However, the fact that the metal-oxygen distances are not constant for each shell shows that the structure is one of a low symmetry. As the last example we mention the study of Blumberger and Sprik42 who were interested in the electron-transfer redox reaction Ru2+ " Ru3+ in an aqueous solution. This reaction can be studied in the framework of the Marcus theory of electron transfer43–46 which provides a simple and effective treatment of the polar solvent. Blumberger and Sprik demonstrated how the ingredients of the Marcus theory could be extracted from electronic-structure calculations on Ru2+ and Ru3+ ions in a water solvent. To this end, they carried through density-functional calculations on a periodically repeated box containing 32 H2O molecules and one metal ion using the Car-Parrinello approach (see, e.g., ref. 1). In order to keep the system neutral, a compensating constant background density was added. Their results indicated that the Ru–O distance for the nearest water shell around the metal ion decreases from 2.18 A˚ for Ru2+ to 2.10 A˚ for Ru3+. These values compare reasonable well with the experimental values of 2.11 and 2.03 A˚, respectively. Ultimately, the authors estimated that the free-energy barrier for the Ru2+ " Ru3+ reaction is 0.38 eV, whereas the experimental value is 0.55 eV. The authors Chem. Modell., 2008, 5, 67–118 | 81 This journal is

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Table 10 The number of water molecules in the first shell around the K+ ions, n1, estimated in two different ways, as well as the average distance between the potassium ion and the oxygen atom of the water molecules, all quantities as a function of temperature. The experimental values were obtained at 300 K. All results are from ref. 40 T (K)

n1

d1 (A˚)

300 350 400 Exp.

6.6/5.7 6.6/5.6 6.5/5.3 5.7

2.770 2.777 2.782 2.732

Table 11 The number of water molecules around a metal M+q ion. n1, n2, and n3 give the number of molecules in the first, second, and third shell, respectively, and d1, d2, and d3 the distances between the metal ion and the oxygen atoms of the water molecules. All results are from ref. 41 M

q

n1

d1 (A˚)

Al Be Mg Na Ti Li Zn Be Li Na

3 2 2 1 4 1 2 2 1 1

6 6 6 6 6 6 6 4 4 4

1.922 1.858 2.099 2.415 1.993 2.191 2.120 1.638 1.956 2.279

0.001 0.010 0.001 0.16 0.28 0.10 0.006 0.018 0.05 0.06

n2

d2 (A˚)

12 12 12 12 12 12 12 8 8 8

3.988 4.103 4.130 4.184 4.092 4.075 4.123 3.672 3.955 4.086

0.010 0.4 0.017 0.19 0.4 0.13 0.07 0.15 0.18 0.28

n3

d3 (A˚)

6 6 6

4.826 0.18 4.808 0.16 4.904 0.17

suggested that the small simulation box is the reason for the discrepancy. That also long-range effects can be important was seen already in the previous subsection. An interesting finding of the authors on this system and, earlier, on Ag+ in water47 is that the volume of the water does not change upon adding the metal ion. The contraction of the liquid around the cation can be ascribed to electrostriction effects. C.

More complex systems

In this subsection we shall concentrate on the solvent-solute structures of more complex systems. We shall not discuss other properties of those systems, but solely consider those recent studies where the structural arrangement of solvent and solute has been the central issue. Other properties will be discussed later in this overview. Methanol-water mixtures were studied by Adamovic and Gordon48 who used the so called effective-fragment-potential method, an approach of the type discussed in section IID, for finite (MeOH/H2O)n clusters with n = 2,3, . . . ,8. Some of their findings were tested by using accurate, parameter-free, electronic-structure methods. The main outcome is the organization of the H2O and MeOH molecules relative to each other, which then is expected to be representative for macroscopic mixtures of water and methanol. Their results for the energetically three lowest structures are summarized in Table 12. Most useful is it to describe the structures in terms of the different types of hydrogen bonds, i.e., the donor and acceptor molecules for the hydrogen bonds. According to Table 12 the most stable structures are found for a large number of H2O–MeOH hydrogen bonds. A related study was presented by Kumar et al.49 who studied alkali-metal hydroxides (MOH, M being Na or K) hydrated by up to six water molecules, i.e., MOH(H2O)n, with n = 1,2, . . . ,6. Their calculations were performed using 82 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 12 Characteristics for the three energetically lowest isomers of (MeOH/H2O)n clusters. m labels the isomer, and DE gives the total energy relative to the most stable isomer. The donor and acceptor entries, X and Y, show the number of hydrogen bonds between the donor X and the acceptor Y. All results are from ref. 48 n Donor Acceptor

m

2

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

3

4

5

6

7

8

DE (kcal/mol) H2O H2O

MeOH MeOH

MeOH H2O

H2O MeOH

0 1 1 1 2 3 1 1 4 1 5 5 4 5 4 2 2 1 5 4 6

0 1 1 1 2 2 0 0 4 1 4 3 2 3 2 1 0 0 1 2 2

2 1 1 2 1 1 4 4 0 4 1 2 4 3 4 6 7 7 7 6 6

2 1 1 3 2 1 6 5 2 8 2 4 7 6 7 11 11 11 10 11 8

0.0 0.0 0.5 0.0 0.6 1.5 0.0 1.5 3.5 0.0 1.2 2.7 0.0 0.2 1.8 0.0 1.8 3.0 0.0 1.6 3.3

different parameter-free, electronic-structure methods. With successive addition of water molecules, both the number of hydrogen bonds and the coordination number of the metal atom increase. By monitoring the M–OH distance (see Table 13) the authors could also identify the change from undissociated, via partly dissociated, to dissociated metal hydroxide upon the addition of water molecules. Tongraar et al.50 studied NO3 in an aqueous solution using QM/MM molecular dynamics simulations for a system consisting of one NO3 ion and 199 water molecules. They found that the oxygen atoms of the NO3 ion participate in weak hydrogen bonds to the water molecules that easily are broken and formed. Parthasarathi et al.51 examined the surroundings of a H3O+ ion in phenol or benzene by studying a complex consisting of one H3O+ ion and three phenol or benzene molecules. They performed parameter-free, electronic-structure calculations on these systems using three different computational methods, i.e., Hartree-Fock calculations, Hartree-Fock calculations with Møller-Plesset perturbational inclusion of correlation effects, and density-functional calculations within the B3LYP framework. They identified three clusters, all shown in Fig. 2. The binding energies (i.e., the energy needed for dissociating the complex in the H3O+ ion and the three phenol or benzene molecules) are shown in Table 14, too, and it is seen that in particular the relative ordering is the same for all computational approaches. Finally, by analysing the topology of the electron density the authors were able to identify the bonding between the H3O+ ion and the surrounding molecules, i.e., for the most stable structure a hydrogen bond was found, whereas for the others they found a p complexation (Table 14). Kokubo and Pettitt52 studied mixtures of urea and water at different concentrations through molecular-dynamics simulations for seven different systems ranging from 1 urea molecule and 1305 water molecules to 530 urea molecules and 0 water Chem. Modell., 2008, 5, 67–118 | 83 This journal is

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Table 13 Binding energy, Eb, for MOH(H2O)n clusters obtained either with Møller-Plesset calculations (MP2) or with density-functional calculations (B3LYP). NHB and NC give the number of hydrogen bonds in the cluster and the coordination number of MOH, R is the distance between the metal atom and the oxygen atom of the hydroxy group, respectively. All results are from ref. 49 M

n

Eb,MP2 (kcal/mol)

Eb,B3LYP (kcal/mol)

NHB

NC

R (A˚)

Na

1 2 3 4 5 6 1 2 3 4 5 6

21.25 39.27 55.08 66.50 76.96 88.75 21.75 40.03 55.46 66.52 78.45 87.68

22.52 41.61 56.83 69.02 79.54 90.12 21.62 39.64 54.18 65.44 75.57 85.22

1 2 3 4 5 9 1 2 3 4 6 7

2 3 4 4 4 3 2 3 4 5 5 4

2.097 2.247 2.674 2.835 2.404 4.031 2.376 2.523 2.734 2.838 2.898 3.017

K

molecules. The interatomic interactions were described using force fields, whereby they examined the performance of two different such models for urea. A main focus of their work was to study how hydrogen bonds are formed and broken, and they

Fig. 2 The four optimized structures of the H3O+ ion surrounded by three phenol or benzene molecules. Reproduced with permission of The American Chemical Society from ref. 51.


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Table 14 Binding energy, Eb, for the complexes consisting of a H3O+ ion surrounded by three phenol or benzene molecules. The results were obtained with Hartree-Fock calculations (HF), Møller-Plesset calculations (MP2), or with density-functional calculations (B3LYP). For each of the four structures (see Fig. 2), also the type of the interaction between the ion and the molecules is shown, respectively. All results are from ref. 51 Isomer

Eb,HF (kcal/mol)

Eb,MP2 (kcal/mol)

Eb,B3LYP (kcal/mol)

Type of bond

1 2 3 4

72.89 51.99 50.16 46.10

83.21 63.06 61.35 55.65

81.18 65.65 63.09 54.73

H-bond OH p OH p OH p

found that both the average number of hydrogen bonds and their average lifetimes do not change very much at different concentrations (see Table 15), but it was also clear that there was some dependence of the results on the model that was used. As a final example in this subsection we mention the recent study of Zillich and Whaley53 who examined LiH solvated in 4He clusters with up to 100 atoms. The authors used a path integral Monte Carlo simulation approach, whose details shall not be discussed further here. The LiH–He interaction potential was found to be highly anisotropic with attractions for He approaching the molecule in a direction parallel to the molecular axis, but with strong repulsions for He approaching the molecule in a direction perpendicular to the molecular bond. Despite these repulsions, the authors found that LiH prefers to occupy central regions of the LiH4Hen clusters for n larger than 10–15. D.

Solvation energies

A fundamental aspect of the theoretical study of solvation is to be able to calculate the solvation energy, i.e., the energy that is required (or released) when a molecule is brought from vacuum into the solution. Therefore, many studies have been devoted to this, but here we shall just briefly discuss three of those. Petrosyan et al.54 have presented a new method for describing solvated systems. It is based on the density-functional formalism and related to the approaches of section IIG. It combines a quantum-mechanical density-functional treatment of the solute Table 15 Various properties related to hydrogen bonds in urea-water mixtures. xu gives the mole fraction of urea, and nx gives the number of hydrogen bonds around molecules of the type x. txy gives the life time (in fs) for hydrogen bonds between molecules x and y, with x being the donor and y the acceptor. The two entries for each concentration shows the results for the two different models for urea. All results are from ref. 52 xu

nurea

nwater

0.00077

3.13 2.27 3.15 2.19 3.11 2.15 3.07 2.10 3.04 2.05 3.00 1.96 2.68 1.37

2.79 2.78 2.77 2.77 2.74 2.75 2.72 2.74 2.69 2.71 2.66 2.66

0.038 0.081 0.129 0.184 0.273 1.0

tuu

100 43 92 32 94 38 90 37 95 39 97 40

tuw

twu

tww

99 37 94 28 97 29 96 30 100 29 99 30

140 210 150 250 150 260 150 270 140 270 150 290

180 180 180 180 180 190 180 200 190 200 190 220


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Fig. 3 Calculated energies of solvation vs. experimental values for (from left to right) water, ethanol, methanol, and methane in water. The diagonal line gives perfect agreement, the open circles results with the joint density-functional theory for electronic structure of solvated systems, and the other symbols results from different continuum approaches. Reproduced with permission from ref. 54.

and a classical density-functional treatment of the solvent, i.e., it is based on minimizing a functional of the electron density of the solute and the averaged density of the solvent. The resulting so called joint density-functional theory for electronic structure of solvated systems was used to calculate the solvation energies for some molecules in water. The results are reproduced in Fig. 3, and it is seen that the new method gives accurate results. The calculation of solvation energies for ions is much more complicated. As discussed, e.g., by Kelly et al.,55 the fact that it is not possible to measure the electrostatic potential between two media56,57 means that the free energy of formation or chemical potential of an individual ion has no operational meaning.58 Instead, it is common practice to set, arbitrarily, the free energy of a proton equal to 0 and, then, to study well-defined free energies of solvation for neutral combinations of anions and cations, whereby those of the individual cations and anions can be estimated. Kelly et al.55 determined theoretically absolute values for the solvation energies of ions as well as of ions surrounded by a finite cluster of water molecules. Their approach was based on the approximation that the difference between the absolute solvation free energy of a positive and a negative cluster ion vanishes when the cluster becomes infinite, lim fDGsol ½ðH2 OÞn M DGsol ½ðH2 OÞn Mþ g ¼ 0:

n!1

ð18Þ

Through this approximation they were able to arrive at an absolute value for the free energy of solvation of the proton by analysing a large number of experimental results. Subsequently, they could determine the free energy of solvation for any other ion, without or with a cluster of water molecules. Some representative results are listed in Table 16, where it is seen that the aqueous solvation energy for clustered ions show a clear convergence trend when the number of water molecules in the cluster exceeds 2. Ultimately, the authors also compared calculated values with experimental values and found that the calculated values in most cases were within 3–5 kcal/mol of the experimental values. Reddy et al.59 used a special method to calculate the differences in the solvation free energies for different molecules. The method was based on continuously transferring one (solute) molecule into the other using the so called thread method.60 Thus, when comparing the A molecule with the B molecule, they introduce a parameter l so that l = 0 corresponds to the A molecule and l = 1 to the B 86 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 16 Selected absolute aqueous solvation free energies (in kcal/mol) for M and M+ ions clustered by n water molecules. All results are from ref. 55 M +

H Li+ Ag+ H3O+ F I OH HO2

n=0

n=1

n=2

n=3

n=4

n=5

n=6

265.9 128.4 118.7 110.3 104.4 59.9 104.7 97.3

105.6 98.3 90.2 87.9 59.0 89.3 84.7

91.1 83.9 81.7 78.8 59.4 82.3

82.3 79.8 76.7 75.2 60.8 78.3

79.2 78.1 75.4 73.9 63.0 77.1

79.1 77.8 75.4 74.0 65.8 77.3

81.0 78.5 76.9 74.9 77.4

molecule. This means that when passing from l = 0 to l = 1, some atoms will disappear and others will show up, bonds will be broken and created, and atoms may move. For the simulations, they used a QM/MM approach. Intermediate values of l were treated in the MM part by scaling the force-field parameters according to l. On the other hand, the quantum-mechanical part is more tricky. Reddy et al. calculated the total energies and forces for the two molecules A and B separately, and combined afterwards these results using a linear interpolation with l being the interpolation parameter. They tested the approach on various systems for which the A - B transition led to a number of different changes in bonding and structural properties. Moreover, they used different approaches for the QM part, but as Table 17 shows, the method is reliable and leads to accurate results. In order to obtain more information on the solvation process Yang and Cui61 performed a so called natural energy decomposition analysis (NEDA)62–64 on monomethyl phosphate ester (MMP) solvated in water. They used a supermolecular approach where the solute plus a number of water molecules (up to 34) were treated quantum-mechanically. A further set of water molecules was treated with a forcefield model. Their results indicate that there is a substantial charge transfer between the solute and the nearest solvent molecules. The interaction energy due to this transfer was found to amount to some 70–80% of that of the electric interactions. Since MMP forms hydrogen bonds with the water molecules, all results together suggests that for such a system it is important to include the nearest solvent molecules in the quantum-mechanical treatment, whereas a continuum approximation or a force field may not be sufficiently accurate.

Table 17 The difference in free solvation energies for the A and B molecules in aqueous solution, DGsol(B) DGsol(A), in kJ/mol. HF, AM1, FF, and exp. marks results from parameter-free Hartree-Fock calculations, semiempirical AM1 calculations, force-field calculations, and experiment, respectively. All results are from ref. 59 A

B

HF

AM1

CH3OH AcCH3 C6H6 CH3CCl3 C6H6 Serine Phenylalanine Cytosine Adenine

(CH3)2 AcNH2 C6H5OH (CH3)2 C5H5N Cysteine Isoleucine Thymine Guanine

29.5 25.9 22.6 9.0 13.7 17.7 11.2 24.8 37.3

1.8 2.1 1.7 2.1 1.6 1.7 3.1 2.1 2.5

29.7 27.9 22.2 9.1 13.2 18.8 10.7 25.6 38.1

FF

1.8 2.3 1.7 2.1 1.5 1.8 3.2 2.3 2.6

31.5 28.8 21.6 9.3 12.6 18.5 10.5 24.1 36.3

Exp.

1.9 2.2 1.7 2.2 1.5 1.8 3.1 2.2 2.6

29.0 24.8 23.5 8.4 14.7 16.0 12.1


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E. Solvation-induced structural changes As should be evident from the discussion in section II, a solvent leads to an extra external potential in which the solute moves. This extra potential is not spatially constant and may, therefore, influence the different parts of the solute differently. Ultimately this means that a solute may change its properties due to the presence of the solvent. In this and the next subsections we shall discuss some recent studies where this issue has been addressed. At first, we shall in this subsection discuss structural changes due to the solvent. Balakina and Nefediev65 studied four different conjugated organic molecules, related to that of Fig. 4. These so called push-pull systems (i.e., conjugated molecules with an electron donor at one end and an electron acceptor at the other) had been studied earlier experimentally by Flipse et al.66 for their non-linear optical responses. Since the experiments were carried through in solutions, the best theoretical approach was to perform calculations for solvated molecules. The three other molecules that were considered by Balakina and Nefediev were obtained by replacing the NH2 group by a NH(CH3)2 group, as well as subsequently replacing the NO2 group by a conjugated (CH)Q(CH)–NO2 group or by a conjugated (CH)Q(C(CN)2) group. Balakina and Nefediev compared the geometries in the gas phase with those found in chloroform or acetone using the polarizable continuum method. It turned out that the structures of the molecules were very robust against solvation. The bond lengths changed less than 0.01 A˚ and the bond angles less than 21. Cucinotta et al.67 studied the equilibrium between the keto and enol tautomers of acetone, cf. Fig. 5, both in vacuum and in an aqueous solution. They used parameter-free, electronic-structure calculations for a periodically repeated supercell (see section IIB) with the Car-Parrinello approach (see, e.g., ref. 1). In static calculations for solely the acetone molecule they found a total-energy difference between the enol and keto forms of 11.8 kcal/mol. The barrier between the two forms was found to be around 58 kcal/mol (Fig. 5). Adding a single water molecule per supercell, the barrier was reduced by somewhat more than 10 kcal/mol. The simulations showed clearly that the water molecule was active in the proton-transfer process that is needed to change the keto form to the enol form. Thus, the solvent does not only provide an external potential in which the molecule is moving, but takes active part in the transition. Having not only one water molecule per supercell but 28, the barrier was reduced by another 8 kcal/mol. In this case, the hydrogen transfer involves several water molecules and can be described as a Grotthuss like mechanism. A related study was undertaken by Zeng and Ren68 who examined the tautomerization of 2-aminothiazole in the gas phase and in solution. They considered more different scenarios, including the isolated molecules, the isolated molecules interacting with 1, 2, or 3 water molecules, and two interacting molecules. In all cases, they considered a gas phase tautomerization as well as reactions occurring in water or

Fig. 4 Schematic representation of one of the donor–acceptor substituted conjugated chromophores studied by Balakina and Nefediev.65 Open circles, crosses, closed circles, and closed triangles mark carbon, hydrogen, nitrogen, and oxygen atoms, respectively.


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Fig. 5 Schematic representation of the keto and enol tautomers of acetone. Open circles, closed circles, and closed triangles mark hydrogen, carbon, and oxygen atoms, respectively.

CCl4. In the latter case, the solvents were treated with a polarizable continuum method (Fig. 6). Similar to the study above, the presence of other molecules (e.g., water or 2aminothiazole) leads to a significant lowering of the reaction barrier, as shown in Table 18. On the other hand, the presence and precise nature of a solvent are both less important, at least as long as it is treated within the continuum approach. The marginal differences when including more water molecules (see Table 18) suggest that in this case a Grotthuss mechanism is not important. As Balakina and Nefediev, also Kar et al.69 studied the properties of donor– acceptor substituted conjugated molecules, this time sesquifulvalene of Fig. 7. They used a polarizable continuum approach in taking into account the effects of a solvent and varied the dielectric constant of the solvent from 1 (i.e., vacuum) to 78.4 (i.e., water). In one case (when the triangle in Fig. 7 represents NH2 and the square represents NO2) they observed a reduction in the bond-length alternation due to solvent effects (i.e., the ‘double’ bonds became longer and the ‘single’ bonds became shorter), which was not the case when the NO2 and NH2 groups were interchanged. Also Perpe`te and Jacquemin70 studied the properties of push-pull systems, in this case NH2(C2H2)nNO2 oligomers for n = 2, . . . ,16. An example of the structure is shown in Fig. 8. The effects of a solvent (water) were included using a polarizable continuum model. Moreover, the authors carried through parameter-free HartreeFock calculations without and with the inclusion of correlation effects. As above, a fundamental quantity describing the structure is the bond-length alternation parameter, Dr, which the authors defined as the difference in the bond lengths of the central units. The evolution of this parameter with the size of the system is shown in Table 19. It is remarkable that the solvent has the most pronounced effects for the shortest chains, whereas for larger chains Dr is almost unaffected by the presence of the water. Meng et al.71 studied another class of conjugated molecules, i.e., the oligothiophenes H(C4H2S)nH for which we in Fig. 9 show the case of n = 2. They also considered cases where some of the hydrogen atoms were substituted by OCH3 groups. Using molecular dynamics simulations for the oligothiophenes in, e.g., water they found that the chains did not stay linear even for quite small values of n. A similar behaviour was found in other solvents like n-hexane, 1,4-dioxane, and chloroform. Furthermore, from parameter-free density-functional calculations with a polarizable continuum approach the authors found that the structures of these

Fig. 6 Schematic representation of the two tautomers of 2-aminothiazole. Open circles, closed circles, closed triangles, and crosses mark carbon, sulphur, nitrogen, and hydrogen, atoms, respectively. The left part shows the amino form and the right part the imino form.


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Table 18 The relative Gibbs free energies (in kcal/mol) of (A) the amino and (B) the imino forms and (TS) the transition state in the tautomerization of isolated, mono-, di-, and trihydrated as well as dimeric species of 2-aminothiazole (cf. Fig. 6). All results are from ref. 68 System

Gas

Water

CCl4

A TS B A + H2O TS + H2O B + H2O A + 2H2O TS + 2H2O B + 2H2O A + 3H2O TS + 3H2O B + 3H2O 2A 2TS 2B

0 50.2 7.5 0 18.3 6.2 0 15.0 5.5 0 17.9 5.2 0 12.6 10.2

0 54.5 7.3 0 19.1 5.7 0 15.0 4.4 0 19.2 5.3 0 13.3 8.3

0 51.3 7.3

0 12.1 9.9

Fig. 7 Schematic representation of sesquifulvalene. Open circles and closed circles mark hydrogen and carbon atoms, respectively, and for the two donor–acceptor substituted structures considered by Kar et al., one of the symbols marked by the closed triangle and square represents a NH2 group and the other a NO2 group.

Fig. 8 Schematic representation of one of the push-pull oligomers that were studied by Perpe`te and Jacquemin. In this case, n = 2. Open circles, closed circles, closed triangles, and closed squares mark hydrogen, carbon, nitrogen, and oxygen atoms, respectively.

systems for n = 2 and 3 are very insensitive to the presence of a solvent, similar to what we have reported above for other conjugated systems. Li et al.72 studied the monochalcogenocarboxylic acids CH3COXH with X being S, Se, or Te (Fig. 10) both in vacuum and in a THF (tetrahydrofuran) solution. They used the polarizable continuum model and calculated the electronic properties of the solute by using Hartree-Fock calculations with a second order Møller-Plesset treatment of correlation effects. They compared the enol and the keto forms and calculated also the transition state between the two. Finally, they considered both 90 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 19 The bond-length alternation parameter (in A˚) for the push-pull oligomers of Fig. 8. The phase shows whether the calculations were performed in vacuum or in the water phase, and HF and MP2 mark Hartree-Fock calculations without (HF) or with (MP2) the inclusion of correlation effects via Møller-Plesset perturbation theory. All results are from ref. 70 n

Phase

HF

MP2

2 4 6 8 10 12 14 16 2 4 6 8 10

Vacuum Vacuum Vacuum Vacuum Vacuum Vacuum Vacuum Vacuum Water Water Water Water Water

0.1089 0.1116 0.1166 0.1193 0.1206 0.1213 0.1216 0.1218 0.0556 0.0873 0.1106 0.1181 0.1205

0.0854 0.0720 0.0682 0.0658 0.0645 0.0638 0.0634 0.0632 0.0542 0.0578 0.0668 0.0673 0.0661

Fig. 9 Schematic representation of one of the oligomers of thiophene that were studied by Meng et al. In this case, n = 2. Open circles, closed circles, and closed squares mark hydrogen, carbon, and sulphur atoms, respectively.

the isolated molecules and ones that were allowed to interact with a dimethyl ether, CH3OCH3, molecule. This molecule was considered as a simpler form of the THF solvent, having, however, similar properties as THF. Table 20 summarizes their findings. It is seen that both in the gas phase and in the THF phase, the keto form is found to be the most stable form. However, when extra molecules are explicitly considered (i.e., the dimethyl ether), solvent effects can change the relative stability of the two forms. In a combined experimental and theoretical study, Borovkov et al.73 examined the properties of n-alkane radical cations in solution. Here, we shall just focus on one single aspect of their work, i.e., the conformers of one of the alkanes, n-nonane, C9H20. In its simplest form of the highest symmetry, its structure can be described as a zigzag chain formed by the 9 carbon atoms. To each of the carbon atoms, 2 hydrogen atoms

Fig. 10 Schematic representation of (left part) the enol and (right part) the keto forms of monochalcogenocarboxylic acids CH3COXH with X being S, Se, or Te. Open circles, closed circles, closed triangles, and closed squares mark hydrogen, carbon, oxygen, and the chalcogen atom, respectively.


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Table 20 The relative total energies in kJ/mol for systems containing n CH3OCH3 molecules plus either the keto or the enol form of the monochalcogenocarboxylic acids of Fig. 10. TS shows the results for the transition state between the two forms. The phase marks whether gas phase or a THF (tetrahydrofuran) solution is considered. All results are from ref. 72 X

n

Phase

Keto

TS

Enol

S S Se Se Te Te S S Se Se Te Te

0 0 0 0 0 0 1 1 1 1 1 1

Gas THF Gas THF Gas THF Gas THF Gas THF Gas THF

0.0 0.0 0.0 0.0 0.0 0.0 18.4 9.1 14.9 7.7 11.3 1.4

117.0 118.8 117.6 118.8 120.4 121.7 82.2 75.0 77.2 63.3 79.0 64.6

15.4 14.8 18.4 14.5 23.3 20.2 16.7 13.8 15.5 15.7 11.2 11.0

are bonded, and one further hydrogen atom is bonded to each of the terminating carbon atoms. More complex structures can be obtained by rotating 1201 about one or more C–C bonds. Using the B3LYP density-functional method for different such structures and the polarizable continuum method for the treatment of a n-heptane solvent, Borovkov et al.73 studied the relative stability of different structures both in the gas phase and in the solution. The results are reproduced in Table 21 and here it is seen that the presence of the solvent leads to an enhanced stabilization of the highsymmetry conformation. In addition, in gas phase the total energy change due to a rotation about one bond depends only very weakly on the bond about which is rotated, but much stronger when the molecule exists in the solution. Finally, the solvent may also be indirectly responsible for structural changes. By solvating a molecule and subsequently applying pressure on the complete system, the solvent molecules transfer the externally applied forces to the solute that, therefore, may change its structure. This was studied by Floriano et al.74 who performed molecular-dynamics simulations on some large biological molecules in aqueous solution for different external pressures. Due to the external pressures, the volume of the system changed and, subsequently, the structure of the solute changed, too. They used different measures in quantifying the structural changes of the solute molecules, but we shall here not discuss this work further.

F. Chemical reactions involving the solvent Already in the previous subsection we saw examples where the solvent molecules take active part in chemical reaction. This was, e.g., the case for proton transfer Table 21 The relative total energies in kcal/mol for different conformations of n-nonane either in the gas phase or in a solution of n-heptane. The energies are given relative to that of the undistorted system (conformation I), and for the other conformations, the bonds about which a 1201 otation has been performed are given through the numbers of the two carbon atoms forming the bond (numbering the atoms from one end). All results are from ref. 73 Conformation

Bonds

DE (vac)

DE (sol)

I II III IV

— 2–3 4–5 3–4, 6–7

0.00 0.83 0.85 1.8

0.00 1.02 4.26 6.56


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Fig. 11 Schematic representation of the pyridine containing molecule studied by De Angelis et al.75 Open circles, closed circles, closed triangles, and closed squares mark hydrogen, carbon, nitrogen, and the halogen atom, respectively.

reactions where one tautomer was changed into another. Here, the proton transfer could take place via water molecules of an aqueous solvent, ultimately resulting in a significant reduction of the energy barrier of the transition state between the two tautomers. In this subsection we shall discuss some further recent theoretical work where an active role of the solvent has been studied. Recently, De Angelis et al.75 studied the b elimination of the compound of Fig. 11. Upon reaction with an OH group, a water molecule and the negatively charged halogen ion will be eliminated. In an earlier work,76 the same group had considered this reaction by modeling the solvent through a polarizable continuum model, thereby obtaining the reaction path shown in Fig. 12. In the more recent work, De Angelis et al.75 performed parameter-free density-functional calculations on a periodically repeated cell containing one solute molecule and 56 water molecules using the Car-Parrinello approach. After an initial molecular-dynamics calculation,

Fig. 12 The variation of the Gibbs free energy (in kcal/mol) for the b-elimination reaction for the molecule of Fig. 11, as well as selected structures along the reaction. Compared to the system of Fig. 11, the NH group has been replaced by a NCH3 group. Reproduced with permission of The American Chemical Society from ref. 76.


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Table 22 The energy of the transition state compared to the initial energy for the hydrolysis of cisplatin in various solvents. All results are from ref. 77 Solvent

HF

CHF3

CH2Cl2

NH3

H2O

CH4

Vacuum

Barrier (eV)

0.54

0.97

1.11

1.11

1.15

1.36

1.59

they selected a single water molecule close to the reagent and replaced that by an OH group, and the simulation was continued. Ultimately, the calculations led to essentially the same results concerning the reaction path, supporting the applicability of both methods. Song and Hu77 studied the hydrolysis of cisplatin, i.e., the reaction of PtCl2(NH3)2 with H2O, in solution. They carried through a supercell calculation in which the two reacting molecules plus a number of solvent molecules were repeated periodically. In the case that the solvent is water, the reacting H2O molecule is part of the solvent, but they considered actually also other solvents. They focused on the reaction energy barrier, i.e., the energy of the transition state relative to that of the initial state, with the ultimate goal of understanding how a solvent can modify this. At first, they examined how many water molecules was needed in the supercell in order to obtain converged results with respect to this parameter. They found that for around 7 molecules this was the case, and this number they used for all their subsequent calculations. Table 22 summarizes their findings. It is clear that a properly chosen solvent can modify the activation energy significantly. A couple of years ago, Kelly et al.78 developed an implicit solvation model for aqueous solutions, related to those we discussed above in section IIE. In a more recent work,79 they tested the model for the calculation of aqueous acid dissociation constants, i.e., the free energy change associated with the reaction AH(aq) - A(aq) + H+(aq)

(19)

with AH and A being an acid and its conjugate base. In aqueous solutions the proton H+ may interact with the water molecules forming, e.g., hydronium H3O+. Therefore, an implicit model where the solvent molecules are not included directly may not be sufficiently accurate. Kelly et al.79 suggested to include one or more water molecules in the quantum-mechanical part of the calculation. They found that in many cases just the addition of a single water molecule can improve the accuracy of the calculations significantly, which could be seen for instance through their calculated pKa values. It may not surprise that the importance of the inclusion of explicit water molecules increases when strong solute–solvent interactions exist. In some cases, with the CO32 ion being special, the inclusion of more explicit water molecules could be important. The case of CO32 was studied in some details by Kelly et al.79 who considered the three arrangements of Fig. 13. In this case, the accuracy depended not only on the number of explicit water molecules but also on the precise form of the solvent model.

Fig. 13 Schematic representation of the CO32 surrounded by 1, 2, or 3 water molecules. Closed circles, open circles, and closed triangles represent oxygen, hydrogen, and carbon atoms, respectively.


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Also Gutowski and Dixon80 studied the reaction of eqn (19) and compared it with the same reaction in the gas phase. Using a polarizable continuum model to describe the effects of solvating the acids and the bases, and setting the free energy of solvation for the proton at 298 K equal to 264.3 kcal/mol, Gutowski and Dixon could calculate pKa values as follows. They first calculate the free energy of reaction for the reaction of eqn (19) in the gas phase, DG298. To this they add the changes in the free energies due to solvation, DGsolv. The sum gives the reaction free energy in the solution, DGsolution = DG298 + DGsolv

(20)

and ultimately pKa is determined from pKa ¼

DGsolution 2:303RT

ð21Þ

with R being the gas constant and T = 298 K being the temperature. The results are listed in Table 23 where they are compared with experimental values whenever possible. It is seen that the two contributions to DGsolution in eqn (20) almost cancel, whereby the main contribution to DGsolv comes from the free energy of solvation of the proton. As we have discussed above, this number is related with some uncertainty. Taking all these issues into accounts, the agreement in the table is nice. In addition, the authors considered also the proton exchange reactions HA + CH3CO2 - A + CH3CO2H and HA + NO3 - A + HNO3. Also from those, values of pKa could be estimated. These are listed in the table, too, as pK 0 a and pK00 a for the first and second reaction, respectively. The fact that these reactions lead to other values of pKa is an indication of the uncertainty of the approach. On the other hand, one may speculate that the inclusion of one or more water molecules from the solvent in the quantum-mechanical treatment would lead to even more Table 23 The free energy of the reaction of eqn (19) in the gas phase, the free energy of solvation, and the free energy of the reaction in an aqueous solution, together with the thereby obtained pKa value in comparison with experimental values (denoted Exp) for the acids given by HA. pKa 0 and pKa00 were calculated by considering other proton exchange reactions. All energies are in kcal/mol and the results are from ref. 80 HA

DG298

DGsolv

DGsolution

pKa

CF3COCH3 (CN)2CH2 (CH3CO)3CH (CF3)2NH (CF3)3COH CH3CO2H CF3SO2NH2 (FSO2)2CH2 (CF3CO)2NH (CF3CO)2CH2 CF3CO2H (CF3SO2)2CH2 CF3COSH HNO3 (CF3CO)3CH CH3SO3H H2SO4 (CF3SO2)2NH FSO3H CH3SO3H (CF3SO2)3CH

343.3 327.9 322.9 323.2 324.0 340.3 320.8 306.0 308.5 309.8 316.9 297.4 312.7 317.5 295.1 312.2 303.8 286.0 294.7 292.4 274.0

319.7 307.8 307.7 310.7 313.3 329.9 311.6 299.3 303.4 305.1 316.9 299.2 313.8 321.8 301.1 320.7 315.8 302.6 312.5 311.7 299.6

23.6 20.1 15.2 12.5 10.7 10.4 9.2 6.7 5.1 4.7 0.0 1.8 1.1 4.3 6.0 8.5 12.0 16.6 17.8 19.3 25.6

17.3 14.7 11.2 9.2 7.8 7.6 6.8 4.9 3.8 3.4 0.0 1.3 0.8 3.2 4.4 6.2 8.8 12.2 13.0 14.2 18.8

pKa (Exp.) 11.2 5.9 5.1 4.76 6.3

0.6, 0.3, 0.5

pKa 0

pKa00

14.4 11.8 7.4 6.4 5.2 4.0 2.3 1.4 0.0 2.8

1.2 1.8 1.0

1.4 2.6, 1.9 3, 10 6.4, r12 5.9, 14

3.8 4.6 7.0 10.6 11.4 12.5 17.4


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Fig. 14 Schematic representation of the different conformers of the fluoroscein anion. Open and closed circles represent carbon and oxygen atoms, respectively. Hydrogen atoms are not shown. For the three isomers the positions 1, 2, and 3 represent –OH, QO, –O (conformation 1), –O, QO, –OH (conformation 2), and –O, –OH, QO (conformation 3), respectively.

accurate results. However, this possibility was not considered by Gutowski and Dixon. As the last example we discuss the study of Friedman et al.,81 who considered proton transfer within a single molecule, fluoroscein, exposed to water. The single anionic fluoroscein molecule can exist in three different conformations, differing in the oxygen atom to which a proton is attached, cf. Fig. 14. According to Friedman et al., the total-energy differences between the different conformations are in the gas phase less than 8 kcal/mol, when using parameter-free electronic-structure calculations. Friedmann et al. carried through QM/MM molecular-dynamics simulations in order to identify possible proton-transfer paths. Subsequently, supermolecule calculations with the fluoroscein molecule plus some extra water molecules were carried through. Here, the water molecules were placed so that a proton transfer could take place through these molecules. According to the authors, two or three water molecules would be involved in the proton-transfer process. The energy barrier they found was some 10–22 kcal/mol, depending on the details of the calculations, which is in good agreement with the experimental value of 11 kcal/mol. Once such a chain of water molecules is involved in a proton-transfer process, it may be speculated whether the process is concerted, semiconcerted, or stepwise. In the first case, the protons along the hydrogen bonds in the water bride move simultaneously, whereas in the last case, they move one by one (i.e., as according to the Grotthuss mechanism). In the case of a semiconcerted process, some but not all protons move simultaneously. The calculations, however, indicated that the process was concerted. G.

Chemical reactions in solution

In this section we shall discuss reactions where the solvent does not take active part but merely provides a medium for the reactions. Of course, in reality the distinction between this type of reactions and the ones we have discussed in the preceding subsection is not sharp. Nevertheless, here we hope that it provides a help for the reader to organize the very many studies that have been devoted to chemical reactions in solution. Improta et al.82 studied large molecules with a well-separated electron donor and electron acceptor pair. In this case, the electron-transfer process may lead to a dissociation of the molecule. Improta et al. applied the Marcus theory43–45 to obtain an expression for the electron transfer rate constant. Subsequently, the values of the parameters entering this expression were calculated using a density-functional approach for the solute together with the polarizable continuum model for the solvent. They applied their approach for a specific system (for the present purpose, the details are less important) and found the results of Table 24. Here, three different 96 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 24 The calculated rate constants for the electron-transfer reaction for three different systems in comparison with experimental values. The two different calculated values correspond to two different basis sets. All energies are from ref. 82 System

log kDET (calc.)

log kDET (exp.)

1 2 3

4.20/5.61 2.86/3.56 0.61/0.39

4.70 0.18

modifications of the system were considered and the results compared to experiment. The agreement is good, but it is also seen that the results depend somewhat on details of the calculations like size of the basis set. A related issue was addressed by Scholes et al.83 who studied how energy is transferred over longer distances in a larger donor–acceptor molecule after this has been photoexcited. The presence of a solvent can influence the rate of this energy transfer, as originally described by Fo¨rster.84 According to the proposition of Fo¨rster, the presence of the solvent will reduce the electronic energy transfer rate by roughly a factor of 4, independent of the relative arrangement of the donor and acceptor. As pointed out by Scholes et al., it may be more correct to expect a distance dependent screening effect. Therefore, they applied a recently developed approach,85,86 that is based on the polarizable continuum model, to study the screening effects due to a solvent. They studied several different donor–acceptor pairs and found an overall exponential behaviour of the screening factor s = A exp(bR) + s0

(22)

(R being the donor–acceptor distance) for which a fit gave A = 2.68, b = 0.27 A˚1, and s0 = 0.54, although the results possessed some scatter around this fitted curve. Kong et al.87 studied the photodissociation products of carbon tetrahalides, CX4, with X being Cl, Br, or I, in different solvents. They used the B3LYP densityfunctional method for the quantum-mechanical treatment of the solute and a polarizable continuum approach for the inclusion of solvation effects. Ultimately, the final outcome of the photodissociation and subsequent reactions is C2X4 and X2. In order to identify possible reaction paths the authors calculated, therefore, the total energies of various species. Some of their results are reproduced in Table 25 for X = Br. It is remarkable that the total energies show only a weak dependence on the presence of the solvent, with the energy of the transition state being the main exception. Ramos et al.88 studied the Hoffmann elimination of (N-Cl),N-methylethanolamine in the gas phase and in aqueous solution, i.e., the reaction CH2OH–CH2–NCl–CH3 Table 25 The calculated relative total energies (in kcal/mol) for various dissociation and recombination products of CBr4. TS marks the energy of the transition state between CBr4 and CBr3 + Br. All energies are from ref. 87 Species

Vacuum

Methanol

Cyclohexane

CBr4 CBr3 + Br CBr2 + Br2 CBr2 + 2Br Br2CBr–Br (C2Br6 + Br2)/2 (C2Br5 + Br + Br2)/2 (C2Br4 + 2Br + Br2)/2 (C2Br4 + 2Br2)/2 TS

0 44.6 50.8 96.4 32.2 1.6 19.2 19.9 2.9 44.6

0 45.6 49.4 97.5 25.9 1.8 19.4 20.1 4.0 36.7

0 44.5 50.5 95.9 30.1 1.6 19.1 19.7 3.0 41.6


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Table 26 The calculated relative total free energies for the Hoffmann elimination reaction. The energies are given in kcal/mol relative to that of the isolated reactants and for the molecules interacting via hydrogen bonds. R, TS, and P denote reactants, transition state, and products, respectively, and n gives the number of water molecules used in the quantum-mechanical treatment. Results for the reaction in the gas phase (gas) and in the solution (sol) are shown. All energies are from ref. 88 n

DG (R, gas)

DG (R, sol)

DG (TS, gas)

DG (TS, sol)

DG (P, gas)

DG (P, sol)

0 1 2

13.89 7.37 9.91

24.20 23.44 22.84

10.14 2.98 2.52

29.02 34.39 23.62

97.20 76.62 79.57

39.95 42.54 43.16

+ OH - CH2OH–CH2–NQCH2 + H2O + Cl. They calculated the total free energies using density-functional methods and treated the solvent within the polarizable continuum method. Moreover, they included n = 0, 1, or 2 water molecules in the density-functional calculations in order to allow for hydrogen bonding. In Table 26 we list the calculated total free energies for the reactants, the transition state, and the products for the structures where hydrogen bonds between the different molecules exist, i.e., the results are not for the well separated molecules. The table shows clearly that the presence of water is very important for the reaction, both for the short-range and for the long-range interactions. A similar conclusion was drawn by Tian and Fang.89 These authors studied the photodissociation of formic acid, i.e., the reaction HCOOH - HCO + OH, H + COOH, or HCOO + H. They compared gas-phase results with those found in aqueous solution when including water molecules both explicitly (i.e., in that part that is treated quantum-mechanically) and via the polarizable continuum model. Hydrogen bonding would modify the properties of the molecules, including explaining a blue shift of an excitation energy. Moreover, long-range effects due to the solvent influence the structures of the molecules.

H.

Electronic properties in solution

After having discussed energetic and structural properties for solvated molecules we shall now turn our attention towards other properties of such systems. These include responses to external perturbations like electromagnetic fields and the topic of the present subsection, the distribution of the electrons. Within several of the models we have discussed in section II, an important ingredient is the ability of the solvent, whether described as a continuum or as a discrete set of atoms or molecules, to provide an external electrostatic potential in which the solute exists. Thus, first of all the charge distribution of the solute will be affected by the presence of the solvent. This may in turn modify other properties, like the structure, of the solute, but changes in the charge distribution of the solute may be considered the immediate response to the presence of the solvent. One way of quantifying this is obtained through the dipole moment. This was considered, e.g., by Balakina and Nefediev65 in the study we already discussed in section IIIE. They studied four different conjugated molecules related to that of Fig. 4. As we have discussed above, the structural changes upon solvation were minor. On the other hand, the dipole moment changes significantly, as shown in Table 27. In this study the solvent was treated as a polarizable continuum. The presence of a solvent may also lead to a reduction in the dipole moment. For the Hoffmann elimination, discussed above, that was studied by Ramos et al.,88 the authors considered the dipole moment for the initial complex consisting of the (NCl),N-methylethanolamine, the OH ion, and 0, 1, or 2 water molecules as well as a polarizable continuum describing the rest of the aqueous solvent. They found that it changed from 6.44, via 5.82, to 6.88 D when including 0, 1, or 2 water molecules. For 98 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 27 Various properties for the molecule of Fig. 4 as well as four modifications of it. m, a, and b denote the dipole moment, the average polarizability, and a certain averaged first hyperpolarizability. The three other headings show the phase. All results are from ref. 65 Molecule

Property

Vacuum

Chloroform

Acetone

PNA

m(D) a (1024 esu) bz (1030 esu) m(D) a (1024 esu) bz (1030 esu) m(D) a (1024 esu) bz (1030 esu) m(D) a (1024 esu) bz (1030 esu)

7.4 13.89 6.67 7.94 17.68 9.75 9.39 23.71 27.87 11.94 33.87 58.69

9.42 17.40 20.3 10.3 22.40 33.14 11.95 30.07 82.16 15.32 42.87 145.26

10.08 18.78 28.15 11.14 24.19 46.17 12.80 32.63 113.61 16.38 46.19 184.57

DMA

DNAP

DACP

the transition state, it changed from 6.31, via 4.26, to 4.08 D in the same three cases. In this case, no clear trend can be identified. These authors studied also the Mulliken atomic charges. Equivalent to the findings above, the inclusion of explicit water molecules in the calculations leads in some cases to an increase, in other cases to a decrease in the atomic charges. In their study of the donor-accepted substituted sesquifulvalene (see Fig. 7), Kar et al.69 found a systematic increase in the dipole moment as a function of the relative dielectric constant of the solvent, which was treated within the polarizable continuum approach. These authors also calculated the hardness, Z = 12(eLUMO eHOMO),

(23)

(eLUMO and eHOMO is the energy of the lowest unoccupied orbital, LUMO, and of the highest occupied orbital, HOMO) which to some extent describes the ability of the system to participate in chemical reactions. Kar et al. observed an only weak dependence of Z on the dielectric constant of the solvent. In the study by Li et al.72 on the tautomerism of monochalcogenocarboxylic acids CH3C(QO)XH, with X being S, Se, or Te, that we discussed above (see, e.g., Table 20 and Fig. 10), the authors calculated also the atomic charges for the different structures in the gas phase. The results are reproduced in Table 28, where it is seen that the presence of a single solvent molecule, i.e., the CH3OCH3 molecule, leads to some modifications. The fact that this molecule takes active part in the proton transfer process explains why the largest effects due to its presence are found for the hydrogen atom at the transition-state structure. Of the atoms listed in the table, the C atom is the one most far away from the CH3OCH3 molecule and, therefore, the one that is the least affected by its presence. Table 28 shows also the total dipole moment for the different systems and structures. Thus, when a CH3OCH3 molecule is included, the dipole moment contains also contributions from this molecule. This explains that in this case the dipole moment shows much larger variations than when the CH3OCH3 molecule is not included. In particular at the transition state some charge transfer between the two molecules is thereby indirectly seen. The electronic chemical hardness, defined in eqn (23), gives a measure of the chemical reactivity of a given compound. Also the chemical potential, m = 12(eLUMO + eHOMO),

(24)

90

is a useful property in this concept. Meneses et al. have therefore studied this for a set of smaller molecules and, in particular, how it changes when the molecule is solvated. They used a supermolecule approach, where the solute was surrounded by Chem. Modell., 2008, 5, 67–118 | 99 This journal is

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Table 28 Results from a natural-population analysis for systems containing n CH3OCH3 molecules plus either the keto or the enol form of the monochalcogenocarboxylic acids of Fig. 10. TS shows the results for the transition state between the two forms. q shows the effective atomic charge and C, O, X, and H represent the carbon atom of the CQO or C–O group, the oxygen atom of that group, the chalcogen atom, and the hydrogen atom that is been transferred, respectively. m is the total dipole moment for the different structures. All results are from ref. 72 X

n

Form

q(C)

q(O)

q(X)

q(H)

m(D)

S

0

Keto TS Enol Keto TS Enol Keto TS Enol Keto TS Enol Keto TS Enol Keto TS Enol

0.543 0.478 0.355 0.539 0.486 0.391 0.502 0.423 0.288 0.495 0.446 0.333 0.448 0.371 0.225 0.441 0.405 0.281

0.648 0.758 0.735 0.680 0.824 0.773 0.645 0.755 0.739 0.673 0.810 0.776 0.643 0.761 0.748 0.669 0.805 0.783

0.025 0.159 0.149 0.054 0.493 0.225 0.106 0.057 0.077 0.074 0.472 0.165 0.296 0.081 0.013 0.276 0.423 0.092

0.121 0.391 0.494 0.168 0.596 0.538 0.033 0.351 0.499 0.082 0.603 0.542 0.093 0.293 0.499 0.051 0.600 0.544

2.23 2.23 2.03 0.80 6.31 1.78 2.30 2.35 2.32 1.16 6.71 2.19 2.59 2.32 2.36 1.76 7.29 2.68

1

Se

0

1

Te

0

1

eight water molecules, and in order to be able to use the definitions of eqns (23) and (24) they identified those orbitals closest to the HOMO and LUMO orbitals of the complete supermolecule that were localized to the solute and for those used the definitions of the chemical potential and chemical hardness. In their calculations they used the B3LYP density-functional method. Table 29 summarizes their findings. It is clear that in all cases the chemical potential gets closer to zero and the hardness becomes smaller when the molecules are solvated. Using the definitions of eqns (23) and (24) this means that both the HOMO and the LUMO moves upward in energy. As a simple approximation (see, e.g., ref. 1), the orbital energy can be interpreted as the energy that is required to move an electron infinitely far away from the molecule (HOMO) or adding an electron from infinitely far away to the molecule (LUMO), assuming that relaxation effects can be ignored. Thus, the energies for these electron-transfer processes are reduced, i.e., the solvent screens the electrostatic interactions. The last two examples of Table 29 are exceptions to these rules, which, however, may be related to the very delocalized nature of the HOMO and/or LUMO orbitals, whereby these orbitals have positive energies both in the gas phase and in the solution. In their study on solvent effects on the properties of oligothiophenes, Fig. 9, Meng et al.71 calculated also the dipole moment for chains consisting of 2–6 thiophene units, either in the gas phase or in either n-hexane or water. The calculations were carried through using the B3LYP density-functional method for the solute and the polarizable continuum method for the solvent. As Table 30 shows, the solvent leads to an increase of the dipole moment, but in all cases a clear even–odd oscillatory pattern is identified. The latter can be related to the zigzag-like structure of the systems, cf. Fig. 9. In their study of the properties of CX4 (with X being a halogen) and their photodissociation products, Kong et al.87 also reported the atomic populations obtained from a natural population analysis for the Br2CBr–Br system in various 100 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 29 The chemical potential (m) and the chemical hardness (Z) for various small molecules in the gas phase (gas) as well as in an aqueous solution (sol). All quantities are given in eV and are from ref. 90 Molecule

m (gas)

m (sol)

Z (gas)

Z (sol)

CH3NH3+ NH4+ H3O+ CH3NH2 NH3 H2O NH2 OH

11.13 14.11 13.40 1.96 2.36 3.11 7.38 7.81

5.94 7.31 5.41 0.08 1.13 2.18 4.92 4.54

11.98 16.75 12.33 8.52 9.01 9.63 5.87 6.87

4.54 6.62 2.40 1.44 1.61 3.70 0.17 1.49

Table 30 The dipole moment, m (in D), for oligothiophenes (cf. Fig. 9) in various solvents as well as in the gas phase. xT represents the oligothiophene containing x units. All results are from ref. 71 Solvent

m(2T)

m(3T)

m(4T)

m(5T)

m(6T)

Gas n-Hexane Water

0.27 0.30 0.32

0.73 0.83 1.05

0.46 0.49 0.54

0.83 0.94 1.09

0.61 0.61 0.60

solvents. The results are reproduced in Table 31. The calculations were performed with a polarizable continuum method for the treatment of the solvent. Thus, the table shows that even this approach leads to significant redistributions of the electronic charges when the system is inserted into the solvent. What may not surprise is that the solvent with the largest relative dielectric constant (water) leads to the largest changes in the atomic populations.

I.

Excitation energies in solution

The determination of responses to external perturbations is the field where experiment and theory meet. Therefore, very much effort has been invested in determining such responses, also for solvated molecules. In this and the following two subsections we shall discuss some of the results that have been obtained when calculating the responses to electromagnetic fields. In this subsection we focus on the (optical) excitation energies. In a work towards the understanding of the optical properties of peptides, Sˇebek et al.91 studied those of an amide-group containing molecule (NMA), N-methylacetamide, CH3–NH–CO–CH3. Using time-dependent density-functional theory Table 31 The atomic net populations on various atoms of Br2CBr–Br in different solvents. The atom is marked in bold face. er gives the relative dielectric constant for the solvent. All results are from ref. 87 Solvent Vacuum Cyclohexane Dodecane 2-Propanol Acetonitrile Propylene carbonate Water

er 1.000 2.023 20.18 36.14 66.14 78.39

Br2CBr–Br

Br2CBr–Br

Br2CBr–Br

Br2CBr–Br

0.244 0.274 0.273 0.355 0.365 0.369 0.370

0.587 0.567 0.568 0.504 0.496 0.493 0.492

0.522 0.515 0.516 0.455 0.446 0.442 0.442

0.422 0.495 0.495 0.662 0.679 0.688 0.689


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Table 32 Transition wavelengths and dipole strengths for N-methylacetamide in the gas phase and in an aqueous solution. Both the COSMO model and a supermolecule approach (denoted clusters) have been applied for treating the solvent. In the latter case, averages over 90 configurations have been calculated, resulting in the error bars for those calculations. All results are from ref. 91 State

l (nm) Vacuum

l (nm) COSMO

l (nm) Clusters

D (debye2) Vacuum

D (debye2) COSMO

D (debye2) Clusters

1A00 2A00 3A 0 4A 0 5A00 6A 0 7A00 8A 0 9A00 10A 0

217 216 214 191 190 183 182 176 175 175

208 202 193 175 183 169 175 180 164 169

208 12 244 23 229 19 183 8 187 9 184 8 189 8 182 9 188 17 182 13

0.03 0.00 0.55 0.57 0.38 0.56 0.28 3.35 0.00 2.68

0.03 0.07 0.90 1.31 0.30 0.33 0.43 9.10 0.01 1.82

0.09 0.05 0.11 1.37 0.86 0.78 0.87 2.29 0.27 1.96

0.16 0.04 0.09 1.61 1.07 0.91 0.90 0.87 0.24 1.84

(TD-DFT) they calculated the transition energies and dipole moments for NMA both in vacuum and in an aqueous solution. Moreover, in the treatment of the solvent they compared two different approaches, i.e., a polarizable-continuum method (COSMO) and a supermolecule approach. For the latter, the authors performed molecular-dynamics calculations using a force-field model and, subsequently, extracted a cluster containing the solute and 3–4 water molecules that form hydrogen bonds to the solute. Averages over 90 such configurations were ultimately determined. Some of their results are reproduced in Table 32. At best, the supermolecule and continuum model give similar trends, but the large scatter in the results for the latter indicates a strong dependence of those on the precise geometry of the supermolecule that is being studied. By calculating the complete absorption spectra, Sˇebek et al.91 confirmed the sensitivity of the results on all kinds of details of the theoretical approach. Thus, these results suggest that the present models for the theoretical treatment of solvation effects are not yet able to provide precise information on such spectra. On the other hand, it has to be mentioned that the calculation of excitation energies (in particular with time-dependent density-functional theory that not yet has reached a mature level) is connected with some uncertainty that has nothing to do with the presence of a solvent. In their detailed study of the photodissociation of CX4 molecules (with X being a halogen), Kong et al.87 reported also various spectroscopic properties. The calculations were performed using a polarizable-continuum model for the solvent and B3LYP density-functional calculations for the solute. In Tables 33–35 we show their results for vibrational properties (Table 33), absorption energies (Table 34), and singlet-triplet excitation energies (Table 35). The tables shall give an impression of what currently is possible, but the tables show also that where a comparison with experiment is possible, there is still some discrepancy, although trends are well reproduced by the calculations. In their study of the photodissociation of formic acid, Tian and Fang89 compared the gas-phase reaction with that of an aqueous solution. For the solvent they used a combined model with a supermolecule treatment of the solute plus the nearest water molecules together with a polarizable continuum model for the long-range solvent effects. They could demonstrate that the explicit treatment of water molecules closest to the solute was important for the theoretical study of the photoabsorption process. The study of Zeng and Ren68 on the tautomerization of 2-aminothiazole that we have discussed above, included also an examination of the excitation properties of the systems in the gas phase and in solutions. Zeng and Ren treated the solvent via a 102 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 33 Calculated and experimental vibrational energies (in cm1) for Br2CBr–Br in different solvents. All results are from ref. 87

Vibration

Acetonitrile with Cyclohexane Cyclohexane Water Acetonitrile traces of water calculation experiment calculation calculation experiment

C–Br str. Br–C–Br wag. Br2–C–Br wag. Br2–C–Br sym. str. Br2–C–Br bend BrCBr bend + Br–Br str. Br–Br str. + BrCBr bend Torsion Br2CBr–Br bend

818 762 344 283 182 175 147 50 36

828

797 775 411 279 185 170 108 48 19

179 155

796 777 408 279 185 170 109 48 20

828

280

polarizable-continuum approach, but included 0–3 water molecules in the quantummechanical part for the aqueous solutions. Their results are summarized in Table 36. For the gas phase, there is a clear dependence on the number of extra water molecules that is included, but in the aqueous solutions the results are quite insensitive to this aspect. This suggests that for optical properties the explicit treatment of (parts of) the solvent may not be needed, although, as we have seen above, the structure may depend on this, and, thereby indirectly, also the optical properties. This finding is partly in contrast to some of those we have found above. Kar et al.69 examined the properties of the sesquifulvalene molecule of Fig. 7. An interesting aspect is whether the molecule can change structure so that the two rings no longer are parallel but perpendicular. This phenomenon is called sudden polarization. One possibility could be to electronically excite the system. Therefore, Kar et al. calculated the relative total energy between the parallel and the perpendicular conformation in different excited states for the two different molecules of their study, both in the gas phase and in a solvent. We show their results in Table 37 for the two different donor–acceptor substituted molecules that they studied. It is interesting that the results depend very sensitive on the substitution. Table 34 Calculated absorption energies (in nm) for Br2CBr–Br in different solvents. For the 6th excitation, the second entry gives experimental values. For the solvents, also the relative dielectric constant is given. All results are from ref. 87 State

Vacuum Cyclohexane er = 1.000 er = 2.023

Dodecane 2-Propanol er = 20.18

Acetonitrile Propylene carbonate er = 36.64 er = 66.14

2 3 4 5 6

883 799 719 651 459

7 8 9 10 11 12 13 14 15 16

417 374 360 357 354 349 342 324 315 310

895 821 749 684 480 480 406 387 365 354 354 346 342 333 326 310

997 955 914 845 650 635 434 404 381 373 364 360 354 349 347 339

897 823 751 685 482 480 406 388 365 354 354 346 342 333 326 310

985 936 893 824 622 540 429 340 382 368 361 358 352 348 347 336

1000 962 922 853 663 635 436 406 381 375 366 360 355 350 347 340


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Table 35 Calculated singlet–triplet energies (in kcal/mol) for three different CX2 molecules in different solvents. All results are from ref. 87 Molecule

Vacuum

Cyclohexane

Methanol

CCl2 CBr2 CI2

17.4 14.3 7.0

17.6 14.4 7.2

18.0 14.7 7.4

Moreover, in one case, the S3 state possesses a sudden polarization when being solvated in a solvent with a not too large dielectric constant. In a study on coumarin derivatives, Nguyen et al.92 compared the performance of different theoretical approaches. They considered different derivatives and different solvents and compared two different polarizable-continuum models for the solvents and four different density functionals for the quantum-mechanical treatment of the solute. In addition, the dependence on the size of the basis set was explored. We show some of their findings in Table 38. We shall not discuss the performance of the different theoretical approaches in detail, but rather emphasize that the spread in the calculated values can be taken as an estimate of the inaccuracies such calculations will have. For more details, the reader is referred to the original study,92 where many more results are reported, too. In their study on oligothiophenes, Meng et al.71 also calculated the lowest excitation energies for the molecules as a function of length, solvent, and sidegroups. They used the time-dependent density-functional approach combined with a polarized-continuum method for the inclusion of solvent effects. Their results are summarized in Table 39, where they are compared with available experimental information. The decrease in the excitation energy upon increased chain length is readily observed. Moreover, the shifts due to the solvent are not as large as are those due to substitution. Finally, even the experimental values show a significant scatter, which most likely is due to the fact that many of the experimental studies were performed on substituted oligothiophenes. Barone et al.93 studied the optical properties of solvated p-benzoquinone, i.e., a benzene molecule in which two opposite C atoms have been replaced by oxygen (cf. Fig. 15). They used time-dependent density-functional calculations in the determination of the optical properties of the solute and a polarizable-continuum approach for the solvent. By using a recent development that allows for optimizing the structure of the excited molecule in solution, they could determine both the ground-state structure and that of the excited state. Then, they could calculate both vertical

Table 36 The calculated vertical excitation energies DE (in eV) and the corresponding oscillator strengths of (A) the amino and (B) the imino forms of isolated, mono-, di-, and trihydrated as well as dimeric species of 2-aminothiazole (cf. Fig. 6). All results are from ref. 68 System

Gas DE

Water DE

CCl4 DE

Gas f

Water f

CCl4 f

A B A + H2O B + H2O A + 2H2O B + 2H2O A + 3H2O B + 3H2O 2A 2B

5.14 5.20 5.07 4.95 5.02 4.94 5.03 4.96 4.85 4.79

5.07 4.98 5.05 4.98 5.02 4.96 5.03 4.96 4.86 4.81

5.09 5.29

0.12 0.002 0.16 0.13 0.16 0.14 0.16 0.13 0.03 0.004

0.15 0.13 0.16 0.15 0.16 0.16 0.16 0.16 0.05 0.05

0.11 0.002

4.86 4.81


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0.04 0.02

Table 37 The relative total energy (in kcal/mol) of the parallel and the perpendicular conformation of the two donor–acceptor substituted sesquifulvalene molecules of Fig. 7. The system denotes the left and right group in the figure, and results are given for the ground and different excited singlet states as a function of the relative dielectric constant of the solvent. (cf. Fig. 7). All results are from ref. 69 System

State

er = 1.0

er = 4.9

er = 20.7

er = 78.4

NH2/NO2

S0 S1 S2 S3 S4 S0 S1 S2 S3 S4

23.7 6.6 18.0 14.1 14.0 38.4 20.5 25.7 15.6 31.9

11.2 28.1 28.8 3.0

8.8 32.7

6.0 25.8

0.16

6.6

38.8 21.7 26.7 15.8 32.7

38.9 22.2 26.9 15.9 33.0

39.1 22.6 27.6 16.1 33.5

NO2/NH2

Table 38 Calculated excitation energies (in eV), oscillator strength, and transition dipole moment (in D) for the coumarin derivative C1 in different media. Two different polarizablecontinuum models (SCRF-S and PCM) and four different density functionals (CAMB3LYP, mCAMB3LYP, PBE0, and B3LYP) have been used. In all cases, the same size of the basis set has been used, except for the B3LYP calculations in vacuum, for which the second and third entry list results for a smaller and a larger basis set, respectively. All results are from ref. 92 Medium

Solvent model

DFT

DE

f

m01

Vacuum

—

CAMB3LYP PBE0 B3LYP

Ethanol

SCRF-S

Water

PCM SCRF-S PCM Experiment SCRF-S

CAMB3LYP mCAMB3LYP PBE0 B3LYP B3LYP

Cyclohexane

PCM Experiment SCRF-S

4.24 4.04 3.94 4.00 3.93 4.00 3.83 3.74 3.69 3.64 3.51 3.99 3.81 3.68 3.63 3.65 4.13 3.97 3.84 3.87 3.77 3.76

0.368 0.363 0.344 0.329 0.338 0.497 0.451 0.477 0.406 0.445 0.318 0.495 0.447 0.401 0.439 0.325 0.477 0.446 0.415 0.495 0.468

4.79 4.87 4.80 4.66 4.76 5.72 5.57 5.80 5.39 5.68 5.54 5.72 5.56 5.36 5.65 4.85 5.51 5.44 5.33 5.81 5.72

CAMB3LYP mCAMB3LYP B3LYP B3LYP CAMB3LYP mCAMB3LYP B3LYP PBE0 B3LYP

PCM Experiment

and non-vertical excitation and emission energies, which are reported in Table 40. Structural relaxation clearly has a significant effects on the optical properties. In a mainly experimental work, Fujihara et al.94 performed photoelectron spectroscopy experiments on Na2 in small water clusters, i.e., Na2(H2O)n with n r 6. In addition, they performed electronic-structure calculations with Hartree-Fock plus 2nd order Møller-Plesset perturbation treatment of correlation effects. Further Chem. Modell., 2008, 5, 67–118 | 105 This journal is

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Table 39 Calculated excitation energies (in eV) for the nT oligothiophenes of Fig. 9. Except for the n = 2 case, hydrogen atoms are attached everywhere to the carbon atoms. For n = 2, two different structures with two OCH3 groups attached have been considered, too. In all cases, the second entry lists experimental results. Also the relative dielectric constants of the solvents are listed. All results are from ref. 71 n

Sidegroups Gas er = 1.000

Acetonitrile n-Hexane er = 36.64 er = 1.89

1,4-Dioxane Chloroform er = 2.21 er = 4.90

Water er = 78.39

2

—

4.00

4.02 4.05, 4.11

4.02 3.96, 4.11

4.00

3.25 3.50–3.51 2.81 3.17 2.53 2.98–3.00 2.34 2.85

3.25 3.50, 3.61 2.80 3.04–3.34 2.53 2.98, 3.16 2.34 2.71–3.06

3.25

4.12

3.84 3.90–3.91 3.70 3.87–3.88

OCH3 OCH3 3

—

3.36

4

—

2.92

5

—

2.64

6

—

2.45

4.03 4.11–4.14 3.86 3.93 3.70 3.89 3.25 3.51–3.55 2.81 3.21–3.22 2.55 3.03 2.35 2.85

2.80 2.51 2.32

calculations on the optimized structures with more advanced methods were carried through in order to calculate the experimental spectra. They were able to identify several different structures for each value of n. For the structures of the lowest total energy for n r 3 the water molecules are bound preferentially to one of the Na atoms by O atoms, so that Na2 becomes asymmetrically solvated. For these structures, the Na–Na+(H2O)n structure grows with n resulting in one Na+ ion and a solvated Na ion. Nikitina et al.95 studied the Stokes shift of some organic chromophores due to the ~0 and R ~1 being the coordinates for the molecule of interest in solvent effects. Letting R ~ and E1(R) ~ its ground state and its first excited state, respectively, and letting E0(R) being the total energy of the ground state and of the first excited state, the Stokes shift is defined as ~0) E0(R ~0)] [E1(R ~1) E0(R ~1)] DESS = [E1(R ~0) E1(R ~1)] + [E0(R ~1) E0(R ~0)] = [E1(R l1 + l0 = (l1,s + l1,v) + (l0,s + l0,v).

(25)

In the first identity, we have written the Stokes shift as a sum of two parts, one from the first excited state and one from the ground state. These two are called the

Fig. 15 Structure of p-benzoquinone. Closed circles, open circles, and closed triangles represent carbon, hydrogen, and oxygen atoms, respectively.


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Table 40 Calculated energies (in eV) for the p-benzoquinone in various media. DEv gives the vertical excitation energy, DEabs is the non-vertical excitation energy, and DEemis is the vertical emission energy. TD-DFT, CASPT2, and SAC-CI represent results obtained with timedependent density-functional calculations as well as with two post-Hartree-Fock wavefunction-based approaches. Properties for the three excitations with the largest oscillator strengths are shown. All results are from ref. 93 Medium Gas

CCl4

DMSO

DEv TD-DFT

DEv CASPT2

DEv SAC-CI

DEabs TD-DFT

DEemis TD-DFT

3.09 3.05 4.11 3.06 3.13 4.09 3.08 3.28 4.08

2.80 2.82 3.56

2.71 3.50 3.74

2.88 2.79

2.67 2.28

2.80 2.87

2.57 2.36

2.67 2.28

2.18 2.48

reorganization energies, l, and they are in the last identity split into one contribution from a solvent, li,s, and another part from the internal structure of the molecule, li,v. In their work, Nikitina et al.95 focused on the solvent contribution, ls = 12(l1,s + l0,s).

(26)

To calculate this they applied a methodology developed by their group earlier.96 Ultimately, they applied it on various larger, organic chromophores and demonstrated a fair agreement with available experimental information. We shall not discuss their results further here. Optical rotation spectroscopy (ORD) can by used in characterizing chiral molecules, like (R)-3-methylcyclopentanone of Fig. 16. This molecule can exist in two conformations, as shown in the figure. Al-Basheer et al.97 studied this molecule experimentally and theoretically in various solvents, including cyclohexane. They calculated the spectra of the two isomers, separately, as shown in Fig. 17. Experimental information on the mole fractions of the two conformers was subsequently used in calculating the spectrum that is shown in Fig. 18, where it is also compared with the experimental results. The agreement between experiment and theory is excellent. A related study was presented by Kundrat and Autschbach.98 They calculated the optical rotation for aromatic amino acids in solution. For each molecule, three primary rotamers, g, h, and t (for ‘gauche’, ‘hindered’, and ‘trans’), may exist. Fig. 19 shows one example from the study of Kundrat and Autschbach.

Fig. 16 Structure of the two different conformations of (R)-3-methylcyclopentanone. Reproduced with permission of The American Chemical Society from ref. 97.


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Fig. 17 Calculated optical rotation spectra for the two isomers of (R)-3-methylcyclopentanone in cyclohexane. Reproduced with permission of The American Chemical Society from ref. 97.

Fig. 18 Calculated and experimental optical rotation spectra for (R)-3-methylcyclopentanone in cyclohexane. Reproduced with permission of The American Chemical Society from ref. 97.

Fig. 19 Optimized rotamers of the phenylalanine zwitterion. Reproduced with permission of The American Chemical Society from ref. 98.

Kundrat and Autschbach calculated the relative total energies of these in the solvent. They used density-functional calculations for the solute and a polarizable continuum method for the solvent. Moreover, they compared the results obtained with two different basis sets. From the relative total energies, Kundrat and Autschbach could calculate the populations for a temperature of 293 K using a Boltzmann distribution, and compare those with available experimental information, see Table 41. Subsequently, by calculating the optical rotation for each rotamer and a wavelength of 589.3 nm (that of the sodium D line) and weighting the results with the Boltzmann factors, they obtained the values for the optical rotations of the 108 | Chem. Modell., 2008, 5, 67–118 This journal is

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Table 41 Different properties from the study of the optical rotation (ORD) in solution. DE gives the relative total energy in kJ/mol, x1 and x2 the Boltzmann populations from the calculations with two different basis sets, and xexp that from experiment. The calculated optical rotations [in (deg cm3)/(g dm)] with the two different basis sets and from experiment is listed, too. Two different molecules are considered and for each three different rotamers. DE has been calculated with the first basis set, and the corresponding Boltzmann populations have been used. The entry ‘mix’ gives the Boltzmann averaged results. All results are from ref. 98 Molecule

Structure DE

Phenylalanine zwitterion

g

7.6

0.03 0.03

0.24

h t mix g

3.0 0.0

0.21 0.19 0.76 0.78

0.27, 0.28 0.48, 0.50

1.8

0.26 0.24

150.9 6.7 22.3 0.24, 0.28, 0.32, 0.36 221.8

h t mix

2.5 0.0

0.19 0.20 0.55 0.57

0.26, 0.27 0.37, 0.42, 0.46, 0.50

Phenylalanine cation

x1

x2

ORD1

xexp

162.4

69.1 25.8 59.0

ORD2 ORDexp 142.8 120.6 18.2 7.7 35.1 210.4 66.7 18.0 52.2

7.4

mixture that are given in Table 41, too. Also for those, a comparison with experimental values is presented. Table 41 contains the results for only two molecules. The original study considered several further molecules, and in some cases a better agreement with experiment was found than what appears to be the case for the results of Table 41. As seen in the table, the ORD is very sensitive to the structure. Therefore, if small inaccuracies occur in the relative total energies, the populations may be quite inaccurate and, consequently, wrong results for the total ORD signal are found. Such inaccuracies could, e.g., have their origin in the lack of explicit solvent molecules in the near neighbourhood of the solute. Mukhopadhyay et al.99 calculated the optical rotation of methyloxirane in water using different approaches and compared the results with experimental information. They considered both explicit and implicit models for the solvent, i.e., they treated the solute as well as some parts of the solvent quantum-mechanically, whereas the long-range effects of the solvent were treated within a polarizable-continuum approximation. They found that without water molecules in the quantum-mechanical treatment, the agreement between experiment and theory was not good. Thus, in this case it is important to include short-range solute–solvent interactions explicitly. In that case, a good agreement was found. J. Linear and nonlinear optical properties in solution The fact that materials respond to electromagnetic fields can be used to manipulate light. In particular the higher-order responses are then of interest. Therefore, many (experimental and theoretical) studies have been devoted to understand, exploit, and optimize these responses. One way of quantifying the responses is to consider the dipole moment of the molecule of interest. In the presence of an external electric field, the ith component of the dipole moment can be expanded as ð0Þ

mi ¼ mi þ

X

aij Ej þ

j

1X 1X b Ej Ek þ g Ej Ek El þ ; 2 jk ijk 6 jkl ijkl

ð27Þ

is the dipole moment in the absence of the external field, aij is the where m(0) i polarizability, and bijk, gijkl, . . . is the first, second, . . . hyperpolarizability. Em is the mth component of the external field. If the external field is frequency dependent, Chem. Modell., 2008, 5, 67–118 | 109 This journal is

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so will the response be. Through the hyperpolarizablities it becomes thereby possible to obtain responses with frequencies that are different from those of the incoming electric field. The conjugated, organic molecules that we have discussed a couple of times during this presentation have turned out to be materials with particularly large values for the hyperpolarizabilities. Therefore, many studies have been devoted to those materials. Among those is the work of Balakina and Nefediev,65 who studied systems like the one of Fig. 4 in the presence of static fields. Assuming that the molecule is lying in the (x, z) plane with the z axis being the main axis of the molecule, they calculated an average polarizability, a = 13(axx + ayy + azz),

(28)

and an average first hyperpolarizability along the molecular axis, bz = bzxx + bzyy + bzzz.

(29)

These two quantities are listed in Table 27 for the four different molecules of their study both in the gas phase and in solution. It is interesting to see that the responses can be increased significantly upon solvation. The authors found that the structure is hardly affected by the solvation, so the effect is mainly a purely electronic one. This was confirmed by calculating the responses in the solution but for the structures of the gas phase. Also this led to significant enhancements of the values of a and gz compared to the gas-phase values, although slightly smaller. Also Perpe`te and Jacquemin70 studied such systems for static fields. They calculated the first hyperpolarizability using a continuum-model for the solvent and Hartree-Fock + Møller-Plesset approaches for the solute. As solute they considered the push-pull systems of Fig. 8. They optimized the structure both in the gas phase and in the solvent. Subsequently, they calculated the properties for the molecules considering both geometries in both phases, whereby they could distinguish between structure and media effects. The results are shown in Table 42 where it is confirmed that the main effects of the solvent are purely electronic, whereas structural changes are less important. It is also interesting to see that bzzz is significantly enhanced in the solution and that bzzz/n goes through a maximum as a function of n. Li et al.100 examined the properties of two organometallic tungsten–carbon complexes, tungsten pentacarbonyl pyridine (TPCP) and tungsten pentacarbonyl trans-1,2-bis(4-pyridyl)-thylene (TPCB), that also had been studied experimentally. They considered the isolated monomers as well as dimers and studied the systems in solutions. They used density-functional methods in order to calculate the linear and nonlinear responses to electric fields, and the solvents were treated with a Table 42 The first hyperpolarizability bzzz (in a.u.) for the push-pull oligomers of Fig. 8 as a function of n. The geometry was optimized in either gas or water phases, and the calculations were carried through for both structures in both media. All results are from ref. 70 n Geometry Medium

bzzz/n Gas Gas

bzzz/n Gas Water

bzzz/n Water Gas

bzzz/n Water Water

2 4 6 8 10 12 14 16

1571 4635 8628 12 301 14 903 16 330 16 824 16 696

7199 29 743 48 727 52 273 48 380 42 864 37 731 33 407

1691 5284 9323 12 851 15 378

6772 39 309 58 041 55 410 50 634


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Table 43 The static dipole moment, the average polarizability, and the average first hyperpolarizability for different tungsten–carbon complexes. The second entry gives the conditions, and the third gives the method. All results are from ref. 100 System

m(0) (D) a (1024 esu) a (1024 esu) b (1030 esu) b (1030 esu) b (1030 esu) static l = 1907 nm static l = 1907 nm l = 1907 nm Calc. Calc. Calc. Calc. Calc. Exp.

TPCP monomer TPCP dimer TPCB monomer TPCB dimer

7.5 5.6 6.1 4.2

28.9 56.5 52.2 102.2

29.2 56.9 53.3 104.2

11.0 5.0 66.0 11.1

15.0 6.5 109.3 21.5

4.4 4.4 7.0 7.0

polarizable-continuum model. In order to compare with experiment, they considered toluene and chloroform for TPCP and TPCB, respectively, as the solvent. Moreover, the experiments were performed for electric fields with a wavelength of l = 1907 nm, and, therefore, both that wavelength and an electrostatic field were considered. Table 43 summarizes some of their findings. It is seen that in particular the hyperpolarizability responds very sensitively to the formation of dimers, although the interaction between the monomers is an only weak p–p stacking effect. The fact that such weak interactions are important suggests that also solvent molecules should be treated explicitly. On the other hand, Table 43 shows that, in the present case, the theoretical results for the dimers are able to reproduce the experimental results, whereas those of the monomers are not. K.

Magnetic properties in solution

Not only the responses of materials to electric fields can be used in characterizing materials as well as be exploited in applications, but also those to magnetic fields. Therefore, theoretical studies of the magnetic responses are as relevant as are those to electric fields. Here, we shall briefly discuss a few recent such ones. Jansik et al.101 calculated the magnetochiral axial birefringence. This effect appears as a change in the optical index of chiral systems when being exposed to a static magnetic field parallel to the direction of the propagation of the light. For a chiral molecule, the axial birefringence due to the presence of a magnetic field can be defined as the difference between the refractive indices for unpolarized light propagating parallel and antiparallel relative to the direction of the externally applied magnetic field, Dn = nmm nmk.

(30)

Here, the first upper index represents the direction of propagation of the light and the second represents the direction of the magnetic field. In an earlier work, the same group had used Hartree-Fock calculations in calculating Dn for the most stable isomer of some chiral organic molecules in vacuum.102 In the more recent work,101 they extended the study in several directions. At first, they applied the B3LYP density-functional method, whereby also correlation effects were included. Second, they compared gas-phase results with those obtained for a solution, whereby they applied the polarizable-continuum method for the treatment of the solvent. And third, for some of the larger molecules they included the effects of having a mixture of more different stable structures in an approach very similar to that we discussed above in section III I for the calculation of the optical rotation. Their results are summarized in Table 44. The table shows that the changes when passing from the Hartree-Fock approximation to the B3LYP approach are large, which the authors interpret as indicating the importance of correlation effects. Moreover, when considering several different isomers with closeby energies, the Chem. Modell., 2008, 5, 67–118 | 111 This journal is

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Table 44 The magnetochiral birefringence, Dn [in 1012 cm3/(T g)], per unit magnetic field and density for various chiral molecules. The theoretical method is either the Hartree-Fock or the B3LYP density-functional method, and the calculations were performed for either the gas phase or a solution. For the last two molecules, not only the most stable isomer was considered but also mixtures of six stable structures with populations according to a Boltzmann distribution. [a] is the specific optical rotation [in deg/(cm3 dm g)]. All results are from ref. 101 Molecule

Method

No. of isomers

Phase

Dn

(R)-C3H6O

HF B3LYP HF B3LYP HF B3LYP HF B3LYP B3LYP B3LYP B3LYP HF B3LYP B3LYP B3LYP B3LYP

1 1 1 1 1 1 1 1 1 6 6 1 1 1 6 6

Gas Gas Gas Gas Gas Gas Gas Gas Solution Gas Solution Gas Gas Solution Gas Solution

7.53 18.50 0.74 3.64 23.36 0.24 0.60 53.91 28.43 1.23 4.61 16.97 33.09 35.96 36.24 33.56

(S)-C3H3OF (R)-C4H6O (S)-C10H14O

(R)-C10H16

[a]

205.1 238.7 41.6 40.5 18.2 18.6 87.5 86.0

results may change significantly. That solvation has a strong effect should be visible in the table, too. NMR spectroscopy is a very useful tool for determining the local chemical surroundings of various atoms. Komin et al.103 studied theoretically this for the adenine molecule of Fig. 20 both in vacuum and in an aqueous solution using different computational approaches. In all cases, density-functional calculations were used for the adenine molecule, but as basis functions they used either a set of localized functions (marked loc in Table 45) or plane waves (marked pw). Furthermore, in order to include the effects of the solvent they used either the polarizable continuum approach (marked PCM) or an explicit QM/MM model with a force field for the solvent and a molecular-dynamics approach for optimizing the structure (marked MD). In that case, the chemical shifts were calculated as averages over 40 snapshots from the molecular-dynamics simulations. Finally, in one case, an extra external potential from the solvent acting on the solute was included, too, marked by the asterisk in the table. The table shows a strong dependence of the nitrogen chemical shifts on the basis set. A similar dependence was not observed for 13C chemical shifts on the same

Fig. 20 Schematic representation of the structure of the adenine molecule. Open circles, closed circles, and closed triangles represent carbon, nitrogen, and hydrogen atoms, respectively. The atom numbering is also shown.


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Table 45 Calculated and experimental values of the 1H and 15N chemical shifts of adenine of Fig. 20 where also the atom numbering is given. For further details, see the text. All results are from ref. 103 Method

Phase

H2

H1

N1

N3

N6

N7

N9

loc loc/PCM pw pw/MD pw/MD pw/MD* Exp.

Gas Sol Gas Gas Sol Sol Sol

8.61 8.63 8.07 8.21 7.99 7.98 8.07

7.79 8.35 7.65 7.74 8.22 8.17 8.11

112.3 124.6 151.8 144.8 162.3 158.5 157.3

121.0 132.8 159.1 154.3 166.7 162.6 158.2

295.3 290.1 271.1 264.8 268.8 273.9 304.0

103.6 119.9 160.9 154.1 180.6 174.3 166.8

212.7 203.4 213.0 213.9 209.0 211.1 209.9

molecule (not shown). Adding solvent effects with the polarizable continuum method brings the chemical shifts in general closer to the experimental values (obtained in solution). The same is true when adding the solvent effects using a QM/MM approach. However, the overall agreement between theory and experiment as seen in Table 45 is not perfect, which actually may suggest that such studies are very sensitive to the accuracy of the theoretical approach and, accordingly, may provide an excellent field for testing new developments. As the final example we mention the recent study of Pollet and Marx.104 They studied the properties of a gadolinium-containing complex in aqueous solution. This complex is used as a contrast agent to enhance the image contrast in magnetic resonance imaging that in turn is used for diagnostic purposes in medicine. In order to be able to use theoretical calculations as a support for the experimental studies, it is mandatory that the theory is able to provide accurate information. Therefore, Pollet and Marx compared different theoretical approaches for studying the complex in water, including the polarizable continuum approach, a QM/MM approach, and a Car-Parrinello supercell approach where also the solvent is treated fully quantummechanically. They found that only the last approach was able to give structural information that was in good agreement with experiment. The authors did, however, not extend the study to the calculation of magnetic responses.

L. Including surfaces Computationally, solvated molecules adsorbed on surfaces represent an enormous challenge. The surface is usually extended in two dimensions, but the presence of the adsorbed molecules breaks the symmetry. Moreover, the material is extended in the third dimension, although the presence of the surface leads to a broken symmetry there, too. Finally, also the presence of the solvent leads to a significant reduction of symmetry. Therefore, theoretical studies of such systems are much less common than those of finite, solvated but otherwise isolated molecules. However, some studies exist, and here we shall briefly discuss two that illustrate the possibilities and approaches. Pool et al.105 studied the adsorption of alkyl thiols on gold surfaces, both without and with a solvent. As solvent they considered n-hexane. For the simulations they used a force field with a so called united-atom model. This means that the thiol heads, the CH2 and CH3 groups, and the gold atoms are treated as individual united ‘atoms’. The interactions between these united pseudoatoms are described in terms of the force fields. Subsequently, molecular dynamics simulations were carried through for the different systems. At first, the authors analysed the adsorption behaviour for a clean perfect Au(111) surface in vacuum. For alkyl thiols with three segments in the alkyl tail they found the structures of Fig. 21. At a temperature of T = 300 K, they found two different structures, depending on whether the simulation was started from a fully loaded Chem. Modell., 2008, 5, 67–118 | 113 This journal is

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Fig. 21 Structure of alkyl thiols adsorbed on a gold surface. Light spheres, dark spheres, and thin lines represent gold atoms, thiol heads, and carbon chains, respectively. The left part shows the two-dimensional, ordered, ‘crystalline’ phase, and the right part the two-dimensional, disordered, ‘liquid’ phase. Top (top) and side (bottom) views are shown in each case. Reproduced with permission of The American Chemical Society from ref. 105.

surface or from a completely empty surface. The former case leads to an ordered, ‘crystalline’ phase, as shown in the left part of the figure, whereas the latter leads to a disordered, ‘liquid’ phase that is shown in the right part of Fig. 21. Similar simulations for finite gold nanocrystals with 561 atoms led to a similar hysteresis between adsorption and desorption. Moreover, the adsorption starts at a lower thiol concentration for the nanocrystals than for the macroscopic crystal surface. Adding the solvent led to significant changes. There results a competitive adsorption between the thiols and the solvent and the interactions between the tails are modified. Thus, as the authors conclude, phenomena that are observed in vacuum may be different from those observed in a solution.

Fig. 22 The model system for the electrochemical double layer above a Pt(111) electrode. The large dark spheres represent the Pt atoms, whereas the smallest spheres represent hydrogen and the intermediate-sized ones oxygen. Hydronium ions, H3O+, and transition states are explicitly marked. The top shows a top view and the bottom a side view. Reproduced from ref. 106.


c


As the last example we discuss the recent study of Sku´lason et al.106 who examined the hydrogen evolution at a platinum electrode. They studied the model system of Fig. 22 that contains a slab of Pt atoms, representing the Pt(111) surface. On top of that, two layers of water molecules plus an excess of H atoms are placed. Thereby, protons become solvated in the water, leading to the formation of H3O+ ions, whereas electrons are transferred to the metal. In their simulations, they use a supercell approach with the cell of Fig. 22 repeated periodically in all three dimensions (with, however, some further layers of Pt atoms). All atoms were treated quantum-mechanically within the density-functional formalism. At the surface, the (fast) so called Volmer reaction H+ + e - Had takes place. Subsequently, either the homolytic Tafel reaction, 2Had - H2, or the heterolytic Heyrovsky reaction, Had + H+ + e - H2, may take place. Using the nudgetelastic-band approach (see, e.g., ref. 2) the authors studied the three reactions on the surfaces. They found that the Heyrovsky reaction may dominate, but the similarity of the calculated activation energies for the Heyrovsky and the Tafel reactions suggests that both reactions may take place in parallel.

IV. Conclusions The theoretical treatment of molecules in solution represents on the one hand a methodological and computational challenge and on the other a possibility for bringing experimental and theoretical studies in closer contact. Since many experiments are performed in solution it is important for theoretical studies to be able to treat this situation. But the presence of a solvent means that the solvated molecules may interact more or less strongly with those of the solvent. Ultimately, the solvent is structured, extended, and able to respond to the presence of the solute. Thus, in the perfect case a full quantum-mechanical treatment of the complete system, solute and solvent, is needed. Such a treatment is not possible and one has to restore to approximations. We discussed some of those in section II. They could, as a first approximation, be split into three different classes. In one set of approximations the solvent was treated within a simplified, parameterized approach. Thereby, the structure of the solvent was retained. As an alternative, the solvent could be approximated as a polarizable continuum that would respond to the presence of the solute but otherwise was without any internal structure. These two types of approaches are coined explicit and implicit methods, respectively. Finally, a full quantum-mechanical treatment of either a smaller, finite supermolecule or an infinite, periodic supercell could be carried through. The first approach suffers from the dependence of the parameterization of the simple model. We saw examples where quite different results could be obtained for different ‘realistic’ models. On the other hand, it has as an advantage that also large, complex systems can be treated, and chemical interactions like the formation of hydrogen bonds can be included. The second approach is computationally simple and efficient, which is a significant advantage. On the other hand, the lack of true atoms of the solvent and of possible chemical or hydrogen bonds between the solvent and the solute may in some cases be serious drawbacks. Also the dependence of the results on the shape of the cavity occupied by the solute may be a problem. The third approach is, in principle, the most accurate one that should be able to give arbitrarily accurate results. However, the computational demands of this put serious limitations on the system sizes that can be treated with this. On the other hand, only this approach allows directly for a proper description of possible charge transfer processes between solvent and solute. Independent of the approach used, we saw several examples of the influences of a solvent on the properties of a solute. Structure, relative stability, optical, and Chem. Modell., 2008, 5, 67–118 | 115 This journal is

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magnetic properties could all be altered due to the presence of a solvent. We also saw many examples where an impressive agreement between experiment and theory could be obtained. Finally, this presentation has hopefully also indicated where the developments in the theoretical treatment of solvated systems will bring us in the not too distant future. This includes for instance that even more complex systems will be treated, including heterogeneous catalysis in solution. Moreover, additional, experimentally accessible quantities will be accessible with calculations, too. And, as our examples have shown, both long-range and short-range interactions are important, which ultimately may lead to the consensus that a combined explicit/implicit treatment of a solvent is most useful. This means that an implicit treatment of the long-range effects of the solvent as a polarizable continuum is combined with an explicit treatment (for instance a full quantum-mechanical one) of the closest solvent molecules to the solute. An increasing number of studies is based on such approaches that appears to lead to some of the currently most accurate results. One may hope that through such multi-scale approaches an optimal accuracy can be obtained and that the arbitrariness of the results due to the parameterization may be reduced as much as possible.

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The solid state E. A. Moore DOI: 10.1039/b801632c

1. Introduction Solid state modelling has long been the poor relation of organic modelling as far as chemists are concerned. There are reasons for this. Firstly it was easier to develop relatively small sets of parameters that would describe a wide range of organic molecules including biologically important molecules such as amino acids and DNA bases, whereas the solid state modeller requires parameters for the entire periodic table. Secondly the initial molecular modellers could work with small molecules and hence make progress with the computer resources available in the 1960s. Solid state modelling requires methods of dealing with infinite arrays. Finally many solids of interest contain open shell atoms which are harder to deal with theoretically and require more computer resource than, the usually closed shell, organic molecules. Ideas for the simulation of solids have been around for a long time—for example the lattice energy calculations such as those of Kapustinskii1 and the electron gas theory,2 but it required the advent of computers for simulation of solid state materials to become widespread. The developments that have led to the recent surge of interest in solid state modelling and the adoption of modelling by bench top chemists have been the vast increases in readily-available computer memory, processor speed and storage space, the development of efficient methods of calculation and the development of graphical interfaces linked to modelling software. It is now possible to run many jobs on a PC, using a graphical interface to construct a crystal and set up a run in a way that our organic chemist colleagues have been able to do for over a decade. As with organic modelling, some knowledge of the programs behind the interface leads to more appropriate use. Solid state modelling is now well-established and this review is long overdue. There have been several issues of journals devoted to the topic, more recent ones include the Journal of Materials Chemistry3 and the Journal of Solid State Chemistry4 and there has been a review on this topic in Annual Reports of Progress in Chemistry.5 The introductory article in the Journal of Solid State Chemistry issue gives a very good summary of the history of quantum mechanics applied to the solid state. In this review I shall look at the different approaches to modelling solids and at some examples of the use of such techniques. The field is now so wide that I am unable to cover all aspects and so I have put more emphasis on areas that interest me while trying to at least mention other aspects that are currently attracting modellers. As in other branches of chemistry there are both ab initio methods based on solving Schro¨dinger’s equation and methods based on classical physics. The latter methods are based around lattice energy calculations and I shall refer to them as interatomic potential methods. In the first section of this review, I shall discuss the development and current position of interatomic potential calculations. The first atomistic simulation programs for solids using interatomic potentials were developed in the 1970s.6 Zeolites and isolated point defects in solids were applications that illustrated the use of such methods early on. Work continues in these areas and other applications have been developed. One such is the use of these methods in molecular dynamics calculations as their efficiency makes possible calculations on large supercells or for relatively long lifetimes. Department of Chemistry and Analytical Sciences, The Open University


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The second section deals with ab initio methods. One way to approach solids is to start from the solid as a structureless box. The solutions to the Schro¨dinger equation for this are of course the well-known ‘particle-in-a-box’ wavefunctions eik r. Physicists have long used such functions to represent the valence electrons of solids, particularly for metals and semiconductors, and consider one of these methods (linear augmented plane waves, described in section 3.1) as the standard against which other methods should be tested. Such methods have not been popular with chemists until this century. A striking current trend is the adoption of plane wave methods by chemists. At the same time physicists are increasingly studying solids of chemical interest. In particular both chemists and physicists are interested in modelling solids with novel magnetic properties such as the manganates which exhibit colossal magnetoresistance. I have therefore spent some time discussing the use of these methods. An alternative approach is to describe the valence electrons by localised functions. In the linear combination of crystalline orbitals (LCCO) which is similar to the LCAO approach to molecules, the valence electrons are represented by sums of atomic orbitals. As well as being in some respects more familiar, this method is capable of all-electron calculations and in principle should be better for properties which need a good description of core electrons. For some of these properties however, extensions to the plane wave method are available which can cope with inner electrons around nuclei of interest. At present, the performance of both approaches is comparable. For some applications, it is useful to treat one part of the system rigorously and employ a more approximate technique for the remainder. This is a technique wellknown for studying active sites of enzymes. Here I consider its application to solid state catalysts where the more rigorous method can be applied to molecules on surfaces and in zeolite cages. In organic/biological chemistry the technique is usually referred to as QM/MM (quantum mechanics/molecular mechanics) and I retain this label with QM referring to ab initio methods and MM to interatomic potential methods. A technique that is increasingly popular is molecular dynamics. This enables the study of free energies and of the effects of changing temperature and pressure. This technique is notoriously computer resource-hungry but increases in storage capacity, memory and processor speed have made it more feasible and it is now possible to combine ab initio and molecular dynamics calculations. The next section is devoted to this and related topics. The now wide-spread use of solid state NMR techniques such as magic angle spinning (MAS) led to a need for calculations of NMR parameters such as chemical shifts and quadrupole coupling constants. A notable achievement this century has been the development of a method to calculate NMR chemical shielding in periodic solids. This will be described in the following section along with developments in the calculation of other properties. Finally I shall present some interesting applications of solid state modelling. Zeolites were one of the early success stories and there is still considerable interest in modelling these and related compounds such as ALPOs. Calculations in this area can predict new structures, determine the distribution of different cations over the available sites and throw light on the use of zeolites as catalysts by investigating the interaction of molecules with the framework. Another area that has been of interest since the development of this field is that of metal oxides. Simple metal oxides such as MgO provide a test bed for new methods. More complex oxides have attracted much interest for their commercially important properties—solid electrolytes, ferroelectrics, catalysts, semiconductors, superconductors, multiferroics. Relatively simple calculations can, for example, track the path of ions through ionic conductors and suggest alternative solids for fuel cells or batteries. Solids with interesting electrical and magnetic properties such as high TC superconductors and solids showing colossal magnetoresistance (CMR) have been 120 | Chem. Modell., 2008, 5, 119–149 This journal is

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the focus of much work in recent years and I shall consider examples of studies in this area in a separate subsection. I shall however take my examples from solids of chemical interest and will not therefore cover metals, alloys and III–V semiconductors. I shall omit some areas which are currently flourishing and of interest to solid state chemists. These are surfaces, polymers, biological materials such as proteins and DNA, metal organic frameworks and organic crystals. These will perhaps be reviewed later. To keep this review to a reasonable length I have regretfully left out references to calculations on nanomaterials, although there is much current work in using solid state methods in this area, particularly on nanotubes. The last application I shall consider is the modelling of minerals. These are often complex solids formed under high pressures or temperatures or in the presence of water and are not easy to model. Nevertheless there is considerable work on minerals both those found on Earth and those on other planets and advances in molecular dynamics and related methods should boost this area. I have been selective in my choice of examples of application and apologise to those whose work I may have omitted either because it did not fall into the categories of compounds I have chosen or because it slipped by me in the huge number of papers on this topic being published. I have tried to include at least some examples of work from most of the large groups in this field. I hope that some of the omissions will be remedied in the next review in this series.

2. Interatomic potential methods Ten years ago when I attended a Faraday Discussion on Solid State Chemistry: New Opportunities from Computer Simulations,7 interatomic potential methods were well developed and the use of ab initio methods starting to become widespread. In his Introductory Lecture Prof. C. R. A. Catlow asked ‘With the continuing growth of the applicability of electronic structure techniques, can we see them as replacing interatomic potential based methods?’ His reply then was ‘there will be a continuing role for interatomic based potential based methods as the field moves to more complex systems.’ Over the last decade, ab initio electronic structure methods have progressed rapidly and for many applications plane-wave ab initio methods are now the first choice for calculations. Nevertheless that reply still holds true. This section will only give a brief overview of interatomic potential methods. The article by Gale and Rohl8 on the widely-used program GULP (General Utility Lattice Program) provides an extensive discussion of the methods used in this and similar programs with a comprehensive list of references. The basis of interatomic potential methods when applied to solids, particularly ionic solids, is the calculation of lattice energies. The starting point is to place each atom on a lattice site and assign it a charge. For simple ionic compounds, the formal charge is used. Perhaps surprisingly the formal charge has also proved successful in calculations on covalently-bonded networks such as zeolites and ALPOs, where the apparently unrealistic charges of +4 on Si and +5 on P are used. However in a study aimed at modelling the effect of radiation damage in solids, Thomas et al.9 found Mulliken charges from ab initio calculations gave a better description of TiO2. The use of formal charges has advantages when studying defects as the defect must necessarily have an integral charge. For covalently-bonded entities within the crystal such as ammonium, nitrate and sulfate ions and water molecules, the atoms within the molecule are assigned partial charges as in organic modelling with the overall charge on the molecule fixed at the ionic charge or at zero. The calculation of the energy due to the electrostatic interaction of all the ions is generally done using the Ewald method.10 Because the electrostatic interaction converges only slowly with distance, the Ewald method splits the summation into two parts. The short range part is evaluated in real space but the long range part is evaluated in reciprocal space. Modifications that accelerate the evaluation of the Ewald summation include the Chem. Modell., 2008, 5, 119–149 | 121 This journal is

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particle-mesh approach11 and the fast multipole method.12,13 Wolf et al.14 have put forward an alternative real space-only summation. In most simulation calculations, the charges on the atoms are fixed. However geometry-dependent charges are used in some cases.15–17 Most such approaches have been based on the concept of electronegativity equalisation.18 The next set of terms usually added allow for van der Waals-type interactions between the ions. The most popular form of the potential in solid state modelling is the Buckingham potential A exp(r/r)C/r6 but there are other forms in use such as the Lennard-Jones rA12 rC6 , the general form A expðr=rÞ rCn and potentials derived rm from splined fits. For molecules and ions containing hydrogen, it is usual to include a Morse potential De((1 ea(rre))2 1) to describe the covalent bond between H and a heavier atom. Such potentials have also been used for molecular anions, for example the carbonate anion in calcite,19 perchlorate and perbromate20 and sulfate.21 Water and OH ions are important in many technological and mineralogical applications but it is difficult to find satisfactory potentials for these species. Chroneos et al.22 have put forward two transferrable new models for hydroxide ions, one using partial charges and one using formal charges. In place of the Morse potential, they employ a screened Coulomb potential for the O–H interaction, qO qH rOH V¼ ð1Þ 1 exp 4pe0 rOH r An idea of the range of available two-body potentials can be obtained from the list of those included in GULP.8 For solids containing covalent bonds, including salts of molecular ions and alumino-silicate framework and similar structures, potential terms akin to those used in molecular mechanics calculations on organic molecules are used. These are bond, angle and torsion terms. A simple harmonic model is often used, V = 12kb(r r0)2 +

1 2

k(y y0)2 + kt(1 S cos nf)

(2)

but there are other options. Many body potentials e.g. Sutton-Chen,23 Tersoff,24,25 Brenner26,27 can be used to describe metals and other continuous solids such as silicon and carbon. The Brenner potential has been particularly successful with fullerenes, carbon nanotubes and diamond. Erhart and Albe28 have derived an analytical potential based on Brenner’s work for carbon, silicon and silicon carbide. The Brenner and Tersoff potentials are examples of bond order potentials. These express the local binding energy between any pair of atoms/ions as the sum of a repulsive term and an attractive term that depends on the bond order between the two atoms. Because the bond order depends on the other neighbours of the two atoms, this apparently two-body potential is in fact many-body. An introduction and history of such potentials has recently been given by Finnis29 in an issue of Progress in Materials Science dedicated to David Pettifor. For a study of solid and liquid MgO Tangney and Scandolo30 derived a many body potential for ionic systems. Polarisability of ions is usually modelled via the shell model of Dick and Overhauser.31 The ion is divided into a point core and a massless spherical shell which interact with each other by a spring force. Two body interatomic potentials such as the Buckingham potential act on the shells. Both core and shell are charged, the sum of their charges being the formal charge of the ion. The distribution of charge between core and shell is purely empirical and can be quite counter-intuitive; it is not intended to represent an inner core and a shell of valence electrons. For example a +3 ion may have a shell charge of 4.97 and a core charge of 1.97. This model makes some allowance for the finite size of ions and can also model distortion of the electron cloud to a limited extent by allowing the shell centre to be non-coincident with the core position. It has been found also that this model can be used to 122 | Chem. Modell., 2008, 5, 119–149 This journal is

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represent a degree of covalency.8 The shell model only deals with dipole polarisability. An alternative approach is to use a point ion dipolar polarisability; this has the advantage that it can easily be extended to higher order polarisabilities. Such an approach has been used for ionic materials by Wilson and Madden.32 An interesting recent development has been a method based on the angular overlap model for the inclusion of ligand field effects.33 The parameters in all these potentials are obtained in one of two ways. The first is by calculation. The earliest method of calculation was the electron gas model popularised by Gordon and Kim.34,35 This model divides the repulsion energy between two closed shell atoms into three terms—electrostatic energy, kinetic energy and exchange energy. Methods of obtaining these terms were reviewed by Wood and Pyper.36 This method has now been superseded by ab initio methods in which the geometry of pairs, clusters or periodic arrays of atoms is varied systematically. The variation of energy with distance is then used to fit the parameters in the chosen interatomic potential model. The second method of obtaining potentials is the empirical method in which the parameters are varied until a good fit to experimental cell constants and, wherever possible, other experimental data is obtained. Typical experimental data used for fitting are elastic constants and dielectric constants. Empirical potentials are often only a realistic approximation to the true potential over a limited range of interatomic distances. Fig. 1 compares energy curves for a typical empirical potential and potentials derived from ab initio calculations on the rock salt structure. The potentials have similar values at typical interionic distances found in crystals but care should be taken in using empirical potentials if the problem involves the ions moving to distances that are much smaller or larger.

Fig. 1 Calculated and empirical potentials for the O2–O2 interaction. From J. Harding, Computer Simulation of Defects in Solids, in Reports on Progress in Physics, 1990, 53, 1403. Reproduced with permission of the publisher, IOP Publishing, Bristol.


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However empirical potentials have proved remarkably successful and a large number of studies of oxides have used empirical potentials based on the O–O potential of Lewis and Catlow38 derived from Hartree–Fock calculations on two O2 ions. Parker et al.39 have shown that quantum mechanical studies can be used to screen possible empirical potential sets and suggest improvements. The potentials obtained by this procedure can then be used with some confidence in problems where ab initio calculations would be prohibitively expensive, for example molecular dynamics runs. An interesting development is the use of a neural network to fit empirical potentials from a large amount of data with a genetic algorithm.40 A review by Harding37 contains a discussion of the advantages and disadvantages of empirical and ab initio fitting and although it was published in 1990 I would still recommend it. While for many applications, ab initio methods are now preferred as they are more accurate and there is no longer such a large penalty in computer resource, one area in which interatomic potentials still have an advantage is in studying the effect of isolated defects. Defect calculations with programs using interatomic potentials generally use the Mott-Littleton approach.41 In this the solid is divided into two spherical regions. In the region immediately surrounding the defect, the defect interacts with the surrounding ions explicitly. The interaction of the ions in region 2 with the defect is approximated. Positions of ions in the bulk solid are allowed to relax to accommodate the defect which can be intrinsic or extrinsic. The concentration of intrinsic defects in crystals is often very small and such calculations are well-suited to determining the energetically most favourable intrinsic point defect. For a Schottky defect for example, this method is used to calculate the energies when one cation is removed from the lattice and when one anion is removed separately from the lattice. Subtracting the lattice energy gives the energy of a Schottky defect for widely-separated cations and anions, but it is of course possible that the vacancies are clustered and this is also easily studied using this method. Early examples of such studies include work by Lewis and Catlow42 on intrinsic defects in alkali halides and a study by Mackrodt and Stewart43 of defects in alkaline earth (Group 2) oxides. Because these calculations are relatively cheap, they are also a good way to produce a first guess for the mode of incorporation of extrinsic ions. The defect structure found this way can then be used as a starting point for Rietveld analysis of powder XRD patterns or to predict the pattern of substitution of ions with very similar XRD response e.g. F and O2. Islam44–46 has for example investigated the position of fluorine in a number of oxyhalides and we have used it in our own work to locate dopant ions in iron oxides.47–50 The Mott-Littleton approach is incorporated into several programs including METADISE,51 GULP.8 Details of the implementation of this scheme in the widely-used program GULP are discussed by Gale.8

3. Ab initio methods The most striking advance in solid state modelling in chemistry in the last decade is the rapid increase in the use of plane wave methods, mostly because there are several readily-available programs and graphical interfaces have made them more userfriendly. I shall discuss both the basis and developments of these methods and of the alternative localised function approach, but first I shall mention some concepts that are common to both approaches. Both approaches treat the solid as an infinite three-dimensional array of unit cells. This enables Bloch’s theorem to be applied so that the electronic wave function of the solid can be written Cnk(r) = unk(r)eik r

(3)

where Cnk(r) is the nth wavefunction for wavenumber k. unk(r) has the periodicity of the unit cell. In the plane wave approach, unk(r) is chosen to be a sum of plane 124 | Chem. Modell., 2008, 5, 119–149 This journal is

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waves eiG r where the wavevectors G are chosen to be commensurate with the cell. Thus the electronic wavefunction in this approach becomes X Cnk ðrÞ ¼ unk ðGÞeiGr eikr ð4Þ G

In the localised function approach, unk(r) is chosen to be a linear combination of functions localised on the atoms in the reference cell X X Cnk ðrÞ ¼ ain ðkÞ ji ðr Ai gÞeikg ð5Þ i

g

where ji(r Ai g) is a local function (such as an atomic orbital) centred on the nucleus whose coordinates are given by Ai in the zero reference cell. I shall introduce here a concept that is common to all methods. The Brillouin zone (BZ) is described in most solid state physics textbooks. It is defined as a reciprocal lattice cell bounded by the planes that are perpendicular bisectors of the vectors from the origin to the reciprocal lattice points. Fig. 2 illustrates the first BZ cell for a hexagonal lattice. To evaluate energies and electron densities in ab initio methods, it is necessary to integrate over the Brillouin zone. The integrations needed are performed over a set of discrete points within the BZ. For insulators, it is only necessary to use a few special points thus cutting down on computational time. The most common method for choosing these points, used in many of the packages described below, is the Monkhorst-Pack method.52 The set of points generated by this approach for a threedimensional system is given by 3 X 2ni Ni 1 Gi 2Ni i

where Gi represents the primitive vectors of the reciprocal lattice, ni = 1,2,3, . . . ,Ni and Ni is selected by the user. For many systems, small values of Ni can produce reasonable results. Another reason for introducing the Brillouin zone is that it is common in physics papers to include diagrams of the change in energy as a function of k along a path from one point in the Brillouin zone to another, for example from G to K. Modern diagrams tend to show several paths and a large number of energy levels giving them the appearance of spaghetti. However gaps between the valence

Fig. 2 Brillouin zone for a hexagonal unit cell showing some of the special points. G is the centre point.


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levels and the conduction levels can be distinguished and give the variation of band gap with k, and partially filled bands can be seen crossing the Fermi level. A recent trend has been to use Wannier functions53 to calculate properties. Wannier functions are orthonormal localized functions spanning the same space as the eigenstates of a particular band or group of bands and are Fourier transforms of the Bloch eigenstates. For one band, i, Wannier functions, wi, are given by Z Vcell dkeikTm cki ðrÞ ð6Þ wi ðr Tm Þ ¼ ð2pÞ3 BZ where Tm is a lattice point in the unit cell associated with wi, Vcell is the volume and BZ refers to integration over the Brillouin zone. Highly localised Wannier functions play the role in periodic solids of localised orbitals, such as Boy’s orbitals, in molecular orbital theory. One widely-used method of obtaining maximally-localised Wannier functions is to minimize the mean square spread, O, where O¼

N X

½hr2 ii hri2i :

ð7Þ

i¼1

N is the number of bands considered in forming the Wannier function. 3.1 Plane wave methods Plane wave methods have long been the preferred approach to solid state calculations for physicists and much of the literature in this field is written for physicists. However it is now becoming popular among the chemical community with plane wave methods included for example in the graphics-led solid state modelling package Materials Studios and so I have included brief descriptions of the various approaches here. A clear and comprehensive textbook in this area is that by Martin.54 There are two main approaches to plane wave methods, both using density functional theory (DFT)—pseudopotential methods and linear methods. Pseudopotential methods are implemented in the program CASTEP55 and in the GNU-licenced ABINIT program.56,57 A good introduction to the background theory and philosophy of implementation is given in the paper describing the previous version of CASTEP.58 An impressive and unusual feature of the 2002 version of CASTEP is that the code was completely redesigned. Most large computational chemistry packages have evolved from codes written in the 1970s or 1980s in Fortran77 (or earlier Fortran versions). As they evolved, more efficient algorithms for parts of the calculation were included, extra properties and the ability to optimise structures added and the code modularised. Added functionality is still dealt with today by adding modules to the existing code. The CASTEP programmers however decided to redesign the code from scratch taking advantage of object oriented design to produce a program that would be easier to debug and update. As in all plane wave methods, the set of plane waves eiG r form the basis set for the valence electrons. Note that they do not contain any reference to the particular elements present in the solid. This has both advantages and disadvantages. One advantage is that there is no need to develop basis sets tailored to each element. The basis set is chosen for any system simply by choosing the number of plane waves to be used. In practice, this number is determined by selecting a cut-off energy EC where EC is the energy of a free electron whose wavefunction has the same energy as the largest wave vector in the basis set, EC ¼

h2 ðGC þ kÞ2 : 2m


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ð8Þ

Sufficient plane waves have to be chosen to cover the whole unit cell including empty volumes. A disadvantage stemming from this is that many plane waves are needed to fill the empty volume for structures with cages, tunnels or irregular shaped ions or molecules, making calculations on such structures expensive. Core electrons are represented by pseudopotentials. These are determined by applying the restraints that (1) the eigenvalues of the valence orbitals in the presence of the pseudopotential must be the same as those obtained in the all-electron case. (2) the continuity of the wavefunctions and their first derivatives must be continuous across the core boundary. In norm-conserving pseudopotentials a third restraint—that integrating the charge in the core region must give the same answer as for the all-electron case— is applied. This ensures that scattering properties remain correct to linear order.59 Pseudopotentials may be derived separately for wavefunctions of different angular momentum quantum number, l. Such sets are known as non-local. In recent years ultrasoft potentials, introduced by Vanderbilt,60,61 have become popular. These are generated by relaxing the norm-conservation restraint and to compensate for the loss of the guarantee that the scattering properties remain correct to linear order, several reference energies are used with a set of projectors for each energy. Such potentials were introduced because of the difficulty of representing transition metals and second row (Li–Ne) elements by norm-conserving pseudopotentials for a plane wave basis. Computational advances which have led to pseudopotential methods becoming efficient are Fast Fourier Transfoms and the approach of Car and Parrinello62 in which wavefunction coefficients are treated as dynamical variables. Parts of the calculations on periodic systems are easier to calculate in real space and others in reciprocal space. Fourier transforms are needed to convert between the two spaces. The introduction of Fast Fourier Transforms substantially reduced the computational costs of this conversion. The key feature of the Car-Parrinello approach is the use of minimisation rather than diagonalisation of the Kohn–Sham equations to reach the ground state. Linear methods originated in the augmented plane wave (APW) method of Slater63 and were developed initially by Andersen.64 Like the pseudopotential method, the linear augmented plane wave (LAPW) approach uses density functional theory, but the ionic cores are not represented by pseudopotentials. Instead each core is modelled by a sphere inside which the wave function has the form of a linear combination of radial functions times spherical harmonics. X jkn ¼ ðAlm ðkn Þul ðr; El Þ þ Blm ðkn Þ½dul ðr; El Þ=dEEl ÞYlm ðrÞ ð9Þ lm

where Alm and Blm are obtained by matching basis functions of the same kn inside and outside the atomic spheres. ul(r,El) is the value at the origin of the solution of the radial Schro¨dinger equation for energy El and the spherical part of the potential inside the atomic sphere. In the full potential linear augmented plane wave method (FLAPW)65 the electron density inside the sphere is replaced by a smoothed density with the exact multipole moments. PAW (projector augmented-wave)66 is a method designed to have the flexibility of FLAPW methods with the simplicity of pseudopotential methods. In the PAW method, all-electron one electron wave functions, Cn, are derived from pseudo-one electron wave ~ n, where n refers to the nth band by means of a linear transformation functions, C X ~ ih~ ~ ni þ ~ jCn i ¼ jC ðjfi i jf ð10Þ i pi jjCn iÞ i

The all-electron partial waves, fi, are chosen to be solutions of the scalar relativistic Schro¨dinger equation for a non-spin-polarised atom with a specific angular Chem. Modell., 2008, 5, 119–149 | 127 This journal is

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~ n are the variational momentum and at a specific energy in the valence region. C quantities and are usually expanded in plane waves. Although described as an allelectron method, the core wavefunctions are usually frozen. A recent paper67 suggests a method of relaxing the frozen core restraint without increasing substantially the computational time. This solves for a self-consistent core charge density while re-establishing the PAW potential parameters at each minimisation step in such a way that the plane wave valence functions remain orthogonal to the core states. This obviates the need to increase the number of electrons in the plane wave part of the Hamiltonian. Mortensen et al.68 have developed a grid-based implementation of PAW. For DFT methods, it is necessary to calculate the electron density and because this is p|C2|, in plane wave methods it involves N2G operations where NG is the number of G vectors used. This can be reduced by obtaining the Bloch states on a grid of NB points in real space when the density can then be found in NB operations. Usually the wave function c(G) is converted via a Fast Fourier Transform to a real space wave function. This is then squared and the density summed over all electrons and all wave vectors, k. A second Fast Fourier Transform converts the electron density back to reciprocal space. Mortenson realised that if the wavefunction is described by a grid in real space then the Fast Fourier Transform can be avoided. By carrying out all calculations in real space and using grids Mortensen et al. were able to put forward a method that could make use of massively-parallel computers and reduce the time for calculations on large systems.

3.2 Methods using localised functions The augmented spherical wave method (ASW) describes the valence electrons by a set of spherical waves centred round ionic cores described by atomic orbital-like functions.69 The linear combination of crystalline orbitals method, as implemented in CRYSTAL06,70 for example, is based on the use of atomic orbitals. In contrast to the methods described in the previous section, both Hartree–Fock (HF) and DFT calculations are possible. The Gaussian basis sets are similar to molecular basis sets but there are crucial differences, one of the most important being that diffuse functions should generally be avoided. Such functions are known to improve molecular calculations especially of negative ions but in a periodic solid they are very time-consuming and can lead to unphysical states.70 The Hartree–Fock (HF) method is not much used these days but HF calculations often converge more easily than DFT calculations and give the correct energy order of spin states so that they can be used for a cheaper preliminary survey of spin states before using a more accurate method to investigate band structure or density of states. BAND71–73 from the Amsterdam Density Functional (ADF) group uses atomic orbital basis sets but these are composed of either Slater or numerical functions rather than Gaussians. The program includes relativistic effects at the zero order relativistic approximation (ZORA) level and it is possible to include spin–orbit coupling. LSDA and GGA functionals are available. This method ought to deal well with strongly-correlated systems but there does not seem to be much work in this area using BAND, possibly because earlier versions lacked the functionality of other solid state programs. The updated BAND2007 has more functionality, for example geometry optimisation, but is too recent for papers to have been published using the results of this program. SIESTA74 is an order-N method that uses norm-conserving pseudopotentials (see above) with strictly localised basis sets. The basis orbitals are products of a numerical radial function and a spherical harmonic and are zero beyond a certain radius from the atomic centre. The radial function is defined by a cubic spline interpolation. The cut-off radius can vary for different radial functions. A more recent reference75 gives a fuller description of the program and some applications. 128 | Chem. Modell., 2008, 5, 119–149 This journal is

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3.3 Density functionals and strongly-correlated solids As in molecular calculations, DFT solid state calculations can use both local density approximation (LDA) and generalised gradient approximation (GGA) density functionals. However in general these methods underestimate the band gap and often do not describe solids where there is strong correlation very well. Familiar examples of this are the monoxides MnO and NiO whose properties are more easily explained by assuming the 3d electrons are localised on the transition metal centres rather than occupying delocalised bands. This was attributed by Liechstenstein et al.76 to the existence of orbital polarisation driven by screened on-site Coulomb interactions (repulsion between d electrons on the same ion) resulting in the survival of atomic characteristics of the electronic states in the region of the transition metal ions. A 2003 review by Matar77 discusses strongly correlated oxides with particular reference to the ASW method. There has been much recent work on ways to overcome this problem. A development in pseudopotential programs has been the inclusion of non-local exchange–correlation functionals. Examples of such functionals are the screened exchange (sX),78,79 exact exchange (EEX or EXX),80–82 weight density approximation (WDA),83–85 metaGGA86–88 and Hartree–Fock.89 A good discussion of these functionals is given in the thesis of M. C. Gibson.90 The method that has been widely adopted to solve this problem for linear methods is the addition of a term describing the Coulomb interaction between d electrons on the same ion.91–94 Such calculations are designated +U. U is a spherically averaged Hubbard parameter describing the energy required for adding an extra d electron to an atom. Filled d orbitals localised on one particular site are moved to lower energies by (U J)/2 and empty orbitals raised by (U J)/2 where J is a parameter describing the screened exchange interaction. The +U method can be used with both LDA and GGA calculations. The optimum value of U varies from system to system and is basis set dependent so that each researcher must determine the best value for their problem. Rather than looking for a universal value of U, it is possible to take the view that the value of U needed provides information on the degree of localisation in the solid. Mosey and Carter95 suggest calculating U and J from HF orbitals and show that this technique gives values that describe Cr2O3 better than conventional DFT. An alternative to GGA (or LDA) +U is the use of hybrid density functionals such as B3LYP.96–98 These seem better at describing band gaps and other properties of highly correlated solids than pure DFT. Franchini et al.99 obtained very good results for the structural, electronic and magnetic properties of MnO using the hybrid functional PBE0100,101 in the PAW-based program VASP.102 PBE0 is derived from the Perdew, Burke and Ernzerhof (GGA) functional103,104 (PBE) which is composed of the Perdew-Wang correlation functional and an exchange functional given by ExPBE ¼

bx2 1 þ ax2

ð11Þ

where a and b are fixed and x ¼ rrr 4=3 with r the electron density. The 0 family of functionals have no adjustable parameters in the expression for the mixing of functionals to form the hybrid functional, that is GGA E0XC = EGGA + 14(EHF ) XC X EX

(12)

This functional which is effectively the PBE functional with 25% of exact Hartree– Fock exchange included, performed better than GGA + U. Alkauskas and Pasquarello105 found that PBE0 increased the calculated band gap of a-quartz to 8.3 eV close to the experimental value of B9 eV and a great improvement on the value of 5.8 eV obtained with PBE. Blaha’s group106 showed that the hybrid functionals PBE0 and B3PW91 gave comparable or superior results to LDA + U for the transition metal monoxides MnO, FeO and CoO. Chem. Modell., 2008, 5, 119–149 | 129 This journal is

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It would be interesting to see how PBE0 compares with GGA + U for other systems and with B3LYP. Hybrid functionals have also proved useful in localised orbital approaches. The B3LYP functional used in CRYSTAL0670 can be written as LSDA + x(EHF ) + yDEGGA Eex = ELSDA ex ex Eex ex

(13)

with x = 0.2 and y = 0.9. Feng and Harrison107 have suggested that a value of x = 0.35 is better for reproducing magnetic coupling constants and work by Catlow’s group5,108 found that x = 0.6 was needed to reproduce the structural and polarisation properties of BaTiO3. In our own work109 we found x = 0.1 reproduced the band gaps of the strongly-correlated oxides a-Fe2O3 and a-Cr2O3. However Montanari et al.110 found that the standard B3LYP functional produced results very close to experiment for the vibrational frequencies of a-Al2O3 and Wilson et al.111 considered this functional good for describing the structure, magnetic moment and band gap of ilmenite (FeTiO3). Hybrid functionals other than B3LYP have also proved successful in localised orbital methods. Yang and Dolg112 found that the hybrid functional B3PW including spin–orbit coupling reproduced the band gap of BiB3O6 well, while Prodan et al.113 found that the HSE (Heyd, Scuseria and Emzerhof)114 screened coulomb hybrid potential described the oxides of uranium and plutonium well. The performance of hybrid functionals is discussed by Cora et al.115 Just as there is no universal value of U for the +U methods, there is not yet an optimised hybrid functional for all solid state properties. In general, though, the use of hybrid functionals does seem to be a way forward for calculating solid state properties of correlated solids. A third approach to the problem of strongly correlated solids is Novak’s Orbital Polarisation Method.116 Finally a method which shows promise for the future is dynamic mean field theory. Dynamical mean field theory uses an approximation to the local spectral density functional (rather than energy density functional) and a set of correlated local orbitals. For solids this local description is combined with a periodic description such as DFT using LDA to provide a method of dealing with both localised and delocalised electrons.117–119 Anisimov et al.117 applied this method to the photoemission spectrum of La1xSrxTiO3. Kotliar120 has recently reviewed the use of dynamical mean-field theory for strongly correlated systems and shown that it models a variety of systems well including metallic Pu, NiO, manganates and fullerenes. Kuchinskii et al.121 have extended DMFT for correlated systems by including a self energy term S(k). This method shows promise for describing cuprate superconductors.

4. QM/MM Combined quantum mechanical/molecular mechanics calculations have become popular over the last decade or so in solid state modelling. Typically a molecule attaching to a surface or the side of a cage structure or an isolated defect is treated quantum mechanically as a cluster embedded in a bulk solid described by an interatomic potential method. One of the earliest examples of this approach was Vail’s work on F-centres which began in 1983122,123 and which treated a cluster around the F-centre quantum-mechanically and the rest of the solid via an interatomic potential method. The quantum mechanical and molecular mechanics methods used are those we have discussed earlier; molecular mechanics in the case of ionic solids and zeolites generally meaning the use of interatomic potential methods. The art of QM/MM is how to deal with the boundary of the two regions. One method used for ionic materials is to have a boundary region which features in both the quantum mechanical and molecular mechanics calculations. The addition boundary scheme 130 | Chem. Modell., 2008, 5, 119–149 This journal is

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which has been adapted widely for ionic materials, gives the total energy as a sum of three terms inner,boundary Etot + Eouter,boundary + Einner,outer,boundary QM/MM = EQM MM QM/MM

(14)

where the first term represents the energy due to interacting quantum particles, the second that due to particles interacting through molecular mechanics and the third the interaction of quantum particles with particles described by molecular mechanics. The third term generally considers the effect of the charge and sometimes the polarisability of one set of particles on the other. It is important to match the charges derived from quantum mechanics with those derived from molecular mechanics. Since programs such as GULP generally use formal charges, this means they work best with highly ionic materials. An alternative to the addition method is the subtraction method. In this the entire system is treated using molecular mechanics and then quantum mechanics calculations and molecular mechanics calculations performed on the inner region or cluster. Usually there is no boundary region but link atoms are added to the inner region or cluster to make a molecule-like entity. The energy is then given by inner,outer Etot + Einner,link Einner,link QM/MM = EMM QM MM

(15)

Link schemes are often used in modelling biological materials such as proteins but have been used for zeolites by, for example, Sauer’s group.124–126 Zeolites are suited to the link method because of their covalent framework. Problems such as reactions of molecules on a catalytic surface are better handled using boundary methods. For all these methods, there are a number of ways of treating the QM/MM interaction. The least sophisticated is to simply regard the QM region or cluster as a free ‘molecule’. Calculations on the MM region are then modified by including point charges derived from QM. More realistic is to solve for the QM region in the presence of a set of point charges representing the MM region. These charges can be partial charges used in the MM scheme employed but better results may require rederivation of the charges. Sauer’s group, for example, in work on zeolites127,128 rederived the charges from a standard force field by fitting to electrostatic potentials. Other groups have used charge equilibration schemes.129,130 Beyond this there are calculations that allow for the polarisation of the MM area due to the QM charge distribution. Finally the QM and MM calculations can be made self-consistent either through iterative solution of both calculations or by matrix inversion (direct reaction field method131,132). Sherwood133 has written a useful guide to QM/MM methods for both solid state and biological systems. The papers134,135 describing the ChemShell program also contain a useful brief description of the QM/MM approach.

5. Molecular dynamics and related methods Static minimization methods, whether ab initio or interatomic potential, strictly describe solids at 0 K. However it is often important to model phase changes and the behaviour of solids under high temperatures and pressures especially in geochemistry. Such modelling requires methods that sample more than one configuration and/or are time-dependent. The least expensive approach is quasiharmonic lattice dynamics. The quasiharmonic approximation136 assumes that the free energy of a crystal can be written as the sum of a static contribution and a vibrational contribution. For periodic structures, this vibrational contribution is given by X 1 ð16Þ Fvib ¼ 2hn j ðkÞ þ kB T lnð1 expðhn j ðkÞ=kB TÞÞ k;j

where nj(k) is the frequency of a vibrational mode with wavevector, k. Chem. Modell., 2008, 5, 119–149 | 131 This journal is

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The equilibrium structure is obtained by minimising the total free energy with respect to a set of internal strain variables. The static potential energy and vibrational frequencies can be calculated using either interatomic potential or ab initio methods. Despite the increasing popularity of molecular dynamics methods, lattice dynamics is still used to investigate solids at high temperatures or undergoing phase transitions. For example, studies in the period 2005-7 included the phase transition of MgSiO3 from perovskite,137 thermal expansion of CaMoO4138 and the origin of negative thermal expansion.139 Monte Carlo methods generate a large number of structures and calculate properties over a weighted average of these. Starting from a reasonable structure, atoms are displaced/removed/added and the energy of the new structure calculated. The most common method for choosing which structures to include in the calculation of properties is Metropolis sampling.140 In this the probability that a structure is selected or rejected depends on a Boltzmann-weighted probability, exp(E/kBT), where E is the energy relative to the initial configuration. Selection of a change from one structure to the next that is neither too small nor too large is crucial to efficient calculation. Molecular dynamics, which is used extensively in modelling biological systems, solves Newton’s equations of motion with the system subject to forces derived from the potentials between atoms. The system is followed as a function of time and as well as time-averaged properties, time-dependent properties such as diffusion can be studied. The coordinates and energy of the system are followed by solving the equations of motion numerically using discrete time steps. The time step is typically of the order of femtoseconds. Using a suitable algorithm such as that due to Verlet,141 simulations remain stable for long runs; these days typically ten or so nanoseconds. The calculations can be run for different temperatures and under constant pressure or constant volume restraints. It is thus possible to obtain free energies. Details of Molecular Dynamics can be found in the classic text book of Allen and Tildesley.142 Molecular dynamics calculations are computationally expensive and are usually performed using interatomic potential methods rather than ab initio methods. For example Martin et al.143 undertook studies of inorganic materials, particularly surfaces, using the DL_POLY144 code. One of their examples is a very nice study of the migration of oxide ions in ceria both in the bulk and in the 111 surface. By setting an oxide ion off towards a vacancy at constant velocity, they obtained free energies of activation and pathways for several mechanisms. The activation energies were considerably modified by the presence of a Ce3+ ion adjacent to the vacancy thus underlining the importance of considering defect clusters as well as isolated defects in these calculations. A study by Kerisit and Rosso145 illustrates how the solid state modeller is not confined to one of the available techniques but can choose different techniques to tackle different aspects of a problem. They used a combination of molecular dynamics with interatomic potentials, periodic Hartree–Fock calculations and ab initio cluster calculations to study charge transfer in FeO. They used molecular dynamics to calculate the reorganisation energy and free energy of reaction and to provide transition state configurations. Ab initio methods were used to calculate electronic coupling. Using molecular dynamics to study reaction rates involves choosing a suitable reaction coordinate. This is not always easy and Dellago et al.146–149 proposed a method based on Monte Carlo sampling of the Molecular Dynamics pathway. van Erp et al.150–152 have more recently introduced an algorithm that makes transition path sampling more efficient, the Transition Interface Sampling technique. Mohn and Stølen153 have suggested a route using a genetic algorithm154 and energy minimizations for dealing with highly disordered systems and show that it leads to accurate calculations of thermodynamic properties when only a small fraction of energy minima are thermally accessible. The hybrid Monte Carlo method of Allan155 combined Monte Carlo and molecular dynamics steps within the same simulation. Allan et al.156 have discussed 132 | Chem. Modell., 2008, 5, 119–149 This journal is

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the relevant merits and limitations of lattice dynamics, Monte Carlo and molecular dynamics, illustrated by examples.

6. Properties Calculated properties of solids include structural, optical, electronic, magnetic and spectroscopic properties. An important aspect of solids is the bulk magnetisation. Many solid transition metal compounds can be considered as having localised net spins on the metal centre. These spins couple ferro- or antiferromagnetically with spins on adjacent metal centres. Spin-polarised ab initio calculations permit the setting up of ferro-, ferri- or antiferromagnetic states by setting the sign of the spin on magnetic centres in the unit cell. In cases where the crystallographic unit cell contains only one atom of a particular type, it becomes necessary to use supercells. The classic example of this is NiO where in order to reproduce the observed ground state antiferromagnetism it is necessary to use the magnetic supercell found from neutron diffraction studies. Electronic structure calculations strictly deal with the ground state at 0 K so that such calculations generally give an ordered magnetic state even though the solid may be above its Curie or Neel temperature at room temperature and hence paramagnetic. The energy difference between ferro- and antiferro-magnetic states can however give information on the temperature at which the onset of paramagnetism occurs. In contrast to this picture of isolated spins interacting via spin–spin coupling (Ising model), is the magnetism of metals in which the magnetic susceptibility arises from alignment of spins in delocalised bands. Magnetic moments can be extracted from population analysis. Hartree–Fock magnetic moments are often closer to experimental values than DFT values are. The nature of the electronic conductivity (insulator, semiconductor, metallic conductor) and band gap values are obtained from the band structure of the solid. For example, a recent paper by Erhart et al.157 used DFT +U calculations to investigate the band gap of indium oxide. Optical measurements suggested an indirect band gap around 1 eV less than the direct band gap at G, however they concluded that this observation could not be explained on the basis of the band structure of the defect free solid. More difficult to calculate are the properties which depend on the response of the solid to an outside influence (stress, electric field, magnetic field, radiation). Elastic constants are obtained by considering the response of the crystal to deformation. Interatomic potential methods often provide good values for these and indeed experimental elastic constants are often used in fitting the potential parameters. Force constants for lattice vibrations (phonons) can be calculated from the energy as a function of atomic coordinates. In the frozen phonon approach, the energy is obtained explicitly as a function of the atom coordinates. Alternatively the deriva2E tive, @R@i @R can be calculated at the equilibrium geometry. j Properties depending on the response to electric and/or magnetic fields such as optical properties and NMR shielding constants have only been implemented relatively recently. A way to calculate electrical polarization in periodic solids, for example, was only found in the 1990s. Spontaneous polarisation, that is polarisation in the absence of an electric field, has been calculated using both a Wannier function approach and a Berry’s phase158 approach. Berry’s phase involves an adiabatic change around a closed loop which results in a change of phase without change in energy. A recent paper by Ferretti et al.159 used the PAW method with ultrasoft pseudopotentials and Wannier functions to calculate the spontaneous polarisation of AlN in its wurtzite phase. Gajdosˇ et al.160 have put forward a method of calculating dielectric tensors using density functional perturbation theory which they extended for the PAW method. Calculations of NMR shielding is difficult for molecules and more so for periodic solids. With the advent of modern solid state NMR techniques such as MAS (magic angle spinning), many experimental data have become available and computational Chem. Modell., 2008, 5, 119–149 | 133 This journal is

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chemists naturally wanted to complement these results. Until a decade ago, however, the only available approach to calculating NMR shielding in solids was to use clusters. The results of course depended strongly on the cluster chosen. In 1996, Mauri et al.161 put forward a method for calculating shielding in periodic solids. Their innovation was to modulate the uniform external field with a finite wave vector, q, giving B(r) - B(r)eiq r and then take the limit of the perturbation energy as q - 0. This group went on to use PAW in conjunction with an approach similar to the gauge invariant (or including) atomic orbital (GIAO) approach used for molecules to calculate NMR chemical shifts in solids.162,163 This approach is now available in NMRCASTEP part of the Materials Studios Package. It should be noted that the approach in NMRCASTEP uses pseudopotentials and therefore does not calculate the contribution to the shielding from core electrons. Sebastiani and Parrinello164 proposed a different approach. They used the R = r variant of the CSGT (continuous set of gauge transformations) method and locally maximised Wannier orbitals. For insulators, the Wannier function decays exponentially and it is therefore possible to choose virtual unit cells whose walls lie in regions where the density for a particular orbital is close to zero. Sebastiani and Parrinello define the position operator as running from l/2 to +l/2 across the virtual cell, then dropping back to –l/2 at the cell wall. Their method is claimed to be more efficient than that of Mauri for systems with large unit cells but again uses pseudopotentials. It has been implemented in the CPMD (Car-Parrinello Molecular Dynamics) package.165 Mauri’s group have published a number of papers on calculated 17O NMR parameters.166–168 Ref. 167 is particularly interesting as it considers vibrational effects on the NMR shielding by averaging the chemical shift over fluctuations of the nuclear positions. They show that vibrational corrections are crucial to reproducing the temperature dependence of the chemical shift. Truflandier et al.169 have published the first periodic calculations of NMR shielding for a transition metal nucleus, albeit on a d0 system (VO43). A number of studies of optical properties of chalcogenides have appeared recently. Weng et al.170 studied magneto-optical Kerr effects in transition metal chalcogenides, Singh et al.171 investigated optical and magneto-optical properties of europium chalcogenides, Reshak et al.172 studied linear and non-linear optical properties of zinc chalcogenides. These studies employed FLAPW calculations and were motivated by the use or potential use of these semiconductors in electronic devices such as switches, light beam addressable memory systems or high density storage. Building on the use of Wannier functions to calculate properties, a very recent paper173 uses both electron Wannier functions and phonon Wannier functions (that is localised orbitals and vibrations) to calculate electron–phonon interaction, a property thought to be important in high TC superconductors.

7. Applications 7.1 Zeolites, ALPOs and other microporous/nanoporous materials Zeolites are now defined as solids that possess a framework of tetrahedra which are all corner-sharing and include a degree of openness such as channels or cavities. Each framework type is issued a unique three letter code by the Structure Commission of the International Zeolite Association.174 At the end of February 2007, there were 176 framework types. This definition does not specify atom types. The original zeolites were alumino-silicate minerals in which the frame work was composed of SiO2 with some Si replaced by Al and the charge balance restored by the presence of other cations within the channels or cavities of the structures. These 134 | Chem. Modell., 2008, 5, 119–149 This journal is

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proved an ideal problem for interatomic potential methods but difficult for ab initio methods. The unit cell is often large thus making calculations computationally expensive and random substitution of Si by Al is easy to represent by interatomic potential methods but difficult for ab initio methods. There have thus been many interatomic potential studies on these solids, particularly by Catlow and his group, and there have been several reviews and books175,176 on this subject. An extensive discussion of the work of Catlow’s group in modelling and predicting zeolite structures is given in a recent Annual Reports of Progress in Chemistry.5 Nonethelesss there have been ab initio studies of these solids and these are likely to increase in future due to the greater computing power available. Early work by the CRYSTAL group employed Hartree–Fock calculations to study zeolites, for example silico-chabazite.177 There have also been a number of DFT studies. Corminboeuf178 has discussed the application of DFT to zeolites and concluded that the way forward was ab initio molecular dynamics. Nachtigall179 reviewed the application of quantum mechanical methods to zeolites. Poulet et al.180 discussed the relative merits of linear (VASP) and localised orbital (SIESTA) methods as applied to the related aluminium phosphate structures (ALPOs). QM/MM methods are useful for considering site occupancy of atoms substituting for Si and positions of molecules and counterions in the cages of these materials. As noted in section 4, QM/MM has been used extensively for zeolites by Sauer’s group. Modelling of zeolites and other microporous materials is mainly focussed on three aspects—structural studies, studies of the interaction of molecules or ions with the materials (particularly in relation to their use as catalysts) and templating. Structural studies can include topological studies, crystallisation and the position of substituted atoms in the framework. One basic question is the number of possible zeolitic frameworks and topological techniques have been used to tackle this, the zeolite framework being viewed as a 4-connected net. A recent paper by O’Keeffe et al.181 provides a clear discussion of the use of tiling in designing materials. Catlow5 in his Annual Report gives references to reviews of earlier work and summarises work in this field from 1990–2005. Foster et al.182 identified possible binodal (two types of tetrahedral sites) structures using tiling theory. These mathematically possible structures were then investigated using interatomic potential methods resulting in the identification of 98 previously unknown toplogies. Some of these were not feasible zeolite structures but a number of very open structures were noted which the authors suggest may be feasible structures for elements other than those normally found in zeolites. An example of these is the D3R (double 3 ring) family containing trigonal prisms (Fig. 3). Double 4 rings (D4R) have also attracted attention. Sastre183 has suggested, based on interatomic potential calculations with the Vessal-Leslie-Catlow184 force field, that D4Rs in some zeolite frameworks are less strained than previously thought. Kamakoti and Barckholtz185 used LDA DFT calculations to study the role of germanium in stabilising D4Rs. Auerbach186 has published a useful, short review of the insights gained from molecular modelling of zeolite formation. Mora-Fonz187 has investigated the role of solvation and pH in the nucleation of silicaceous zeolites. An article by Agger et al.188 in an issue of Journal of Crystal Growth dedicated to new advances in crystal growth and nucleation, uses interatomic potential methods to study the crystal growth of analcime by considering low energy hydrated surfaces. van Erp et al.189 propose the use of transition interface sampling (see section 5) for theoretical studies of the synthesis of zeolites as the efficiency of this method is less sensitive to the reaction coordinates chosen than conventional free energy techniques. This could be an advantage in the study of zeolite synthesis because the reaction pathway is usually difficult to find. Modellers often use silica zeolites but most zeolites contain other cations as well as silicon in the framework. Recent work by Gale190 uses ab initio methods and molecular dynamics to study the location of titanium within TS-1. To et al.191 Chem. Modell., 2008, 5, 119–149 | 135 This journal is

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Fig. 3 The most feasible structure of the D3R family [adapted from ref. 182. Reproduced with permission of the International Union of Crystallography (http://journals.iucr.org/)].

employed QM/MM methods to investigate the active site in this zeolite. They concluded that a tetrapodal site (Ti bonded to four O–Si bonds) would be the substitutional site under anhydrous conditions, but that on hydration tripodal structures would prevail and act as the active site for catalysis. Bulanek et al.192 used DFT to investigate the position of Cu+ in Cu-FER. Because zeolites are much used as catalysts, there have been many studies on the interaction of molecules with the framework. Recent studies (2005–2007) include the effect of incorporation of impurity anions into the zeolite DSP,193 adsorption of NO in iron-exchanged ferrierite,194 adsorption of CO in Li+-FER,195 adsorption of CO on Cu+ in Cu-FER,192 adsorption of methane in zeolites A and Y,196 methane permeation through zeolite membranes,197 reactions of 1,3-dimethylbenzene in zeolites198 and the separation of CO2 and CH4 in silicalite.199 Thompson et al.200 used Monte Carlo methods to study the absorption of HCN and methyl ethyl ketone in silicalite, mordenite and zeolite beta. Their motivation was the desire to explore the use of zeolites in removing methyl ethyl ketone (an irritant found in glues and varnishes) from industrial gases. The absorption was found to depend heavily on pore size and cation loading. In an extension of the QM/MM approach, Tuma and Sauer201 treat the protonation of isobutene in ferrierite by using MP2 calculations on the reaction site and hybrid DFT for the zeolite. Sholl202 has written a review of molecular dynamics applied to diffusion of molecules in nanoporous materials including zeolites and carbon nanotubes. Bougeard and Smirnov203 have reviewed the important, but difficult to model, area of water in nanoporous materials. They discuss both quantum mechanical and molecular dynamics studies of water in clays and zeolites. Ruiz-Salvador et al.204 used interatomic potential methods to study the effect of water, framework atoms and extra-framework cations on the structure of the zeolite Goosecreekite (GOO). A modification of the potential for water developed by de Leeuw205,206 was found to be suitable. This zeolite (GOO) contains an ordered arrangement of Al3+ on the framework sites and Ca2+ in the cavities. All water in the zeolite is coordinated to Ca2+ making modelling more tractable. Their model confirmed that the extra-framework cations were the structural driving force for collapse on dehydration. Organic molecules are traditionally used as structure-directing agents for zeolites and work in this area continues. Huang et al.207 studied organic cations in a 136 | Chem. Modell., 2008, 5, 119–149 This journal is

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borophosphate network. Gomez-Hortiguela et al.208 studied fluorinated dibenzyldimethylammonium as a structure-directing agent for ALPOs. Sastre209 has used interatomic potential calculations to compare the roles of framework atom substitution, extra-framework anions and organic molecules as structure directing agents. 7.2 Metal oxides Simple metal oxides were the subject of much early work and provide test systems for new methods. There is still work to be done though as evidenced by recent papers on simple solids such as MgO,5,210,211 and ZnO.210,212–215 MgO remains popular as the test crystal for new methods; ZnO has received renewed attention due to increasing exploitation of its semiconducting properties in applications. Transition metal oxides remain problematic. These oxides are strongly correlated and as explained above these systems are not well described by pure DFT calculations; indeed in some cases such calculations predict a metallic state when the true ground state is an insulator. As an example of an apparently simple oxide that is difficult to model, Table 1 gives some results of ab initio calculations on the oxide a-Fe2O3. Both plane wave and LCCO methods can now reproduce the band gap and nature of the valence and conduction bands but only if recent improvements are used. Neither have yet managed to correctly predict both the band gap and the equilibrium structure. A further point to note is that the calculations in Table 1 refer to a-Fe2O3 at low temperatures. At the Morin transition the alignment of the spins with respect to the crystal axes changes. This effect can only be modelled by including spin–orbit coupling. Table 1 Calculated band gaps for a-Fe2O3 Method

a/A˚

Experimental

5.035 13.747 4.9 B2 5.4241 (55117 0 ) 5.112 13.820 4.74 11

HFa LSDA/fully relativistic ASWb LSDAc LSDA + Uc U = 2 eV LSDA + Ud U = 5 eV GGA (PAW)e GGA + Ue (PAW) U = 4 eV B3LYPf B3LYPf 10%

c/A˚ (a)

m/mB Eg/eV Nature of valence band

3.72 0.75 5.0347 13.7473 3.43 0.51 (fixed) (fixed) 3.84 1.42 5.345 (55113 0 ) 4.1

1.88

5.007 13.829

3.45 B0.3

5.067 13.882

4.11 B2

5.057 13.883

4.21 3.31

5.055 13.939

4.04 1.93

Fe3d/O2p upper edge dominated by O2p states Weak Fe3d/O2p mixing. Upper edge dominated by O2p Strong mixing of Fe3d/O2p. Fe3d concentrated at top of band Strong mixing of Fe3d/O2p. Increase in O2p states close to band edge. Mixing of Fe3d/O2p. Fe3d shifted strongly to bottom of valence band. Strong mixing of Fe3d/O2p. Fe3d and O2p at upper edge in comparable amounts. Weaker Fe3d/O2p mixing. O2p states at band edge. Fe3d/O2p mixing. Fe3d and O2p at band edge O2p dominant Fe3d/O2p mixing. Fe3d and O2p at band edge O2p dominant

Rhombohedral cell. a M. Catti, G. Valerio and R. Dovesi, Phys. Rev. B: Condens. Matter, 1995, 51, 7441. b L. M. Sandratskii, M. Uhl and J. Ku¨bler, J. Phys. Condens. Matter, 1996, 8, 983. c M. P. J. Punkkinen, K. Kokko, W. Hergert and I. J. Vaÿrynen, J. Phys. Condens. Matter 1999, 11, 23421. d A. Bandyopadhyay, J. Velev, W. H. Butler, S. K. Sarker and O. Bengone, Phys. Rev. B: Condens. Matter, 2004, 69, 174429. e G. Rollman, A. Rohrbach, P. Entel and J. Hafner, Phys. Rev. B: Condens. Matter, 2004, 69, 165107. f E. A. Moore, Phys. Rev. B: Condens. Matter, 2007, 76, 195107.


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Oxides are important industrially and rather than list all the work on oxides, I shall consider three classes of compounds—perovskites, the layered componds AMO2 where M is a transition metal and A an alkali metal, and solid ionic conductors. Perovskites have a wide range of uses because of their electric (e.g. ferroelectrics such as barium titanate), optical (e.g. lithium niobate) and magnetic properties. Studies of perovskites with interesting magnetic or electrical properties such as superconductivity and colossal magnetoresistance are dealt with in a later subsection. Interatomic potential methods continue to play a useful role in structural studies and there have been a number of studies of defects in perovskites using this approach. Jackson216,217 has studied the ferroelectric LiNbO3 (used, for example, in wave guides where carefully-controlled doping is crucial) , Thomas218,219 has studied defects in SrTiO3 (used in capacitors) , Diniz and Paschoal220 have studied the compounds Ln(TaTi)O6 where Ln is a lanthanide. Fisher et al.221 used molecular dynamics to study oxide ion diffusion in Ba1xSrxCo1yFeyO2.5 and conclude that the conductivity is stongly dependent on the values of x and y. Other groups have used ab initio methods to study MgTiO3,222 PbTiO3,223 Ba0.5Sr0.5TiO3,224 BaCd1/3Ta2/3O3 and BaZn1/3Ta2/3O3,225 BiAlO3,226,227 BiGaO3,226 CaGeO3,228 KNbO3,229 KTaO3,230 and Bi2FeCrO6, Bi2MnNiO6 and Bi2CuCrO6.231 The layered compounds AMO2 where M is a transition metal and A an alkali metal, Cu or Ag have attracted interest because of their electrical properties. LiCoO2 is used in rechargeable lithium ion batteries and NaxCoO2 is metallic except at x = 0.5 and for values of x of about 0.35 becomes superconducting in the presence of intercalated water. Li et al.232 and Xu and Zeng233 have studied Na0.5CoO2. A FLAPW GGA study of the structure and optical properties of AgFeO2 and CuFeO2 has been carried out by Blaha’s group.234 Ogata235 has reviewed band models of NaxCoO2. This solid has also been the subject of a study by the recently-developed dynamic mean field theory (DMFT).236 This study used VASP with an LDA functional, a value of U = 3 eV for Co and an on-site potential for Na. It was found that the Na potential was the key to explaining the stronger correlation for x = 0.7 than for x = 0.3. Idemoto et al.237,238 have used FLAPW to obtain calculated electron density in the solids LiMn1xMxO2, where M = Mn, Co, Ni, Zn, which they were investigating as cathode material for Li ion batteries. This is primarily an experimental paper and is an example of an increasing trend whereby experimentalists are using modelling to complement their practical results. Park et al.239 have used VASP with a GGA density functional to calculate the structure, density of states and ionic charge of Li(M1/3Mn1/3Ni1/3)O2 with M = Al, Ti, Cr, Fe and Mo. Using LDA + U calculations, Mazin240 has shown that in LiCrO2 the antiferromagnetism arises from Cr–Cr overlap rather than superexchange. Ion migration is important for solids with applications in solid oxide fuel cells, gas sensors and batteries. Interatomic potential methods are particularly useful in this type of work as the migrating ion can be moved along a suggested path and its local environment minimised at each step thus producing an energy profile for the migration. Islam’s group has been particularly productive in this area. An account of much of the group’s work up to 2000 is given in an article in Journal of Materials Chemistry241 and a 2002 paper242 presents work on ion transport in perovskite oxides. Initially this work calculated the potential energy surface for the migration of oxygen vacancies. Molecular dynamics was then used to calculate the rate of diffusion. Finally dopant ions were introduced and the effect of these on the migration calculated. More recent work by this group includes oxide ion migration in apatites,243–247 ionic conduction in Sc2(WO4)3,248 lithium ion transport in LiFePO4249 and oxide ion transport in CaTiO3,250 and in Sr2Fe2O5 and Sr4Fe6O13.251 A number of other groups have recently used modelling to study ion migration in ceria, CeO2. The reaction pathway for migration of oxygen vacancies in ceria has been studied using the PAW method with GGA + U.252 The results confirm the 138 | Chem. Modell., 2008, 5, 119–149 This journal is

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pathway found by interatomic potential methods.253 This ab initio study was used to test the potentials in the molecular dynamics study on CeO2143 described in section 5. Gotte et al.254 used large scale molecular dynamics to study reduced CeO2 at high temperatures and find that only a h100i oxygen vacancy diffusion occurs. Stølen et al.255 used GGA calculations in VASP on doped Ba2In2O5 to support their hypothesis that a high density of low energy local structures was a prerequisite for high ionic conductivity. They predicted that the Sr-doped solid would be a better ionic conductor than the un-doped solid in contrast to the effect of doping with Ga and in agreement with experimental results. 7.3 Solids exhibiting superconductivity or novel magnetism The discovery of the high TC superconductors naturally sparked much interest among solid state modellers. More recently interest has been aroused by the discovery of solids displaying colossal magnetoresistance. Perhaps surprisingly, since these properties are electronic properties, interatomic potential methods have proved of some use. Such modelling can investigate ordering of atoms, charges or vacancies and the ordering may provide clues as to the conditions necessary for solids to possess interesting electronic and magnetic properties. Allan and Mackrodt,256 for example, argued that since the nature and distribution of defects apparently played a crucial role in determining TC, studies of defects might indicate why some cuprates were high temperature superconductors and others not and studied a series of compounds M2CuO4 plus La2NiO4. They concluded that La2CuO4 had a unique combination of lattice and defect properties. Islam257 studied HgBa2Ca2Cu3O8+d and found interstitial oxygen defects to be the most favourable hole-doping mechanism. Two groups have used interatomic potential methods to study ordering in the colossal magnetoresistance material La1xCaxMnO3.258,259 Ab initio methods as well as looking at the structure can of course consider band structure. In the first rush of excitement following the discovery of high TC superconductors, a number of studies on the band structure of cuprates were undertaken. The results of these up to 1989 are reviewed in detail by Pickett.260 These calculations established that the bands that crossed the Fermi level were antibonding Cu 3d /O 2p bands. Most of them used an LDA functional but these compounds are expected to be strongly correlated and there is experimental evidence for local Coulomb repulsion on Cu(II) sites. Larbaoui et al.261 found that using a GGA functional led to considerable improvement for YBa2Cu3O7 but such systems really need methods such as +U or the use of hybrid functionals. Blaha et al.262 showed that the use of LDA +U with a value of 6eV for U on Cu(II) for YBa2Cu3O7 improved the calculated value of the electric field gradient at Cu. Electron–phonon interactions are important in conventional superconductors and there have been suggestions that this is true for high TC superconductors as well. Zhang et al.263 found that the use of LDA + U enhances electron–phonon coupling of a Cu–O stretching mode. However Ohkawa264 in a study of Bi2Sr2CaCu2O8d concludes that electron– phonon interaction plays little or no role in cuprate d wave superconductivity and that Cu–Cu superexchange is responsible. Kuchinskii et al.265 have employed DMFT with an LDA + U functional and an additional self energy term s(k) to study Bi2Sr2CaCu2O8d and find good semiquantitative agreement with angular resolved photoemission (ARPES) results. The discovery of colossal magnetoresistance (CMR) in alkaline-earth-doped rare earth manganates, Ln1xAxMnO3, naturally inspired a rush to model these solids. The earliest explanation was in terms of double exchange, although there are now known examples of solids exhibiting colossal magnetoresistance in which double exchange is not thought to be present. In the undoped solids manganese is present as Mn3+, high spin d4, with three electrons occupying t2g bands and one occupying eg bands. The eg bands are close to the Fermi level. On doping, a fraction of the Chem. Modell., 2008, 5, 119–149 | 139 This journal is

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Fig. 4 Unit cell of double perovskite. Small open spheres M/M 0 , large open spheres Sr, shaded spheres O.

managanese ions are oxidised to Mn4+ leaving these ions without any eg electrons. The eg electrons migrate through the solid and because on-site repulsion causes them to line up with the t2g electrons with Mn3+ couples ferromagnetically to Mn4+. Ramakrishnan266 has reviewed the problems of modelling the manganates and discusses the interactions involved. Structurally these solids are often found to form several phases close in energy and it has been suggested that regions (on the nanometre to micron scale) of different ‘phase’ exist in a single sample. The dopant atoms may also be ordered or disordered. Other problems are that there is evidence of strong on-site correlation and the behaviour of interest is not that at 0 K. However examining the strength of the on-site interaction, double exchange coupling and Jahn–Teller effects is important in understanding these solids. Ramakrishnan himself reports results of DMFT calculations with large values of U. The interatomic potential calculations mentioned above explore cation ordering. Akhtar et al.259 find, using a Monte Carlo method that at low temperatures for x = 0.5 there is ordering of the La3+ and Ca2+ ions in Ln1xCaxMnO3such that along the b axis there are alternate planes of La3+ and Ca2+. However at temperatures typically used to prepare these solids, they find that random occupation is favoured. Tang and Zhang267 also favour a layer ordering. Birsan268 has used a Monte Carlo method to explore the influence of superexchange in CMR manganates. Experimentally there has been much recent interest in solids based on the perovskite SrFeO3. SrFeO3 itself is metallic and shows no Jahn–Teller distortion although it formally contains the d4 ion Fe4+. SrFeO3 is rarely synthesised as stoichiometric; its formula being better described as SrFeO3d. Shein et al. have calculated the magnetic moments and band structure for both stoichiometric SrFeO3269,270 and SrFeO2.875.271 The calculations for the stoichiometric solid using VASP with an LSDA + U functional give a ferromagnetic ground state. Substitution of Fe by other metals gives a series of double perovskites, Sr2MM 0 O6, Fig. 4. The properties of these solids depend on the nature of M and M 0 . Shein’s group has considered the mixed double perovskites ASn1xMxO3272 where A = Ca, Sr, Ba and M = Fe, Mn, Co. They find that Sr2SnMnO6 is a ferromagnetic semiconductor, Sr2SnCoO6 a ferromagnetic metal and Sr2SnFeO6 a ferromagnetic semimetal. However although they use the accurate FLAPW method, the density functional employed is LSDA. A very recent intriguing result from this group273 is the prediction that substitution of B, C or N for O in the perovskites SrMO3 (M = Ti, Zn, Sn) yields a spin-polarised impurity band. FLAPW + lo was used for this study. Stoeffler and Colis274 studied the effect of oxygen vacancies in Sr2FeMoO6, a half-metallic ferromagnet that they suggest may be a good candidate for use in 140 | Chem. Modell., 2008, 5, 119–149 This journal is

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magnetic tunnel junctions. Again they use FLAPW but employ a GGA density functional. It might well be worth seeing if +U, hybrid functional or other methods aimed at strongly-correlated systems give the same results. An interesting approach to spin ordering and spin glasses is the Monte Carlo analysis used by Gibb.275 This simply considers the spin interactions, Js1 s2. Applied to Sr2FeRuO6,276 it shows that the next nearest neighbour interaction, although small, can lead to spin frustration.

8. Minerals Minerals are particularly challenging for modellers as they are often formed under conditions of high temperature and/or pressure and may contain several randomlydistributed cations and water. Nevertheless modelling solids at the temperatures and pressures found in the Earth’s core and mantle and in the core of other planets is possible whereas it is not always possible to reproduce these conditions experimentally. Molecular dynamics is particularly suited to disordered systems and high temperature simulations and the advances in computer hardware offer an opportunity for an expansion in modelling of minerals. However it is sometimes possible to obtain useful insights from static interatomic potential calculations. Parker et al.277 discuss examples of applications of interatomic potentials to minerals using HADES, METADISE and PARADISE. Other studies using interatomic potentials include the diffusion of sodium and potassium ions in nepheline,278 the interaction of heavy metals with dolomite surfaces,279 the dissolution of jarosite [KFe3(SO4)2(OH)6],280 properties of lizardite.281 de Leeuw’s group have conducted a number of simulations on apatites.282–284 An interesting recent set of studies, which show that simulations can be of use to the mining industry are those by Ngoepe285,286 on platinum and palladium sulfide minerals. Vinograd et al.287 use lattice energy calculations to study carbonate solid solutions. Breu has made a number of studies on clays. Natural clays are inhomogeneous and severely disordered but by considering ordered structures he was able to gain insights into the factors determining long range order of pillars.288 This led to the preparation of artificial ordered clays which may prove better in applications than natural clays. Many minerals occur as solid solutions and to understand these, it is necessary to consider the enthalpy and, preferably, free energy of mixing of the components. Allan et al.289,290 describe how Monte Carlo and lattice dynamics methods can be combined to study solid solutions and highly disordered systems and apply this to garnets289,290 and spinels and carbonates.291 Reich and Becker292 combined Monte Carlo and ab initio methods to study the solution of arsenic in pyrites and concluded that for a wide range of geologically relevant temperatures, a mixture of two phases (FeS2 and FeAsS) is more stable than a solid solution. Ab initio methods have been used to calculate vibrational frequencies of kaolinite293 and katoite.294 It is not only minerals in the Earth’s crust that have been studied. There have been a series of studies of planetary interiors. Gillan et al.295 review work on the interiors of Earth, Jupiter, Saturn, Uranus and Neptune. The materials of interest are magnesium silicate in the Earth’s mantle, hydrogen in the interior of Jupiter and Saturn and methane, ammonia and water in the interiors of Uranus and Neptune. Calculations on the post-perovskite phase of (Mg, Fe)SiO3 explained aspects of the behaviour of the D layer, the lowermost part of the Earth’s mantle.296 Most of these studies exploited the increase in computer power to use molecular dynamics or lattice dynamics in conjunction with ab initio calculations. Study of the Earth’s mantle has continued. Carrez et al.297 studied dislocation structures in the D layer starting from generalised stacking faults which they generated using the ab initio package VASP. Blanchard et al.298,299 have derived new interatomic potentials to Chem. Modell., 2008, 5, 119–149 | 141 This journal is

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study magnesium silicate. Jung and Oganov300 used PAW + GGA and fitting of the calculated energy–volume curve to study CaSiO3 at high pressure. The eMinerals project was set up to exploit grid computing in the simulation of minerals. This project links computers in London, Bath and Cambridge enabling workers in the consortium to undertake demanding simulations, using both interatomic potential and ab initio methods. An issue of Molecular Simulation301 contains a number of papers resulting from this project.

9. Conclusions Solid state modelling is currently a flourishing and exciting area. Simple lattice energy-based calculations based on interatomic potentials have been used to model solids for the past three decades and continue to provide useful insights into the structures of materials often of industrial (e.g. ionic conductors, catalysts) or geological importance. A marked trend in recent years is the use of plane wave ab initio methods for solids of chemical interest. The use of these methods by chemical modellers and even bench top chemists has been encouraged by graphics-led packages. Until recently, most calculations were of static solids at 0 K but increasing computer performance now means that molecular dynamics is an increasingly popular tool and it is even feasible to combine this technique with ab initio calculations. This enables modellers to tackle problems such as highly-disordered solids and phase diagrams. The properties that can be modelled have increased in recent years, a notable addition being NMR chemical shifts. The use of localised orbitals (Wannier functions) has helped in this area. Zeolites were one of the earliest types of inorganic solids to be modelled and are still a popular area for modelling with the approach moving from simple interatomic methods to QM/MM and ab initio and the systems studied expanding into related areas such as ALPOs. Strongly-correlated transition metal oxides have been of much interest experimentally due to their novel electronic and magnetic properties. These pose a particular problem to modellers in that it is necessary to allow for localisation of spins on the transition metal ions. Currently the two most popular approaches to this problem are the use of hybrid DFT functionals and the addition of an on-site Coulomb term +U. Minerals, being often disordered and formed under high pressures and temperatures are ideal candidates for molecular dynamics or similar approaches. There has been some exciting work on the minerals found in the Earth’s mantle and this is an area which should expand.

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Density functional theory studies of alloys in heterogeneous catalysis John R. Kitchin,*ab Spencer D. Millera and David S. Shollzab DOI: 10.1039/b608782p

1. Introduction Heterogeneous catalysis is used broadly today in industry. Catalysts crack crude oils into the fuels we use daily. They are used in catalytic converters to minimize dangerous gas emissions from internal combustion engines. Many chemicals are made in large quantities using heterogeneous catalysts. Ethylene oxide is one well known example of a commodity chemical made in this way. Heterogeneous catalysts can be made of many materials, including carbon, metal oxides, transition metals, and a variety of promoters such as alkalis or halogens. The catalytic properties of these materials depend on their detailed morphological structure and in particular on the properties of their surfaces. These properties may additionally depend on the reaction environment. These complexities can make it difficult to understand how changes in catalyst composition and structure are related to the catalyst activity or selectivity. This situation is exacerbated by limitations in our ability to characterize catalysts at atomic scales under reaction conditions. Computational studies of heterogeneous catalysts can often provide additional insight into them, because the detailed atomic structure in these simulations is completely specified. Decades of heterogeneous catalysis research have found empirically that combinations of transition metals are often better catalysts than the pure metal components by themselves.1 It remains challenging in many cases, however, to understand why these multi-metallic materials are better catalysts than the pure components. We focus in this review on combinations of transition metals where mixing of the components has occurred on atomic length-scales (as opposed to the conceptually simpler situation where the two components co-exist in pure but separated forms). We will refer to these combinations generically as alloys, although below we define a more precise nomenclature for the range of possible materials that exist. The composition of an alloy surface is often very different than the alloy’s bulk composition due to segregation effects. The overall activity of a catalyst is determined by the distribution of active sites. This distribution may be very heterogeneous both in terms of the local environments that define each site and their chemical reactivities. The reactivity of any specific active site can be affected by contributions from strain, ligand and ensemble effects. Computational methods are well suited to exploring these effects because one can simulate model systems where only one effect dominates as well as model systems where multiple effects are important. Quantum chemical simulations based on density functional theory (DFT) are widely regarded as reaching the appropriate compromise between chemical accuracy and the need to study structurally complex extended materials in order to tackle problems associate with heterogeneous catalysis involving alloys. A review of DFT and heterogeneous catalysis can be found in the previous SPR2 and that review also listed several general reviews of applications and foundations of DFT. Experimental and theoretical studies of monolayer bimetallic surfaces were recently reviewed.3 In a b

Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh PA 15213, USA. E-mail: [email protected]; Fax: +1 412 268 7139; Tel: +1 412 268 7803 National Energy Technology Laboratory, Pittsburgh PA 15236, USA

{ Address from January 2008: School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA


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this review, we examine recent applications of density functional theory in the study of alloy heterogeneous catalysis, limiting our attention to papers published between approximately mid-2005 and mid-2007. 1.1 Nomenclature conventions in this review We use the term alloy or bimetallic to mean any combination of two or more metals that are mixed to some degree on atomic length-scales. In bulk materials, this term encompasses a broad range of structures, including disordered alloys in which the chemical species form a solid solution by being randomly distributed on the lattice sites of the material’s crystal structure, ordered alloys in which the chemical species have a well defined ordering, and of course many intermediate structures with greater or lesser degrees of short and/or long range ordering. As emphasized above, heterogeneous catalysis is largely a surface effect. There are several examples mentioned later in the review where behavior of surfaces or nanoparticles is completely different than what is expected from bulk behaviors. It is therefore useful to use terms that distinguish between various types of alloy surfaces. These bimetallic structures include sandwiches, pseudomorphic monolayers and near surface alloys. These structures are illustrated and briefly defined in Fig. 1. In describing the catalytic properties of an alloy there are three primary mechanisms responsible for the modification of an alloy surface’s catalytic properties at the atomistic scale: strain, ensemble and ligand effects. An alloy typically has a lattice constant that differs from that of the pure metal components, resulting in a different average metal-metal distance in the alloy compared to that in the pure metal components. The electronic structure of an alloy is largely determined by the overlap of neighboring atomic orbitals. The change in metal-metal distances results in a change in orbital overlap and consequently a change in the electronic structure and the chemical properties. This effect is termed a strain effect, with strain defined relative to some reference system, usually one of the pure components in a similar structure. One way to probe strain effects computationally is to change the lattice constant of the system, or to examine pseudomorphic monolayers as illustrated in Fig. 1b. It can often happen that the active site in an alloy is composed of more than one species of metal atom. Adsorbates then bond to different metals simultaneously, and consequently the adsorption energy or reaction barriers are different than if the active site was composed of only one type of metal. This effect is called an ensemble effect. Fig. 1c and Fig. 2b shows one example where ensemble effects can be probed in a DFT calculation. We note that another somewhat different use of the term ensemble effect is to describe scenarios where a critical number or geometry of

Fig. 1 Side view of slab models of various bimetallic structures often used in computational studies. In each case, the bottom layers of the material are defined using the structure of a specified bulk material. The number of surface and bulk layers varies in different studies. (a) In the sandwich structure the surface is one component, often the same component as the bulk material and the second layer is another component. This structure is often used to determine ligand effects. (b) The pseudomorphic monolayer structure combines strain and ligand effects in one structure by placing a second component on top of a bulk material. (c) The near surface alloy combines strain, ligand and ensemble effects in one structure by considering an alloy film defined by just a few atomic layers on top of an ordered bulk material.


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Fig. 2 Illustration of strain, ensemble and ligand effects. (a) Adsorbate bonding is affected by a change in metal-metal bond distances. (b) Adsorbate bonding is affected by bonding to two different metals instead of one type of metal. (c) Adsorbate bonding is affected by heterometallic bonding of the active site atoms with neighboring atoms.

adsorption sites is required for a reaction to occur, and the critical number and geometry of sites is termed the ensemble. Finally, the electronic structure of a metal atom that is bonded to unlike metals is typically different than the electronic structure of the metal atom in its pure state. The change in electronic structure is again related to overlap of neighboring orbitals; the presence of different atomic orbitals changes the orbital overlap of the reference atom and thus, its electronic structure. This effect is termed a ligand effect and it is an indirect but significant effect. This effect is illustrated in Fig. 2c. In reality, all three effects usually take place on any practical catalyst and they are not independent. Strain and ligand effects in particular are both related to orbital overlaps of neighboring metal atoms. As a result, it can be difficult to disentangle the significance of each effect. Efforts aimed at separating out the effects usually rely on model surfaces such as those in Fig. 1. Additional effects that can be significant in alloy catalysis include site distributions and bifunctional mechanisms. Since a catalyst may be comprised of a distribution of active site compositions and geometries, there will be a distribution of adsorption energies and reaction rates. In nanoparticles, edge and corner sites may form a significant fraction of the active sites. These sites will have different properties than more highly coordinated sites on the terraces. The true measure of catalytic activity is the average of these rates, weighted by the distribution of sites. This average rate can be substantially different than the rates associated with an individual site. In 152 | Chem. Modell., 2008, 5, 150–181 This journal is

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bifunctional mechanisms, one process occurs on one type of site, and another process (perhaps involving the products of the first process) occurs on a different type of site. The electrocatalysis reactions reviewed in section 6 include several examples of bifunctional mechanisms. With regards to the sign of adsorption energies we have used the convention that an exothermic adsorption energy has a negative sign. That is, more negative adsorption energies correspond to more energetically favored states. In some papers the opposite convention was used, and in those cases we changed the sign of their energies when reporting results in this review. We used the units presented in each paper throughout the review. 1.2 Review organization The review is organized in the following manner. We first discuss modeling studies of segregation in alloys including the effects of reactive environments in section 2. We follow that with a review of the adsorption of small molecules on alloy surfaces in section 3. Neither of these topics alone offer a complete description of the catalytic properties of bimetallic catalysts, but a thorough understanding of these phenomena is a prerequisite to serious efforts to model these catalysts. Simulations of reactions on alloy surfaces are presented in section 4. Section 5 briefly examines two topics closely related to bimetallic catalysts, namely metals supported on metal oxides and the magnetic properties of alloys. Studies of alloy surfaces that are related to electrocatalysis are reviewed in section 6. We conclude in section 7 with some comments about future directions and needs in the application of theoretical methods to bimetallic heterogeneous catalysis.

2. Segregation It is difficult to understand the catalytic behavior of a surface if the actual composition and structure of that surface is unknown. Segregation is crucial to understanding what the surface of a bimetallic material actually looks like, because a simple termination of the bulk composition is generally not the most energetically favorable configuration. In fact, very dramatic differences can be seen in the composition and organization of the first few layers of a surface of a material compared to its bulk configuration, and this can be further complicated by the presence of adsorbates on the surface. In section 2.1 we discuss how DFT has been used to help understand experimental alloy surface characterization. We then move on to using DFT to study segregation in nanoparticles in section 2.2, and alloy bulk surfaces in section 2.3. The potentially powerful effect of adsorbate induced segregation is discussed in section 2.4, and statistical mechanical simulations of segregation in section 2.5. 2.1 Alloy surface characterization Although the focus of this article is catalysis on alloy surfaces, we recognize the importance of characterizing alloy surfaces as an essential aspect of research in this area. Here we briefly mention papers that focus solely on using DFT to help interpret experimental characterization of alloy surfaces. X-ray photoelectron spectroscopy (XPS) is commonly used to examine alloy structures because the binding energy of core electrons is sensitive to the chemical environment around the atom. Thus, formation of alloys may cause shifts in the XPS spectra that allow one to say an alloy has been formed. However, many factors in the alloy affect these shifts and some of them may cancel each other so a shift that is near zero does not necessarily imply no change in the chemical environment. In a disordered alloy the wide range of chemical environments can cause disorder broadening in the signals, further complicating the analysis. Computational tools based on DFT can simulate Chem. Modell., 2008, 5, 150–181 | 153 This journal is

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these spectra for different alloy models and facilitate the interpretation of these experiments. Olovsson and co-workers have been most active in this area. They have presented a broad and detailed framework for calculating core level shifts including an extension to Auger spectroscopy.4 Their approach has been applied to many alloy systems including the following random fcc alloys: CuPd, AgPd, CuNi, NiPd, CuAu, PdAu, CuPt, and NiPt.5 They have used their methods to follow the layer composition profile in film growth of PdCu and PdAg alloy films on Ru(0001).6 In a related work they studied how the electronic structure of embedded alloy layers is related to the bulk alloys6 and developed a method to characterize deep interfaces using computed core-level shifts.7 The effect of disorder broadening and the contributions from initial and final states was examined.8 Significantly they found that the importance of these effects seems to be metal dependent. 2.2 Nanoparticles Nanoclusters are of great interest because of their high surface area to mass ratio, which allows an exposure of a large catalytic surface per volume of catalyst. Because many of these catalytic compounds are costly noble metals, such as platinum, it may be advantageous to create bimetallic nanoclusters which expose an even higher number of catalytically active noble metal atoms per volume of the more expensive component. Fernańdez, Balba´s, et al. studied two such nanoclusters, CuAu and PtPd, finding the lowest energy isomers of clusters containing 5–22 pairs of atoms using the many body Gupta potential with parameters from the bulk alloys, followed by energy minimization using the DFT GGA method.9 They found that the ordering of these lowest energy isomers by the Gupta potential and DFT were not in good agreement, however some structural trends found for the nanoclusters using the Gupta potential were the same as those found via DFT. Specifically they found CuAu structural properties corresponding to the existence of icosahedral patterns in the geometry of the lowest lying isomers of the clusters using both methodologies. Additionally, the segregation of PtPd nanoclusters, which form Pt cores with Pdenriched surfaces, were reproduced using both methodologies despite the disagreement in energy ordering of the lowest level isomers. This suggests that although the Gupta potential parameterized for the bulk materials may not definitively identify the lowest energy isomers, it may be capable of identifying overall trends in the structure of nanoclusters. Following on this work, Paz-Borboń, Johnston, Barcaro, and Fortunelli performed similar analysis for 34 atom Pd-Pt clusters using a genetic search algorithm and the many body Gupta potential followed by energy minimization by DFT.10 The general conclusions of their work were the same: the ordering of the lowestenergy configurations of the clusters at a given composition was not consistent between the Gupta potential minimization and DFT calculations. They concluded that although the Gupta potential predicted several nearly degenerate lowest energy configurations, reoptimization by DFT showed that the configuration known as Dhcp(DT), which for the 20–14 Pd-Pt composition involves a core that is a double tetrahedron of Pt atoms, is the lowest energy state for all compositions. They additionally confirmed that the results of the Gupta potential and DFT calculations are in agreement that Pd segregates to the surface, forming Pt cores at the center of the nanoclusters. Also considering bimetallic nanoclusters, Sahoo, Rollman and Entel have studied segregation and ordering in Fe-Ni nanoclusters of 13 and 55 atoms using DFT.11 They studied every different composition of 13 atom nanoclusters and determined that in every case the lowest energy geometry is a distorted icosahedra where the Ni atoms occupy surface sites. They also studied the mixing energy of these nanoclusters, a measurement of the configuration’s stability. They found that the most stable composition in the 13 atom nanocluster was Fe3Ni10. For the 55 atom cluster they 154 | Chem. Modell., 2008, 5, 150–181 This journal is

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only considered two compositions: Fe12Ni43 and Fe43Ni12. In both cases they found that Ni atoms are found on the surface in the lowest energy geometry with an Fe core. In the Fe-rich case the Ni atoms on the surface formed a very particular configuration with no Ni–Ni bonds. An examination of the mixing energy vs. the number of Fe atoms on the surface confirmed that the most stable configurations are those where the surface is dominated by Ni atoms. 2.3 Bulk surfaces Although nanoclusters are of great interest, due largely to their high surface area, bimetallic surfaces remain an active field of catalysis research. Similar to nanoclusters the inclusion of multiple metals can serve to lower the cost of the catalyst while maintaining its activity, to improve the activity of the catalyst, or to protect it from corrosion and poisoning. Key to understanding how a bimetallic catalyst will operate in a chemical environment is understanding what surface is actually present for a given bulk composition. Ropo presented an ab initio study of AgPd surface segregation and how it depends on the surface orientation.12 DFT calculations were performed using the exact muffin-tin orbitals method using the coherent potential approximation on 8 layer slabs, the bottom 6 of which were fixed at bulk alloy compositions. He found that the surface layer relaxation played a minimal role in surface segregation for (111), (110), and (100) surfaces, but that this surface relaxation did play an important role in segregation in the second layer (the first subsurface layer) for the (110) and (100) surfaces, with an impact of up to 9% in Ag concentration from the non-relaxed state for a 4% inward movement on the (100) surface. He also found results that matched with previous theoretical and experimental results which show that Ag strongly segregates to the surface, with Pd only appearing on the surface for Pd-rich alloys. Yuge et al. explored the Cu75Pt25 (111) surface using DFT calculations and the cluster expansion technique, with Monte Carlo simulations to study segregation behavior at finite temperature.13 DFT calculations were carried out on 9 layer slabs, selected to insure that bulk behavior would be reproduced by the lower layers. Great care was taken in the selection of the cluster expansion parameters from the bulk alloy, which resulted in an estimated predictive error of 4.0 meV/atom. The cluster expansion results were then used in a Monte Carlo simulation to study the segregation of the alloy surface at 800 K, near a phase transition to a L12-ordered alloy. MC showed that the segregation on the alloy surface is driven by a competition between segregation energies and a slight ordering tendency, with the end result of platinum segregation to the first and third layers, copper segregation to the second layer (with bulk-like behavior on the 4th and lower layers) and no long range order in the top layer, despite the presence of some short-range order. Investigation of surface mixing enthalpies identified two surface ground-state atomic arrangements (for layers 1–3) for the bulk L12-ordered structure for Pt compositions of 0.167 and 0.333, suggesting short-range order. They studied the electronic structure of these configurations to produce their density of states (DOS) band structures, and the top layer for both cases shows a significant peak around 3.0 eV. This matches with experimental results, showing significant ordering in the first layer surface electronic states of the alloy. Løvvik studied the segregation of Pd based alloys by performing DFT and bandstructure calculations on Pd(111) slabs alloyed with 5% of Ag, Au, Cd, Cu, Fe, Mn, Ni, Pb, Pt, Rh, Ru, and Sb.14 The goal of these calculations was to find the segregation energy of the alloy atoms, and how this varies with the depth of the substituted atom. The results of these calculations showed that Mn, Ru, Fe, Rh, Ni, Cu, and Pt all have positive segregation energies suggesting that they tend to segregate into the lower levels leaving Pd on the surface while Ag, Cd, Au, Sn, and Pb all had negative segregation energies suggesting that they segregate to the surface. This matches well with experimental results, except in the case of Cu which Chem. Modell., 2008, 5, 150–181 | 155 This journal is

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has been seen to segregate to the surface in Pd–Cu nanoparticles. Cu and Ni were also found to have oscillatory depth profiles, a feature that has also been observed experimentally for the Ni alloy. Lastly the relationship between atom size (based on the covalent radii) and experimental surface energies with segregation energies were tested, showing that although there are trends (larger atoms tend to segregate to the surface as do those with lower surface energies) they are unfortunately not strong enough to be of predictive value. Ni3Al is an alloy of interest due to its high temperature strength, high hardness, wear resistance, low density, high melting point, and high temperature corrosion resistance. Because of these properties Ni3Al is used in gas turbine and cutting tool applications. In order to improve its oxidation-resistance it can be alloyed with Pt, and it is believed that the surface segregation of the Pt may serve an important function in this improvement. Jiang and Gleeson studied this segregation using wellconverged DFT results for the low-index (100), (110), and (111) planes with one platinum per (2 2) surface unit cell.15 The results confirmed what is known from experiments: namely that Pt strongly segregates to the surface layer. They also found that Ni segregates to the second layer for both the (100) and (111) surfaces, but for the (110) surfaces with its narrower spacing they found that Pt segregates to both the first and second layers. These results were confirmed by Qin et al.16 with DFT calculations and experimental methods. Additionally Qin added evidence that size effects are responsible for this segregation behavior. Pt segregation to the surface helps to explain why Pt improves the oxidation-resistance of the alloy: by displacing Ni to the lower levels it increases the surface Al:Ni ratio, allowing the formation of Al2O3 to be more favorable to the NiO oxidation. By forming a scale exclusively composed of Al2O3, further oxidation is prevented. 2.4 Adsorbate-induced segregation Segregation, in addition to being caused by the properties of the alloy surface, can be strongly influenced by the presence of adsorbate molecules on the surface. This can have a powerful effect on the surface composition, which can thus vary depending on the conditions in which the alloy is manufactured. Han, Van der Ven, Ceder, and Hwang studied these effects for oxygen adsorption on PtRu alloys (111) surfaces using DFT and the cluster expansion method, as well as Monte Carlo simulations.17 Standard cluster expansion calculations could account for either the alloy composition at the surface or for the presence of adsorbates, but not for both simultaneously, so a special approach was required to perform these calculations. A ‘‘coupled cluster-expansion’’ was therefore used which was composed of two interwoven lattices, one representing the surface composition, and the other representing the empty or occupied adsorption sites. The results of the coupled cluster expansion were then utilized in Monte Carlo simulations to calculate the surface properties at finite temperatures from 100–1100 K and different Ru chemical potentials in the bulk beneath the surface (corresponding to different alloy compositions). What they found was that although in a vacuum Pt would be expected to fully saturate the surface, oxygen adsorption on the surface could dramatically increase the amount of Ru on the surface (because the strong binding between Ru and O easily overcomes the higher surface energy of Ru), where it forms island structures. The size and structure of these Ru islands could be crucial in the effectiveness of the alloy as a catalyst in fuel-cell applications, confirming that these adsorbate-driven segregation effects must be seriously considered. Adsorption driven segregation is not necessarily only an issue in the manufacture of an alloy, but also in understanding how operating conditions may effect segregation compared to what is seen by studies performed in ultra-high vacuum (UHV) conditions. Vestergaard et al. examined this issue for the Au/Ni (111) surface by using both experimental techniques and DFT calculations.18 They found that when the AuNi alloy system is exposed to high pressures of CO, the surface is heavily 156 | Chem. Modell., 2008, 5, 150–181 This journal is

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modified, exposing Au clusters on the surface. By studying STM movies of the process occurring on the surface under these high CO pressure conditions, and utilizing relevant DFT calculations to help illuminate the underlying physical driving forces, they found that CO selectively adsorbs to the Ni atoms on the surface, and hypothesize that it forms Ni(CO)3 at step edges which subsequently desorb from the surface. This desorption causes the step edges to move, and Au atoms diffuse along the step edge to form clusters that are left behind on the resulting terraces as the step edge moves away. Although this is not traditional segregation behavior, this chemical attack on the bimetallic surface under operating conditions helps to highlight the importance of understanding the limitations of UHV approaches and the utility of computational and experimental methods to better understand surface behaviors under more complicated conditions. We have already seen how Pt alloyed with Ni3Al can strengthen the alloy against oxidation, and how studying the segregation of the surface can help explain why the alloy surface is more oxidation-resistant. Could the presence of oxygen have an effect on this surface segregation? A similar oxidation-resistance benefit has been postulated for silver addition to copper. Kangas et al. addressed this issue with a study of Ag segregation in the Ag/Cu system when exposed to oxygen adsorbates.19 They used a DFT approach, calculating segregation energies for silver on the first, second, and third layers with different oxygen coverage on the surface. It was found that at 0.25 monolayer (ML) coverage, segregation is not strongly favored by the energies, but at 0.5ML and higher O coverage the segregation energies suggest that Cu strongly segregates to the surface. This suggests that what may form under actual conditions is a Cu/O phase on the surface with an unoxidized subsurface Cu/Ag alloy covering the remaining bulk Cu. The underlying Cu/Ag alloy layer prevents further oxidation of the Cu bulk. 2.5 Statistical mechanical simulations Ruban et al. used the generalized perturbation method (GPM) based on the coherent potential approximation (CPA) to study segregation on the surfaces of Ag-Pd and isoelectronic alloys (Cu-Pd, Au-Pd, Cu-Pt, Ag-Pt, and Au-Pt).20 Although the longperiod superstructures they identified are unlikely to actually occur, they concluded that the ordering behavior of Ag-Pd that they predicted can be partly observed on alloy surfaces. They found that these alloys tend toward a formation of (111)-type ordered structures, a tendency which originates from a relatively strong ordering interaction at the second coordination shell. In the case of Pd-rich Ag-Pd alloys this leads to an oscillating concentration profile for the (111) surface. Other tendencies were also identified such as the hindrance of long range order caused by Pt-Pt interactions in Ag(Au)-Pt alloys. Mu¨ller approached the problem of surface segregation and interface properties of bimetallic alloys using a combination of DFT, a cluster expansion, and Monte Carlo. The two systems studied were the Zn-Al solid solution21 and the Pt-Rh (111) surface.22 DFT calculations were performed to parameterize the cluster expansion (CE) that was then used to produce a predicted ground state line. Configurations near this line that had not yet had DFT calculations performed for them were then used in new DFT calculations to produce a new CE until the entire ground state line was characterized by DFT calculations. This CE was then used in Monte Carlo simulations to simulate the segregation behavior of the surface at finite temperatures. For PtRh(111) the results of MC showed that for temperatures of roughly 1000 K Pt composed 70% of the surface, with no long range order present. These results are in excellent agreement with experimental results showing 69% Pt on the surface, and no long range order. For the Zn-Al solid solution he studied the formation of precipitates in the Al rich alloy, reproducing experimental results and identifying the correct precipitate formations ([111]-soft precipitates of Zn embedded in a matrix of Al that has an elastically soft [100] direction). These calculations also Chem. Modell., 2008, 5, 150–181 | 157 This journal is

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allowed for the identification of the driving forces behind these behaviors with interfacial energies stabilizing the (111) oriented interfaces in the Al-Zn solid solution, while the Pt-Rh segregation was explained by the on-site energies of the near-surface layers.

3. Adsorption properties on alloy surfaces Heterogeneous catalysis involves adsorption of intermediates on surfaces. Alloy surfaces pose particular challenges in modeling. The bonding to alloy surfaces is different than that on the pure metal surfaces due to heterometallic bond formation (ligand effects), multifunctional adsorption sites (ensemble effects), lattice constant changes (strain effects) and coverage effects. Disentangling these effects is non-trivial and can typically only be done clearly in model systems. In this section we review studies of atomic and molecular adsorbates on alloy surfaces. The section is organized from simplest adsorbates to most complex. Within each subsection we have organized the papers reviewed from most general to most specific. We begin with hydrogen in section 3.1 and oxgyen in 3.2. Carbon monoxide, carbon dioxide and other small molecules such as NO are presented in 3.3. Hydrocarbons such as ethylene and acetylene are discussed in 3.4. Finally, carbon and sulfur are presented in section 3.5. 3.1 Hydrogen Greeley and Mavrikakis have investigated the adsorption of surface and subsurface hydrogen and the diffusion of hydrogen on a wide range of near-surface alloys (NSAs).23 Near-surface alloys are defined simply as alloys where the surface composition differs from the bulk composition, even if the compositions are unstable. Remarkably, they found that hydrogen adsorption on some NSAs was as weak as the adsorption on noble metals, while simultaneously dissociating hydrogen more easily (i.e. lower dissociation barriers). Subsurface hydrogen was generally found to be unstable in NSAs with respect to gas phase hydrogen,23 with the exception of Pd. The ability of some NSAs to simultaneously bind hydrogen weakly and easily dissociate hydrogen makes them interesting candidates for fuel cell anode electrocatalysts.24 Carbon monoxide poisoning is a serious problem in hydrogen fuel cells because the hydrogen fuel contains trace amounts of CO. One strategy for mitigating the effect of CO poisoning is to design new electrocatalysts that bind CO more weakly. Surfaces that bind CO weakly tend to bind hydrogen weakly too, and weak binding of H suggests high dissociation barriers. If that were always true, there would be little hope for solving this problem. The discovery of new NSAs that avoid this problem is remarkable. In particular, Pd and Pt-terminated NSAs were shown to have weak CO binding energies and facile hydrogen dissociation barriers. These types of surfaces are among the best known anode electrocatalysts to date. Substrate interactions and strain effects are two of the primary mechanisms that modify the adsorption properties in alloy surfaces. In pseudomorphic Pd layers on Cu(111) the Pd atoms are compressed by the smaller Cu lattice constant. In addition, strong Cu-Pd bonds cause rehybridization of the Pd d-band, resulting in weaker H adsorption energies on the Pd monolayer than on Pd(111).25 In contrast, for a Cu monolayer on Pd(111) the tensile strain of the Cu monolayer should increase the H adsorption energy, while the Pd-Cu bonds should reduce the adsorption energy. The net result is coincidentally that the Cu monolayer has the same H adsorption energy as Cu(111).25 The adsorption energies of hydrogen were changed by less than 60 meV when a water bilayer was added to the surface compared to the vacuum reference. This suggests that trends in gas-phase calculations may be sufficiently representative of electrochemical environments to identify trends in those environments. 158 | Chem. Modell., 2008, 5, 150–181 This journal is

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The adsorption energy of hydrogen can play a crucial role in the selective hydrogenation of unsaturated bonds.26 However, hydrogen can induce changes in the composition of an alloy surface. Ag-Pd alloys have been used to tune the adsorption energy of hydrogen. Under vacuum conditions, Ag segregation is observed, but with increasing coverage of hydrogen atoms Pd segregation gradually occurs. This work highlights one of the challenges in interpreting experimental results on alloy surfaces: the composition of alloy surfaces can typically only be measured on clean surfaces, but the composition under reaction conditions may be different than that of the clean surface. In addition to developing alloy surfaces with tunable adsorption energies to modify selectivity, another approach is to modify the diffusion of hydrogen on alloy surfaces. Fearon and Watson have found that differences in the diffusion of hydrogen on Pt and PtSn(111) surfaces may be responsible for changes in the catalytic activity of these surfaces.27 Hydrogen adsorption and vibrational features on three-fold adsorption sites on Pt and PtSn surfaces are nearly identical. The diffusion of hydrogen on these surfaces is very different. On Pt, the diffusion barriers are typically 4–5 kJ/mol. In contrast, on the PtSn surfaces the barriers for diffusion are at least five times higher than those on Pt due to the very weak interactions of the Sn atoms with the hydrogen atoms. These differences are likely to be important in catalytic hydrogenations. 3.2 Oxygen Computational screening studies are often expected to be more efficient than an equivalent experimental study. Still, the expense of running large numbers (thousands) of DFT calculations is not trivial and the combinatorial space for computational work is still just as large. Consequently, there is a significant need for simpler methods to calculate surface properties to help narrow the focus of computational studies. Greeley and Nørskov have developed a simple way to rapidly estimate the oxygen adsorption energy on alloy surfaces based on DFT calculations.28 Their scheme relies on the fact that oxygen adsorption energies tend to correlate strongly with the d-band center of the clean metal surface. They found that the d-band center of an alloy surface can be reasonably approximated from linear combinations of pure overlayer data from their clean surface electronic structure database. Thus, with a reasonable estimate of the d-band center, they could use the correlation of oxygen binding energy to estimate the binding energy on an alloy surface without performing additional DFT calculations. With their simple scheme they predicted the oxygen binding energy on several hundred binary alloy surfaces and compared them directly to DFT calculations. They found that in the majority of cases the prediction error was less than 0.2 eV. The model had large prediction errors for a few systems involving Pt-group overlayers on noble metals. This work points to the utility of simple chemical descriptors of alloy surfaces such as the ‘‘effective’’ d-band center of a multi-component adsorption site in making reasonable correlations for narrowing the scope of computational studies. Oxygen bonds to transition metals strongly. In alloy systems, this may have an unintended consequence of inducing segregation of an undesirable alloy component under reaction conditions. The stability of Pt-3d-Pt(111) sandwich structures (3d = Ti, V, Cr, Mn, Fe, Co or Ni) in the presence of oxygen was investigated using DFT to determine whether oxygen could induce a segregation of the subsurface 3d metals to the surface.29 It was found that as little as 12 a monolayer of oxygen could destabilize the sandwich structure and cause segregation of the more reactive 3d metal to the surface. These results were confirmed experimentally for Pt-Ni-Pt(111) and Pt-Co-Pt(111) structures, where the activation barrier to segregation of subsurface Ni was found to be approximately 15 kcal/mol and for Co to be 7 kcal/mol. The authors noted that the apparent activation energy was temperature dependent, Chem. Modell., 2008, 5, 150–181 | 159 This journal is

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indicating the possibility there are two barriers: one for inward diffusion and one for outward segregation. Surface composition heterogeneity can also strongly affect the adsorption properties of alloy surfaces. Ji and co-workers have investigated the adsorption and diffusion of OH on Pt(111) and PtMo(111) surfaces.30 They approximated the PtMo(111) surface in a 7 layer, 3 3 Pt surface supercell with a single Mo atom in each layer, yielding a Mo concentration of 11%. The presence of Mo in the surface layer caused a bigger relaxation of the surface layers than that found in Pt(111), probably due to the strong Pt-Mo bond strength. The Mo atom tends to sink deeper in the surface than Pt atoms do. OH preferentially adsorbs on the top site of both surfaces with the most stable adsorption energy of 2.35 eV on Pt(111) and 3.32 eV on the top Mo site of the PtMo(111) surface. On Mo(110) (the most close-packed bcc surface) OH prefers a three-fold site with an adsorption energy of 4.43 eV. Consequently, the dramatic increase in adsorption energy of OH on the alloy surface is due primarily to the bonding of OH to the Mo atom. The stronger adsorption energy of OH on the Mo atom has consequences on reactions that have OH as a product, for example, water dissociation. In the same work the authors found that the dissociation barrier for water on Pt(111) to be 0.89 eV, whereas on the alloy surface with OH ending on the Mo atom the barrier is only 0.63 eV.30 Thus, water dissociates more readily on the alloy surface than the Pt(111) surface. Of course, if a catalytic cycle is to continue, the OH fragment must be able to diffuse away from the Mo atom. On Pt(111) they calculate a diffusion barrier of only 0.13 eV from a top site to a top site through a bridge site transition state. In contrast, on the PtMo surface the barrier of OH leaving a Mo top site is around 1.2–1.3 eV, suggesting that once OH forms on the site, it will stay there a long time.30 There could still be important repulsive effects that weaken OH adsorption energies on neighboring sites, but this is an area of research that has not been fully explored. In the oxygen reduction reaction, OOH is postulated to be one of the main products in the first step of some mechanisms. Seminario et al. have investigated the adsorption and dissociation of OOH on three atom Pt-X-X and Pt-Pt-X (X = Co, Cr, and Ni) bimetallic clusters.31 They found these X atoms are better adsorption sites for OOH than Pt, but they hypothesize that the strong adsorption may not facilitate dissociation. Instead, the X atoms appear to transfer electrons to the Pt atom, making the Pt more negatively charged, and more able to transfer electrons to oxygenated species. Thus, the authors speculate that subsurface X atoms may contribute favorably to facilitating OOH dissociation. However, the authors did find exceptions to that suggestion; clusters with two Cr atoms were able to dissociate OOH with no barrier.31 Larger alloy clusters have also been investigated. Lacaze-Dufaure et al. have investigated the adsorption and dissociation of oxygen on Cu-doped Al clusters up to 31 metal atoms.32 They found that the addition of Cu stabilizes the aluminum clusters differently depending on where the Cu atom is. If the Cu atom is at the surface, the cluster is more stable by 0.31 eV, but if it is in the center of the cluster the cluster is stabilized by 1.18 eV. This indicates that Cu anti-segregates in these clusters, preferring to be in the bulk, more highly coordinated, environment rather than at the low-coordinated surface. Furthermore, electron transfer from the Al cluster to the Cu atom is significant (0.7 e) at the surface, and negligible in the bulk. Interestingly, the Cu 3d orbital filling remains nearly constant, and all the electron transfer changes are observed in the 4s and 4p orbitals. The authors found that incorporation of Cu atoms into the oxygen adsorption site on the Al clusters results in a weakening of the oxygen adsorption energy. They obtained an oxygen adsorption energy of 5.99 eV on a pure Al31 cluster and 5.64 eV on the same site with a Cu atom in it for an Al30Cu cluster. On another type of site they calculated an adsorption energy of 5.58 eV on a pure Al31 cluster, and 5.32 and 5.28 eV for the same site depending on where the Cu atom was placed for an Al30Cu cluster.32 Cu incorporation into the active site reduces the oxygen adsorption energy 160 | Chem. Modell., 2008, 5, 150–181 This journal is

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by reducing the charge transfer from the cluster to the oxygen atom. The authors note a large sensitivity of their results to relaxation effects. 3.3 Carbon monoxide and other small molecules 3.3.1 Carbon monoxide. Carbon monoxide is one of the most commonly used probe molecules in the study of the chemical properties of metal surfaces. CO represents a step in the direction of complexity compared to atomic adsorbates and diatomic molecules. On one hand, the bonding involves molecular orbitals and it is sensitive to the detailed electronic structure of the metal surface. This allows one to use the CO bonding properties as a probe of changes in surface electronic structure. Yet at the same time, in many cases CO retains aspects of the simplicity that atomic adsorbates have. The effect of alloy formation within a supported monolayer on the adsorption of CO on these surfaces was investigated by Gonzalez and Illas.33 In that work, the authors used model monolayers of Pd, Rh, and Cu as well as mixed monolayers of these metals supported on Ru(0001) to determine the role of alloy formation in modifying the CO adsorption properties. Generally, they found that the variations in adsorption energies were largely explained by changes in the ability of a metal to back donate charge into the CO(2 p*) orbital, as indicated by changes in the d-band center and density of d-states at the Fermi level. Although the adsorption energies varied from 21 kcal/mol (Cu/Ru(0001)) to 44 kcal/mol (Rh/Ru(0001)), the adsorption geometry was practically the same on these surfaces. In the previous work, the authors found that CO bonds more weakly to Pd monolayers supported on Ru(0001) than on Pd(111).33 Remarkably, incorporating Cu atoms into the supported Pd monolayer was found to increase the CO adsorption energy on the Pd atoms by up to 12 kcal/mol, depending on the Cu concentration. In contrast, CO bonds slightly more strongly to Cu/Ru(0001), and incorporation of Pd into the Cu monolayer tends to decrease the CO adsorption energy on the Cu atoms by up to 6 kcal/mol. These effects are attributed to an increase in the back-donation to CO on these surfaces. In related work, Sakong et al. have used CO adsorption on ordered Cu-Pd alloys and surface alloys to help distinguish between these ligand, ensemble and strain effects, and in some cases show they are related.34 They attribute the increased reactivity of Cu atoms with increasing Pd content in a surface alloy as due to an ‘‘effective compressive strain’’ due to the larger size of the Pd atoms compared to the Cu atoms. Despite observed variations in the CO adsorption energy on the Cu atoms, the CO vibrational frequency was hardly changed. On Pd surfaces, in contrast, large blueshifts were observed on the compressed Pd overlayers. The interaction of CO with alloy impurities is significantly different if the impurity is a metal adatom on the surface, or a substitutional impurity in the surface. The adsorption properties of CO on Au/Ni(111) surfaces has been studied with Au in the form of adatoms35 and substitutional impurities.36 In the case of adatoms, the adsorption energy of CO in sites that include a nearest neighbor of the Au adatom is reduced by up to 1.2 eV. At larger distances from the adatom the adsorption energies are nearly unchanged. The Au adatom was found to bind CO much more strongly than a flat Au(111) surface. In contrast, the substitutional Au impurity bonds CO more weakly than a flat Au(111) surface. A substitutional Au impurity also reduces the CO adsorption energy on Au/ Ni(111) surfaces on the sites immediately surrounding the impurity.36 In this case, the incorporation of Au into the Ni(111) surfaces by exchanging with a Ni atom was found to be endothermic, and consequently surface alloy formation must be entropically driven. The Au atom center of mass sits about 0.5 A˚ above the average location of the Ni atoms due to its larger size. The adsorption energy in the threefold sites that include the Au atom are only 1.18 eV, compared to 2.16 eV in neighboring three fold sites that only include Ni. On pure Ni(111), the CO Chem. Modell., 2008, 5, 150–181 | 161 This journal is

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adsorption energy is 2.13 eV. The effect of the Au impurity is significant, but highly localized. In the preferential oxidation of CO in the presence of hydrogen it is desirable to use an alloy catalyst with weak CO adsorption energies. One strategy for modifying Pt to weaken the CO adsorption energy is to alloy it with Sn. The adsorption properties of CO on Pt3Sn alloys have been calculated for the (111), (110) and (001) surfaces and compared to the same surfaces of pure Pt.37 In general, the alloy surface bonds CO more weakly than the corresponding Pt surface. For alloy sites that contain Sn or with neighboring Sn, the adsorption of CO is always weaker than sites that do not contain the Sn atom. In contrast, subsurface Sn atoms increase the adsorption energy of CO to the Pt atoms directly above the Sn atom. One of the main challenges in modeling alloy surfaces is finding ways to unambiguously relate the results to experimental data. The majority of comparisons between theory and experiments have used the adsorption energy directly. An alternative approach is to compare computed and experimental vibrational fingerprints of CO on well-defined alloy surfaces. Dupont et al. have used DFT to calculate the anharmonic vibrational frequencies and intensities and systematically compared them to high-resolution electron energy loss spectroscopy measurements (HREELS).38 Their approach to calculating the anharmonic frequencies is to first calculate the harmonic frequencies and modes and then to use these to sample the potential energy curve of each vibrational mode. They fit a Morse potential to each of these curves to calculate the anharmonic vibrational frequency. The average corrective anharmonic frequency was 17 cm1. Finally, the frequencies were scaled by 1.022 (the ratio of the experimental gas-phase stretching frequency to the calculated stretching frequency). The authors are able to simulate HREELS spectra using these frequencies for direct comparison with the experiments. Using this hybrid experimental/computational approach the authors concluded that CO preferentially adsorbs on atop sites of Pt, but that multifold adsorption can coexist at room temperature, in contrast with Pt(111). CO does not bond to the Sn atoms. The presence of a second metal in the surface can actually change the reaction coordinate for some reactions. Ji and Li demonstrated that a Mo atom in a PtMo alloy surface changes the reaction coordinate for CO oxidation.39 They found the barrier for CO oxidation on Pt(111) to be 1.2 eV, but on PtMo(111) the barrier was only 1.0 eV. They observed a correlation between the reaction barrier and CO adsorption energy: stronger adsorption energies had higher oxidation barriers. This correlation is due to the desorption of the product CO2. Examination of the reaction coordinates on Pt(111) and PtMo(111) showed that the two reaction coordinates were not the same. On Pt(111) oxygen atoms prefer to adsorb in three-fold sites, where as on the PtMo(111) surface oxygen atoms prefer the atop site of the Mo atom. In the transition state the oxygen atom is practically on an atop site, which means additional energy must be used on Pt(111) to get the atom out of a three-fold site. The authors suggest that it is this geometric effect on PtMo where the oxygen atom is already near the position it occupies in the transition state that results in the lower reaction barrier. Methanation of CO is interesting for a number of applications including cleaning up hydrogen gas of CO, and synthetic fuel synthesis. Industry prefers Ni catalysts, but there is still substantial room for improvement in activity. Andersson and coworkers have applied a computational screening approach to optimize multiple objectives such as activity, selectivity and cost in finding a better catalyst.40 There are two important surface properties that determine the rate of the methanation process: the CO dissociation barrier and the stability of the C and O products. They showed these two properties are related through a Brønstead-Evans-Polanyi relation and consequently only one descriptor is needed to characterize the surface activity. Furthermore, they identified a volcano-shaped dependence of the activity of pure metal surfaces on the CO dissociation energies on those surfaces. This allowed them to identify a narrow range of desirable CO dissociation energies that have activities 162 | Chem. Modell., 2008, 5, 150–181 This journal is

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at least as good as the best pure metals. By developing a simple interpolative scheme to estimate the CO dissociation energies on alloy surfaces based on combinations of pure metal surface properties and mixed overlayer bimetallic structures they were able to rapidly screen 117 potential alloy catalyst candidates for activity. It is not simply the most active catalyst that is necessarily the best in an application, the authors showed that the most active catalysts are the most expensive ones, and that there can be a compromise in cost and activity. They identified a series of Ni-Fe surfaces that were ‘‘Pareto-optimal’’ in cost and activity. These surfaces were examined in a set of experiments and were shown to be better than pure Ni catalysts.41,42 Nanoparticles have different morphologies than flat, bulk surfaces. Perez et al. have considered the activation of water and COads + OHads reactions on Pt and PtRu clusters including the effects of solvation.43 They found that the presence of under-coordinated Ru adatoms on the Pt cluster surfaces enhances the production of OHads from water compared to Ru alloyed into the nanoparticle surfaces. More significantly, they found that the presence of an aqueous environment simulated by up to six water molecules dramatically stabilized the transition state and products of the reactions. For example, in a gas-phase environment they calculated a water dissociation barrier of 20 kcal/mol whereas in the solvated environment the barrier was reduced to 4.5 kcal/mol on the alloy surface. The barrier for water dissociation on the Ru adatom in the aqueous environment was only 0.9 kcal/mol. Although their results are for an adatom on a near flat (111) surface, they may have significance in describing the catalytic properties of undercoordinated Ru atoms at edge and corner sites on nanoparticles. The phase behavior of alloy nanoparticles may differ from that in the bulk due to the increased significance of surface effects. Ge and coworkers examined the heats of formation of Pt-Au nanoparticles up to 13 atoms and compared them to bulk heats of formation.44 For bulk systems they found the heats of formation of Pt-Au alloys were all small and positive indicating a tendency to phase separate at low temperatures. In contrast, the heats of formation of some of the alloy atom clusters were significantly negative indicating that alloy nanoparticles may readily form. The adsorption of CO on the Pt-Au nanoparticles was larger than that on the bulk surfaces, and CO preferentially adsorbs on Pt sites that are adjacent to Au atoms. The effect of coadsorption of oxygenated species on the adsorption and reactions of adsorbed CO on Pt and PtRu(111) surfaces has been investigated by Han and Ceder.45 They found that relaxation effects in the slab calculations were significant (40.1 eV) and must be considered in quantitative work. The coadsorption of O or OH weakens the CO adsorption energy on Pt(111) by about 0.2 eV. They found the effect of alloying Ru in the surface to be minimal on the adsorption energy of CO on neighboring Pt, unless the Ru is oxidized by adsorption of O or OH, and then it significantly reduces the CO adsorption energy. They note that OH bonds more strongly to Ru than CO does, and in an aqueous environment one expects Ru to be oxidized.

3.3.2 Other small molecules. Carbon dioxide interacts weakly with most surfaces, and promoters are typically required to activate the molecule to induce chemisorption. Vines et al. examined the role of sodium promoters in CO2 adsorption on W(011) surfaces in a combined experimental and computational effort.46 On clean W(011) CO2 physisorbs at temperatures below 120 K. In the presence of at least 0.5 ML of Na, CO2 chemisorbs in a mechanism that involves charge transfer from the Na atoms, converting them to Na+. At low coverages the authors found the adsorption energies of Na decreases in the range of 0 to 0.5 ML due to repulsive interactions between the positively charged alkali adsorbates (the valence electron on Na is ‘‘lost’’ by incorporation into the W slab valence band). Above 0.5 ML metallic bonding of the Na to W was observed. They observed very little difference in the Chem. Modell., 2008, 5, 150–181 | 163 This journal is

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adsorption energy (o0.1 eV) of Na on atop, bridge and 4-fold hollow sites at low coverages. The charge donated to the W valence band from Na is ‘‘back donated’’ to adsorbing CO2 molecules which stabilizes the adsorption.46 On clean W(011) the most stable adsorption energy of CO2 is only 0.06 eV, and the CO2 molecule retains its linear geometry. In the presence of Na, the adsorption energy increases dramatically to 5.23 eV and the geometry of the chemisorbed CO2 is bent, closely resembling a gas-phase CO2 anion. Predicting the adsorption energy of molecules in heterogeneous adsorption sites remains a challenge. Tang and Trout illustrated some of the short-comings in simple adsorption models that only utilize the d-band center and introduced a new weighting scheme that improves predictions of NO adsorption on Pt(111), Rh/ Pt(111) and Pd/Pt(111).47 The main issue they identified is that simple models tend to fail when adsorbates bond unequally to multiple surface atoms. In this model, in addition to weighting the d-band center by a d-band coupling matrix element which partially accounts for ligand effects on the electronic structure of the atom due to neighboring atoms, they also weight the bond strength contributions of each metal atom in the active site according to an electrostatic bond model. Although the simpler model and previous d-band weighting schemes48 qualitatively showed the same predictive trends the prediction errors were typically greater than 5 kJ/mol. The more sophisticated weighting scheme developed in this paper had a significantly lower prediction error than the simpler models with an average error on the order of 5 kJ/mol or less. 3.4 Hydrocarbon molecules As molecules get physically larger and chemically multifunctional the challenges in modeling their adsorption and reactions increase rapidly. Zellner et al. have used DFT in conjunction with surface science measurements to investigate the adsorption properties of ethylene on Pd/Mo(110) and Pd/W(110) monolayer bimetallic surfaces.49 Hydrogen and ethylene both adsorb more weakly to these bimetallic surfaces than to the corresponding pure metal surfaces. A simple d-band correlation was unsuccessful in predicting the adsorption energy of ethylene on the metal surfaces used in this work (Pd, Pd, Mo, W, and combinations thereof). This occurs because one must consider the Pauli repulsion between the p electrons in the ethylene molecule and the metal surface. The amount of repulsion depends on the spatial extent of the d-orbitals. Since 5d orbitals are larger than 4d orbitals which are larger than 3d orbitals, the authors observed separate correlations between the d-band center and adsorption energy depending on surface metal. Their experiments showed that ethylene is p-bonded to the bimetallic surfaces, but di-s bonded to the parent metal surfaces. Goda et al. examined reaction pathways of C2 hydrocarbons on bimetallic surfaces using a hybrid DFT/bond order conservation (BOC approach).50 They examined hydrogen, ethylene, acetylene, ethyl and vinyl on a variety of monometallic and bimetallic surfaces. DFT was used to examine the adsorption energies and correlations with the electronic structure. BOC theory was then used with these adsorption energies to calculate activation energies for ethyl and vinyl dehydrogenation. The authors observed separate adsorption energy/d-band center correlations for 3d, 4d and 5d metal surfaces as seen by Zellner et al.49 Acetylene tends to bond more strongly to the surfaces than ethylene does. They also found that the dehydrogenation reaction barriers correlate reasonably with the surface d-band centers. This is consistent with a linear free energy relationship between an initial state and transition state. CuPt clusters have been found to have high ethylene selectivity in the hydrogen assisted dechlorination of 1,2-dichloroethane. Avdeev et al. have used DFT to investigate the adsorption of ethylene on Cu12Pt2 clusters to help explain these 164 | Chem. Modell., 2008, 5, 150–181 This journal is

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observations.51 They examined three types of clusters with different symmetry. Each cluster had an active site for ethylene adsorption that consisted of a Pt atom surrounded by Cu atoms. Ethylene bonds in a p configuration on all of the clusters. Ethlylene was found to bond more strongly to the bimetallic cluster than a pure Cu cluster, but more weakly than to a pure Pt cluster. The cluster with C3v symmetry had the strongest adsorption energy, likely because the Pt atom was the most undercoordinated. The weaker bonds to the CuPt bimetallic cluster are due to the lack of Pt-Pt adsorption sites where ehthylene could form a di-s bond. The authors also calculated vibrational frequencies of the adsorbed ethylene molecules and observed shifts in the features that could be used to identify p-bonded ethylene species on real surfaces. In the partial hydrogenation of multifunctional adsorbates such as acrolein, which contains unsaturated CQO and CQC bonds, the selectivity with which each functional group is hydrogenated can be important. Murillo and coworkers investigated the partial hydrogenation of acrolein on two Pt/Ni bimetallic surfaces in ultra high vacuum and using DFT.52 The authors use pseudomorphic overlayers of Ni on Pt(111), denoted Ni-Pt-Pt(111) and sandwich structures with a subsurface monolayer of Ni in Pt(111), denoted Pt-Ni-P(111) as model surfaces to perform calculations on. On Pt(111) and Ni-Pt-Pt(111) they only observed hydrogenation of the CQC bond. On the Pt-Ni-Pt(111) surface, however, they saw a higher activity and production of 2-propenol and fully hydrogenated 1-propanol. DFT calculations suggest that on Pt(111) and Ni-Pt-Pt(111) the acrolein can only adsorb in a di-s C-C geometry which leads to hydrogenation of the CQC bond. On Pt-Ni-Pt(111), however, the di-s C–C and di-s C–O geometries are close in energy. That suggests either bond could be hydrogenated under reaction conditions. Finally, the authors showed a correlation between different adsorption geometry energies and the surface d-band center. It may be possible to identify more selective bimetallic surfaces using this correlation. 3.5 Carbon and Sulfur Carbon poisoning, or coking, occurs at the anodes of solid oxide fuel cells that utilize Ni as electrocatalysts. In a series of papers Nikolla et al. analyze the origin of this effect and propose a novel alloy that is shown to suppress coking.53,54 Using DFT calculations they showed that C–C bond formation and C–O bond formation activation energies are comparable on Ni surfaces. They further found that C atoms are quite mobile on these surfaces at temperatures relevant to fuel cell operation. By formulating the problem of coking in terms of C–C bond and C–O bond selectivity they identified that a Ni-Sn surface has a higher selectivity towards C–O bond formation. Consequently, Ni-Sn is resistant to coking. The mechanism for the enhanced selectivity is in the relative rates of diffusion of C and O atoms. They found that C atom diffusion was substantially more hindered by Sn surface alloys than O was. They further analyzed the stability of Ni-Sn surfaces using DFT and thermodynamic arguments and found that Sn was stable in the surface under typical operating conditions. Their findings were confirmed experimentally by synthesizing Ni-Sn catalysts and running them under steam reforming conditions where coking was not observed. Sulfur is typically a poison in transition metal catalysts, either by site-blocking or formation of inactive bulk sulfides. Alloys are one route to improving the sulfur tolerance of catalysts. Sulfur can be present in many forms including H2S, sulfurcontaining hydrocarbons and SO2. Zhao et al. investigated the chemisorption of SO2 on Cu/Au(111) surfaces using synchrotron techniques and DFT.55 They found using DFT that Cu preferentially segregates to the second layer down from the surface by comparing the total energies of the alloy slabs in different compositional configurations. SO2 bonds more strongly to Cu than to Au, and the more Cu atoms there were in the surface the stronger the SO2 adsorption energy was. The Cu-SO2 bonds were Chem. Modell., 2008, 5, 150–181 | 165 This journal is

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not strong enough, however, to induce Cu segregation to the surface. On Cu(111) SO2 dissociates and the resulting atomic oxygen oxidizes the Cu surface, corroding it. In contrast, on the Cu/Au(111) surface the Cu atoms dissolve into the second layer and SO2 desorbs molecularly. Thus, a Cu-Au alloy may be protected from corrosion due to the lack of active sites for SO2 dissociation. Pd alloys are often used in dense metal membranes to separate hydrogen. However, most hydrogen streams that need purification also contain sulfur compounds such as H2S. These sulfur compounds can poison the surface, reducing the effectiveness of the membranes. Jiang et al. found that alloying a small amount of Pt into Pd improved the sulfur resistance of the alloy.56 Incorporation of Pt increased the adsorption energies of both H2 and H2S, but the adsorption energy of H2 was increased more. Furthermore, they found that H2S prefers to bond to an atop Pt site if it is available, while H2 always dissociates in three-fold sites. Therefore, in addition to changing the adsorption energies differently the alloy surface also changes the preferred sites for the different adsorbates. The interactions of S and S-compounds with transition metal surfaces and alloy surfaces are not trivial. Whereas simple correlations between the adsorption energy of H and O and the surface d-band center of many alloy structures have been observed in many cases, Hyman and co-workers report that the Fermi energy and density of states at the Fermi level must also be considered to understand and predict trends in H2S reaction energetics on alloy surfaces where strain and ligand effects are important.57 In the absence of ligand effects, they found that the d-band center and density of states at the Fermi level were strongly correlated and only one of them was necessary to predict adsorption properties. However, ligand effects in many alloys destroy the correlation and then it requires both parameters to understand trends in adsorption properties. The density of states at the Fermi level is an indicator in bonding because these electrons participate directly in bonding, and they indicate the density of states just above the Fermi level which are necessary to reduce antibonding repulsion. The authors also found that correlations between changes in the adsorption properties of H2S were well correlated with changes in the adsorption properties of S on all the surfaces. Thus, correlations between adsorbate properties may be more useful for predicting changes in other adsorbates than electronic structure correlations. In a solid oxide fuel cell, fuel is electrochemically oxidized at the anode. The fuel is typically hydrogen or a hydrocarbon, and very often the fuel contains trace amounts of sulfur-containing compounds. The highly active Ni-based catalysts that are preferred are not sulfur tolerant due to nickel sulfide formation, and the more sulfur-tolerant Cu catalysts are not sufficiently active. Choi and coworkers have used DFT to investigate the sulfur tolerance of Ni-Cu alloy surfaces.58 They found that S bonds more strongly to Ni than to Cu, and that alloying Cu into the Ni surface reduces the adsorption energy of the S making the alloy surface more sulfur tolerant than a pure Ni surface but not as tolerant as a Cu surface.

4. Reaction kinetics The ultimate test of whether a bimetallic catalyst is practically interesting is of course how the catalyst performs a chemical reaction of interest. Characterizing the complete mechanism and rate of a surface catalyzed reaction is certainly possible with DFT calculations, but this process remains time consuming even for structurally simple surfaces and chemically simple reactions. In this section, we review DFT calculations that have approached this task for a variety of reactions. Perhaps the key quantity of interest in most catalyzed reactions is the activation energy of the rate determining step(s), so we have only included work in this section that explicitly computed activation energies by finding transition states on the potential energy surface of the reacting species. The section is organized according to the reactions that were studied in roughly increasing order of chemical complexity. 166 | Chem. Modell., 2008, 5, 150–181 This journal is

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4.1 CO oxidation It is probably reasonable to say that there have been more DFT studies of CO oxidation on metal surfaces than for all other surface catalyzed reactions combined. The popularity of this reaction is driven in part by the simplicity of the chemical species it involves, but it is also driven by technological demands. In past years, much activity focused on developing and understanding catalysts for use in automobile catalytic converters. More recently, attention has shifted to catalysts for fuel cells, where catalyst poisoning by CO can be a severe issue. Because bimetallic catalysts can potentially reduce the cost of fuel cell catalysts relative to the ‘‘standard’’ material, Pt, and also mitigate the effects of CO poisoning, a larger number of experimental and theoretical groups are actively studying CO chemistry on bimetallic surfaces. Ji and Li studied CO dissociation on Pt(111) and in Mo-doped Pt(111) where 1/4 of the surface atoms are Mo.39 The bimetallic surface weakens the chemisorption of CO to the surface and also reduces the activation energy for CO dissociation. Jiang and Carter reported similar calculations for CO dissociation on a monolayer of Al on Fe(100).59 CO was found to bind weakly on this surface and to dissociate with a barrier of B0.35 eV, compared to the barrier of B1 eV on Fe(100). Wang, Jiao, and Bu analyzed several possible mechanisms for CO oxidation on ordered Au3Ni(111) surfaces.60 When both CO and O2 are chemisorbed on the surface, a pathway to formation of CO2 with a barrier of only 0.13 eV was found. In this reaction, the chemisorbed O2 has a di-s configuration on the surface with an adsorption energy of 1.3 eV. The existence of this pathway is interesting, but it gives an incomplete description of steady state formation of CO2 for at least two reasons. First, Wang et al. also showed that chemisorbed O2 can dissociate into two adsorbed O atoms with a negligible barrier. This process is highly energetically favored, indicating that the population of chemisorbed O2 on the surface will be extremely low. Moreover, the pathway outlined above for CO2 formation also creates strongly bound atomic O on the surface. This work therefore gives interesting hints regarding the overall activity of ordered Au3Ni(111) surfaces, but doesn’t give a complete picture of the steady state reaction kinetics of this surface. Dupont, Jugnet and Loffreda carefully characterized the kinetics of CO oxidation on O-precovered PtSn surfaces, comparing their results with pure Pt.61 Two kinds of Pt-Sn surfaces were considered, one in which the top layer of a Pt(111) surface is replaced with Pt3Sn, and separately the (111) termination of an ordered Pt3Sn crystal. Using previous results for Pt(111), the authors make the plausible argument that the dissociation of O2 into adsorbed atomic oxygen is not rate limiting, so understanding the reaction between adsorbed CO and O is sufficient to characterize steady state reaction kinetics. An especially useful feature of this work was that the vibrational frequencies (including surface phonons) were computed for each energy minimum and transition state within the harmonic approximation. These frequencies allow zero point energy corrections and transition state theory prefactors to be calculated. The existence of a surface layer of Pt3Sn on Pt(111) was found to lower the barrier to CO oxidation by 0.18 eV relative to Pt(111) and also to increase the transition state theory prefactor by an order of magnitude. The activity of Pt3Sn(111) was also shown to be higher than that of Pt(111), but this material is predicted to be less active for CO oxidation than the material with a single surface layer of Pt3Sn. Analysis of CO oxidation on surfaces precovered with O was also performed by Gonza´lez, Sousa, and Illas for bimetallic RhCu(111) surfaces.62 Because Cu has very low miscibility in Rh, the Rh-rich bimetallic surface was modeled using a slab with one Cu atom in a (3 3) surface unit cell and no Cu atoms in the subsurface layers. Chem. Modell., 2008, 5, 150–181 | 167 This journal is

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A moderate reduction in the activation energy for CO oxidation from B1.4 eV on Rh(111) to B1.1 eV on some sites on the bimetallic surface was found. Other paths with considerably lower activation energies were also found, but the authors were careful to point out that these paths are unlikely to be thermodynamically relevant. These authors also examined NO dissociation on the same surface and an analogous Cu-rich surface.63 Similar to CO, the activation energy for NO dissociation is lower on the Rh-rich bimetallic surface than on Rh(111), while the opposite was found for the Cu-rich bimetallic. Wang and Hammer approached the challenging problem of describing catalytic oxidation on a metal surface in a low temperature fuel cell by considering CO oxidation on Pt2Mo(111) covered by a bilayer of water.64 Materials that readily oxidize CO in this environment are desirable because they would help make a fuel cell tolerant to CO. The binding energy of CO on the bimetallic surface is lower than on Pt(111) in the presence of adsorbed water. The oxidation of CO in the presence of water can occur via formation of surface hydroxyl species and subsequent oxidation of CO via a COOH intermediate. The bimetallic surface was calculated to be much more favorable for OH formation than Pt(111), and the overall formation of CO2 from the water bilayer and adsorbed CO can take place with an overall barrier of B0.3 eV. It is especially interesting to note that the activation energies relevant to COOH intermediates in the absence of the water layers are considerably higher than when the water is present. This trend is also observed in studies of electrocatalytic systems which are reviewed later in section 6.

4.2 Water gas shift (WGS) catalysis The water gas shift (WGS) reaction, conversion of CO and H2O to CO2 and H2, is of enormous importance in future scenarios for generating large amounts of H2 from fossil fuel sources. This reaction is closely related to CO oxidation—in fact the work discussed above regarding CO oxidation in an aqueous layer could also be thought of as an example of WGS chemistry. Knudsen et al. performed a clever combination of experiments and DFT calculations to identify a near-surface alloy of Cu/Pt as a promising bimetallic WGS catalyst.65 First, they used STM and XPS measurements and DFT calculations to show that Cu deposited on a Pt(111) crystal preferentially segregates to the subsurface layer, with much smaller amounts of Cu appearing in layers further from the surface. Having established the stability of this NSA, DFT calculations were used66,67 to probe three key characteristics of the surface of a WGS catalyst: (i) chemisorption of CO, (ii) chemisorption of formate, and (iii) activation of H2O. The first two species are known to block sites on WGS catalysts, so reductions in these bond strengths are desirable. The calculations indicate that both CO and formate are destabilized on the NSA relative to Pt(111). This effect was confirmed directly in experimental measurements of CO desorption. Water activation is useful as a proxy for WGS activity because for a number of well characterized catalysts this is the rate limiting step. DFT calculations indicate that the activation energy for H2O dissociation is lower on Pt(111) than on the Cu/Pt NSA or Cu(111). Superficially, this may appear to make the NSA less attractive. The bond strength of OH on the NSA, however, is considerably weaker than on Pt(111), which would reduce poisoning of the catalyst by OH groups. An obvious direction for future work motivated by this study would be the synthesis and testing of a high surface area catalyst based on the Cu/Pt NSA. Another study that is relevant for WGS chemistry used a DFT cluster approach to describe water dissociation on bimetallic Pt/Ru clusters.68 A cluster in which a single Ru atom sites on top of a Pt(111) plane was found to be more active for water dissociation and CO oxidation than a flat alloy surface, but it is not clear that this type of material is thermodynamically stable. 168 | Chem. Modell., 2008, 5, 150–181 This journal is

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4.3 Methanol steam reforming Catalytic reforming of methanol with water, CH3OH + H2O - 3 H2 + CO2, is an attractive candidate reaction for fuel cells because methanol is relatively inexpensive and is easy to store and transport. One kind of catalyst that holds promise for this reaction is a combination of Pd and ZnO. Several lines of evidence point to the 1:1 PdZn alloy as being the active component in this catalyst. Ro¨sch and co-workers have performed detailed DFT calculations to understand a series of relevant processes on this alloy.69–71 Calculations of the surface energies of various surfaces of PdZn (in the L10 structure) indicated that the (111) surface has the lowest surface energy and that at low temperatures surface segregation is not a strong effect.71 Activation energies for the dehydrogenation of formaldehyde were calculated on PdZn(111) and also PdZn(211), a stepped surface.69 For comparison, similar calculations were performed on Cu(111) and Pd(111). This dehydrogenation reaction was calculated to be favorable on Pd(111) (relative to formaldehyde desorption), but not on the flat Cu or PdZn surfaces. The reaction is more favored on the stepped PdZn surface than the flat bimetallic surface, but not as strongly as on Pd(111). Lim, Moskaleva, and Ro¨sch also tackled the important task of assessing the surface coverage of adsorbed species under conditions relevant for practical catalysis.70,71. Specifically, they used ab initio thermodynamics to describe PdZn(111) in the presence of H2O and H2 at 500 K. The general principles underlying ab initio thermodynamics have been reviewed in an earlier SPR2 and elsewhere72–77. These methods have limitations in systems where the coverages of adsorbed species are coupled via reactions, Lim et al. also derived a kinetic model for the surface coverages of the relevant surface species. The main outcome of this analysis is that the surface is expected to be free of all water-related adsorbates, including surface oxygen, at 500 K whenever the partial pressure of H2 exceeds a very small value, estimated to be B104 atm. This result is a strong indication that the previous calculations that had been performed to understand the catalytic chemistry of formaldehyde and related species on this surface are relevant to realistic catalysts.

4.4 Other reactions Albenze and Shamsi used DFT to characterize H2S dissociation on Ni(111) and on two bimetallic surfaces that mimic Ni-Mo solid solutions with o20% Mo.78 The existence of Mo atoms in the surface layer was found to reduce the energy barriers that exist to H2S dissociation, but this reaction is predicted to be facile on all three surfaces at elevated temperatures. Although some calculations on the effect of surface S coverage were performed, it is not clear how relevant these calculations are to experimental observations with Ni/Mo catalysts, which tend to sulfide rapidly. Most of the papers we have discussed apply DFT to a small number of adsorbed species on one or two specific surfaces. An ambitious effort to explore a larger variety of surfaces with a reduced level of rigor was developed by Kieken, Neurock, and Mei.79 In this work, extensive DFT calculations were performed for the energies and activation barriers for NO decomposition in the presence of excess O2 on Pt(111). These results were then used to parameterize a bond order conservation (BOC) description of each intermediate and reaction. The important advantage of this description is that it could be extended with relative ease to a large number of PtAu alloy surfaces without performing detailed DFT calculations on these alloys. Kinetic Monte Carlo simulations were used to simulate the steady state kinetics on each surface considered by defining a lattice gas model of the reaction kinetics from the BOC description. By considering a large variety of Pt-Au(100) surfaces with random distributions of Pt and Au atoms, surface compositions that increased the turnover frequency relative to pure Pt by up to a factor of two were found. Chem. Modell., 2008, 5, 150–181 | 169 This journal is

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Independent DFT calculations of the key steps on surfaces with this composition would have provided a useful test of the BOC approach, but at least in this paper no calculations of this kind were reported. This general approach of a limited dataset of DFT results with more empirical descriptions that give rapid access to a large number of material compositions should receive more consideration. In general, it is of course desirable to examine the most promising materials found in a screening exercise of this kind through more rigorous means.

5. Miscellaneous In this section, we discuss two topics that are not direct applications of bimetallic catalysts but for which a number of theoretical studies have appeared that also give useful insights into the properties of bimetallic catalysts. 5.1 Metals supported on metal oxides The striking catalytic properties of small Au clusters on metal oxide supports has attracted enormous amounts of experimental and theoretical attention. This area was reviewed in detail in an earlier RSC SPR.2 Here, we simply mention several recent theoretical studies of small metal clusters on metal oxides to give the interested reader an entry into this literature. Most of this work has focused on MgO substrates, since the structural simplicity of this material is advantageous for both theoretical and experimental studies. A key observation that is now well established is that the catalytic activity of metal clusters is often quite different on defect-free and defective MgO surfaces. A particularly interesting example of this phenomenon was analyzed by Yoon et al., who performed experiments and calculations for CO oxidation by Au8 clusters on MgO.80 In this instance, the presence of a surface defect, specifically, an oxygen vacancy, underneath the cluster is critical to the cluster being catalytically active. The combination of experimental and theoretical analysis given by Yoon et al. indicates that charging of the clusters adsorbed on these defects is crucial. The catalytic activity of Au particles supported on metal oxide supports was reviewed in depth by Molina and Hammer.81 They were able to conclude that lowcoordinated edge and corner sites are crucial to understanding the catalytic activity of Au particles. Similarly, the Au/oxide interface periphery can be highly active, with the precise type of interface having an additional effect on activity. The exact nature of the CO oxidation reaction that is catalyzed by Au nanoparticles supported on TiO2 was not well understood at this time, but later work by Wang and Hammer was able to reveal a mechanism by which atomic oxygen adsorbs at perimeter sites near the support and CO further away from the support, and are then able to react with small energy barriers.82 A number of papers have examined the adsorption of metal atoms other than Au on MgO. Fernandez et al. systematically studied all first row transition metals (KZn) on defect-free MgO(100), at coverages corresponding to epitaxial overlayers and at coverages representative of isolated atoms on the surface.83 The trends in adsorption energy for these species were dominated by the interaction between the adsorbed atoms and O atoms in the surface, although interactions with adjacent surface Mg atoms also contribute for the larger adsorbed species (e.g. K and V). It is not clear from these results whether these trends would transfer in a simple way to defects on the oxide surface. Two studies using cluster models for MgO(100) and localized basis sets have given insight into the complexities of understanding small clusters of metal atoms on metal oxide supports. Giordano et al. examined the stability of Pd dimers on a range of defects on MgO(100), including neutral vacancies (also known as F centers), monoatomic steps, and surface OH groups.84 To characterize the stability of these adsorbed dimers, the energy of the dimer relative to having a single Pd atom 170 | Chem. Modell., 2008, 5, 150–181 This journal is

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adsorbed at the defect and another Pd atom adsorbed on a defect-free MgO(100) terrace was calculated for each type of defect. For each kind of defect, this comparison showed that it was energetically favorable to move the Pd atom from the defect-free surface to form a dimer at the defect site. More interestingly, the F center showed the weakest energy for dimer formation of the three kinds of defect sites. As Giordano et al. point out, however, this does not necessarily mean that step or OH sites will dominate nucleation of Pd clusters on MgO(100), since in many situations the density of these sites will be far lower than the density of F centers. Barcaro and Fortunelli have tackled the difficult problem of probing the structure of bimetallic clusters adsorbed on defect sites on MgO(100).85 In particular, they studied Pd1AgN clusters for N = 1–8 adsorbed on F centers on MgO(100) using a cluster model of MgO(100) and localized basis sets. A basin-hopping algorithm was used to search for the lowest energy structure of each cluster. Multiple local minima among the cluster configurations found with this method that lie within 0.3 eV of the lowest energy structures. One striking conclusion from these calculations is the prediction that Pd1Ag6 is a magic cluster on the surface, with a large HOMO–LUMO gap. In the preceding studies, the metal oxide defines the substrate upon which metals are deposited. Ferrari, Roetti, and Pisani studied the complementary situation of thin MgO islands deposited on Ag(100) using B3LYP DFT in a periodic slab geometry.86 The dissociation of water on these surfaces was found to be dominated by the edge sites of the MgO islands. An additional complication in understanding the properties of small metal clusters on metal oxide supports is that controlled growth of these clusters is often achieved in practice by creating a thin metal oxide layer on top of another substrate. This situation can in some cases give the resulting metal clusters properties that are different from those adsorbed on a bulk metal oxide. A recent example that provides a good overview of this interesting area is the work of Honkala and Ha¨kkinen, who reported calculations for adsorption of isolated Au atoms on thin MgO(100) films supported on Mo.87 The adsorption energy of an Au atom on a defect-free film is significantly different from the outcome for a bulk MgO substrate, even when the film is five layers thick. In contrast, if the Au atom was located at a defect on the MgO film, its properties were very similar to those of the same defect on a bulk MgO substrate. Other aspects of this problem have been reported by Pacchioni and co-workers.88–90 Similar calculations were performed by Wang, Fan, and Liu for Au clusters on the (101) surface of anatase TiO2 that had been modified by depositing either AlOOH or Al2O3.91 These calculations included assessments of the oxidation of CO on the surface. The binding energy of Au atoms on the Al2O3-modified anatase was computed to be far larger than the equivalent energy on bare anatase, but the boundary between the adsorbed Au clusters and anatase was determined to be the active site for CO oxidation in both cases. Taniike et al. used DFT calculations to probe NO-CO reactions on a dimer of Co atoms on a g-alumina support.92 These dimers had been shown to exist in previous experiments that also suggested an unusual mechanism in which NO is reduced to N2O via interaction with gas phase CO molecules. The calculations indicated that Co dimers allow the formation of an adsorbed cis-(NO)2 species that has an unusually large reactive cross section for reaction with gaseous CO via an Eley-Rideal mechanism. The dynamics and growth of small metal clusters on substrates is of course important for realistic catalysts, but is also relevant in other areas. One recent example is the work of Wu et al., who examined Cu atoms and small Cu clusters (up to 4 atoms) on WN(001) as a model for Cu agglomeration in barrier materials for semiconductor devices.93 A noteworthy aspect of this work is that it explicitly considered the pathways for diffusion and growth of small clusters. Chem. Modell., 2008, 5, 150–181 | 171 This journal is

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5.2 Magnetic properties Several studies have examined the magnetic properties of bimetallic surfaces. Canto and Ordejoń performed GGA calculations for the (001) surface of TiFe, an ordered intermetallic compound with the CsCl structure.94 If this surface is defect-free it has two possible terminations, one with a Fe layer and another with a Ti layer. The calculations indicate a strong enhancement in the surface magnetic moment for the Fe-terminated surface, but a much weaker effect for the Ti-termination. Achilli et al. used Green’s function-based DFT calculations to study a perfect monolayer of Fe on a Cu(100) substrate.95 To understand the relative roles of bonding of this monolayer to the substrate and the strain induced by the substrate, calculations were also performed for an unsupported Fe layer. The magnetic moment of the Fe atoms was larger than the bulk value for both the supported and unsupported layers, with the effects of the substrate being fairly weak. Pick and Demangeat reported DFT calculations of thin Mn films on Co(001) substrates.96 In this system, earlier experiments had suggested that surface oxidation could play a decisive role in the surface magnetic ordering, so calculations for several possible states of adsorbed oxygen were also performed. These calculations confirm that the precise Mn film thickness and organization of O atoms can alter the characteristics of the surface magnetism, while at the same time suggesting that the atomic configurations that have been observed experimentally are more complex than the ones examined theoretically. Qian used a composite elastic description to assess the surface stress in W(001) with and without an overlayer of Fe.97 This approach uses DFT calculations at several different lattice constants as input into a description of the surface stress based on linear elasticity. This framework may provide useful insight into the roles of surface stresses in other similar situations.

6. Electrocatalysis Electrocatalysis is receiving increasing levels of attention in the computational community due to the recent interest in fuel cells and electrochemical energy conversion technology. The presence of an electrolyte and electric potential substantially complicates modeling efforts of these systems. Simple models to account for these phenomena were developed by Nørskov and co-workers,98,99 while more sophisticated approaches were developed by Neurock and co-workers.100–106 There are other approaches as well, but these two are the ones most often used in the work reviewed here. Other notable approaches include that by Alavi and co-workers107– 109 , and the Anderson group’s approach.110 In this section, we review approaches to predicting dissolution potentials in 6.1. Next we discuss the hydrogen evolution reaction and studies to understand it and predict new electrocatalysts in 6.2. Electrochemical oxidation reactions that are typical in fuel cells are discussed in 6.3. Finally, the studies on the increasingly important oxygen reduction reaction are reviewed in 6.4. 6.1 Dissolution Alloy stability is always of concern in heterogeneous catalysis, but in electrocatalysis there are new mechanisms for destabilizing alloys, namely electrochemical dissolution or corrosion. Greeley and Nørskov developed an intuitive and simple thermodynamic framework for estimating the stability of alloy surfaces in electrochemical environments.111 Their scheme is essentially an extension of an atomistic thermodynamic approach that uses chemical potentials to determine stability to one that uses electrochemical potentials to determine stability. They estimate the electrochemical potentials using total energies calculated within DFT and ideal solution behavior of the ions to consider concentration and pH effects. Within this formalism they are able to estimate the dissolution potential of metals in alloys. They further compared the trends in dissolution behavior to trends in segregation behavior and 172 | Chem. Modell., 2008, 5, 150–181 This journal is

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found they tend to follow each other. The simple and intuitive reasoning for this is related to the stability of the surfaces. Solutes that are more stable in the surface are more resistant to dissolution. The most significant results in this work are the prediction that Pt-skins from Pt3X (X = Fe, Co and Ni) are more stable to dissolution than pure Pt surfaces are. However, when water splitting also occurs, the dissolution potentials of Pt are reduced because the OH adsorption energy is greater on pure Pt than it is on the alloy surface. Consequently, there is a thermodynamic driving force to form pure Pt surfaces. Gu and Balbuena also predict that most solutes dissolved in Pt reduce its stability towards dissolution.112 They investigated the role of oxygenated intermediates in the oxygen reduction reaction on the dissolution mechanisms of Pt surfaces. They found that the most favorable mechanism for dissolution is an electrochemical mechanism (as opposed to a purely chemical mechanism) that involves oxygenated metal surface atoms. Their investigation focused on single metal atoms or dimers, however, and used small clusters of water to include some solvation effects. They compared the relative stabilities of Ir, Pd, Rh, Ni and Co and found that Pt-Ir was the most stable. 6.2 Hydrogen evolution In order to predict the electrocatalytic properties of an alloy surface, for instance, its hydrogen evolution activity, we must understand the factors that contribute to electrocatalytic activity. Greeley and coworkers developed a simple view of activity trends in hydrogen evolution on Pd-monolayers supported on other metals based on the free energy of hydrogen adsorption.113 With this descriptor they observe a volcano curve in activity on a series of Pd monolayer bimetallic systems with the maximum activity occurring at a DGH* of 0 eV, where H* represents an adsorbed H atom. For metals with greater DGH* hydrogen evolution is limited by a high barrier to get H* on the surface, whereas for bimetallic surfaces with DGH* much less than zero the reaction is limited by getting H* off the surface. The variations they observed in DGH* are largely explained by changes in the electronic structure (the dband center) in the Pd monolayer due to interactions with the support metal.114 Once a good chemical descriptor has been identified, such as DGH* or the d-band center in the previous work, then large-scale screening approaches to identifying new alloy electrocatalysts can be attempted. Greeley et al. have used DFT to screen over 700 binary alloy surfaces to identify candidate surfaces with DGH* near 0 eV.115,116 They further screen the candidate structures with the right hydrogen adsorption properties to exclude candidates that are not stable with respect to segregation, dissolution or that would be expected to be poisoned by hydroxide intermediates. Using this approach they identified PtBi as a promising hydrogen evolution electrocatalyst and demonstrated experimentally that it was indeed more active than pure Pt. The significance of their approach is that Bi is known to be a bad hydrogen evolution catalyst on its own, and the primary reason it was tried experimentally is that the computational approach ‘‘found’’ it. Finally, there is a substantial need for non-Pt based electrocatalysts. Pt is very expensive and not particularly poison tolerant. Jaramillo et al. used the concepts described above to identify MoS2 as a potential hydrogen evolution electrocatalyst.117 They observed that the DGH* on MoS2 was only 0.08 eV. This is sufficiently close to 0 eV that they hypothesized it could be a good hydrogen evolution electrocatalyst. They demonstrated this by synthesizing MoS2 nanoparticles on a Au(111) surfaces and found indeed MoS2 is a reasonable electrocatalyst for this reaction that is better than pure Mo, but not as high in activity as Pt. 6.3 Anode reactions At the anode in a polymer electrolyte membrane (PEM) fuel cell, the fuel is typically oxidized to create protons and electrons and CO2 in the case of carbon containing Chem. Modell., 2008, 5, 150–181 | 173 This journal is

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fuels. One major concern in hydrogen fuel cells is that residual CO in the fuel stream could poison the anode electrocatalyst. One approach to solving this problem is to develop alloy anode materials that either weaken the CO adsorption energy or make it easier to oxidize, and preferably without lowering the hydrogen oxidation activity. Shimodaira et al. have used DFT to screen a large number of Pt5M5 alloy clusters with these goals in mind.118 Specifically, they sought alloy surfaces where the CO adsorption energy was weaker than that on Pt, and where the activation energy of hydrogen scission was not higher than that on Pt. Among the systems they examined Co-Pt, Pt-Ni, Pt-Au, Pt-Fe, Pt-Mo and Pt-Ru were identified as meeting these criteria. They also investigated the stability of these surfaces with respect to segregation and found their results were in reasonable agreement with what is expected from the difference in experimental surface tension data. That is to say, the metal with the lowest surface tension tends to segregate to the surface. CO oxidation is critically important at the anode. If the CO can not be removed from the anode during operation it will block all the sites and poison the surface. Davies et al. have examined the ligand effect in Pt/Ru catalysts on the desorption of CO using DFT and steady state isotopic transient kinetic analysis.119 They found the primary effect of the alloy surface is to reduce the CO coverage due to the reduced adsorption energy of CO on the alloy surface. The exchange rate of CO on the surface was unaffected by the alloy surface. However, in the presence of hydrogen gas the exchange rate is influenced on the alloy because hydrogen can compete for adsorption sites, resulting in lower desorption/exchange rates for CO in the presence of hydrogen gas. 6.4 Oxygen reduction From a catalysis standpoint the oxygen reduction reaction at the PEM fuel cell cathode is one of the biggest challenges for fuel cells. The main problem is the reaction rate is too slow, resulting in high overpotentials and low performance in fuel cells. One solution to this problem has been the use of alloy surfaces to increase the reaction rates. Wang and Balbuena developed thermodynamic guidelines for designing oxygen reduction catalysts based on DFT calculations of elementary reactions steps on seventeen transition metal atoms from groups V to XII and on clusters of three atoms for ten of those metals.120 In this highly simplified model system they found a significant correlation between the reaction energy in the formation of OOH by hydrogenation and the reductions of O and OH. This correlation showed that metals that were good at formation of OOH were not good at reduction of the intermediates and vice versa. Furthermore, these reactivities correlated well with the number of vacancies in the d orbitals. Metals with unfilled dorbitals were good at forming OOH, and metals with filled d-orbitals were better at reducing the intermediates. They propose that coupling metals from these two groups would result in a superior bifunctional electrocatalyst. Calvo and Balbuena examined the structure and reactivity of Pd-Pt nanoclusters with 10 atoms in the oxygen reduction reaction.121 In contrast with what is expected in a periodic slab calculation, they found that ‘‘mixed’’ states with randomly distributed Pd atoms in a Pt7Pd3 cluster was more stable than an ‘‘ordered’’ cluster structure due to more effective charge transfer in the ‘‘mixed’’ state. They found that increasing the concentration of Pd in the surface favors formation of the OOH species in the first step of the reaction, but Pt atoms were needed to promote the second stage of the oxygen reduction reaction. They report that due to charge transfer effects the Pd atoms have an intermediate reactivity between pure Pd and Pt, and in the mixed cluster the Pd atoms the Pd atoms act more similarly to Pt than in the ordered cluster. There are two mechanisms that are considered in the oxygen reduction reaction. In the series mechanism oxygen molecules are protonated twice generating hydrogen peroxide as an intermediate which can then be further reduced. In the direct 174 | Chem. Modell., 2008, 5, 150–181 This journal is

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mechanism the O–O bond is broken and then the atomic adsorbates are each protonated twice to form water. Lamas and Balbuena investigated these two mechanisms on Pd-Co alloy surfaces by calculating the binding energies of oxygen reduction intermediates on Pt, Pd, Pd0.75Co0.25, and Pt0.75Co0.25.122 In this work they showed that both mechanisms are likely to be operating in parallel and that the highest thermodynamic barriers were in the first hydrogenation step in both mechanisms. Although they used an electrochemical potential formalism to estimate the thermochemical barriers of electrochemical reactions, they did not include solvent effects and their results are for oxygen coverages of 1/4 ML. Hyman and Medlin considered relationships between the adsorption energies of oxygen reduction intermediates and the surface electronic structure of Pt surface alloys.123 In this work they used model surface structures to disentangle strain and ligand effects in the alloys. They found that compressive strain destabilized the adsorption energies of all intermediates due to the broadening of the surface d-band. For the OH, O2 and OOH species they found strong correlations between the changes in adsorption energies of these species on all the surfaces including both strain and ligand effects. In contrast, they found poor correlations between these intermediates and the O adsorption energy. The d-band center was only an effective descriptor of adsorption energies for the O atom, and there were separate correlations for the 3d, 4d and 5d alloy components. They found that they could not predict the changes in adsorption energy due to strain and ligand effects by separately estimating the strain effect and ligand effect and then summing them. These effects are not independent of each other; the magnitude of the ligand effect depends on the amount of strain.124 Peroxide formation at the cathode is problematic for two reasons. First it is only a two electron process and not as thermodynamically favorable as formation of water, which results in a lower fuel cell potential. Second, it may decompose into oxidizing radicals that can degrade the membrane and catalyst support. Balbuena and coworkers used three atom clusters of Pt with Ni, Co or Cr to investigate the catalytic decomposition of hydrogen peroxide.125 H2O2 binds weakly to the Pt top positions in all of the bimetallic clusters and much stronger on the 3d metals. H2O2 dissociates in the hollow sites of these clusters. The three atom clusters appear substantially more reactive than a 10 atom cluster and more reactive than periodic surfaces of similar composition. Solvation effects are not negligible in this system, and the authors showed that water molecules destabilize adsorbed H2O2 by hydrogen bonding. Roques and Anderson used a different approach to understanding the increased reactivity of Pt-Co electrocatalysts in the oxygen reduction reaction.126 Based on the observation that OH species adsorb strongly to the surface and block sites for further reactions they set out to determine the electrochemical stability of these adsorbates on different alloy surfaces. They developed a model where the potential at which the intermediate is stable on the surface, and more importantly the change in that potential due to alloying, is directly related to the change in adsorption energy of the OH species. This allows them to compare potential shifts on surfaces with different compositions. Relative to Pt(111) they found that incorporation of Co into the subsurface layers weakens the OH adsorption energy and increases the potential at which adsorbed OH is stable. The effect stabilizes above 75% Co in the second layer. The changes in adsorption energies are again largely explained by shifts in the d-band center, but only for Pt atoms directly bonded to sub-surface Co. They reported a good correlation between the Pt surface charge and the OH adsorption energy for the alloys (not including a pure Co surface). Fernandez et al. used a combined high throughput electrocatalyst study and DFT study to examine Pd-Co electrocatalysts for oxygen reduction.127 They screened an array of electrocatalysts using a scanning electrochemical microscope to assess the activity of each element. The DFT study was performed to help identify the role of Co in the system. They examined oxygen molecule adsorption, dissociation and Chem. Modell., 2008, 5, 150–181 | 175 This journal is

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oxygen atom diffusion at and around a Co impurity in a Pd surface. They found that Co atoms in the surface substantially reduce the dissociation energy of O2 on the surface, but that the Co atom is likely poisoned by the strong Co-O bond. However, their preliminary calculations suggest O2 can still dissociate more easily near a poisoned Co atom than on a pure Pd surface. They estimated that on a clean surface Co would prefer to anti-segregate to the second layer, but in the presence of oxygen the strong O-Co bonds make it stable in the top surface layer. Ternary alloy surfaces offer another dimension in tuning the catalytic properties of oxygen reduction electrocatalysts. Zhang et al. used a novel synthetic approach combined with DFT studies to identify active electrocatalysts and explain trends in ternary alloy electrocatalysts.128 They used galvanic displacement of underpotential deposited Cu on Pd surfaces to create mixed Pt-M alloy surfaces supported on Pd nanoparticles as electrocatalysts. They chose M based on DFT calculations so that M would strongly bond O or OH resulting in lateral repulsion of other O and OH species near the Pt surface atoms. They are able to tune the Pt properties both by the interactions with Pd in the subsurface and by ligand effects with M in the surface. Re and Os were found to destabilize the OH on Pt the most, and had the highest activity. In contrast, Au and Pd were not effective at destabilizing OH on Pt and they had very low activity. Stamenkovic et al. used the adsorption energy of O on alloy surfaces as a catalytic activity indicator.129 They use this approach to develop a volcano model of activity for the oxygen reduction reactivity and compare their computed predictions of activity to experimental results. They showed that Pt-3d alloy surfaces are near the top of the volcano, but more significantly identified the chemical descriptor (d-band center or O adsorption energy) that could be used to identify new alloy surfaces that could have higher activity.

7. Conclusions The last 5 years have seen considerable advances in the use of quantitative computational models to understand (and in a limited number of cases, predict) the reactivity of active sites with relevance for practical bimetallic heterogeneous catalysts. The bibliography below is an indication of the high level of activity in this area by research groups around the world. There are multiple issues that are still challenging to address computationally in this area, so we conclude by outlining these areas and pointing to topics with the greatest probability for making large impacts in the field. Perhaps the central issue that runs through all of the areas we identify below is the need to sample a large range of active sites in order for theoretical methods to make physically meaningful predictions about the behavior of bimetallic materials. Significantly more work needs to be done in identifying the relevance or statistical probability of the range of sites that can exist in real materials and in relating atomistic scale properties to meso- and macro-scale properties of alloy surfaces with site heterogeneity. This is a very challenging problem that will require linking computational approaches that address different length and time scales as well as discrete to continuum descriptions of the physics at these scales. This problem will require considerable methodological development; a ‘‘business as usual’’ model in which individual surface sites are probed with plane wave DFT calculations cannot be expected to completely solve this problem. The general problem we just outlined can be reiterated for almost every property that controls the catalytic performance of bimetallic systems. Quantitatively describing segregation behavior in alloy surfaces remains challenging. On one hand it is relatively easy to compare the relative stabilities of two surface alloy configurations using DFT. However, the phase space is so rich that we cannot yet compare the relative stabilities of enough configurations to adequately sample the free energy landscape. Furthermore, the poor scaling of DFT calculation time with system size 176 | Chem. Modell., 2008, 5, 150–181 This journal is

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limits the size of the calculations to only a few hundred, perhaps up to a thousand transition metal atoms. Methods that use simple models (potentials) to identify structures to examine with DFT9,10 or use DFT to parameterize simpler and faster models are under development in several groups9,10,17,45,130 to address this problem. These models can be used to evaluate the role of entropy in determining composition profiles because they are substantially faster to calculate with than DFT. In other fields of computational materials design, such as bulk alloys, new methods based on genetic programming are being developed for sampling complex potential energy surfaces to build simple potentials.131–133 Another approach used in the bulk alloy community is to use datamining techniques on large databases to extract trends and correlations that can be used to predict new behavior.134–138 These machine learning approaches could certainly result in new understanding and insights into alloy catalysis simply by expanding our ability to analyze complex systems and identify non-trivial correlations between them. Some of these approaches are starting to be used in studies of alloy catalysis. The presence of a reactive environment can also influence the segregation profile in an alloy surface, i.e. the types and numbers of active sites. The complexity of the interactions of adsorbates with alloy surfaces makes it difficult to predict the effects of the environment on the alloy surface. By implication, it is difficult to predict a meaningful or relevant model of the alloy surface and its active sites under reaction conditions for use in simulating reactions. Future developments in this area will certainly advance the field. Modeling kinetic phenomena on alloy surfaces is an even more challenging area of research. The topics listed above can be largely described by thermodynamic approaches, and while not at all trivial, the concepts are essentially well understood. Reaction conditions, in contrast, often operate away from equilibrium and consequently thermodynamic approaches are only an approximation to real systems. Direct inclusion of reactive environments, such as bilayers of water, is increasingly common and is shown in many cases to significantly impact reaction barriers and adsorption energies on alloy surfaces. This adds additional challenges in identifying relevant configurations because these solvent configurations may have many local minima, making it difficult to get an ‘‘average’’ solvent effect. We did not see any examples of alloy calculations with an effective solvent medium while preparing this review, although this approach is very common in quantum chemistry calculations for other types of physical systems. Kinetic simulations that involve calculating reaction barriers or adsorbate vibrational modes are substantially more computationally time consuming than ‘‘simple’’ geometry optimizations. The high probability of heterogeneity in site composition and distribution further add to the challenges of choosing relevant active sites and reaction pathways that can be related to experimental results. New algorithms for exploring the manifold of possible reactions and identifying the most kinetically relevant reactions in alloy systems are critically needed to advance the field beyond the current situation where a ‘‘best guess’’ for reaction coordinates is often used. It seems likely to us that DFT will be used to parameterize simpler and faster models that can more rapidly search the phase space for relevant reaction pathways to study in more detail with DFT. While all of these challenges point to the need for new methodological developments, it is also important to remember that quantitative modeling of complex materials such as bimetallic catalysts must continually make contact with experimental reality. Innovative ways to combine theoretical and experimental observations that can jointly speed up the search for new catalysts are in the long run likely to make the largest impacts in the field. Heterogeneous catalysis will play a central role in the development of new technologies to address many of the pressing energy concerns facing society. Computational modeling is poised to make important contributions to this area, Chem. Modell., 2008, 5, 150–181 | 177 This journal is

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and we expect that this will be a vibrant area for both fundamental and applied research for the foreseeable future.

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Fluctuation relations, free energy calculations and irreversibility Debra J. Searlesa and Denis J. Evansb DOI: 10.1039/b608839m

1. Introduction Over the past 15 years a number of important fundamental theorems in nonequilibrium statistical mechanics of many-particle systems have been proved. These proofs result in a number of important relations in nonequilibrium statistical mechanics. In this review we focus on three new relationships: the dissipation function or Evans-Searles fluctuation relation (ES FR),1–5 the Jarzynski equality (JE)6–9 and the Crooks fluctuation relation (Crooks FR).10,11 These relations not only provide a better understanding of the behaviour of nonequilibrium systems (in particular their irreversibility), but provide new practical results that apply to manyparticle systems including derivations of transport properties (viscosity, thermal conductivity and shear viscosity) and completely new ways of calculating free energy differences between states and determining potentials of mean force. The purpose of this review will be to outline these three relations, discuss their links with irreversibility and calculation of free energy changes, and to provide a critical review of some recent work on these topics (i.e. on work published BJuly-2005–June-2007). The first of these relations, the ES FR, is also referred to as the transient fluctuation relation (transient FR) or the dissipation function fluctuation relation (O-FR). It gives an analytic expression for the probability that for a finite system observed R for a finite time, the average of the so-called ‘dissipation function’, t ¼ 1 t OðGðsÞÞds, takes on a positive or negative value. That is it states: O t 0 t ¼ AÞ PðO At t ¼ AÞ ¼ e PðO

ð1:1Þ

t takes on a value between A dA and t = A) is the probability that O where P(O A + dA. Since there are few analytic relationships that describe the behaviour of many-particle, nonequilibrium systems, this is an important result in itself. However, in many cases the dissipation function can be shown to be related to an important physical property, and it leads to some simple, fundamental results. In a wide class of nonequilibrium steady states, close to equilibrium (where entropy can be defined) the average time-integral of the dissipation function becomes the entropy production,12 and therefore it expresses the probability that for a finite time the entropy will be consumed rather than produced, as is normally the case in thermodynamic (macroscopic) systems. The ES FR can thus be viewed as a generalisation of the Second Law of Thermodynamics since it applies to finite systems and finite times. As well as providing a derivation of the Second Law of Thermodynamics, it predicts fluctuations in the sign of the entropy production do occur in small systems, and (1.1) provides a relationship to described their probability. It therefore has potential utility in the study of small systems and development of technology—including those of interest in nanotechnology and the development of nanomachines—where entropy consumption is predicted for measurable times. The importance of taking thermal fluctuations into account when measuring properties of small systems is a

Nanoscale Science and Technology Centre and School of Biomolecular and Physical Sciences, Griffith University, Brisbane Qld 4111, Australia b Research School of Chemistry, Australian National University, Canberra ACT 0200, Australia


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indicated by (1.1).13 The result can also be applied to Hamiltonian systems, giving relationships that are consistent with Le Chatelier’s principle for large systems.14 It can be used to derive the Green-Kubo relations for transport coefficients4,15–17 and shows how thermodynamic irreversibility emerges from time reversible equations of motion.17–19 Finally, fluctuation relations for arbitrary properties that are odd with respect to time reversal can be obtained. These expressions are more complex than (1.1), but have been verified numerically.20 The JE and the Crooks FR, in contrast, predict free energy differences between different equilibrium systems. Free energy differences provide us with crucial information on the relative stability of states and the direction of change of a process. If free energy differences are known, important information including phase diagrams and the stability of nanoscale structures for example, may be determined. Furthermore, these relations allow free energy differences to be calculated in both molecular simulations and experiments. Through simulation studies, insight into the behaviour of these systems can be obtained at the molecular level, extending our understanding of important phenomena such as phase transitions, nonequilibrium response and the behaviour of nanoscale systems. The Crooks FR has a very similar form to the transient FR, P1 ðw1!2 ¼ AÞ ¼ ebðDF1!2 AÞ P2 ðw2!1 ¼ AÞ

ð1:2Þ

where Pi(wi-j = A) is the probability that the work done along a path from an equilibrium state i to a second equilibrium state j is between A dA and A + dA, and DFi-j is the free energy difference between states i and j. Furthermore, the arguments of the two fluctuation relations (work and the dissipation function) can be shown to be closely related.11,17,21 The novelty of the Crooks FR lies in the fact that it allows calculation of differences in free energies of two equilibrium states using the work along irreversible paths between them. Integration of (1.2) gives the Jarzynski equality, hebw1-2i1 = ebDA1-2.

(1.3)

However, we note that this relation was derived prior to (1.2), in a different way.6,7 We also note that like the ES FR, the JE can be extended to consider other phase variables.22 Another important FR (the L-FR or GC FR) describes fluctuations in the phase space compression rate, L: L¼

@ _ C @C

ð1:4Þ

in a thermostatted nonequilibrium steady state, where C (q, p) is a point in phase space. It can be written, t ¼ AÞ 1 PðL lim ln ¼ A; A oAoA t PðLt ¼ AÞ

t!1

ð1:5Þ

where A* provide the bounds on the magnitude of A to which it applies. This relation was first derived heuristically by Evans, Cohen and Morriss23 in 1993 for an isoenergetic system. Indeed this was the first statement of any Fluctuation Relation. Eqn (1.5) was first rigorously proven by Gallavotti and Cohen24,25 and the FR is widely referred to as the Gallavotti-Cohen fluctuation theorem (GCFT). This relationship is described in section 2.1, along with its connection to the relationships that are the main subject of this review. Historically there has been a lot of confusion about nomenclature. Many authors have confused the GC FR with the ES FR. Because these relations refer to fluctuations there is a profound difference between the two. Nevertheless this confusion persists in the literature to the present day, as discussed in section 5. Chem. Modell., 2008, 5, 182–207 | 183 This journal is

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In recent years, a number of reviews on these topics have appeared.4,5,17,26–41 Bustamante et al.29 published a comprehensible Physics Today article that summarises some of the main features of the fluctuation relations, discusses their practical implications for small systems and highlights experimental work in the area. We note that in order to make the paper accessible to the wide audience of Physics Today, some subtleties associated with the different arguments of the ES FR and of the GC FR (see discussion in section 5) are not addressed. In particular, it should be noted that the 2002 experiment42 discussed on page 47 of ref. 29 does not test eqn (3) of that paper (as stated), but rather the fluctuation relation where the argument is the work and not the phase space compression. Bustamante30 and Tinoco et al.31 also review experimental studies on RNA unfolding experiments that have been carried out to test fluctuation relations in their more general reviews on single-molecule unfolding experiments. Ritort28 discusses experiments designed to test the various FRs. Evans and Searles4 reviewed the transient fluctuation relations, how steady state relations can be derived from them, their implications and applications, and experimental tests available in 2002. A shorter review paper highlighted the main results, and a derivation of the second law inequality.5 Recently, Sevick et al.17 reviewed the ES FR, the Crooks FR and the Jarzynski equality, highlighting the similarities and differences between the two FRs and also discussing experimental work that has been carried out to test these results. Rondoni and Mejı´ a-Monasterio32 give a thorough, and current review of the various FRs. They discuss their theoretical basis as well as applications. Kurchan33 presents a concise review including the ES FR, Crooks FR and JE, emphasising the generality and simplicity of the relationships. Several reviews have focussed on stochastic dynamics. Harris and Schu¨tz27 discuss a range of fluctuation theorems (the Jarzynski equality, the ES FR and the GC FR) in the context of stochastic, Markov systems, but in a widely applicable context. They investigate the conditions under which they apply, including an analysis of the conditions under which the GC FR is valid. In 2006, Gaspard26 reviewed studies where a stochastic approach to the treatment of boundaries is used to obtain FRs for the current in nanosystems, with a focus on work from his group, and also published a review on Hamiltonian systems34 that includes a discussion on FRs and the JE. Astumian35 presents a pedagodic discussion on FRs and their treatment in terms of Onsager-Machlup theory close to equilibrium. Jarzynski36 provides a comprehensive review of the JE, discussing the practical application of the relationship, the importance of carefully defining the function used to calculate the work, and also its application in single molecule experiments. Kofke37 discusses different approaches to determination of free energy differences, and compares their accuracy for model systems, and in reference38 compares the use of the JE with other routes to the free energy difference. Xiong et al.39 also review the JE, discussing examples where it had been applied. Hummer and Szabo40 review work on the JE with particular emphasis on its application to the creation of free energy profiles using single-molecule pulling experiments. Meirovitch’s recent review41 on methods for calculation of free energies for biological systems, includes a discussion on the JE.

2. Fluctuation relations 2.1 Background Development of the ES FR was motivated by the seminal work by Evans, Cohen and Morriss.23 This paper focussed on constant energy dynamics and, based on results for the 2 dimensional Lorentz model, proposed that the measure of a trajectory was related to the exponential of the sum of the positive finite-time Lyapunov exponents of that trajectory. In this way they obtained a relationship for 184 | Chem. Modell., 2008, 5, 182–207 This journal is

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the probability ratio of the entropy production. Evans and Searles approached the same problem from a different point of view1,2,4—using the Liouville measure to derive the fluctuation relations. The work of Evans et al.23 also motivated Gallavotti and Cohen to develop the well-known Gallavotti-Cohen Fluctuation Relation for the phase space compression (GC FR, eqn (1.5)) that is valid for a special class of dynamical systems known as ‘‘Anosov’’ systems.43 While the usual chemical and physical systems cannot be classified as ‘‘Anosov’’, it was proposed that this relation is applicable to such systems under particular conditions referred to as the ‘‘chaotic hypothesis’’24,25 and it was originally expected that it might be very widely applicable. However, recently it has been found that there are various restrictions on the system related to the boundedness of the interaction potential44 or state space;45 and the values of the phase space compression to which it can be applied,16,43,44 which seem to go to zero as O(Fe2) in some cases, and are even more restrictive in some others.46 The proof also breaks down at large fields, and a modified relation has been proposed to treat this system.47,48 It has also been found that the convergence times of the GC FR can be very large, and much larger than those of the ES FR.49–51 The types of systems to which it is expected to apply, and its convergence time are therefore difficult to characterise. Due to these difficulties, the GC FR will not be discussed in detail in this review, and only studies that make comparisons with the relations that are the main subject of this manuscript will be considered. Readers are referred to references16,44,45,48,49,52–58 for recent work in this area. When applied to systems relaxing from an equilibrium state to a nonequilibrium steady state, the ES FR takes on the form given by eqn (1.1). This equation is very general—it only requires the assumption of reversibility of the equations of motionw (and in fact this can be relaxed to the assumption that for any trajectory, the time reverse trajectory is also a solution of the equation of motion), and the assumption of ergodic consistency.60 Ergodic consistency means that the combination of dynamics and ensemble are selected so that phase points along the nonequilibrium trajectories are represented in the initial distribution of states with a nonzero probability density. The dissipation function, O, is defined through its time-integral as: Z t t t ln fðCð0Þ; 0Þ O KðCðsÞÞds: ð2:1Þ fðCðtÞ; 0Þ 0 were f(C,0) is the equilibrium phase space density at the point C and f(C(s1),s2) is the phase space distribution function at a point C(s1) according to a distribution function that has evolved from the equilibrium distribution for a period s2. Using this notation, the condition of ergodic consistency can be expressed as f(C(t),0) a 0 for all C(0) for which f(C(0),0) a 0. As mentioned above, the time-integral of the dissipation function takes on the value of the extensive generalised entropy production, St, over a period, t under suitable circumstances.12,17 The main requirement is that the dynamics satisfies the condition know as ‘the adiabatic incompressibility of phase space’.60 In this case, St = JtFebV, where Jt is the dissipative flux caused by the field, Fe, b = 1/(kBT) where T is the temperature of the corresponding initial system and V is the volume of the system. An example where such a relation can be applied is if a molten salt at equilibrium was exposed to a constant electric field. In that case the entropy production would be directly proportional to the current induced, and the FR would describe the probability that it would be observed to flow in the +ve or ve { By time reversibility here, we mean that the time reversed trajectory exists. This therefore includes, for example, systems subject to a sinusoidal field where the phase at t = 0 is variable. It also allows inclusion of some classes of dynamics where a stochastic term is added.59


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direction. The FR can then be written: PðJt ¼ AÞ ¼ eAFe bVt PðJt ¼ AÞ

ð2:2Þ

and a negative dissipative flux implies a positive current. If Fe and the current are chosen to be positive (i.e. A o 0 and Fe 4 0) then the right hand side of this equation will increase exponentially with both the volume and time. That is, in the limit V N, the probability ratio will increase exponentially and it will become impossible to observe trajectories with the corresponding negative value of the current. In the limit t - N the same effect will be observed. So in the large system or long time limits, negative currents will not be observed. This is consistent with the 2nd Law of Thermodynamics. It demonstrates how irreversibility emerges, despite the assumption of the reversibility of the equations of motion in derivation of (2.2). However, (2.2) also tells us that when V and t are small, negative fluctuations in the current are possible. The ES FR can be used to provide a simple proof of the Second Law Inequality:5 ti 4 0 hO

(2.3)

that the entropy production cannot be negative. It should be noted here that the inequality does not apply to the instantaneous values of the dissipation function or the entropy production rate, but to total entropy production over the period t.61,62 These relations apply to systems with adiabatic dynamics as well as those that are thermostatted, say using artificial, reversible thermostats such as the Gaussian thermostat or Nose´-Hoover thermostat. In consideration of steady state dynamics, it is necessary to use such thermostats or ergostats. However, Williams et al.63 have recently provided a theoretical argument showing that using thermostats that are applied remotely (i.e. so that the driven particles are Newtonian but some ‘wall’ particles, separated from the main system by other Newtonian particles, are thermostatted, as illustrated in Fig. 1), the ES FR, which provides information about fluctuations in the field-driven system, is insensitive to the details of the thermostatting mechanism. They used MD simulations to support their argument. 2.2 Recent results for deterministic dynamics In the earliest paper on FRs, a footnote remarked that it is possible to obtain the Green-Kubo relations from them if the fluctuations have a Gaussian distribution.23

Fig. 1 Schematic diagram of a unit cell in a simulation where artificial thermostatting is applied remotely to the particles that are being driven by a field.


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A published derivation of the Green-Kubo or fluctuation-dissipation expressions from the combination of the FR and the central limit theorem (CLT) was finally presented in 2005.16 This issue had been addressed previously and the main arguments presented,15 but subtleties in taking limits in time and field that lead to breakdown of linear response theory at large fields, despite the fact that both the FR and CLT apply,15,64 were not fully resolved.15,17 In the derivation of the ES FR, time-reversibility of the equations of motion is required. Thus time reversibility is a sufficient condition for the ES FR. The question arises as to whether time reversibility is a necessary condition. In numerical calculations where irreversibility was introduced by employing an applied field that had no definite parity under time reversal, it was difficult to observe the breakdown of the ES FR. Careful numerical experiments have finally shown that a breakdown indeed occurs and identifies how this comes about.65–67z This work confirms that time reversibility and ergodic consistency are necessary and sufficient conditions for ES FR. In the early work of Evans and Searles, it was presumed that: (i) provided a valid transient version of the ES FR holds for a given system (i.e. where the initial distribution is an equilibrium distribution); (ii) and provided the system is able to relax to a unique steady state at long times under the application of a dissipative field and a thermostat (Note: this is certainly not always the case.); (iii) and provided negative fluctuations in the dissipative flux can be observed in the steady state, then a steady state version of the FR (with the same dissipation function) would be valid. These arguments are described in references,4,17 and the resulting steady state ES FR is, t ¼ AÞ 1 PðO lim ln ¼A t PðOt ¼ AÞ

t!1

ð2:4Þ

However, despite numerical, and recent experimental evidence,68,69 that the steady state relation does hold under these assumptions, it must be said that there are some subtleties regarding convergence times and the decay of fluctuations in nonequilibrium steady states. Recently a much more rigorous derivation of the steady state ES FR has been given. This derivation demonstrated that the underlying assumption necessary to obtain (2.4) is that time-correlations decay sufficiently quickly.32,70 As discussed in more detail in section 5 below, there has been some confusion in the literature as to the similarities and differences between the GC FR and the ES FR. In steady isoenergetic dynamics O(t) L(t), but under other conditions they are not instantaneously equal. There has therefore been some work has also been carried out to clarify the situation.16,32 An aspect of the ES FR that has not been fully exploited as yet is the fact that the dissipation function is sensitive to the choice of the distribution function. Therefore, if a system is presumed to have a particular equilibrium distribution function, with an associated dissipation function, then a field is applied, the transient ES FR should be satisfied for all time. This provides a way of testing if a system is equilibrated, for example. If the FR is not satisfied for the presumed O, then it indicates that the equilibrium state is not what was expected. This fact has recently been used to establish that domains of the nondissipative, nonequilibrium distributions of glassy systems can be described by Boltzmann weights.71 Apart from the reversibility of the dynamics, the other key assumption in the derivation of the transient ES FR is that the initial distribution and the dynamics are ergodically consistent.4,60 In the same paper,71 Williams and Evans demonstrated that away from the actual glass { We note that in some cases use of symmetry in the experiment allows what would seem to have irreversible equations of motion to be set in a reversible form or to use time-reversal mappings other than the standard (q, p) - (q, p).3


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transition, the domains of the glassy systems were robust—that is, they did not change when the system was subject to small changes in the pressure or temperature—by testing the FR when this change was imposed. If the FR was not satisfied, it would indicate that ergodic consistency had broken down and that the domains had changed when the system was subject to the changes in the state variables. Williams and Evans72 have tested the transient steady state ES FR on glassy systems. They verified that while the transient ES FR is obeyed at all times, the ES FR converges slowly, and the convergence time increases as the glass transition point is reached. Since the steady state ES FR in conjunction with the central limit theorem can be used to derive the linear response expressions for transport properties, this indicates that the strength of the field for which linear response theory applies becomes smaller as this point is approached. The slow convergence of the ES FR is in accord with its derivation which relies on decay of correlation times for convergence.70 It was noted in that at sufficiently large fields, the derivation of the GC FR following Gallavotti and Cohen would break down.47 A modified version of the FR was proposed in that case. This was a curious result because the steady state ES FR derived by Evans and Searles becomes equivalent to the GC FR under some circumstances (constant energy dynamics), and therefore this appeared to be a contradiction. It has been difficult to test this result because at large fields the probability of observing negative fluctuations decreases, and for large systems the correction is quite small. However, recently a simple model was developed where the correction term would be significant.48 The model was a 1-particle system, thermostatted using Nose´-Hoover thermostats. It could be driven to a sufficiently high field that the proposed correction term was significant, but fluctuations could still be observed. No correction term was found to apply, and in fact the GC FR appeared to describe the data better at larger fields, as opposed to the prediction. It would seem that this system did not satisfy the chaotic hypothesis on which the GallavottiCohen derivation of the GC FR was based. Giardina` et al.73 point out the difficulty of testing or numerically predicting large deviation functions because they require sampling of rare events. A similar issue arises in testing the FRs. They devise a numerical procedure that allows the rare events to be sampled in a biased manner and therefore allow more efficient determination of the probability functions in the rare event regime. We should also point out that in some recent work on nonlinear response theory, it has been shown that there is an intimate and deep connection between the dissipation function of the fluctuation relation, and the general expressions for the exact nonlinear response of systems.74 Nonlinear response theory, originally developed for homogeneously thermostatted or ergostatted nonequilibrium steady states60 can now be applied to a much wider class of systems through this use of the dissipation function.74 Evans et al.74 show how the dissipation function is the generally applicable variable in the nonlinear response theory expression. Andrieux and Gaspard use stochastic FRs to obtain non-linear response coefficients,75 and Harada and Sasa have formulated an expression for the energy dissipation in the nonlinear regime for Langevin systems.76 Intricate experimental studies have also recently enabled nonlinear response expressions (formulated in a different way than that in reference60 by Speck and Seifert77), to be tested and verified.78 The Fluctuation Relation has been used by Dewar79 to assist in establishing a basis maximum entropy production principle. The difficulties with this outside the linear response regime have recently been discussed by Bruers.80 2.3 Recent results for stochastic and quantum dynamics Although the transient FR (1.1) is usually applied to systems that are initially at equilibrium, the consideration of the derivation shows that this is not necessary, and the underlying requirement is just that the initial distribution is known. For many-particle deterministic dynamics, the nonequilibrium distribution function is 188 | Chem. Modell., 2008, 5, 182–207 This journal is

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not known (at least not in a closed form), and therefore the derivation is restricted and only an asymptotic relation (2.4) can be derived. However, for stochastic dynamics, initial distributions are sometimes known, so then FRs can be developed for nonequilibrium steady states that are valid at all times—these are known as the ‘exact fluctuation theoerm’.68,81 For deterministic systems, the FR takes on the form (1.1) and the dissipation function is given by (2.1). However, in the case of stochastic dynamics, the same process might be modelled at different levels with different dynamics, and for each model a different fluctuation relation may be obtained. Therefore there are more papers on stochastic systems than on deterministic dynamics, as derivations for new dynamics allow new systems to be treated. This is particularly true for the Jarzynski relation, as discussed in section 3.2. Extensive theoretical work on FR for stochastic systems has been carried out by van Zon et al.82–84 In particular they considered fluctuations in the properties work, w, and heat, Q, which can be shown to correspond to the dissipation function and phase space contraction, respectively.53 They found that FRs of the form (1.1) and (2.4) were obeyed when the argument was w, but not for Q. One of the systems that they analysed was a particle dragged through a medium at constant temperature. Baiesi,81 also considered work and heat FRs for this system, describing the behaviour by a Langevin equation with a time-dependent potential and derived a modified heat FR. This study differed from that of van Zon et al., because they did not assume a harmonic potential and used an arbitrary protocol for movement. The results were tested computationally. Dhar85 modelled the stretching of a polymer using the stochastic Rouse model, for which distributions of various definitions of the work can be obtained. Two mechanisms for the stretching were considered: one where the force on the end of the polymer was constrained and the other where its end was constrained. Dhar commented that the variable selected for the work was only clearly identified as the entropy production in the latter case. In the former case they argue that the average work is non-zero for an adiabatic process, and therefore should not be considered as an entropy production, however we note that the expression is consistent with a product of flux and field as used in linear irreversible thermodynamics. In understanding experimental studies where a particle in an optical trap could be considered as a Brownian particle, FRs based on the stochastic Langevin equations were developed.86 This allowed analytic expressions for the entropy production and its probability to be obtained, and numerical predictions to be made. A similar approach has been used to study a Brownian particle diffusing in a periodic potential under steady state conditions and useful information characterising the fluctuations have been obtained analytically and from numerical calculations.70 Recently, there has been interest in considering FR for systems coupled to heat reservoirs at different temperatures.46,87,88 The relevant FR for systems modelled using deterministic dynamics was determined and tested numerically in 2001,89 however interest in heat flow for simple models is important for studies aimed at determining the nonequilibrium temperatures and studying Fourier’s Law, and therefore consideration of FR in these systems has received some attention. Gomez-Marin and Sancho88 and were unable to verify an FR proposed by Jarzynski and Wo´jcik,87 presumably due to neglect of boundary terms in the theoretical analysis that were relevant for their numerical calculations. Visco46 has studied the GC FR in a model heat flow problem. Fluctuation relations for stochastic models of granular gases have been studied and the ES FR was found to be verified, but the GC FR was not satisfied on the timescales considered.51,56,57 Golinelli and Mallik90 have examined FRs in mathematical models that have been developed for studying general properties of nonequilibrium systems, and which are analytically tractable. Tanaiguchi and Cohen91 consider the relationship between Onsager-Machlup theory and the FRs. Chem. Modell., 2008, 5, 182–207 | 189 This journal is

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The first quantum versions of the FR appeared in the early 2000s.92,93 Unlike the classical, deterministic FR, in quantum systems there is no clear concept of a single trajectory,94,95 and there is some flexibility in the choice of the argument that appears in the FR, and therefore a number of FR have been obtained using different approaches.93,95 For some cases the FR takes on the form of (1.1),92,94 while in others a corrected formula is obtained.93 The formulation of FRs for quantum systems, and development of experiments to test them, is therefore ongoing. Recently De Roeck and Maes96 obtained a quantum FR for a system undergoing heat conduction due to coupling with two reservoirs at different temperatures. Jennings et al.97 note that in studying photosynthetic processes, thermodynamic assumptions are assumed, preventing negative entropy changes. They then derive results which indicated that there can be a situation where negative entropy production occurs. Their arguments were based on relations based on thermodynamic (averaged) properties and were therefore rightly disputed,98,99 however the work highlighted the point that for these small systems, for finite times finite-time negative entropy producing events might occur. 2.4 Experiments The first conclusive experimental tests of the fluctuation relations were on a colloidal particle in water, held by an optical trap.19,42 The trap was either translated by movement of the optical trap relative to the system, or used to ‘capture’ the particle by increasing the strength of the optical trap.19 More recently, Carberry et al.100 were able to experimentally demonstrate that the so-called ‘‘Kawasaki function’’, that is heOtti = 1 for any time, t, when the system is initially at equilibrium. This follows directly from the fluctuation relation and is also discussed in ref. 70. This relationship suffers from similar difficulties in convergence as do the integrated fluctuation relation and Jarzynski relation (see discussion t needing to be properly below), with rare trajectories with negative values of O sampled. It therefore serves as a useful experimental control to provide an indication of the level of sampling required for the integrated fluctuation relation and Jarzynski relation to converge. Similar elegant experiments have subsequently been carried out. Wang et al.68,69 verified the steady state version of the ES FR, firstly in an integrated form:68 t o0Þ PðO t 40Þ ¼ hexpðOt tÞiO t o0 t!1 PðO lim

ð2:5Þ

t o 0, where h . . . iO to0 implies that the ensemble average is over trajectories with O 69 and then in the complete form given by eqn (2.4). This was a particularly important result due to controversy that surrounded the validity of applying this relation to steady state dynamics at the time. In all these experimental studies, the particle was in a viscous fluid and therefore the equations of motion of the particle were well approximated by a stochastic Langevin equation. In 2007, a capture experiment was carried out in a viscoelastic solvent where this approximation no longer applies. It was shown that despite this, the experiments validated the ES FR, and therefore could not be consider just a special property of Brownian dynamics. Blickle et al.101 verified the fluctuation relation for the work (or dissipation function) for a system where the trap potential was not harmonic. Narayan and Dhar102 demonstrated the importance of choosing the correct expression for the entropy production (see discussion in section 5), by demonstrating that an FR for heat (corresponding to the GC FR53) is not obeyed in their experimental studies, whereas the FR for work (corresponding to the ES FR53) is. 190 | Chem. Modell., 2008, 5, 182–207 This journal is

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Garnier and Ciliberto103 verified an FR for work (equivalent to O in this paper53) of the form given by eqn (1.1), and a heat fluctuation relation as predicted by van Zon et al.82–84 They studied fluctuations in the power injected to an electrical dipole that is subject to current. Recently, Douarche et al.104 verified the transient ES FR and steady state ES FR for a harmonic oscillator (a brass pendulum in a water–glycerol solution, that is driven out of equilibrium by an applied torque). They also developed a steady state relation for a system with a sinusoidal forcing, and showed that the convergence time was considerably longer in this case. Schuler et al.105 and Tietz et al.106 excite a single defect in an diamond by a laser in a periodic manner. They test the Kawasaki function, which is obeyed, and the ES FR and find that for symmetric protocols it is obeyed. Andrieux et al.107 develop stochastic FRs from which the change in entropy can be obtained, and consider two experiments to verify the FR: a dragged Brownian particle and electric circuit with imposed mean current.107 Seitaridou et al.108 have examined a flux FR for particles undergoing diffusion. In this case the dissipation function will be proportional to the flux. They find a linear relationship, consistent with eqn (1.1), however, it is unclear if the slope is consistent with the predictions of (1.1) as the proportionality constant is not known. It would be interesting to examine this experiment further.

3. Free energy relations 3.1 Background In the past few years, development of new theories have led to completely new ways of determining free energy changes. Traditionally, the difference in the free energy of two equilibrium state is (DF1-2) and the free energy change of a process can be obtained directly from the statistical mechanical definition of the free energy, F, Rin terms of the partition function. For the canonical ensemble F = kBT ln dCebH(C) = kBT ln Z, where kB is Boltzmann’s constant, H(C) is the phase variable corresponding to the internal energy, and also the Hamiltonian of the dynamics that preserves that internal energy, T is the temperature of the system, and b = 1/(kBT). That is, DF1-2 = kBT ln (Z1) kBT ln (Z2) = kBT ln heb(H2(C)H1(C))i1

(3.1)

where the initial state has Hamiltonian H1 and the final state has Hamiltonian H2. The notation h . . . i1 implies that the equilibrium distribution is taken with respect to the distribution function of the initial state. This approach is referred to as free energy perturbation (FEP) and was introduced by Zwanzig. Due to numerical issues related to determination of the final expression directly when the free energy difference is large, it is usual to break the procedure into tractable (often nonphysical) stages that take the system step-wise from state 1 to state 2, and then to use thermodynamic integration to obtain an overall result. Bias potentials or reweighting schemes (e.g. umbrella sampling methods) have been used to improve the sampling. Alternatively, if the change from state 1 to state 2 is carried out so slowly that the process can be considered ‘reversible’ the well known relationship between the change in free energy and the reversible work done in transforming from state 1 to 2, w1-2;rev, applies: DF1-2 = w1-2;rev.

(3.2)

In simulations, the change from one state to another is achieved by slowly and continuously varying a control parameter that produces state 1 initially and state 2 finally. A completely reversible situation can only be approached, and the procedure is often referred to as a ‘slow switching’ approach. Chem. Modell., 2008, 5, 182–207 | 191 This journal is

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From differentiation of the partition function and carrying out a reversible change in the internal energy using parameter, l, (H = H0(1 l) + H1l) that goes from 0 to 1, it can be shown that: R (3.3) DF1-2 = 10hH1 H0irev,ldl. Evaluation of free energy differences using eqn (3.3) is referred to as thermodynamic integration. Development of the Jarzynski equality (also known as the Work Relation or the Nonequilibrium Work Relation) in 1997, and the Crooks FR (or Crooks Identity or Crooks Fluctuation Theorem) have provided new approaches to the calculation of free energy differences, enabling their determination by following irreversible (and therefore possibly fast) pathways. The approaches have not yet resulted in a generally more efficient algorithm for calculating macroscopic free energy differences. Indeed it is probable that these new approaches will not lead to more efficient schemes for macroscopic systems. However for nano-scale systems these new approaches offer much promise and have already proved their worth. These new methods are often referred to as a ‘fast switching’ method. The Crooks FR, eqn (1.2), compares the probability that the work done in going from state 1 to 2 takes a value of A, with the probability that it takes a value of A in going from state 2 to 1. The ratio of these probabilities is directly related to the difference in free energy of states 1 and 2 through eqn (1.2). As mentioned above, the JE (1.3) is obtained by integrating (1.2), and it gives another expression from which the free energy difference between the states can be obtained. The change is made parametrically varying H(C,l(s)) over a period, t, with the parameter, l, changing so that H1(C) = H(C,l(0)); H2(C) = H(C,l(t)). The initial distribution is assumed to be canonical, f¼R

ebHðCÞ ebHðCÞ ¼ bHðCÞ Z e dC

ð3:4Þ

Originally the equations of motion for the process were selected to be the Hamiltonian equations of motion obtained using the time-varying H. That is, @HðC; lðsÞÞ @p @HðC; lðsÞÞ p_ ¼ @q q_ ¼

ð3:5Þ

and the work in this case is, w1!2 ¼

Z 0

t

@HðC; lðsÞÞ _ lðsÞds @l

ð3:6Þ

In contrast to relation (3.2), the path over which the work is measured does not have to be thermodynamically reversible, so the system can be out of equilibrium during the process. The time-dependence of l (often referred to as the ‘protocol’) is arbitrary from a theoretical perspective, so the nonequilibrium dynamics can have enormous variety. However, as discussed below, some protocols are more feasible from a numerical point of view. Theoretically, the only restriction is that the dynamics be selected so that if f1(C(0)) a 0, then f2(C(t)) a 0. During the ‘reverse’ trajectory from state 2 to 1, the time-evolution of l should be reversed. Recently it has been recognised that the equations of motion can be further modified and can even be non-Hamiltonian. This is discussed below. Eqns (3.1) and (3.2) can be considered as direct consequences of (1.3). Eqn (3.1) is obtained when l is a step function and the change is applied instantaneously, whereas (3.2) is obtained by taking the infinitely slow path. Furthermore, using (1.3), it can be shown that the work is maximised in the reversible process and the work 192 | Chem. Modell., 2008, 5, 182–207 This journal is

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done in an irreversible process is always greater than the difference in free energy between the states:7 DF1-2 Z w1-2;irrev

(3.7)

Both the Crooks FR and JE have been verified experimentally and used to obtain free energy changes. A number of recent papers discuss the relationship of the average value of the irreversible work that appears in the Crooks and Jarzynski relations with the irreversible entropy production.109,110 In fact Ben-Amotz and Honig define an excess entropy (difference between the irreversible and reversible entropy change) in terms of averages of the irreversible work providing a relationship which they refer to as a ‘rectified 2nd law’, although they make some strong assumptions on the temperature that they choose to use in the processes. In the past there has been some discussion on the validity of the JE in real systems due to assumptions based on the coupling between the system and the bath.9,111 However, the relationship has now been derived in various settings, including deterministic, fully thermostatted systems,112,113 as well as tested experimentally. The controversy now seems to have settled and the JE it is widely accepted and applied.114 It is implemented in molecular dynamics programmes the such as the widely used NAMD software.115 A recent paper116 notes that the JE is an identity, but points out that if an adiabatic experiment is carried out that starts at one temperature, is altered parametrically over a finite period, and then is allowed to relax, the difference in free energy between the initial state and the final state will not be that predicted by the Jarzynski equality. This is because even if the final system reaches equilibrium, it will not be at the required temperature. They also note that some processes will not even relax to a new equilibrium state. However, we note that it is also possible to carry out the whole process under thermostatted conditions,112 leading to the same expression, and in that case the difference in the free energy of the state at the beginning and end will be equal to that predicted by the Jarzynski equality, provided the system relaxes to equilibrium. As recognised in ref. 116, the JE was not supposed to give the difference in free energy along such a path, but the difference in free energies between two states at the same temperature, and the free energy obtained is equal to the work done in a fictitious, reversible path between those two states. Therefore the observations in ref. 116 have no impact on the validity of the JE. Recent reviews on the Jarzynski equality and related theorems discuss the method and its application.17,27–33,36,37,39–41 In section 3.2 we describe recent developments associated with the Crooks FR and JE, citing papers where the different approaches have been used and compared. A special issue of Molecular Simulation ‘‘Challenges in Free Energy Calculations’’ published in Jan–Feb 2002 includes articles on the different approaches to free energy calculations, including the Jarzynski approach. A number of reviews consider alternative approaches to calculation of free energy differences, and the ‘traditional’ methods are continually being update and improved (see, for example, refs. 117–121.) Fowler et al. describe how calculations can be sped up using a computational grid.122 Here we will focus on the JE and Crooks FR, and for more general reviews on recent advances in the calculation of free energies refer the reader to reviews by Rodinger and Pome`s,123 Kofke,37 Adcock and McCammon124 and Hu and Yang.125 3.2 Recent theoretical results An instructive study of the JE for an exactly solvable model of an ideal gas under expansion (where the probability distribution of the work can be derived) verifies the relationship and shows how its predictions compare with numerical results, providing insight into its application.64 With similar motives, a model for effusion of the ideal gas is considered,126 and more recently, this group has extended this analysis to Chem. Modell., 2008, 5, 182–207 | 193 This journal is

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include momentum transfer during the effusion.127 The effusion is considered as a stochastic process, and good agreement with simulations of hard spheres is obtained. Recently, Crooks and Jarzynski128 consider a dilute classical gas of interacting particles. They obtain the work distribution function and verify the JE and Crooks FR. They also test the relationship obtained when the distribution of the work is approximated as a Gaussian, showing the limitations of this approach. Dhar85 has considered a stochastic Rouse model for the stretching of a polymer where the distribution of the work is shown to be Gaussian and the JE is verified, and Piana studied the stretching of a DNA dodecamer to obtain the free energy curve.129 Imparato et al.130 use an Ising model for protein unfolding and generate the free energy landscape using a JE-like equality. Chvosta et al.131 consider another case where the work probability distribution function can be determined. They study a two energy-level system, modelled as a stochastic, Markovian process, where the transition rates and energies depend on time. Like the previous examples it provides an exact model that can be used to assist in identifying the accuracy of approximate, numerical studies. Ge and Qian extended the stochastic derivation for a Markovian chain to a inhomogeneous Markov chain.132 Recently, Kawai et al.133 define the time average of the dissipative work for systems undergoing adiabatic dynamics using a very similar setup-up to that used by Evans and Searles4 to give eqn (2.1) for the dissipation function (which is equivalent to the dissipative work, and is simplified for adiabatic dynamics where L = 0 at all times) and consequently derive (1.1). However, in determining the probability of a trajectory, rather than considering the distribution functions at the initial point of trajectories and their time-reversed conjugates, they consider them halfway along, and they also allow the initial equilibrium distribution functions to be different giving: wdiss ¼ ln

f1 ðCðt=2Þ; 0Þ f2 ðC ðt=2Þ; 0Þ

ð3:8Þ

where C* is the time-reversed map of C (q, p), C* (q, p). Futhermore in generating C*(t/2), the protocol is reversed (note that the Evans-Searles treatment D E is

restricted to protocols that are time-reversible). The property

ðCðt=2Þ;0Þ ln ff21ðC ðt=2Þ;0Þ

is

recognised as the Kullback-Leibler distance, or relative entropy, between forward and backward probability distributions, and therefore the average dissipation function or average dissipative work in this case can be considered to be a relative entropy. This can then be related to the difficulty of distinguishing forward and backward trajectories and to the breaking of reversibility or detailed balance.133 Imparato and Peliti134,135 show how the work distribution for a given protocol can be obtained using a joint probability distribution for the work and the initial state, and apply it to system represented by a coordinate in a mean field. They show that it leads to the JE, and demonstrate some numerical issues when applied to large systems where it becomes difficult to observe fluctuations. They also provide an analytical treatment of a driven Brownian particle, obtaining the probability distribution for the work and FR for this system.136 Paramore et al.22 show that the JE can be extended to consider variables other than the work, and thereby derive the expression for the determination of the free energy of a system where a reaction coordinate is fixed. Horowitz and Jarzynski21 examine the connection between a result by Bochkov and Kuzovlev137 that can be used to obtain an FR, the Crooks FR and the ES FR. They show that the relationships differ in their definition of the work and range of applicability. Recently Jayannavar and Sahoo138 have obtained an FR for a stochastic model of a charged particle that is translated or oscillated in a magnetic field and show that the free energy is not affected by the magnetic field, in agreement with the Bohr–van 194 | Chem. Modell., 2008, 5, 182–207 This journal is

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Leeuwen theorem. Marathe and Dhar66 study an Ising spin in a time-varying magnetic field. They verify the Crooks FR and JE, and confirm that time reversibility is required for the transient ES FR. They also observe that the steady state ES FR is not obeyed, confirming that this is an asymptotic result and indicating that observation times were insufficient for convergence. Originally the JE treated processes that were weakly coupled with a thermostat6 implying a theory for adiabatic processes. In subsequent work Jarzynski7 developed a stochastic approach where this was not required. Recently, Baule et al. considered the isothermal expansion of a gas as a stochastic process, and showed that the JE applies.139 Evans112 also considered general deterministic processes that were directly thermostatted, deriving the Crooks FR and JE in a way that closely parallels the derivation of the ES FR.4 The details of the thermostat were somewhat arbitrary and the treatment could include Gaussian and Nose´-Hoover thermostatted systems. A very similar approach was later used by Cuendet140 but only for systems specifically thermostatted with a Nose´-Hoover thermostat. Cuendet later extended it to other thermostatting (also barostatting) mechanisms.141 Scho¨ll-Paschinger and Dellago113 also obtained an FR for systems whose dynamics preserved a canonical distribution when the parameter, l, was fixed. Adib142 determined a fluctuation relation for the free energy difference between two constrained configurations (i.e. with q fixed at qA and then at qB) by allowing the two different constrained configurations to relax to equilibrium. As noted in that work, this relationship is fundamentally different to Crooks (Jarzynski) because no work is performed during the process. The FR is given by PðqA ! q; tÞ ¼ ebDFA!B PðqB ! q; tÞ

ð3:9Þ

where P(qA - q; t) denotes the probability that a system whose configuration was fixed at qA relaxes to a configuration q at time t. The relation is considered in the context of experiments that allow single molecules to be constrained in particular configurations. The JE and Crooks fluctuation relations were originally developed to treat systems in canonical equilibrium states. Adib143 extended the JE and thermodynamic integration schemes to obtain DS of isoenergetic systems that originate in a microcanonical ensemble. The FR is obtained by designing constant energy equations of motion and is given by: heLtiE = eDS1-2/kB

(3.10)

where the subscript E is used to indicate that the ensemble average with R t is taken @ respect to the microcanonical distribution with energy E, and Lt 0 ds @C C_ is the phase space expansion. In a very general treatment for arbitrary ensembles and both deterministic and stochastic dynamics, Reid et al.144 discuss the relationship of the ES FR with the Crooks FR and a new ‘conjugate’ version of the Crooks FR. They tabulate the arguments of the these FR for the micro-canonical, canonical and isothermal–isobaric ensemble, indicating that the Crooks FR is in accord with the JE given by eqn (3.10): PE ðLt;1!2 ¼ AÞ ¼ eAþDS1!2 =kB PE ðLt;1!2 ¼ AÞ

ð3:11Þ

In a similar study by Cleuren et al.,145 samples from an initial microcanonical ensemble are considered (on an energy shell with energy E), but the dynamics in not isoenergetic, and therefore each trajectory moves off the energy shell. The Crooks Fluctuation relation is then given by: PE ðw ¼ AÞ ¼ eðSEþw SE Þ=kB ¼ eDS1!2 =kB PEþw ðw ¼ AÞ

ð3:12Þ


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which contrasts to other versions of the Crooks FR where the probability distribution considered in the denominator is the same for all values of w. In this paper the authors also attempt to explain how the GC FR can be obtained from this result, however the thermodynamic limit is taken and the fluctuating term (w) is equated to the average change in energy (DE) in their argument, and therefore this argument is not generally applicable. The JE has also been extended to cases where there is a change in the temperature of the system146 for Markovian systems. Recent work has shown that this is also readily derived for deterministic systems.147,148 In both cases, the relationship becomes Rt _ _ dsðbðsÞHðCðsÞ;lðsÞÞþbðsÞ HðCðsÞ;lðsÞÞ Þ 0 e ¼ eðb2 A2 b1 A1 Þ ð3:13Þ 1

and in the particular case where there is no work done on the system due to change in the Hamiltonian, and the change in b is linear

he(b2b1)Hti1 = e(b2A2b1A1)

(3.14)

In ref. 146 this approach is applied to determine the surface tension of a 3D Ising model. Other derivations for free energy differences in ensembles other than canonical NVT ensemble have been presented,141,148–150 as well as the ratio of partition functions for systems that are in different ensembles.148 The results have also been tested numerically.151 In our group, the difference in free energy has also been derived and verified in numerical simulations of realistic systems where the system of interest is thermostatted and/or barostatted externally.150 One of the interesting aspects of the FR is that the protocol used is rather arbitrary (at least in a theoretical sense). As well as varying the protocol, however, external fields can be applied during the period over which l is changing.17,21 These fields will do additional work on the system, and this must be included in the expression for work. In addition, the system may be thermostatted. A thermostat will not do work on the system, but will contribute to Q. For example the equations of motion could be written, @HðC; lðsÞÞ þ C Fe @p @HðC; lðsÞÞ þ D Fe ap p_ ¼ @q

q_ ¼

ð3:15Þ

where C and D represent the coupling of the field to the system, and a is the thermostat multiplier that extracts heat. In this case the work that appears in the Crooks FR and JE is, Z t @HðC; lðsÞÞ _ @H @H lðsÞ þ C Fe þ D Fe ds w1!2 ¼ ð3:16Þ @l @q @p 0 Use of appropriately chosen fields may improve the convergence properties of the JE, and this is currently under investigation. In deterministic systems, it is the dissipation function that is the subject of the FR (1.1). In contrast, it has been demonstrated that for stochastic systems, there can be more than one property that satisfies a fluctuation relation.86,152 There have been a number of quantum versions of the Crooks FR and JE. In 2005, Monnai153 presented a paper that clearly demonstrated how the FR, as presented by Crooks, could be readily extended to quantum systems in general. They provide a definition for the argument of the FR and show that in the quantum system there is some flexibility in this argument, depending on the choice of observables made in determining the probability ratio of paths. Also in 2005, 196 | Chem. Modell., 2008, 5, 182–207 This journal is

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Allahverdyan and Nieuwenhuizen154 consider Thomson’s formulation of the second law and argue that the variable of the quantum FRs in previous studies should not be considered to be the work. They discuss different definitions of the work in quantum systems and argue for introduction of a new definition. Esposito and Mukamel95 present transient and steady state FRs as well as JE by developing a quantum mechanical trajectory, and then carrying out a derivation in a similar way to that used for stochastic dynamics. Talkner and Ha¨nggi155 recently derived a quantum version of the Crooks FR using a characteristic function for the work obtained by Talkner et al.156

3.5 Applications in simulations There have been many applications of the nonequilibrium free energy calculations in the literature, and here we outline a number of them. The Crooks FR has been using in simulations and experiments, but presumably because of the simplicity of evaluation of the free energy using JE, this relationship is much more widely used. In some particular examples, advantages of the Crooks approach have been noted.157,158 JE has shown considerable promise in steered molecular dynamics studies of the free energy landscape, and in particular it has become a popular choice for protein unfolding dynamics probably partly due to the exciting experiments carried out soon after its discovery. In this regard, the unfolding of deca-alanine has provided a relatively simple, yet realistic model that has been used to compare and test different approaches.118–120,159–161 However, like other approaches to determining free energy differences, the JE can be troubled with numerical accuracy. In slow switching techniques, thermodynamic integration and FEP, problems associated with equilibration are met. While these are largely avoided in the JE, adequately sampling the distribution to obtain a reliable ensemble average, that is also an issue in FEP methods, can be problematic. The difficultly lies in the fact that rare events when the work is large and negative, will give large contributions to the ensemble average. Depending on the shape of the distribution, this can cause errors that are difficult to eliminate. Recently several studies have considered the performance of the JE and Crooks FR with other methods. It has been observed that in some cases the new approaches are more efficient157 and in others the traditional approaches are still preferable. In order for JE to be accurate one must sample the extremely rare events which if investigated individually would appear to violate the second law (the waterfall running backwards or the jet engine running backwards generating oxygen and kerosene). If these time reversed events cannot be sampled both JE and Crooks will not yield accurate results. Moreover in the case of the JE, the results will appear converged, so some control is required to ensure that they have converged to the correct result. A number of approaches have been used to try to improve the convergence of the JE in cases where it is problematic. Some of these are analogues of processes introduced to improve sampling in the FEP approaches. For example, Adjanor et al.162 introduce path-biasing schemes to improve convergence. They consider a stochastic system, and carry out tests with simulations on clusters of LJ particles. Kofke38 has provided a useful analysis to determine when numerical problems will occur in the calculation of the JE, and considers a pedagogical example. It is demonstrated that convergence will be problematic when the probability distributions of the work for the forward and reverse protocols are well separated. This will generally increase as the number of particles, rate of change of the parameter and size of the perturbation are increased. Kofke proposes using the relative entropy to assist in the assessment of the accuracy of the results obtained. Chem. Modell., 2008, 5, 182–207 | 197 This journal is

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Wu and Kofke163,164 discuss the significance of the overlap of the phase space domain of different states in the testing and use of FRs. They also introduce Rosenbluth sampling to assist in convergence of the JE to the correct value.165 MacFadyen and Andricioaei166 used as scheme whereby the initial distribution is modified by increasing the variance of the momentum distribution. They show that the results obtained significantly improve the accuracy of the results for a simple model of molecular isomerization. Lechner and Dellago167 compare the efficiency of various approaches for evaluation of the free energy using the JE for a particle pulled through a viscous liquid and for the expansion of an ideal gas. In both cases the analytic result for the probability distribution of the work is known. They describe and compare thermodynamic integration, fast switching methods and variations of the JE: (i) a direct method; (ii) a method based on thermodynamic integration where the sampling of important rare events is increased by introduction of a biasing parameter; (iii) a method based on umbrella sampling with a weighting function based on the path. It is found that the indirect methods can significantly increase the efficiency of the calculations. Lua and Grosberg168 also treat an ideal gas expansion and show that the number of samples required to give accurate results increases exponentially with system size. Shirts and Pande169 also compare methods, noting that for simple models there is no universally preferred approach and then examine the different methods for particle insertions. Ytreberg et al.170 consider the free energy change due to a change in the size or charge of a molecule in a solvent in their study on the efficiency and precision of various approaches. Using the fact that, in theory, the JE approach is insensitive to the path taken, Lechner et al.171 showed that the timestep in the numerical integration of the nonequilibrium trajectory can be made quite large, thus improving speed without losing accuracy. A low mean value of the work distribution seems consistent with a distribution that samples the negative fluctuations more regularly, however this is not certain as the variance and higher moments might shrink at the same time and reduce negative fluctuations. Nevertheless, assuming that the minimum mean work leads to the best sampling path, Schmiedl and Seifert172 determined the optimal path by minimising the mean work using different protocols. They found that in two cases, the optimal path involved step functions in the protocol. Oostenbrink and van Gunsteren173 study processes that are often met in biomolecular studies—redistribution of charges, creation and annihilation of neutral particles and conformational changes-comparing calculations of free energy changes using thermodynamic integration and the JE. They find that using the standard approaches, thermodynamic integration is more efficient, but that if slow conformational changes occur the JE offers an advantage. Li et al.174 compare different approaches, in the study of the conversion of ethane to methanol. They employ a l weighted histogram analysis method (WHAM) in order to improve statistics. Minh175 uses WHAM to develop a multi-dimensional potential of mean force. Lelievre et al.176 use a projection of the dynamics, rather then a constraint to follow the reaction coordinate. Bastug and Kuyucak177 compare the success of the JE for a simple and complex system for a restricted range of parameters. They showed that it was necessary to carry out the change slowly to get good results in both cases. They also carried out a simulation of double occupancy ions in a channel using the JE to determine the free energy profiles with the aim of examining biological process.178 Recently Calderon159 introduced a ‘surrogate process approximation’ (SPA) to improve the sampling in calculation of the JE. The scheme is applied to the study of the unravelling of deca-alanine at constant temperature in a steered molecular dynamics simulation. The distribution of the work is approximated by developing a model for the dynamics using a relatively small number of real trajectories in conjunction with stochastic differential equations selected to model the process. The 198 | Chem. Modell., 2008, 5, 182–207 This journal is

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scheme avoids the need to make assumptions about the distribution of the work, and takes advantage of the fact that there is generally a lot of flexibility in the choice of the nonequilibrium process that takes one from the initial to the final state. Min et al.,179 consider the free energy change during a stochastic model of an enzyme reaction scheme. They consider the Crooks FR, the JE (via the Kawasaki function) and the results from a Gaussian approximation (fluctuation dissipation theorem) and show that in this case the Crooks FR is more efficient. They also consider the errors associated with the schemes for finite sampling. In order to circumvent problems associated with sampling, in some situations where the distribution is Gaussian it is possible to use limited data in conjunction with JE or the Crooks FR to obtain a simpler expression for the free energy difference.180,181 A cumulant expansion of the JE is also sometimes used, with a second order cumulant expansion: b DF1!2 ¼ hw1!2 i1 ðhw21!2 i1 hw1!2 i21 Þ 2

ð3:17Þ

being exact if the distribution is Gaussian. We would be cautious with such approaches unless it was known that the distribution is indeed Gaussian. This is because the convergence difficulties occur in the wings of the distribution where the noise is large, and the central limit theorem does not necessarily predict a Gaussian distribution. This departure from a Gaussian will be difficult to observe and erroneous assumptions on the distribution will lead to poor results. This view is supported by work on a system where an analytical expression for the work is known.128 If the distribution is approximated by a Gaussian or a truncated cumulant expansion, preliminary studies should be carried out to compare the results with other approaches or a full JE or Crooks FR evaluation to ensure that the distribution is well approximated by a Gaussian over the relevant range of the values of work. Recent work by Presse´ and Silbey discuss the application of cumulant expansions to different systems.182 Schettino et al.183 demonstrate this point in their protein-folding/unfolding simulations. They show that the use of a Gaussian approximation for distributions (in situations where a Gaussian distribution has been assumed to be a good approximation) can lead to inaccurate results. Liu et al. also carry out simulations on RNA unfolding, and create a free energy landscape for the process. They find that there is no barrier to unfolding, and question models based on a two-level system.184 Mihn185 demonstrates how data obtained using various pulling velocities can be combined to improve free energy estimates. Orzechowski and Cieplak186 carry out steered molecular dynamics studies using the JE to look at formation of a DNAdrug complex and stretching of molecules, however the number of samples used in the ensemble averages were only 10, which by far too small to give useful results. Gao et al.187 give a clear summary of the use of the JE in steered MD on proteins, discussing the reconstruction of the potential of mean force using the JE, and also problems that can occur. They provide some solutions to these problems in the case of a distribution of work that is Gaussian. West et al.188 study protein folding and point out that care needs to be taken when comparing the predictions of the JE to those of experiments, as the JE (1.3) normally considers averages over an initial equilibrium state, whereas the experiments constrain the initial configuration. In ref. 189 a potential of mean force is calculated and used to determine diffusion of ubiquinone through a light-harvesting complex determined using MD simulations. The complete JE and JE with a Gaussian approximation are used and compared.189 Bidon-Chanal et al.190 have used JE in steered MD studies of free-energy profile for ligand diffusion through a tunnel system of truncated haemoglobin, and Amaro et al.191 used steered MD simulation and calculation of free energy using JE to Chem. Modell., 2008, 5, 182–207 | 199 This journal is

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determine the passage of ammonia through a barrel enzyme. They used a 2nd order cumulant expression for the free energy. Hwang et al.192 use the second order cumulant expression to examine the potential of mean force for ion transport in a channel—specifically the transport of Na+ and K+ ions in a cyclic peptide nanotube in water. They show how the de-solvation of the ions on entering the tube produces a free energy barrier and an attractive interaction between the ions and the functionalised tube provide free energy minimum in the tube. A second order cumulant expansion of the JE is also used to study the transport of sugar across lactose permease,193 and the potential of mean forces as the distance between dimers of cyclic peptides was decreased.194 In the latter case, a full average was also performed to check the accuracy of the assumption of the results obtained using the Gaussian approximation, and it was found that it was necessary to increase the transformation time in some cases in order to get reasonable agreement. Wang et al.195 use a Gaussian approximation to examine the potential of mean force of glycerol passing through an aquaporin. Classical simulations are used to examine the transport of di-oxygen across to the active site of the heme protein.196 An expansion was not used, and different scanning times and agreement of the profiles were used to assess the convergence of the results. Vemparala et al.197 use the JE and methods based on the JE to examine the transport of a halothane molecule across a lipid/water interface. QM-MM simulations were used to generate trajectories for the steered molecular dynamics calculation of the free energy profile for the conversion of two molecules (chorismate to prephenate) involved in the biosynthesis of amino acids.198 The results showed good agreement with those obtained from umbrella sampling. Only fair agreement with experiment was attributed to the accuracy of the density function theory calculations. The JE is used by Mu and Song199 to calculate the interfacial free energies of different crystal orientations for a Lennard-Jones system, and they find it more efficient than thermodynamic integration. Due to the changes in the dynamics, a general relationship for stochastic dynamics is not available like it is for deterministic dynamics. However, for mesoscopic systems, a mesoscopic FR is useful.200 Therefore, there has been much work on developing stochastic models with different conditions. Andrieux and Gaspard developed a stochastic fluctuation relation for nonequilibrium systems whose dynamics can be described by Schnakenberg’s network theory (e.g. mesoscopic electron transport, biophysical models of ion transport and some chemical reactions).201 Due to early experimental work on protein unfolding and related molecular motors, and their ready treatment by stochastic dynamics, a number of papers have appeared that model these systems and test the FR200,202 or JE for these.202 FR have also been obtained for stochastic models of chemical reaction networks.203,204 A number of fluctuation relations have also been developed with the view to design of experiments (see section 3.6) or devices. Recently a stochastic model of a ‘molecular refrigerator’ has been developed and the relevant expression for the FR derived.205 It is a Brownian particle with magnetic moment that is captured by an optical trap, tethered to another particle using a polymer to prevent rotation, and then subject to magnetic field that acts to counter the motion due to thermal fluctuations. The magnetic field thereby serves to ensure that the kinetic energy of the bead is less than its surroundings. The subject of the JE is no longer just the work due to the movement of the optical trap, but also includes a contribution due to the feedback mechanism. 3.6 Experiments The first tests of the JE and Crooks FR were by Liphardt et al.206 who used optical tweezers to extend a DNA-RNA hybrid chain, measuring the work required as the 200 | Chem. Modell., 2008, 5, 182–207 This journal is

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extension proceeded. As well as demonstrating the ability of observing fluctuations that would allow the JE and FR to be applied, it led to the use of the JE as an experimental tool for studying protein folding and for generating free energy landscapes. More recently, Collin et al.158 carried out an experiment using the Crooks FR to determine the difference in free energies of an RNA molecule and a mutant that differs by one base pair. The Crooks FR was shown to be useful far from equilibrium where insufficient sampling hampers convergence of the JE. Hummer and Szabo40 demonstrated that in single molecule stretching experiments, the JE provides an expression for the work at different times, whereas from an experimental point of view it is of more interest to know the free energy difference between states at different extensions of the molecule. They show how this can be obtained and apply it in experiments. Blickle et al.101 carried out an experiment resembling the optical trap ‘drag’ experiment discussed in section 2.3 to verify the JE in a system with a non-harmonic optical trap where there was no free energy change. Noy has used207 the JE to benefit in interpretation of experimental results of chemical force microscopy where the probes of atomic force microscopy are functionalized, and Douarche et al.208 have verified the Crooks FR and JE for fluctuations in the work of a mechanical oscillator that is in contact with a reservoir and driven by a large external field.

4. Fluctuation relations and irreversibility Several works have discussed the implication of the FR and JE on thermodynamic irreversibility.17–19,86,144,209–213 It should be mentioned, that underlying the derivation of these relationships is the assumption of causality.18 That is, in determining the probability of sets of trajectories we assume that they are sampled from an initial known phase space density, not a final one. Should we assume the reverse, an ‘antiFR’ would be obtained which would predict that, on average, the entropy decreases. However, given this, the reversible equations of motion lead to relations such as the Second Law Inequality (2.3), and the maximum work principle (3.7) that clearly shows irreversibility. Furthermore, in the form given by eqn (2.2), the way in which irreversibility emerges with increasing observation time or volume is evident. Recent papers have noted the connection of the FR to Kullback-Leibler distance, or relative entropy133,212 which measures the irreversibility of the system and Sevick et al.17 consider a similar property—the departure of the average of the timeaveraged dissipation function from zero—as a measure of irreversibility. The effect of system on size on reversibility is discussed in ref. 213.

5. Comment on the interpretation of the fluctuation relation The arguments of the deterministic fluctuations relations are well defined (see eqns (1.4), (2.1) and (3.6) above). In the ES FR, O is the subject of the fluctuation relation, in the Crooks FR it is the work and in the GC FR it is L. However, considerable confusion has developed in the past due to the temptation of equating fluctuations in properties when they have the same mean. For example, in field driven nonequilibrium system discussed above, the argument of the transient fluctuation relation is O = J FeVb, where J is the dissipative flux, Fe is the applied field, V is the volume of the system, and b = 1/(kBT) where T is the temperature. For a thermostatted system, the ensemble average value of hOi (which can be interpreted as work multiplied by b) is equal to the average phase space contraction hLi = h3Nthermai where Ntherm is the number of thermostatting particles and a is the thermostat multiplier and which can be interpreted as heat. However, the fluctuations in these properties are very different, and if O satisfies a fluctuation relationship the fact hOi = h3Nthermai alone clearly does not imply that 3Ntherma does, even in the long time limit. Nevertheless, this is often wrongly assumed. A clear numerical Chem. Modell., 2008, 5, 182–207 | 201 This journal is

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demonstration of the difference in the fluctuations of the two properties is given by Dolowschia´k and Kovaćs49 who study the Nose´-Hoover thermostatted periodic Lorentz gas. The problem is possibly perpetuated by the fact that the argument of the fluctuation relations is often referred to as the entropy production, although the entropy production is only defined in the thermodynamic limit and therefore many properties (with different fluctuations) will average to give an entropy production. The argument is also often referred to as ‘work’ or ‘heat’ when these properties are also not uniquely defined (see ref. 21, for example). However, another reason that this result has led to some confusion in the literature is that in the original works on the fluctuation theorems,1,2,23 isoenergetic systems were considered. In this particular case J(t) FeVb = 3Ntherma(t) instantaneously (i.e. the dissipation function is equal to the phase space contraction), and therefore the fluctuations in these properties are identical in this special case. This has led to authors mistakenly suggesting that, even in thermostatted systems, the ES FR refers to phase space contraction8 and/or that the GC FR refer to the dissipative flux when only the reverse is true in general.16 It has even resulted in the misleading adoption of the term ‘Gallavotti-Cohen Fluctuation Theorem’ for the fluctuation relations irrespective of the argument involved, although the derivation by Gallavotti and Cohen refers specifically to fluctuations in the phase space contraction. It should also be noted that different approaches or theorems can lead to FR that apply to the same property, but under different conditions. For example FR can be derived using the approach of Evans and Searles or Crooks that apply at all times, in contrast to the approach of Gallavotti and Cohen who require a long time limit (see discussion on heat flow where this distinction was important.88,214,215 The Crooks derivation does not require that the protocol or applied field be time-symmetric (or time-antisymmetric), whereas the approach of Evans and Searles does. While the arguments of the fluctuation relations are well defined theoretically, in some complex physical and chemical systems it might not be easy to identify the specific relationship for that property or to directly measure the correct property. This has led to many papers where a relationship for the property that appears in the argument is proposed (say B), and then the fluctuation relation is tested. In some PðBt ¼aÞ cases a linear relationship for the variation of ln PðB vs. a is obtained over a t ¼aÞ PðBt ¼aÞ restricted range of values of |a| r amax, but this is hardly surprising as ln PðB is t ¼aÞ an odd function of a. At best, these papers serve to test whether or not the property is appropriate and should not be interpreted as failure of the fluctuation theorems, or if a linear relationship with a slope other than the predicted unit slope is obtained, it does not provide evidence that the fluctuation relation is obeyed.

7. Conclusions We have presented brief descriptions of the relationships obtained from theorems about nonequilibrium systems: the ES FR, the Crooks FR and the JE. We also mention the GC FR in regard to its relationship to the ES FR. Over the years these relationships have increased our fundamental understanding of nonequilibrium systems and can explain how thermodynamic irreversibility emerges from underlying reversible equations of motion. This review describes recent work on the development and extensions of the relationships, tests of the relationships using simulation and experiment, and studies that have been carried out to understand physical and chemical phenomena. This area of research is growing rapidly. The relations are now well established and much work is being carried out on examining the utility and limitations. In the case of the JE, in particular, many adaptations have been proposed to improve its efficiency. Although it is highly unlikely to replace other techniques for free energy calculations in general, in some circumstances, particularly small systems where equilibration is slow, they will prove to be the method of choice. 202 | Chem. Modell., 2008, 5, 182–207 This journal is

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As noted in several papers, the FR are particularly relevant for small systems and are therefore expected to play an important role in nanoscience and understanding biological, single-molecule processes. However, at this point the practical application in these areas has been limited,207 with the most common use being the use of the JE to calculate free energy landscapes for transport in biological systems, for which other methods are available. It will be of considerable interest to see how the FR can be used in more novel applications—predicting rare events, design of nanomachines, development of nanofluidics, etc. A new area that is starting to attract is the application of these relations and those derived from them to nonergodic systems such as glasses.71,147,216 An important spin-off from the work on fluctuation relations is that it has been clearly demonstrated that experimental techniques have developed to such an extent that we can now test nonequilibrium relations directly.30,78,217 It is anticipated that such experiments will continue to improve to provide a tool that can be used as an alternative, or complement to computer simulations in testing new relationships.

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Many-body perturbation theory and its application to the molecular structure problem S. Wilsonab DOI: 10.1039/b608836h

1. Introduction At its most fundamental, chemistry is a quantum mechanical subject. The properties of molecules and interactions between molecular systems are determined by the solutions of the Schro¨dinger equation (or its relativistic generalizations) describing the component electrons and nuclei. Approximate solutions to the Schro¨dinger equation are developed by first decoupling the motions of the electrons from those of the nuclei by invoking the Born-Oppenheimer model. The development of approximations to the solutions of the resulting electronic Schro¨dinger equation are the central problem of molecular electronic structure theory. This problem is most usually attacked in two stages. In the first stage, some independent electron model is assumed in which each electron is assumed to move in some mean field. HartreeFock theory is the most widely used independent electron model. The second and most challenging stage of a molecular electronic structure study takes account of electron correlation effects, that is, the corrections to independent electron models attributable to instantaneous interactions between electrons. The ‘traditional’ approach to the electron correlation problem is the method of configuration interaction, C.I., (sometimes called configuration mixing). Over the past twenty years, the configuration interaction approach has been largely superceded by ‘manybody’ methods. This report covers applications of and developments in the theory of the many-body perturbative approach to the molecular structure problem during the period June 2005 through to May 2007. Applications of many-body perturbation theory in its lowest order form, which is often designated MP2 w, continue to make it the most widely used of the ab initio approaches to the molecular electronic structure problem which go beyond an independent particle model and take account of the effects of electron correlation. During the reporting period, applications have been reported in an ever-increasing range of research areas and the main focus of this review is on some of the emerging fields in which MP2 calculations are being carried out. Obviously, within the limited space available it is not possible to cover all of the fields of application. To some extent, the choices made reflect my personal taste, but it is to be hoped that they provide a ‘‘snapshot’’ of the range of contemporary applications of chemical modelling using many-body perturbation theory. This review continues four earlier biennial reviews published in this series1–4 in 2000, 2002, 2004 and 2006. In turn, these reviews built on my report, written almost three decades ago, for a previous Specialist Periodical Reports series (Theoretical Chemistry) and published in 1981.5 This earlier review was one of the first to survey the application of ‘many-body’ methods, and, in particular, many-body perturbation theory to the molecular structure problem. a

Physical & Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ b Department of Chemical Physics, Faculty of Mathematics, Physics and Informatics, Comenius University, 84215, Bratislava, Slovakia. E-mail: [email protected] { Møller-Plesset theory through second order, after Møller and Plesset who first considered the application of second order Rayleigh-Schro¨dinger perturbation theory with respect to a Hartree-Fock model reference function


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We begin this report with a very brief overview of my previous reports to this series since, unlike the present report, previous reports have concentrated on the development of new ‘many-body’ theory and its practical realization through computation. This overview is given in section 2. The present review is almost exclusively focussed on applications of many-body perturbative methods in chemical modelling and it is to this application-orientated research that we turn our attention in section 3 and the sections which follow. In developing new methodology, the primary focus of research is often, quite rightly, on the introduction of improved approximations. Approximations may be improvements on existing techiques because they support higher accuracy or because they display greater computational efficiency allowing the study of larger and/or more complex systems. In supporting new application areas, attention is focussed on the development of ‘‘theoretical model chemistries’’ or what may be described as a ‘‘numerical spectrometer’’. This ‘‘numerical spectrometer’’ places computational quantum chemistry along side other probes of matter on an atomic and molecular scale. It is to the development and successful exploitation of these ‘‘numerical spectrometers’’ z that we devote section 3. In section 4, we turn our attention to the many applications of second order manybody perturbation theory in chemical modelling. We provide a brief synopsis of the applications of ‘‘MP2’’ theory during the reporting period. In the following four sections, sections 5, 6, 7 and 8, we present more detailed descriptions of four specific application areas. In section 5, we describe the application of many-body perturbation theory to periodic systems. In section 6, we turn our attention to the application of many-body perturbation theory to biochemical molecular systems, in particular, DNA bases and amino acids. The study of the structure and function of biological macromolecules is, at present, dominated by molecular mechanics techniques which are invariably built on empirical force field. A third application area, which we consider in section 7, is in the ‘‘benchmarking’’ of DFT-based methods. DFT-based methods do not handle the two-electron interactions explicitly and thus have lower computational cost and therefore a wider range of applicability. However, practical forms of DFT involve a considerable amount of empirical parameterization. Furthermore, there is no clear defined procedure leading to the convergence of DFT methods to the right answer for the right reason. The final application area considered in some detail is basis set extrapolation and the calibration of general energy models. This is the focus of section 8. A summary is given in section 9 where future directions are briefly surveyed.

2. An overview of previous reports This, being the fifth in a series of biennial reports to the series Chemical ModellingApplications and Theory, it brings to ten years the period for which detailed reviews of the literature of many-body perturbation theory and its application to the molecular structure problem have been presented. (The first report, published in 2000, updated a previous report written some twenty years earlier for another series. For the first and second reports only applications of many-body perturbation theory to the problem of molecular electronic structure were considered, but with the third and subsequent reports the word ‘‘electronic’’ was dropped to indicate a broader remit). After a decade, it seems appropriate to begin this report by identifying the key advances in applications and in theory over the past decade before turning to the primary purpose of this present report which is to review the literature for the period June, 2005, to May, 2007. { The plural is used here since different ‘‘theoretical model chemistries’’ give rise to different ‘‘spectrometers’’. In rough terms, the methods and approximations which constitute different ‘‘theoretical model chemistries’’ can be liken to the different component parts of a spectrometer.


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The first report in this series was published in 2000 and reviewed the literature up to June 1999. The developments considered included: the analysis and evaluation of higher order terms in the many-body perturbation series. the formulation of many-body perturbation theory within the framework of relativistic quantum mechanics. the systematic implementation of the algebraic approximation with particular reference to many-body perturbation theory studies using finite basis set approximations. the exploitation of concurrent computing techniques in electron correlation calculations based on many-body perturbation theory. Published in 2002, the second report covered the period June 1999 to May 2001 and provided an opportunity to review the wide range of applications to which many-body perturbation theory in its simplest form, that is Møller-Plesset perturbation theory through second order, was being put at the turn of the millennium. The main development considered in this second report were: ‘‘MP2’’ theory and its current status in the landscape of many-body perturbation theory-based techniques. the remainder term in low-order perturbation theory the choice of zero order Hamiltonian operator in many-body perturbation theory. the status of theory for handling problems requiring a multireference formulation of the electron correlation problem based on Brillouin-Wigner perturbation theory. The period between June 2001 and May 2003 was reviewed in the third report which was published in 2004. This was the report for which the word ‘‘electronic’’ was dropped from the title and the application of ‘many-body’ methods to nuclear motion considered. The developments considered in this report were: the development of many-body perturbation theory to simultaneously describe both the electronic and the nuclear motion in molecular systems. the use of low order approximants (Pade´ and quadratic Pade´ approximants) in many-body perturbation theory studies. the development of local formulations of the electronic structure problem based on second order many-body perturbation theory for handling large molecules. A review of the wide range of applications of ‘‘MP2’’ theory-based methods was also given for the period covered by this review. The fourth report, immediately prior to this, was published in 2006 and covered the molecular many-body perturbation theory literature for the period beginning June 2003 and ending May 2005. The report focussed on two important and related areas of early twentieth century science: computation and supercomputation, and complexity. The former are becoming increasingly important in the realization of practical routes to useful chemical information from the basic equations of quantum mechanics. The latter is recognized as an important characteristic of the molecular 210 | Chem. Modell., 2008, 5, 208–248 This journal is

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entities and materials which are of interest in modern chemistry. In the discussion of computation and supercomputation, we gave particular emphasis to literate programming techniques and the potential that they have for a new approach to computational quantum chemistry, providing as they do a firm bridge between theoretical developments and their practical realization. The complexity of molecular systems was considered under the headings: large systems, relativistic formulations, multireference formalisms, and the theoretical description of multicomponent systems. This fourth report also contained a synopsis of applications carried out during the period under review.

3. Applications The main emphasis of this report is on the applications of many-body perturbation theory in chemical modelling, whereas previous reports have concentrated on new theory and its practical realization through computation. Application-orientated research using many-body perturbation theory adopts a different perspective to that assumed in methodological work. As was stated in the introduction, in supporting new application areas, attention is focussed on the development of ‘‘theoretical model chemistries’’ or what may be described as a ‘‘numerical spectrometer’’. This ‘‘numerical spectrometer’’ places computational quantum chemistry alongside other probes of matter on an atomic and molecular scale. It is to the development and successful exploitation of these ‘‘numerical spectrometers’’ that we devote this section whilst in the following sections we consider the many applications of second order many-body perturbation theory in chemical modelling and some specific application areas. We begin this section by considering the concept of a ‘‘theoretical model chemistry’’ and the development of a ‘‘numerical spectrometer’’ in section 1. In section 2, we briefly consider the complementary probes of matter which are used in modern science to elucidate the structure and properties of matter on an atomic scale. In section 3, we discuss how the very concept of atoms and molecules depends on the probes employed and how in modern science, computers provide one of the most powerful probes. The complementarity of different probes of matter is briefly described in section 4. The different perspectives given by complementary probes are emphasised and the greater potency acquired when probes are used in conjunction in a ‘‘problem based’’ environment is underlined. We turn our attention to the ‘knowledge base’ of quantum chemistry and, in particular, quantum chemical many-body perturbation theory in sections 5, 6 and 7. In section 5, we review the various texts into which our knowledge of the many-body problem in molecules has been distilled. These text books are complemented by the computer software and hardware which form essential ingredients of the ‘knowledge base’, in that they translate from the printed page to practical schemes of computation. A computer program6 ‘‘often possesses a structure and is characterised by details not adequately described by conventional mathematical notation. This is largely a consequence of the fact that the execution of a program is a dynamic process, in which the sequence of operations is of prime importance.’’ In section 6, available computer software for carrying out many-body perturbation theory calculations for molecular systems is described. Whereas a printed page can be copied with little regard to the nature of the material, usually paper, on to which it is copied, the correct execution of computer software is usually highly dependent on the hardware upon which it is implemented. In section 7, we discuss the hardware requirements for computational many-body perturbation theory for molecules. 3.1 Computational quantum chemistry: numerical spectrometer The approximate solution of the electronic Schro¨dinger equation is central to the study of molecular structure. Following the advent of the electronic digital computer Chem. Modell., 2008, 5, 208–248 | 211 This journal is

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in the mid-twentieth century, it was recognised that the computer offered a route to approximate solutions of the Schro¨dinger equation for molecular systems and a sustained effort to deveop the necessary theoretical and computational apparatus was undertaken. In a famous after dinner speech delivered at the Boulder Conference in 1959, the leading British Theoretical Chemist of the day, Professor Charles A. Coulson,7 predicted that ‘‘the whole group of theoretical chemists is on the point of splitting into two parts. . . . In its simplest form this difference was associated with the large-scale use of electronic computers, though . . . I think there is a deeper aspect of it than just this.’’ He named the two groups ‘‘group I (electronic computors) and group II (nonelectronic computors)’’ but suggested that the alternative designations ‘‘ab initio-ists and a posteriori-ists’’ might be more appropriate. Today, these ‘‘ab initio-ists and a posteriori-ists’’ have evolved into two groups who both use computers in their research. Computing is now ubiquitous in chemical modelling. The modern groups differ in their approach to the development, calibration and exploitation of quantum chemical models. In contemporary research, the study of molecular structure using ab initio methodology can be divided into two conceptually different approaches. In their well-known text ‘‘Ab inito Molecular Orbital Theory’’,8 published in 1986, J. A. Pople et al. describe the first of these two approaches as that in which ‘‘each problem is examined at the highest level of theory currently feasible for a system of its size’’ In the second approach, ‘‘a level of theory is first clearly defined after which it is applied uniformly to molecular systems of all sizes up to a maximum determined by available computational resources.’’ ‘‘Such a theory, if prescribed uniquely for any configuration of nuclei and any number of electrons, may be termed a theoretical model, within which all structures, energies, and other physical properties can be explored once the mathematical procedure has been implemented through a computer program.’’ This approach results in what Pople and his co-workers term a ‘‘theoretical model chemistry’’. Systematic comparison of the results obtained by application of this model with corresponding experimental data allows the model to acquire some predictive capability in situations where experiment is either difficult or impossible, or simply too expensive. In recent years, a range of theoretical methods have become available which provide a hierarchy of techniques for the molecular electronic structure problem. For example, full CI (configuration interaction) provides an exact solution of the electronic Schro¨dinger equation within a given basis set. The computational demands of the full CI method dictate that only basis sets of very modest size can be employed and the basis set truncation errors are very large indeed. On the other hand, calculations using systematic sequences of basis sets, an approach pioneered in the late 1970s, provides a technique for estimating the basis set limit for theoretical methods which are less demanding than full CI. The development of a theoretical model chemistry can involve the comparison of some ‘high-level’ (i.e. more accurate, but usually more computationally demanding) method with some ‘lower-level’’ technique which might be less accurate but which is also less 212 | Chem. Modell., 2008, 5, 208–248 This journal is

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demanding of computer resources and can therefore be applied more widely and also to larger systems. The past twenty years have witnessed a relentless increase in the power of computing machines. It has been observedy that the processing power of computers seems to double every eighteen months. This has been dubbed ‘‘Moore’s Law’’. As J. M. Roberts9 points out ‘‘No other technology has ever improved so rapidly for so long.’’ This ever-increasing computing power has led to higher accuracy in molecular electronic structure calculations, often because larger basis sets can be utilized. It has also has opened up the possibility of applications to larger molecules and molecular systems. Moore’s ‘law’ coupled with advances in theory and computer algorithms has led to molecular modelling becoming increasingly competitive as well as providing a complementary probe of matter. The understanding of the structure, behavior, and interactions of atoms and molecules is crucial to progress in modern science. Over the past thirty years, computational quantum chemistry has become part of the broader field of computational chemistry. Computational chemistry is itself one of the computational sciences. Today’s computers and supercomputers are revealing the intricacies of matter in unprecedented detail.

3.2 Complementary probes of matter Computational science is one of a suite of powerful techniques capable of elucidating the structure and properties of matter on an atomic scale. The value of computational science lies in the unique capabilities of the computer as a non-destructive and non-invassive probe of matter. Computational science is often complementary to probes employed in laboratory experiments. When applied in a ‘‘problem based’’, as opposed to a ‘‘technique based’’, approach to a given challenge, each technique affords a different perspective. Different techniques give complementary information about the studied system; the sum of the information gained is greater than that obtained from the techniques separately. The complexity inherent in many new forms of matter suggests that a single probe is unlikely to provide a complete understanding of a given problem. This complexity carries with it the need for complementary probes. Computational science often relates information obtained from different experimental probes of matter.

3.3 Probes of matter The concept of atoms and molecules emerged in the earliest days of modern chemistry from meticulous measurements of mass and volume. These were the only probes of matter available to early workers. The reality of atoms and molecules was established both by the explanatory power of the ‘atomic’ model and by experimental evidence of microscopic entities, with Einstein’s famous interpretation of Brownian motion in 1905 providing irrefutable confirmation of the atomic hypothesis. Today, science has an armoury of powerful probes to study matter at a range of scales from viral proteins to sub-atomic quarks. In spectroscopy, these probes are electromagnetic radiation, whilst in scattering experiments they are particles. Waveparticle duality allows electrons, for example, to be used in either mode. Each probe affords a unique perspective on the nature of the sample under investigation and different probes interact with the sample in different ways. } by Mr G. E. Moore, co-founder Intel, Electronics, 38, No. 8, April 19, 1965.


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The the use of a computational approach provides a unique and non-invasive probe of matter. The computer is often a more cost-effective and/or less hazardous probe. However, the complexity of the systems studied coupled with the complexity of the equations which describe them always necessitates the introduction of approximations and the construction of models. This limits the applicability of the computational approach, but, as we shall demonstrate in this report, this is a limitation which is continually being push back both by developments in computational quantum chemistry itself and advances in computing technology.

3.4 Complementarity Whilst computational science on its own provides a unique and powerful probe of matter on an atomic scale, it often acquires greater potency when used in conjunction with other techniques, such as synchrotron radiation, nuclear magnetic resonance, transmission electron microscopy or neutron scattering. Complementarity between experimental techniques is well established. For example, both X ray and neutron scattering scattering are immensely valuable in the study of the structure, bonding, and function of molecules. However, there are important differences between the two methods, which make them complementary. X-ray photons interact primarily with electrons whereas neutrons interact primarily with nuclei. Heavier atoms, with more electrons, scatter X-rays more strongly than light atoms. X-ray diffraction is not, therefore, so well suited to locating light atoms, such as hydrogen. Yet the identification of hydrogen atoms and hydrogen bonds is fundamental to the understanding of many materials, for example, the folding, stability and function of biomolecules. Neutron scattering from nuclei varies almost randomly across the Periodic Table and is particularly strong for hydrogen and, in particular, its isotope deuterium. Neutrons are therefore able to ‘‘see’’ hydrogen atoms. Computational studies complement experimental methods. Complementarity’ is at the heart of a ‘problem-based’ approach to the study of matter at a microscopic level, which is increasingly seen as more productive than traditional ‘technique-based’ approaches. Complementary experimental techniques can each give a different perspective on a specific ‘problem’, since they probe the studied system in different ways. Computer simulation and modelling can provide an alternative route to information which may be difficult and/or expensive and/or hazardous to obtain experimentally. Equally, they can also help to interpret and relate results obtained from different experimental techniques. Increasingly modern science is concerned with complex systems, for example, the layered copper-oxide compounds displaying high temperature superconductivity, the folding and interaction of biomacromolecules which is influenced not only by the solvent but also by intra- and inter-molecular hydrogen bonding. With complexity, materials may gain new emergent properties which are not possessed by their component parts. However, the complexity inherent in many new forms of matter suggests that a single probe is unlikely to provide a complete understanding of a given system. Complexity carries with it the need for complementarity. Other techniques including X-ray and optical spectroscopy, NMR, and electron and proximal probe microscopies (STM-scanning tunneling microscopy, SFM-scanning force microscopy, NSOM-near-field scanning optical microscopy) are required to build up a more complete picture. With this requirement comes the need for an integrated, problemsolving approach. The great flexibility of the computational approach makes it a most powerful probe of the microscopic nature of matter which will continue to reveal key insights. 214 | Chem. Modell., 2008, 5, 208–248 This journal is

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3.5 The knowledge base: text books Any application of the methods of computational quantum chemistry and the broader field of computational chemistry is based on the available knowledge and although this is contained in the primary literature it is at its most accessible in textbooks. There are a number of textbooks at various levels from undergraduate to research frontier. In reverse chronological order, these are: S. Wilson, Electron Correlation in Molecules, Dover Publications, New York, 200710 F. Jensen, Introduction to Computational Chemistry, 2nd Edition, John Wiley, Chichester, 200611 C. J. Cramer, Essentials of Computational Chemistry, 2nd Edition, John Wiley, Chichester, 200412 P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, Oxford University Press, 200413 G. H. Grant and W. G. Richards, Computational Chemistry, Oxford University Press, 200414 A. Hinchliffe, Molecular Modelling for Beginners, John Wiley, Chichester, 200315 Handbook of Molecular Physics and Quantum Chemistry, eds. S. Wilson, P. F. Bernath and R. McWeeny, John Wiley, Chichester, 200316 D. Young, Computational Chemistry: A Practical Guide for Applying Techniques to Real World Problems, John Wiley, Chichester, 200117 T. Helgaker, P. Jorgensen and, J. Olsen, Molecular Electronic Structure Theory, John Wiley, Chichester, 200018 A. Hinchliffe, Modelling Molecular Structures, John Wiley, Chichester, 200019 D. B. Cook, Handbook of Computational Quantum Chemistry, Oxford University Press, 199820 Encyclopedia of Computational Chemistry, eds. P. von Rague´ Schleyer (Editorin-Chief), N. L. Allinger, H. F. Schaefer III, T. Clark, J. Gasteiger, P. Kollman and P. Schreiner, John Wiley, Chichester, 199821 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, 2nd Edition, Dover Publications, New York, 199622 F. E. Harris, D. L. Freeman and H. J. Monkhorst, Algebraic and Diagrammatic Methods in Many-Fermion Theory, Oxford University Press, 199223 R. McWeeny, Methods of Molecular Quantum Mechanics, 2nd Edition, Academic Press, 199224 Methods in Computational Chemistry: Electron Correlation in Atoms and Molecules, ed. S. Wilson, Kluwer Academic/Plenum Press, Dordrecht 198725 Chem. Modell., 2008, 5, 208–248 | 215 This journal is

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I. Lindgren and J. Morrison Atomic Many-Body Theory, Springer-Verlag, Berlin, 198626 W. J. Hehre, L. Radom, P. V. Schleyer and J. A. Pople, Ab initio Molecular Orbital Theory, John Wiley, Chichester, 198627 S. Wilson, Chemistry by Computer: An Overview of the Applications of Computers in Chemistry, Kluwer Academic/Plenum Publishers, Dordrecht, 198628 A. C. Hurley, Electron Correlation in Small Molecules, Academic Press, 197729 A. C. Hurley Introduction to the Electron Theory of Small Molecules, Academic Press, 197730 The following series contain reviews of aspects of comutational quantum chemistry: Computational Chemistry: Reviews of Current Trends, ed. J. Leszczynski, World Scientific, 1996-31 Reviews in Computational Chemistry, eds. K. B. Lipkowitz, D. B. Boyd, [except vol. 21–22] R. Larter [vols. 19–21] and T. R. Cundari [vols. 19–23], V. J. Gillet [vol. 22], Wiley, 1990-32 There are also a number of volumes dealing with the problem of describing atoms and molecules within a relativistic formalism. Again in reverse chronological order these are: K. G. Dyall, and K. Faegri, Introduction to Relativistic Quantum Chemistry, Oxford University Press, 200733 I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation, Springer-Verlag, New York, 200634 Relativistic Electronic Structure Theory. Part 2. Applications, ed. P. Schwerdtfeger, Elsevier, Amsterdam, 200435 Recent Advances in Relativisitic Molecular Theory, eds. K. Hirao and Y. Ishikawa, World Scientific Publishing, Singapore, 200436 Theoretical Chemistry and Physics of Heavy and Superheavy Elements, eds. U. Kaldor and S. Wilson, Kluwer Academic Publishers, Dordrecht, 200337 Relativistic Effects in Heavy Element Chemistry and Physics, ed. B. A. Hess, John Wiley, Chichester, 200238 Relativistic Electronic Structure Theory. Part 1. Fundamentals, ed. P. Schwerdtfeger, Elsevier, Amsterdam, 200239 K. Balasubramanian, Relativistic Effects in Chemistry: Theory and Techniques and Relativistic Effects in Chemistry: Theory and Techniques Pt. A, John Wiley, Chichester, 199740 K. Balasubramanian, Relativistic Effects in Chemistry: Applications: Applications Pt. B, John Wiley, Chichester, 199741 216 | Chem. Modell., 2008, 5, 208–248 This journal is

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The Effects of Relativity in Atoms, Molecules and the Solid State, eds. S. Wilson, I. P.Grant and B. L. Gyorffy, Kluwer Academic/Plenum Press, Dordrecht, 199142 Methods in Computational Chemistry: Relativistic Effects in Atoms and Molecules, ed. S. Wilson, Kluwer Academic/Plenum Press, Dordrecht, 198943 Relativistic Effects in Atoms, Molecules, and Solids, ed. G. L. Malli, 198344

3.6 The knowledge base: computer software The text books, encyclopedias and handbooks listed in the previous section are complemented by the computer software and hardware through which practical schemes of computation are realized. In this section, we consider the computer software for carrying out molecular many-body perturbation theory calculations. The computer hardware appropriate for such calculations is considered in section 3.7. That computation would play a central role in practical applications of quantum chemistry was recognized in the 1950s, when, for example, in a now famous article entitled ‘‘Broken Bottlenecks and the Future of Molecular Quantum Mechanics’’, Mulliken and Roothann recognized that in ref. 45 ‘‘A major and indeed crucial step beyond the development of formulas for molecular integrals was the programming for large electronic digital computers of the otherwise still excessively time-consuming numerical computation of these integrals, and of their combination to obtain the desired molecular wave functions and related molecular properties; The pioneering work in this field was that of S. F. Boys at Cambridge, England.’’ Mulliken and Roothaan continued with the following prophecy which has largely been fulfilled45 ‘‘It can now be predicted with confidence that machine calculations will lead gradually toward a really fundamental quantitative understanding of the rules of valence theory and the exceptions to these; toward a real understanding of the dimensions and detailed structures, force constants, dipole moments, ionization potentials, and other properties of stable molecules and equally of unstable radicals, anions, and cations, and chemical reaction intermediates; toward a basic understanding of activated states in chemical reactions, and of triplet and other excited states which are important in combustion and explosion processes and in photochemistry and in radiation chemistry; also of intermolecular forces; further, of the structure and stability of metals and other solids; of those parts of molecular wave functions which are important in nuclear magnetic resonance, nuclear quadrupole coupling, and other interactions involving electrons and nuclei; and of very many other aspects of the structure of matter which are now understood only qualitatively or semiempirically. From the wave functions of ground and excited states, spectroscopic transition probabilities and lifetimes of excited states can also be computed; these and other items of information which can be obtained for molecules and radicals and are especially difficult to obtain experimentally are important in an understanding of upper atmosphere phenomena and in astrophysics.’’ In our introduction to section 3, we emphasized that a computer program6 ‘‘often possesses a structure and is characterised by details not adequately described by conventional mathematical notation.’’ The algorithms employed in quantum chemistry are central to its practical application and therefore should be regarded as an integral part of the literature of the field. In a previous report, we have considered Chem. Modell., 2008, 5, 208–248 | 217 This journal is

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the effective communication of ideas and concepts embodied in quantum chemistry computer software. We have emphasized, following Ziman46 that ‘‘Research results do not count as scientific unless they are reported, disseminated, shared, and eventually transformed into communal property by being formally published.’’ Ziman47 recognizes that ‘‘ . . . what is distinctive about formal scientific communication is neither the medium nor the message: it is that it is published’’ the essential point being that ‘‘ . . . it is fully and freely available for open criticism and constructive use.’’ Unfortunately, as we explained in our previous report,4 ‘‘much contemporary quantum chemistry software does not satisfy these requirements’’ of transparency, accountability and, therefore, accessibility. The Quantum Chemistry Program Exchange (QCPE)48 was an early attempt to formalize the distribution of quantum chemistry software. It was created in 1962 by H. Shull ay Indiana University and offered a small collection of some 33 pieces of software. Another attempt to formalize the distribution of software for atomic and molecular physics is the journal Computer Physics Communications49 K. V. Roberts, Comput. Phys. Commun., 1969, 1, 1 which published both scientific papers and computer programs. In spite of the continued success of this journal, it has nevertheless has to be observed that few complete quantum chemical packages have been published via this mechanism, although there are exceptions. For example, the HONDO code, ‘‘The general atomic and molecular electronic structure system HONDO: version 7.0’’, was published in Computer Physics Communications in 1989.50 In some of our recent work,51–53 in our previous report4 and in the volume by Cook,54 the use of literate programming to provide a mechanism for introducing higher standards of code documentation and thus greater levels of collaboration in quantum chemistry code development and distribution is discussed. Literate programming combines the theoretical development of a particular model with the associated computer code. In 1984, D. E. Knuth published his seminal paper55 entitled ‘‘Literate programming’’ in The Computer Journal. However, this approach has not been widely adopted in spite of its widely recognized advantages in the computer science and computational science communities. In this section, we restrict our attention to the available computer software for carrying out many-body perturbation theory calculations for molecular systems. The first published quantum chemical code for carrying out molecular many-body perturbation theory calculations was due to D. M. Silver56,57 and the present author.58 The package, which was originally published in Computer Physics Communications in 1978, was written in FORTRAN code for an IBM 360/91 computer. The Computer Physics Communication library numbersz are:

} http://cpc.cs.qub.ac.uk/cpc/


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Together these programs calculate the components of the correlation energy corresponding to all second and third order diagrammatic terms. The code was heavily overlaid and restricted in the size of problem that could be handled. They were:56 ‘‘ . . . restricted to non-degenerate, closed-shell ground- states of atoms and molecules. The reference wavefunction must be closed-shell matrix Hartree-Fock single-determinantal wavefunction. Program dimension statements limit the basis set size to 10 occupied spatial orbitals (20 virtual electrons) and 25 unoccupied spatial orbitals (50 virtual states): these dimensions can easily be increased if necessary.’’ Today, quantum chemical code for carrying out molecular many-body perturbation theory calculations is made available as a component of many ‘‘black box’’ computer packages. Although, it is obviously possible to build many-body perturbation theory code from routines taken from ‘‘standard libraries’’ such as the NAG Library8 or the ‘‘Numerical recipes’’59 volumes* (and this may well be done in development work), most contemporary applications exploit careful tailored packages which can handle the whole molecular electronic structure problem by means of an interfaced suite of routines for carrying out the various stages of the calculation. Often the complexity of quantum chemical calculations demands the use of code specifically tailored to the algorithm being invoked and the hardware platform being employed. We note, however, that these carefully tailored packages often make use of optimized code for certain computational kernels. For example, the Basic Linear Algebra Subroutines (BLAS) are often exploited. Some use is made of software tools such as MAPLE ‘‘an advanced, high performance mathematical computation engine with fully integrated numerics & symbolics, all accessible from a WYSIWYG technical document environment’’ is made in development work.ww Some quantum chemistry packages make using of more specialized tools such as the Tensor Contraction Engine (TCE)zz. Hirata60 describes how complex symbolic algebra, such as the manipulation of second-quantized operators, Slater determinants, Feynman diagrams, which is inevitable in quantum chemistry, can be carried out by machine. This approach faciliates the automation of the algebraic transformation and computer implementation of ‘many-body’ methods for electron correlation calculations. The development of ‘‘black box’’ computer programs began in the early 1960s. The POLYATOM system was described by M. P. Barnett in 1963 in a paper published in Rev. Mod. Phys. entitled ‘‘Mechanized Molecular Calculations The POLYATOM System’’,61 although it was revealed in an historical review62 published in 1997 by B. T. Sutcliffe, one of Barnett’s coworkers, that ‘‘the report was more about what was planned to happen rather than what by that time had happened’’

|| The NAG Library was orginally developed as collaborative project led by Dr B. Ford at the University of Nottingham, where he had worked on numerical algorithms for carrying out quantum chemistry calculations in association Professor G. G. Hall. ‘‘The Nottingham Algorithm Group’’ became ‘‘The Numerical Algorithms Group’’ providing ‘‘general purpose’’ numerical software. For further details consult the company website at http://www.nag.co.uk/. * For details consult the ‘‘Numerical recipes’’ website at http://www.nr.com/ {{ For details, see the website http://www.maplesoft.com/. See also MATHEMATICA http://www.wolfram.com/ {{ For discussion of the Tensor Contraction Engine see http://www.csc.lsu.edu/gb/TCE/ and the recent publication of S. Hirata60 entitled ‘‘Symbolic Algebra in Quantum Chemistry’’


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Sutcliffe continues: . . . it is nevertheless an accurate account of what eventually was to happen. By the end of 1963 the program had become widely distributed, without guarantee, and though it was not quite bug-free initially, it rapidly became so on use . . . ’’ In this paper, Barnett61 explains ‘‘The system will consist of several major blocks which are listed below. It will be possible to use these in a fully automatic manner, or under manual control, by the provision of the appropriate master programs and linking parameters. The system is being coded in FORTRAN for the IBM 709/90 computers. It is designed to allow independent modification and refinement of the individual blocks of the system, and to allow easy expansion to include further blocks as these are developed.’’ The initial structure of the POLYATOM system corrresponded to the procedure described earlier by Boys and Cook63 and in similar work by Reeves and Harrison64 entitled ‘‘Use of Gaussian Functions in the Calculation of Wavefunctions for Small Molecules’’. The use of Gaussian functions, rather than the Slater (exponential) functions favoured by Mulliken and his coworkers at that time, was to be crucial to the success of practical computational quantum chemistry and, in particular, its application to arbitrary polyatomic molecules. The main blocks of the initial POLYATOM system were as follows: 1. ‘‘an input program, to store the nuclear coordinates and orbital parameters, with their transformation properties’’ 2. ‘‘an integral list generator’’ 3. ‘‘an integral evaluation program’’ 4. ‘‘an

SCF

program’’

5. ‘‘a program to transform one- and two-electron integrals from one basis to another’’ 6. ‘‘a codetor list generator to read or generate a list of functions that are to used in a configuration interaction calculation’’ 7. ‘‘a formula generator . . . for the matrix elements between codetors’’ 8. ‘‘a matrix generator to compute the matrix elements between codetors’’ 9. ‘‘a program to solve the secular equation’’ 10. ‘‘a program to form the density matrices, dipole moments,’’ etc The POLYATOM project was not the only one of its type. Indeed, in the same issue of Rev. Mod. Phys. as Barnett’s paper, R. K. Nesbet65 of IBM Cooporation’s San Jose Laboratory describes ‘‘Computer Programmes for Electronic Wave Function calculations’’ and F. E. Harris66 discusses ‘‘Gaussian Wavefunctions for Polyatomic Molecules’’. A fuller account of the POLYATOM code was published in 1966 by Csizmadia, Harrison, Moskowitz and Sutcliffe67 in a paper entitled ‘‘Non-Empirical LCAO-MOSCF-CI Calculations on Organic Molecules with Gaussian Type Functions. Part I. Introductory Review and Mathematical Formalism’’. This was followed immediately 220 | Chem. Modell., 2008, 5, 208–248 This journal is

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by one of the first applications of the POLYATOM program68 ‘‘Part II. Preliminary Investigations on Formyl Fluoride’’. In their historical review, Smith and Sutcliffe62 recount how ‘‘In 1963 the Chicago group and Enrico Clementi were taking the first steps toward what was to become IBMOL, and for a time there was a friendly rivalry between them and the developers of POLYATOM, but as IBMOL began its IBM-based development it drew steadily ahead of POLYATOM in the level of support and the facilities provided. But even today it is possible to find programs employing POLYATOM ideas and, occasionally, the vestiges of POLYATOM code can be recognised. But POLYATOM and IBMOL began a line of program systems for molecular electronic structure calculations whose current representatives are GAUSSIAN, GAMESS, and so on.’’ Initially, the POLYATOM program was coded to ‘‘limit the basis set to 50 elements’’.61 By 1966, the IBMOL program could (in principle) ‘‘handle a maximum of 800 Gaussian functions distributed on 50 centers.’’69 This program also used contracted Gaussian basis functions which had been introduced earlier by Clementi.70 Many other codes followed. The most well known of these was the GAUSSIAN code which led to the award of the Nobel Prize for Chemistry to the late Sir John Pople in 1998 ‘‘for his development of computational methods in quantum chemistry’’. GAUSSIAN 7071 was published a program number 236 by the Quantum Chemistry Program Exchange.48 Other codes included the HONDO program developed by Dupuis and his collaborators72 and also published through the Quantum Chemistry Program Exchange, and the ATMOL program by Saunders and coworkers.73 In the 1970s, development of the ATMOL code was undertaken as part of the UK’s Collaborative Computational Project (CCP1) on Electron correlation in molecules. During the 1980s and early 1990s the collaborative element of CCP1 was replaced by a more competitive approach for we saw the emergence of a number of rival electronic structure packages in the United Kingdom beginning with CADPAC and GAMESS(UK), and subsequently MOLPRO and contributions to Pople’s GAUSSIAN. There is much duplication in the functionality of these codes which will be discussed further below. Many of the ideas and methods incorporated in these codes had, as Smith and Sutcliffe pointed out62 in their 1997 review on ‘‘The development of Computational Chemistry in the United Kingdom’’, ‘‘ . . . been grown at public expense and by various hands’’ and, therefore, their ‘‘ . . . position as property, to be disposed of or traded, was deeply ambiguous.’’yy Today, many quantum chemistry program packages are available which contain code which can perform ‘many body’ calculations using a perturbative formalism. Some of these packages are freely available, some are marketed as commercial products. For some the source code is freely available, for others only the binary code is distributed. Some are well documented, others only provide documentation for the user. Our list is probably not complete, but should serve to illustrate what is available. The list is arranged alphabetically by the name of the quantum chemistry package. }} Smith and Sutcliffe continue ‘‘The obvious solution to it is to expropriate the codes and market them, using any profits to further the work. But this would undoubtedly cause tensions in the groups involved and might well call into question the basic idea of a publicly funded collaborative computational enterprise.’’


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ACES

II-Advanced Concepts in Electronic Structure

ACES II is a program product of the Quantum Theory Project, University of Florida. Authored by R. J. Bartlett and his co-workers,74 this program was developed for calculations using ‘many-body’ techniques to treat electron correlation. The authorship is as follows: J. F. Stanton, J. Gauss, S. A. Perera, J. D. Watts, A. D. Yau, M. Nooijen, N. Oliphant, P. G. Szalay, W. J. Lauderdale, S. R. Gwaltney, S. Beck, A. Balkov 0 a, D. E. Bernholdt, K. K. Baeck, P. Rozyczko, H. Sekino, C. Huber, J. Pittner, W. Cencek, D. Taylor, and R. J. Bartlett. Integral packages included are VMOL (J. Almlof and P. R. Taylor); VPROPS (P. Taylor); ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, and P. R. Taylor); HONDO/GAMESS (M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. J. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis and J. A. Montgomery). Further details can be found at the website: http://www.qtp.ufl.edu/Aces2/ The ACES II Mainz-Austin-Budapest version was released in 2005. Details of this version can be found at the following website: http://www.aces2.de/

CADPAC

- The Cambridge Analytical Derivatives Package

This code was original developed during the 1980s under the auspices of the UK CCP1. The current version is cited as follows: R. D. Amos with contributions from I. L. Alberts, J. S. Andrews, S. M. Colwell, N. C. Handy, D. Jayatilaka, P. J. Knowles, R. Kobayashi, K. E. Laidig, G. Laming, A. M. Lee, P. E. Maslen, C. W. Murray, J. E. Rice, E. D. Simandiras, A. J. Stone, M.-D. Su and D. J. Tozer, The Cambridge Analytic Derivatives Package-A suite of quantum chemistry programs, Issue 6, Cambridge, 1995. Further details can be found at the website: http://www-theor.ch.cam.ac.uk/software/cadpac.html

COLUMBUS

originated in 1980 in the Department of Chemistry of the Ohio State University, this code was developed by I. Shavitt (Ohio State), H. Lischka (University of Vienna) and R. Shepard (Battelle Columbus Laboratories. The key publications describing the COLUMBUS package are H. Lischka, R. Shepard, F. B. Brown and I. Shavitt, Int. J. Quantum Chem., Quantum Chem. Symp., 1981, 15, 91. R. Shepard, I. Shavitt, R. M. Pitzer, D. C. Comeau, M. Pepper, H. Lischka, P. G. Szalay, R. Ahlrichs, F. B. Brown, and J. Zhao, Int. J. Quantum Chem., Quantum Chem. Symp., 1988, 22, 149. H. Lischka, R. Shepard, R. M. Pitzer, I. Shavitt, M. Dallos, Th. Mu¨ller, P. G. Szalay, M. Seth, G. S. Kedziora, S. Yabushita and Z. Zhang, Phys. Chem. Chem. Phys., 2001, 3, 664. The current release is cited as H. Lischka, R. Shepard, I. Shavitt, R. M. Pitzer, M. Dallos, Th. Mu¨ller, P. G. Szalay, F. B. Brown, R. Ahlrichs, H. J. Bo¨hm, A. Chang, D. C. Comeau, R. Gdanitz, H. Dachsel, C. Ehrhardt, M. Ernzerhof, P. Ho¨chtl, S. Irle, G. Kedziora, T. Kovar, V. Parasuk, M. J. M. Pepper, P. Scharf, H. Schiffer, M. Schindler, M. Schu¨ler, M. Seth, E. A. Stahlberg, J.-G. Zhao, S. Yabushita, Z. Zhang, 222 | Chem. Modell., 2008, 5, 208–248 This journal is

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M. Barbatti, S. Matsika, M. Schuurmann, D. R. Yarkony, S. R. Brozell, E. V. Beck, and J.-P. Blaudeau, COLUMBUS, an ab initio electronic structure program, release 5.9.1, 2006 Further details can be found at the website: http://www.univie.ac.at/columbus/

DALTON

Named after John Daltonzz the father of modern atomic theory, this package is described by the authors (in the program manual) as ‘‘in many respects an ‘expert’s’ program’’ The current version is cited as follows: DALTON, a molecular electronic structure program, Release 2.0, 2005 Further details, including the list of contributors, can be found at the website: http://www.kjemi.uio.no/software/dalton/dalton.html

HONDO

This code was first developed during the 1970s and distributed by QCPE. It featured very fast integral evaluation based on the work of Dupuis, Rys and King.72 HONDO 78 was an early package containing procedures to compute the energy gradient in geometry optimization and force constant matrix evaluation.76 The current version is cited as follows: M. Dupuis, A. Marquez, and E. R. Davidson, HONDO 99.6, 1999, based on HONDO 95.3, M. Dupuis, A. Marquez, and E. R. Davidson, Quantum Chemistry Program Exchange (QCPE), Indiana University, Bloomington, IN 47405. Additional capabilities are included in the HONDOPLUS package, which is cited as follows: HONDOPLUS-v.5.1, by H. Nakamura, J. D. Xidos, A. C. Chamberlin, C. P. Kelly, R. Valero, J. D. Thompson, J. Li, G. D. Hawkins, T. Zhu, B. J. Lynch, Y. Volobuev, D. Rinaldi, D. A. Liotard, C. J. Cramer, and D. G. Truhlar, University of Minnesota, Minneapolis, 2007 based on HONDO v.99.6.

GAMESS-General

Atomic and Molecular Electron Structure System

This package is based on code originally developed by the National Resource for Computations in Chemistry The citation for the original code is: M. Dupuis, D. Spangler, and J. J. Wendoloski National Resource for Computations in Chemistry Software Catalog, University of California: Berkeley, CA, 1980, Program QG01 The current version is cited as follows: M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis and J. A. Montgomery, J. Comput. Chem., 1993, 14, 1347–1363. General Atomic and Molecular Electronic Structure System M. S. Gordon, M. W. Schmidt, in ‘‘Theory and Applications of Computational Chemistry: the first forty years’’ eds. C. E. Dykstra, G. Frenking, K. S. Kim, G. E. Scuseria, Elsevier, Amsterdam, 2005, pp. 1167–1189, Advances in electronic structure theory: GAMESS a decade later Further details can be found at the website: http://www.msg.ameslab.gov/gamess/ There is also an independent user group which can be found at the website: http://groups.google.com/group/gamess }} 1766–1844


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There are a number of programs based on or derived from GAMESS. These include: – PC-GAMESS A freely available code derived from GAMESS which has been ‘‘developed to offer high performance on Intel-compatible x86, AMD64, and EM64T processors’’ The code contains ‘‘very efficient MP2 energy and gradient modules’’. Further details can be found at the website: http://classic.chem.msu.su/gran/gamess/ – CHEMBIODRAW This is a powerful chemistry and biology drawing application delivering drawing, publishing and analytical features on a Windows/Mac environment. GAMESS is provided in the latest release ChemBioDraw Ultra 11.0, but although the underlying code is the full version, there is no way for users to access any features not available in the graphical user interface. Further details can be found at the website: http://www.cambridgesoft.com/ – GAMESS (UK) A version of GAMESS developed and made available to the acedemic community in the UK. Further details can be found at the website: http://www.nsccs.ac.uk/

GAUSSIAN

Certainly GAUSSIAN is today the most well known and widely used quantum chemistry program. GAUSSIAN70 was distributed by QSCP as program number 23671 entitled GAUSSIAN 70: Ab initio SCF-MO Calculations on Organic Molecules and authored by W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton and J. A. Pople. The current version is cited as M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr, T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. AlLaham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, GAUSSIAN 03, (Revision C.02), Gaussian, Inc., Wallingford CT, 2004. and is marketed by Gaussian Inc. Further details can be found at the Gaussian Inc. website: http://www.gaussian.com/ HYPERCHEM A PC-based computational chemistry code with very efficient routines for optimization of MP2 geometries using the method ofconjugate directions. The current version, HYPERCHEM 8.0, is marketed by HYPERCHEM Molecular Modeling Software, Hypercube, Inc., 224 | Chem. Modell., 2008, 5, 208–248 This journal is

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1115 NW 4th St., Gainesville FL, USA Further details can be found at the website: http://www.hyper.com/

JAGUAR

JAGUAR is described as a ‘‘a high-performance ab initio package for both gas and solution phase simulations, with particular strength in treating metal containing systems’’. It is marketed by a company called Schro¨dinger with head office at Schro¨dinger Inc., 101 SW Main Street, Suite 1300, Portland, OR 97204, USA Further details can be found at the website: http://www.schrodinger.com/

MOLCAS

MOLCAS, like DALTON, is something of an ‘‘experts code’’. It is described as a code for the ‘‘ab initio treatment of very general electronic structure problems for molecular systems in both ground and excited states’’. It is particularly for electronic structure problems requiring a multi-reference treatment. G. Karlstrom, R. Lindh, P.-E. Malmqvist, B. O. Roos, U. Ryde, V. Veryazov, P.-O. Widmark, M. Cossi, B. Schimmelpfennig, P. Neogrady, L. Seijo, Comput. Mater. Sci. 2003, 28, 222. Further details can be found at the website: http://www.teokem.lu.se/molcas/

MOLPRO

The MOLPRO quantum chemistry package has an ‘‘emphasis is on highly accurate computations, with extensive treatment of the electron correlation problem’’. It is maintained by H.-J. Werner and P. J. Knowles, and contains contributions from a number of authors: MOLPRO is a package of ab initio programs written by H.-J. Werner and P. J. Knowles, with contributions from R. D. Amos, A. Bernhardsson, A. Berning, P. Celani, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, C. Hampel, G. Hetzer, T. Korona, R. Lindh, A. W. Lloyd, S. J. McNicholas, F. R. Mandy, W. Meyer, M. E. Mura, A. Nicklass, P. Palmieri, R. Pitzer, G. Rauhut, M. Schutz, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson. Further details can be found at the website: http://www.molpro.net/

MPQC

a quantum chemistry package written for massively parallel computers and available as open-source software. The program reference is: The Massively Parallel Quantum Chemistry Program (MPQC), C. L. Janssen, I. B. Nielsen, M. L. Leininger, E. F. Valeev, E. T. Seidl, Sandia National Laboratories, Livermore, CA, USA, 2004 More details of the approach used in this package can be found in: J. P. Kenny, S. J. Benson, Y. Alexeev, J. Sarich, C. L. Janssen, L. Curfman Mcinnes, M. Krishnan, J. Nieplocha, E. Jurrus, C. Fahlstrom and T. L. Windus, Component-based integration of chemistry and optimization software, J. Comp. Chem. 2004, 25, 1717 Further details can be found at the website: http://www.mpqc.org/ The open-source software is distributed by sourceforge.net http://sourceforge.net/projects/mpqc Chem. Modell., 2008, 5, 208–248 | 225 This journal is

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NWCHEM an ab initio quantum chemistry program written specfically for parallel computers. It was developed by the Molecular Sciences Software group of the Environmental Molecular Sciences Laboratory (EMSL) at the Pacific Northwest National Laboratory (PNNL). The current version, NMCHEM 5.0, is cited as: E. J. Bylaska, W. A. de Jong, K. Kowalski, T. P. Straatsma, M. Valiev, D. Wang, E. Apra`, T. L. Windus, S. Hirata, M. T. Hackler, Y. Zhao, P.-D. Fan, R. J. Harrison, M. Dupuis, D. M. A. Smith, J. Nieplocha, V. Tipparaju, M. Krishnan, A. A. Auer, M. Nooijen, E. Brown, G. Cisneros, G. I. Fann, H. Fru¨chtl, J. Garza, K. Hirao, R. Kendall, J. A. Nichols, K. Tsemekhman, K. Wolinski, J. Anchell, D. Bernholdt, P. Borowski, T. Clark, D. Clerc, H. Dachsel, M. Deegan, K. Dyall, D. Elwood, E. Glendening, M. Gutowski, A. Hess, J. Jaffe, B. Johnson, J. Ju, R. Kobayashi, R. Kutteh, Z. Lin, R. Littlefield, X. Long, B. Meng, T. Nakajima, S. Niu, L. Pollack, M. Rosing, G. Sandrone, M. Stave, H. Taylor, G. Thomas, J. van Lenthe, A. Wong, and Z. Zhang, NWChem, A Computational Chemistry Package for Parallel Computers, Version 5.0 Pacific Northwest National Laboratory, Richland, Washington 99352–0999, USA. 2006. An overview of NWCHEM can be found in: High Performance Computational Chemistry: An Overview of NWChem a Distributed Parallel Application, R. A. Kendall, E. Apra, D. E. Bernholdt, E. J. Bylaska, M. Dupuis, G. I. Fann, R. J. Harrison, J. Ju, J. A. Nichols, J. Nieplocha, T. P. Straatsma, T. L. Windus, A. T. Wong, Computer Phys. Comm., 2000, 128, 260–283. Further details can be found at the website: http://www.emsl.pnl.gov/docs/nwchem/nwchem.html

PQS

Parallel Quantum Solutions is a commercial organization offering ‘‘parallel computers with integrated software for high performance computational chemistry’’. The hardware offered is based on personal computer technology. The authors of the software, P. Pulay, J. Baker and K. Wolinski, argue that: ‘‘The high computational demands of most ab initio calculations requires parallel processing to achieve reasonably fast turnaround. General purpose parallel computers are very expensive and are usually shared between many users, negating most of the advantages of parallelism. Parallel computational chemistry programs are expensive and difficult to set up.’’ PQS can be found at Parallel Quantum Solutions, 2013 Green Acres Road, Suite A, Fayetteville, Arkansas 72703 USA Further details can be found at the website: http://www.pqs-chem.com

PSI

3.3.0

an open-source ab initio electronic structure package, details of which were published recently in: PSI3: An Open-Source Ab initio Electronic Structure Package, T. D. Crawford, C. D. Sherrill, E. F. Valeev, J. T. Fermann, R. A. King, M. L. Leininger, S. T. Brown, C. L. Janssen, E. T. Seidl, J. P. Kenny, and W. D. Allen, J. Comp. Chem. 2007, 28, 1610–1616. Further details can be found at the website: http://www.psicode.org/index.html The open-source software is distributed by sourceforge.net http://sourceforge.net/projects/psicode 226 | Chem. Modell., 2008, 5, 208–248 This journal is

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QCHEM 3.1 ‘‘a comprehensive ab initio quantum chemistry package’’. Details can be found in: J. Kong, C. A. White, A. I. Krylov, D. Sherrill, R. D. Adamson, T. R. Furlani, M. S. Lee, A. M. Lee, S. R. Gwaltney, T. R. Adams, C. Ochsenfeld, A. T. B. Gilbert, G. S. Kedziora, V. A. Rassolov, D. R. Maurice, N. Nair, Y. Shao, N. A. Besley, P. E. Maslen, J. P. Dombrowski, H. Daschel, W. Zhang, P. P. Korambath, J. Baker, E. F. Byrd, T. van Voorhis, M. Oumi, S. Hirata, C.-P. Hsu, N. Ishikawa, J. Florian, A. Warshel, B. G. Johnson, P. M. W. Gill, M. Head-Gordon, J. A. Pople, J. Comput. Chem., 2000, 21, 1532 The latest version is QCHEM 3.1. The QCHEM package is marketed by Q-Chem, Inc., The Design Center, Suite 690 5001 Baum Blvd. Pittsburgh, PA 15213 Further details can be found at the website: http://www.q-chem.com/index.htm

SPARTAN

aims to provide ‘‘easy-to-learn and easy-to-use molecular modeling tools for chemistry researchers and educators’’. In conjunction with QCHEM, SPARTAN provides a full range of post-Hartree-Fock methods. Further details can be found at the website: http://www.wavefun.com/index.html

TURBOMOLE

a general purpose code originally designed for high efficiency on workstation by the Ahlrichs group in Karlsruhe. Useful references are Electronic Structure Calculations on Workstation Computers: the Program System TURBOMOLE, R. Ahlrichs, M. Ba¨r, M. Ha¨ser, H. Horn and Ch. Ka¨lmel, Chem. Phys. Letters, 1989, 162, 165 S. Brode, H. Horn, M. Ehrig, D. Moldrup, J. E. Rice, R. Ahlrichs, J. Comput. Chem., 1993, 14, 1142 Further details can be found at the website: http://www.cosmologic.de/QuantumChemistry/ UT-CHEM a program suite for ab initio quantum chemistry. Details can be found in: T. Yanai, H. Nakano, T. Nakajima, T. Tsuneda, S. Hirata, Y. Kawashima, Y. Nakao, M. Kamiya, H. Sekino, K. Hirao, Computational Science-International Conference on Computational Science 2003, Lecture Notes in Computer Science (Springer) 2003, pp. 84–95 Further details can be found at the website: http://utchem.qcl.t.u-tokyo.ac.jp/

3.7 The knowledge base: computer hardware In the introduction to this section, we remarked that ‘‘Whereas a printed page can be copied with little regard to the nature of the material, usually paper, on to which it is copied, the correct execution of computer software is usually highly dependent on the hardware upon which it is implemented.’’ The performance of the software packages described in the previous section is usually strongly dependent on the hardware platform on which it is implemented and this in turn determines the range of application that can be considered. Chem. Modell., 2008, 5, 208–248 | 227 This journal is

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In this section, we turned our attention to the hardware requirements for computational many-body perturbation theory for molecules. As Hennessy and Patterson explain in their well known treatise on computer architecture77 ‘‘Computer technology has made incredible progress in the roughly 60 years since the first general-purpose electronic computer was created. Today less than $500 will purchase a personal computer that has more performance, more main memory, and more disk storage than a computer bought in 1985 for 1 million dollars. This rapid improvement has come both from advances in the technology used to build computers and form innovation in computer design.’’ Today, quantum chemistry calculations are carried out on a wide range of machines and architectures from a personal computer to a state-of-the-art supercomputer. To date, the vast majority of computational chemists have been content to use general purpose computers in their work designing software to exploit the capabilities of commercially available machines. However, as long ago as 1977 in a volume entitled Personal Computers in Chemistry, published by the American Chemical Society, Neil Ostlund of Hypercube Inc., expressed the view that chemists should be involved in the construction of the machines that they use:78 ‘‘Computational chemistry is to established and important a field to leave its only significant piece of apparatus to disinterested computer scientists. Chemists in other fields have commonly built very sophisticated instruments—molecular beam machines, electron scattering spectrometers, ion cyclotron resonance spectrometers, etc. . . . If computational chemists are to take full advantage of the current revolution in micro-electronics, which offers great prospects for cost-effective high-performance computation, it will be necessary to become involved in the design and building of highly parallel architectures specific to specific chemical applications.’’ Some computational chemists design their own systems by buying commercially available components and thereby tailoring their system to their specific needs. As we described in the previous section, P. Pulay and his coworkers have established Parallel Quantum Solutions marketing ‘‘parallel computers with integrated software for high performance computational chemistry’’. These authors write88: ‘‘The high computational demands of most ab initio calculations requires parallel processing to achieve reasonably fast turnaround. General purpose parallel computers are very expensive and are usually shared between many users, negating most of the advantages of parallelism. Parallel computational chemistry programs are expensive and difficult to set up.’’ Typically, quantum chemistry packages are suppied for a range of hardware platforms. For example, GAUSSIAN 03 is available for machines from a a range of vendors with different CPU and operating systems: AMD with the – AMD64 (Opteron, Athlon 64) CPU and the * SuSE Linux 9.0, 9.3, 10.1–10.3, * SuSE Linux Enterprise 10, or * Red Hat Enterprise Linux 4 (Updates 3,4) and RHEL 5 (initial release) operating system. |||| http://www.pqs-chem.com


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Apple with the – IA32 (32-bit Intel Mac) CPU and the * OS X 10.4.9 and later Tiger releases of the operating system Apple with the – EM64T (64-bit Intel Mac) CPU and the * OS X 10.4.9 and later Tiger releases of the operating system Fujitsu with the – Prime Power CPU and the * Solaris 9 and 10 operating system HP with the – Alpha CPU and the * Tru64 5.1B operating system and so on . . . A full list can be found in the GAUSSIAN 03 documentation available on the website http://www.gaussian.com. In distributing any quantum chemistry packages, each machine specific implementation has been tailored to the given hardware platform taking account of the computer architecture. In brief, computer architecture comprises at least three main subcategories:77 Instruction Set Architecture—the abstract image of a computing system that is seen by a machine language (or assembly language) programmer. This inlcudes the instruction set, memory address modes, processor registers, and address and data formats. The Microarchitecture is a lower level, more concrete, description of the system. It involves such details as how the constituent parts of the system are interconnected and how they interoperate in order to implement the instruction set architecture. For example, the size of a computer’s cache memory is an organizational issue which, in general, has nothing to do with the instruction set architecture. System Design which includes all of the other hardware components within a computing system such as: 1. system interconnects such as computer busses and switches 2. memory controllers and hierarchies 3. CPU ‘off-load’ mechanisms such as Direct Memory Access (DMA) 4. issues such as multi-processing. The computers used in ab initio quantum chemistry range from desktop machines through to supercomuters. As an example of a state-of-the-art supercomputer, we briefly consider the NEC SX-9 lauched by NEC Corporation** in October 2007. When completed this will be the world’s fastest vector supercomputer with a peak processing performance of 839 TFLOPS (one trillion floating point operations per ** Press release from NEC Corporation, 25 October 2007, see http://www.nec.co.jp/press/en/ 0710/2501.html for details.


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second). The NEC SX-9 will have CPUs capable of a peak vector performance of 102.4 GFLOPS (one billion floating point operations per second) per single core. This will combine large-scale shared memory of up to 1TB (terabyte) and ultra highspeed interconnects achieving speeds up to 128GB/second. The largest NEC SX-9 can have 8,192 CPUs and will achieve peak processing performance of 839 TFLOPS, thus approaching the PFLOPS (one quadrillion floating point operations per second) range.

4. An overview of applications of second order theory 4.1 Incidence of the string ‘‘MP2’’ in titles and/or keywords and/or abstracts In previous reports to this series, the increasing use of many-body perturbation theory in molecular electronic structure studies was measured by interrogating the Institute for Scientific Information (ISI) databases. In particular, I determined the number of incidences of the string ‘‘MP2’’ in titles and/or keywords and/or abstracts. This acronym is frequently associated with the simplest form of manybody perturbation theory. This assessment of the use of second order many-body perturbation theory will undoubtedly miss many routine applications but should serve to convey both the extent and the breadth of contemporary application areas. In my report for the period up to 1999, I noted that the string ‘‘MP2’’ had occurred in the title and/or keywords of just 3 publications in 1989 but that this number had risen to 854 in 1998. In my report for the period June 1999 to May 2001, I measured 821 ‘‘hits’’ for 1999 and 883 for the year 2000. For my report covering the period June 2001 to May 2003, I found a total of 757 incidences for 2001 and 828 for 2002. For the period June 2003 to May 2005, the ISI database contained 834 and 819 ‘‘hits’’ for the years 2003 and 2004, respectively. In the present reporting period, June 2005 to May 2007, the ISI database contained 919 and 904 ‘‘hits’’ for the years 2005 and 2006, respectively, being, of course, the last two years at the time of writing for which data is available. Over the nine year period 1998–2006 inclusive, the average number of publications with the string ‘‘MP2’’ in the title and/or keywords has been 847, ranging between a minimum of 757 and a maximum of 919. Our analysis is summarized in Table 1. 4.2 Synopsis of applications of second order many-body perturbation theory As in my three most recent reviews,2–4 I have attempted to provide a snapshot of the many applications of many-body perturbation theory in its simplest form, i.e. Møller-Plesset theory designated MP2, during the period under review by performing a literature search for publications with the string ‘‘MP2’’ in the title only. During the period covered by the present report, a total of 80 papers were discovered satisfying this criterion. Table 1 Incidence of the acronym ‘‘MP2’’ in the title and/or keywords and/or abstract or publications over the period 1998 to 2006 Year

‘‘MP2’’

1998 1999 2000 2001 2002 2003 2004 2005 2006

854 821 883 757 828 834 819 919 904


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We list the titles of these papers here: 1. MP2 study of substituent effects of 2-substituted alkyl ethyl methylcarbamates in homogeneous, unimolecular gas phase elimination reaction79 2. Activation energies of pericyclic reactions: Performance of DFT, MP2, and CBS-QB3 methods for the prediction of activation barriers and reaction energetics of 1,3-dipolar cycloadditions, and revised activation enthalpies for a standard set of hydrocarbon pericyclic reactions80 3. G3(MP2) enthalpies of hydrogenation, isomerization, and formation of extended linear polyacetylenes81 4. Basis set superposition error in MP2 and density-functional theory: A case of methane-nitric oxide association82 5. Parametric representation of the low-energy portion of the water-carbon dioxide G3(MP2)//B3LYP electronic energy surface. Variations on the theme developed by Professor B. Widom in his PhD dissertation83 6. Structure and stability of thiourea with water, DFT and MP2 calculations84 7. G3(MP2) ring strain in bicyclic phosphorus heterocycles and their hydrocarbon analogues85 8. MP2 static first hyperpolarizability of azo-enaminone isomers86 9. The molecular structures and electron distributions of the 1,8-bis(dimethylamino)-naphthalenes, studied by density functional and ab initio MP2 calculations87 10. MP2 and DFT studies of the DNA rare base pairs: the molecular mechanism of the spontaneous substitution mutations conditioned by tautomerism of bases88 11. DFT and MP2 study on the redistributing preparation of dichlorodimethyl89 silane catalyzed by AlCl(II) 3 12. G3(MP2) study of the C3H6O+ isomers fragmented from 1,4-dioxane+90 13. First-principle DFT and MP2 modeling of infrared reflection-Absorption spectra of oriented helical ethylene glycol oligomers91 14. Energetics and bonding study of hexamethylenetetramine and fourteen related cage molecules: An ab initio G3(MP2) investigation92 15. An analysis of the vibrational spectra of N,N-dimethylformamide isotopomers and unsubstituted N,N-dimethylcabamoyl chloride with scaling the force field calculated by the MP2 method using a basis set including f functions93 16. Characterization of the conformational probability of N-acetyl-phenylalanylNH2 by RHF, DFT, and MP2 computation and AIM analyses, confirmed by jet-cooled infrared data94 17. Insights into the role of the aromatic residue in galactose-binding sites: MP2/6-311G++** study on galactose- and glucose-aromatic residue analogue complexes95 Chem. Modell., 2008, 5, 208–248 | 231 This journal is

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18. Comparison of large basis set DFT and MP2 calculations in the study of the barrier for internal rotation of 2,3,5,6-tetrafluoroanisole96 19. DFT and local MP2 study of switching process in a pH controllable molecular ‘‘shuttle’’97 20. A combined DFT, MP2 and CASSCF study on the properties of Tropolone(H2O)n, (n = 0–2) complexes98 21. Novel 1,5-difluoropenta-1,4-diyn-3-one, its sulfur and selenium analogues: MP2 and DFT gas phase study of their molecular structures and vibrational spectra99 22. Berberine alkaloid: Quantum chemical study of different forms by the DFT and MP2 methods100 23. Modeling of the ‘hydration shell’ of uracil and thymine in small water clusters by DFT and MP2 methods101 24. Permanganate oxidation of alkenes. Substituent and solvent effects. Difficulties with MP2 calculations102 25. Treating dispersion effects in extended systems by hybrid MP2: DFT calculations—protonation of isobutene in zeolite ferrierite103 26. A DFT and MP2 study of luminescence of gold(I) complexes104 27. MP2 study of cation–(p)n–p interactions (n = 1–4)105 28. A new nonsymmetric As(OH)3 species. Comparison with the known C3 species and themochemistry at the HF, DFT(B3LYP), MP2, MP4, and CCSD(T) levels of theory106 29. Electron affinities and ionization potentials of 4d and 5d transition metal atoms by CCSD(T), MP2 and density functional theory107 30. The p-type hydrogen bond with triple C–C bond acting as a proton-acceptor. A gradient-corrected hybrid HF-DFT and MP2 study of the phenol-acetylene dimer in the neutral S-0 ground state108 31. Benchmark database of accurate (MP2 and CCSD(T) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs109 32. Ab initio base-pairing energies of uracil and 5-hydroxyuracil with standard DNA bases at the BSSE-free DFT and MP2 theory levels110 33. MP2 theory investigation on the halides of D6h C36:C36Xn (X = F,Cl,Br; n = 2,4,6,12)111 34. Estimated MP2 and CCSD(T) interaction energies of n-alkane dimers at the basis set limit: Comparison of the methods of Helgaker et al. and Feller112 35. The MP2 quantum chemistry study on the local minima of guanine stacked with all four nucleic acid bases in conformations corresponding to mean B-DNA113 232 | Chem. Modell., 2008, 5, 208–248 This journal is

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36. Structures and properties of cytosine-BX3 (X = F, Cl) complexes: an investigation with DFT and MP2 methods114 37. B3LYP and MP2 calculations of the enthalpies of hydrogen-bonded complexes of methanol with neutral bases and anions: Comparison with experimental data115 38. Benchmark RI-MP2 database of nucleic acid base trimers: performance of different density functional models for prediction of structures and binding energies116 39. Assessment of the MP2 method, along with several basis sets, for the computation of interaction energies of biologically relevant hydrogen bonded and dispersion bound complexes117 40. Investigation of the intermolecular proton transfer in the supersystems adenine-methanol/ethanol/i-propanol: MP2 and DFT levels study118 41. New parallel algorithm for MP2 energy gradient calculations119 42. How many conformers determine the thymidine low-temperature matrix infrared spectrum? DFT and MP2 quantum chemical study120 43. MP2/6-311++G(d,p) study on galactose-aromatic residue analog complexes in different position-orientations of the saccharide relative to aromatic residue121 44. DFT and MP2 investigations on the interaction of furan homologues C4H4Y (Y = O, S) with BX3 (X = H, F, Cl)122 45. Insufficient description of dispersion in B3LYP and large basis set superposition errors in MP2 calculations can hide peptide conformers123 46. Modeling of the ‘hydration shell’ of uracil and thymine in small water clusters by DFT and MP2 methods124 47. Ab initio base-pairing energies of an oxidized thymine product, 5-formyluracil, with standard DNA bases at the BSSE-free DFT and MP2 theory levels125 48. Comprehensive conformational analysis of the nucleoside analogue 2 0 -b-deoxy-6-azacytidine by DFT and MP2 calculations126 49. Density functional static dipole polarizability and first-hyperpolarizability calculations of Nan (n = 2,4,6,8) clusters using an approximate CPKS method and its comparison with MP2 calculations127 50. Structure and vibrational spectra of vinyl ether conformers. The comparison of B3LYP and MP2 predictions128 51. MP2 study of anion–p complexes of trifluoro-s-triazine with tetrahedral and octahedral anions129 52. New nonsymmetric P(OH)3 species. Comparison with the C-3 isomer and themochemistry at the DFT, MP2, and CCSD(T) levels of theory130 53. Structure and bonding differences in C3N4 and Si3N4 isomers-A comparative study of [Si-3,N-4] and [C-3,N-4] potential energy surfaces using DFT and MP2 methodologies131 Chem. Modell., 2008, 5, 208–248 | 233 This journal is

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54. Analytical RISM-MP2 free energy gradient method: Application to the Schlenk equilibrium of Grignard reagent132 55. MP2 study of cooperative effects between cation-pi, anion-pi and pi-pi interactions133 56. Conformational study of isolated pindolol by HF, DFT and MP2 calculations134 57. Comparison of the conformational stability for several vinylhalomethanes and silanes with experiment using MP2 perturbation theory and DFT135 58. Stabilisation energy of C6H6 C6X6 (X = F, Cl, Br, I, CN) complexes: complete basis set limit calculations at MP2 and CCSD(T) levels136 59. A new non-symmetric N(OH)3 species: Comparison with the C3 species and thermochemistry at the HF, DFT, MP2, MP4 and CCSD(T) levels of theory137 60. Pt-bridges in various single-strand and double-helix DNA sequences. DFT and MP2 study of the cisplatin coordination with guanine, adenine, and cytosine138 61. Hydrogen-bonded nucleic acid base pairs containing unusual base tautomers: Complete basis set calculations at the MP2 and CCSD(T) levels139 62. Predicting potential stable isomers on the singlet surface of the [H,P,C,S] system by the MP2 and QCISD(T) methods140 63. Probing ab initio MP2 approach towards the prediction of vibrational infrared spectra of DNA base pairs141 64. C–H functionalisation through singlet chlorocarbenes insertions-MP2 and DFT investigations142 65. Alternative formulation of the matrix elements in MP2-R12 theory143 66. General orbital invariant MP2-F12 theory144 67. Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn145 68. Computational aspects of a local-MP2 treatment of electron correlation in periodic systems: SiC vs. BeS146 69. Fast local-MP2 method with density-fitting for crystals. I. Theory and algorithms147 70. Fast local-MP2 method with density-fitting for crystals. II. Test calculations and application to the carbon dioxide crystal148 71. Ab initio asymptotic-expansion coefficients for pair energies in MP2 perturbation theory for atoms149 72. MP2 basis set limit binding energy estimates of hydrogen-bonded complexes from extrapolation-oriented basis sets150 234 | Chem. Modell., 2008, 5, 208–248 This journal is

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73. A general efficient implementation of the BSSE-free SCF and MP2 methods based on the Chemical Hamiltonian Approach151 74. Augmented Gaussian basis sets of triple and quadruple zeta valence quality for the atoms H and from Li to Ar: applications. in HF, MP2, and DFT calculations-of molecular dipole moment and dipole (hyper)polarizability152 75. Calculations with correlated molecular wave functions. HF, MP2 and DFT calculations on second-row diatomic hydrides153 76. HF and MP2 calculations on CN, N2, AlF, SiO, PN, SC, CIB, and P2 using correlated molecular wave functions154 77. Hybrid correlation models based on active-space partitioning: Correcting MP2 theory for bond-breaking reactions155 78. A new parallel algorithm of MP2 energy calculations156 79. DIESEL-MP2: A new program to perform large-scale multireference-MP2 computations157 80. Array files for computational chemistry: MP2 energies158 These papers were published in a total of 30 different journals which are listed alphabetically in Table 2. The number of relevant papers appearing in each journal is given in parenthesis. The largest number of papers appeared in J. Molec. Struct.Theochem, followed by J. Phys. Chem. A and Chem. Phys. Lett. 4.3 Comparison with other methods In volume 1 of this series, I compared the use of second-order many-body perturbation theory in its ‘‘MP2’’ form with that of density functional theory and coupled cluster theory. I recorded how the number of ‘‘hits’’ in a literature search on the string ‘‘MP2’’ rises from 3 in 1989 to 854 in 1998. The corresponding results for DFT, the most widely used semi-empirical method, are 7 in 1989 growing to 733 by 1998. By 1998, the number of ‘‘hits’’ recorded for CCSD stood as 244. I extended this comparison for the period 1998–2002 in volume 3 of this series, and to 2004 in volume 4. In Table 3, the comparison is continued through to 2006, the last complete year for which data is available at the time of writing. The most striking observation about this table is the growth in the use of ‘‘DFT’’ which first exceeded that of ‘‘MP2’’ in 1999, stood at roughly a factor of two greater at the time of my last report and now, according to the 2006 data, is approaching a point where its use will be a factor of four more. It is undoubtedly the demand for methods that can be deployed in the description of larger systems that is fueling the growth in the use of density functional theory. A significant number of papers with the string ‘‘MP2’’ in the title and/or keywords and/or abstract report comparative studies in which practical applications of ‘‘MP2’’ theory are compared with applications of other methods. In order to measure the number of papers of this type in recent years, literature search were carried out to determine:1. the number of papers containing both the string ‘‘MP2’’ and the string ‘‘DFT’’ in the title and/or keywords and/or abstract 2. the number of papers containing both the string ‘‘MP2’’ and the string ‘‘CCSD’’ in the title and/or keywords and/or abstract 3. the number of papers containing both the string ‘‘DFT’’ and the string ‘‘CCSD’’ in the title and/or keywords and/or abstract Chem. Modell., 2008, 5, 208–248 | 235 This journal is

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Table 2 Journals in which papers appeared containing the acronym ‘‘MP2’’ in the title and/or keywords and/or abstract over the period June 2005 to May 2007 Abs. Am. Chem. Soc (1) Acta. Chimica Sinica (1) Biochem. (1) Bull. Korean Chem. Soc. (1) Chem. Phys. (2) Chem. Phys. Lett. (8) Collect. Czechoslovak Chem. Commun. (1) Inorganica Chimica Acta (1) Intern. J. Quantum Chem. (4) J. Am. Chem. Soc. (1) J. Biomolec. Struct. & Dynam. (1) J. Chem. Phys. (4) J. Chem. Theory & Comput. (1) J. Comput. Chem. (5) J. Molec. Modelling (2) J. Molec. Struct. (1) J. Molec. Struct.-Theochem (12) J. Organic Chem. (1) J. Phys. Chem. A (9) J. Phys. Chem. B (4) J. Serbian Chem. Soc. (1) J. Theoret. & Comput. Chem. (1) Lect. Notes Comp. Sci. (1) Molec. Phys. (4) New J. Chem. (1) Organic & Biomolec. Chem. (2) Phys. Chem. Chem. Phys. (4) Phys. Rev. B (2) Russian J. Phys. Chem. (1) Theoret. Chem. Accounts (2)

Table 3 Incidence of the acronyms ‘‘MP2’’, ‘‘DFT’’ and ‘‘CCSD’’ in the title and/or keywords and/or abstract of publications over the period 1998 to 2006 Year

‘‘MP2’’

‘‘DFT’’

‘‘CCSD’’

1998 1999 2000 2001 2002 2003 2004 2005 2006

854 821 883 757 828 834 819 919 904

738 923 1221 1528 1723 2101 2433 3393 3576

244 263 283 318 303 354 320 446 480

4. the number of papers containing the string ‘‘MP2’’, the string ‘‘DFT’’ and the string ‘‘CCSD’’ in the title and/or keywords and/or abstract The results are collected in Table 4. In 2006, the most recent year for which complete data is available at the time of writing, 228 publications referred to both ‘‘MP2’’ and ‘‘DFT’’. This represents over 25% of the 904 publications with ‘‘MP2’’ in the title and/or keywords and/or abstract of publications in 2006. 172 publications in the same year refer to ‘‘MP2’’ and ‘‘CCSD’’; about 19% of those with ‘‘MP2’’ in 236 | Chem. Modell., 2008, 5, 208–248 This journal is

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Table 4 Incidence of two or more of the acronyms ‘‘MP2’’, ‘‘DFT’’ and ‘‘CCSD’’ in the title and/or keywords and/or abstract of publications appearing in 2003–2006 Methods

2003

2004

2005

2006

‘‘MP2’’ & ‘‘DFT’’ ‘‘MP2’’ & ‘‘CCSD’’ ‘‘DFT’’ & ‘‘CCSD’’ ‘‘MP2’’ & ‘‘DFT’’ & ‘‘CCSD’’

204 109 50 25

189 103 46 18

267 137 86 39

228 172 83 40

the title and/or keywords and/or abstract. 83 publications refer to ‘‘DFT ’’ and ‘‘CCSD’’. 40 publications in 2006 refer to ‘‘MP2’’, ‘‘DFT ’’ and ‘‘CCSD’’.

5. Application area 1: periodic systems Some progress has been made during the reporting period towards incorporating a description of correlation effects in periodic solids using second order many-body perturbation theory. The aim research in this area is to provide a powerful and general-purpose computational tool, which can be used to study a variety of applications in condensed matter physics and solid state chemistry. Over the past twenty years, the available software for Hartree–Fock (HF) and for density-functional calculations on crystals has undergone very significant progress in terms of effectiveness and reliability. For example, the CRYSTAL program package159,160 is today applied routinely to a wide range of problems. The incorporation of correlation effects in calculations for periodic solids requires the use of a many-body formalism. Second order many-body perturbation theory, in its MP2 form, should provide the basis of an efficient computational approach to this problem. In particular, the local MP2 methods originally developed for large molecules can be adapted for the treatment of periodic solids. In a previous report,3 we have emphasized that ‘‘ . . . the steep scaling of algorithms for describing electron correlation effects in molecular systems is often an artifact of the orthogonal canonical basis, i.e the solutions of the matrix Hartree-Fock equations, used to construct post-HartreeFock correlation theories’’ For the local MP2 algorithm described by Hetzer et al.161 in 2000 ‘‘ . . . the calculation of the MP2 energy is less expensive than the calculation of the Hartree-Fock energy for large systems’’ Pisani et al.146,162 have developed a new computer program which they call for describing correlation effects in periodic solids. This program essentially combines the accurate HF solution of the crystal structure problem in an atomic orbital (AO) representation available from the CRYSTAL code, with local correlation techniques, and, in particular, those implemented the MOLPRO program of the Stuttgart group.163 In considering the computational aspects of a local-MP2 treatment of electron correlation in periodic systems, Pisani et al.146 point to number of distinctive features including: (i) full exploitation of point symmetry; (ii) proper definition of the ‘localvirtual’ basis set; (iii) assessment of the ‘locality of excitations’ concept; (iv) proper use of different levels of treatment of 2-electron integrals; (v) extrapolation of local results to infinity. Pisani et al. examine the relative importance of different kinds of local excitations and their dependence on the prevailingly covalent or ionic character of the crystal. They also demonstrate the usefulness of a multipolar approximation for the evaluation of the majority of 2-electron integrals which arise. CRYSCOR


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In subsequent work, Pisani et al.147,148 demonstrate the use of density-fitting techniques in the evaluation of the two-electron Coulomb interaction integrals between product distributions. They also show that use of density fitting techniques avoids the bottleneck of the four index transformation for neighbouring distributions.

6. Application area 2: DNA bases and amino acids The structure of proteins, deoxyribonucleic acids (DNA), ribonucleic acid (RNA), proteins and other biomacromolecular species is determined by noncovalent interactions among the building blocks-the DNA and RNA bases and amino acids. The methods of ab initio quantum chemistry have a key role to play in gaining an understanding of the structure and properties of biomacromolecules since over recent years accurate calculations for molecules of the size and complexity of these building blocks have become computationally tractable. For example, models of the four nucleic acids, adenine, cytosine, guanine and thymine, are shown in Fig. 1–4. Today, systems of this size are amenable to high quality ab initio quantum chemical studies and many-body perturbation theory is often the method of choice when incorporating the effects of electron correlation in such studies. The building blocks of biomacromolecules may be either electrically neutral or charged. Among the various energy terms which contribute to their overall stabilization, hydrogen bonding and electrostatic contributions are expected to be the most important. Hydrogen-bonding is highly specific and directional and thus responsible for the stabilization of biomacromolecules. Electrostatic interactions are is especially important in proteins, and ion pairs or salt bridges are known to play an important role in stabilizing selected protein structures. (Induction and charge-transfer terms do not play a decisive role but nevertheless should be taken into account.) It is now evident that stacking plays a key role in many biomacromolecules. Stabilization energies for stacked DNA base pairs as well as for stacked amino acids can be surprisingly large.

Fig. 1 Model of the adenine molecule (7H-purin-6-amine), one of the two purine nucleobases forming the nucleotides of the nucleic acids. The molecule has the formula C5H5N5 making it a 70-electron system. Today, systems of this size are amenable to high quality ab initio quantum chemical studies.


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Fig. 2 Model of the cytosine molecule (4-amino-1H-pyrimidine-2-one), one of the two pyrimidine nucleobases forming the nucleotides of the nucleic acids. The molecule has the formula C4H5N3O making it a 58-electron system. Today, systems of this size are amenable to high quality ab initio quantum chemical studies.

The size and complexity of extended biomacromolecules makes the understanding of the various energy contributions which contribute to their stabilization difficult, since only calculations using simple empirical potential calculations are tractable. Fortunately, the most importance biomacromolecules, DNA and proteins, consist of characteristic building blocks-the nucleic acid bases and amino acids-interacting through noncovalent interactions. The system can therefore be fragmented into smaller components, each of which can be described by means of ab initio quantum chemical methods.

Fig. 3 Model of the guanine molecule (2-amino-1H-purin-6(9H)-one), one of the two purine nucleobases forming the nucleotides of the nucleic acids. The molecule has the formula C5H5N5O making it a 78-electron system. Today, systems of this size are amenable to high quality ab initio quantum chemical studies.


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Fig. 4 Model of the thymine molecule (5-methylpyrimidine-2,4(1H,3H)-dione), one of the two pyrimidine nucleobases forming the nucleotides of the nucleic acids. The molecule has the formula C5H6N2O2 making it a 76-electron system. Today, systems of this size are amenable to high quality ab initio quantum chemical studies.

Jurecˇka et al.109 have published a database of accurate ‘‘benchmark quality’’ interaction energies of small model complexes, DNA base pairs, and amino acid pairs obtained from MP2 and CCSD(T) calculations together with estimates of the complete basis set limit. They examine interaction energies and geometries for more than 100 DNA base pairs, amino acid pairs and model complexes. Extrapolation to the complete basis set limit was carried out by these researchers using a two-point method in conjunction with correlation consistent basis sets. In related work, Riley and Hobza117 have provided an assessment of the MP2 method in conjunction with several medium and extended basis sets in the study of various hydrogen bonded and dispersion bound complexes of biological relevance. They report that the MP2/cc-pVTZ method supports the most well balanced description of noncovalent interactions. However, the MP2 method does not provide reliable results for the cyclic hydrogen bonds found in nucleic acid base pairs in combination with any of the basis sets considered by these authors. A new database for nucleic acid base trimers obtained by using RI-MP2 theory with a ‘‘triple zeta’’ quality basis set has been reported by Hobza and his coworkers.116 This database has been used to assess various density functional models. The heterogeneity of the interactions in these trimers, particularly the simultaneous hydrogen bonding and stacking interactions, makes this a demanding test of approximate methods. The authors suggests that ‘‘correct descriptions of all energy terms are unlikely to be accomplished by fortuitous cancellations of systematic errors’’. Rejnek and Hobza139 used MP2 and CCSD(T) theory together with procedures to estimate the complete basis set limits in the study of hydrogen-bonded nucleic acid base pairs containing unusual base tautomers. In particular, they focussed on the stabilization energies of adenine-thymine and guanine-cytosine base pairs containing various tautomeric forms. They found that the interaction energies for the optimized H-bonded tautomeric base pairs can be obtained with sufficient accuracy by the MP2/CBS approach, since the the CCSD(T) correction is negligible. They used a continuum model to describe bulk water. Rejnek and Hobza139 concluded that ‘‘The results clearly show how precisely the canonical building blocks of DNA molecules were chosen and how well their stability is maintained.’’ 240 | Chem. Modell., 2008, 5, 208–248 This journal is

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MP2 theory and DFT was used by Danilov et al.88 to study DNA rare base pairs. In particular, these authors examined the molecular mechanism of the spontaneous substitution mutations conditioned by tautomerism of bases Of course, most biochemical reactions occur in aqueous solution. MP2 theory was used by Danilov et al.101,124 to model the hydration shell of uracil and thymine using small water clusters. The structure of the hydration shell is found to be determined by competition between the water-water and base-water interactions. It is known that exposure to reactive oxidation species and subsequent DNA damage can be linked to diseases, such as cancer and rheumatoid arthritis, and to aging.164,165 Volk et al.110 used MP2 theory and DFT to study the oxidized cytosine product, 5-hydroxyuracil. Oxidized cytosine products have been shown to be major chemical precursors in DNA for GC to AT transition mutations.166–168 In related work, Volk et al.125 studied the highly mutagenic 5-formyluracil which results from oxidation of the thymine methyl group. The choice of papers above necessarily represents a selective sample of the literature published during the period reviewed.

7. Application area 3: DFT benchmarking At the end of his 1998 Nobel Prize lecture, Professor Sir John Pople commented on the molecular applications of density functional theory. Of DFT he writes ‘‘Such methods do not handle the two-electron interactions explicitly but rather allow for them using properties of the one-electron density. This leads to lower cost and therefore a wider range of applicability. Recent forms of DFT have also introduced a considerable amount of empirical parameterization, sometimes using the same set of experimental data. At the present time, the principal limitation of DFT models is that there is no clear route for convergence of methods to the correct answer . . . ’’ He concluded that ‘‘Interaction between these two groups of theoretical chemists is a hopeful direction for future progress’’ The groups he mentions are essentially the ‘‘ab initio-ists and a posteriori-ists’’ discussed by Coulson back in 1959. The ‘‘ab initio-ists’’ demand a ‘‘clear route for convergence’’. The ‘‘a posteriori-ists’’ are willing to accept ‘‘a considerable amount of empirical parameterization’’ in order to facilitate a particular application. It is not surprising that considerable effort has been devoted to comparing ab initio methods and DFT. Second order many-body perturbation theory is the most efficient of ab initio methods and it is not surprising, as we have already noted in section 3, 228 of the 904 papers for which the string ‘‘MP2’’ occured in the title and/ or keywords and/or abstract—that is 25%—also contained the string ‘‘DFT’’. Comparisons of MP2 and DFT abound in almost all areas of application. For example, the work of Hobza et al.116 which we discussed in section 6 above, seeks to establish the validity of the DFT approach in studies of the nucleic acid base pairs. Other examples of comparative work on MP2 and DFT are: Gordon and Palmer87 who study the structure and electron distribution of the 1,8-bis-(dimethylamino)naphthalenes and Kieninger et al.96 who investigate the barrier for internal rotation around the -OCH3 bond in 2,3,5,6-tetrafluoroanisole. The switching process in a pH controllable molecular ‘‘shuttle’’ was investigated by Fomine et al.97 and Malysheva et al. studied the absorption spectra of oriented helical ethylene glycol oligomers containing self-assembled monolayers to gain a understanding of their infrared reflection absorption spectra. Four Au(I) mixed thiolate and phosphine complexes were examined by Costa and Calhorda;104 Ignatyev et al. studied the structure and vibrational spectra of vinyl ether conformers and Maron and A. Ramirez-Solis130 Chem. Modell., 2008, 5, 208–248 | 241 This journal is

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focussed on a new nonsymmetric P(OH)3 species as well as a new nonsymmetric N(OH)3 species. This is inevitably a small sample of the many publications in which MP2 theory and DFT are compared during the reporting period.

8. Application area 4: basis set extrapolation and the calibration of general energy models The use of the algebraic approximation, that is the expansion of one-electron functions in terms of a finite analytic basis set, underpins much of the success of modern quantum chemistry particularly when the basis functions are chosen to be Gaussians. The integro-differential Hartree-Fock equations are replaced by a set of algebraic equations which can be cast in matrx form. However, the introduction of the algebraic approximation is not without consequences and molecular calculations carried out using a finite basis set necessarily suffer from basis set truncation errors. Such errors can often be as large or even larger than errors associated with the truncation errors arising in the correlation energy expansion. Quantum chemical calculations are usually defined in terms of the model used, including, except in the case of Hartree-Fock theory, the correlation treatmant used, and the basis set employed. The situation is summarized in Fig. 5. The column on the far right, headed full CI (FCI), records the exact solution within the finite space defined by the basis set. The entry at the bottom of the column on the far right represents the exact solution of the Schro¨dinger equation, which, of course, is not obtainable in practice. The importance of gaining some measure of the significance of basis set truncation effects, leads to many studies using sequences of basis sets constructed so as to approach the basis set limit. In addition, this approach also gives reliable procedures for performing extrapolations to the basis set limit from calculations employing smaller basis sets. Such extrapolation techniques necessarily introduce an empirical element into otherwise ab initio calculations. A typical example of this approach is the study of the interaction energies of n-alkane dimers using MP2 and CCSD(T) by Tsuzuki et al.,112 which compares the extrapolation method of Helgaker et al.169 with that of Feller170 using Dunning’s correlation consistent basis sets171. Similar studies using the two-point extrapolation formula of Helgaker et al. for the stabilisation energy of C6H6 C6X6 (X = F, Cl, Br, I, CN) complexes were reported by Pluhackova et al.136 The target accuracy in a given application extends beyond exploring the basis set limit for a given model to the complete basis set limit for full configuration

Fig. 5 General model table or model chart. The columns are labelled by the various methods for describing the effects of electron correlation arranged in order of increasing sophistication from left to right. HF denotes the Hartree–Fock model in which no account is taken of electron correlation. MP2 denotes second order many-body Møller–Plesset perturbation theory whilst MP3 and MP4 denote this theory taken through third and fourth order, respectively. QCI denotes the quadratic configuration interaction approach. The column on the far right which is headed FCI contains the full configuration interaction result within a given basis set. Basis sets are displayed vertically becoming more flexible down the chart. The bottom row of the chart corresponds to an infinite basis. On the right-hand side of this row we have FCI in an infinite basis set, i.e. the solution of the Schro¨dinger equation (S-eqn).


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Fig. 6 The G3 model table, an example of a composite model in which some compromises are made in order to obtain a wide range of applicability. The columns are labelled by the various methods for describing the effects of electron correlation arranged in order of increasing sophistication from left to right. HF denotes the Hartree–Fock model in which no account is taken of electron correlation. MP2 denotes second order many-body Møller–Plesset perturbation theory whilst MP4 denote this theory taken through fourth order and QCI denotes the quadratic configuration interaction approach. Basis sets are displayed vertically becoming more flexible down the chart. G3large is a large and flexible basis set for which only MP2 calculations are tractable. The composite model is labelled ’’model-1//model-2’’ where calculations are carried out using model-1 using the optimized geometry determined by using model-2. In the G3 table, the geometry is optimized using an MP2/6-31G(d) level of theory. All other calculations are carried out at this geometry. An estimate of the QCI/G3large energy is obtained by assuming additivity of corrections according to: ? = 2 + (3 2) + (5 2) + (7 2) + (8 1) (4 1) (6 1). Vibrational frequencies can be determined from a HF calculation at the same geometry together with a scaling factor.

interaction. In practice, this cannot be achieved with finite computing resources, except for the very smallest systems. Some compromises have to be made to achieve a wide range of applicability. For example, geometry optimization may be carried out using some ‘lower level’ theory followed by a ‘more accurate’ calculation using a ‘higher level’ carried out at the optimized geometry. Again, this approach introduces a degree of empiricism into an otherwise ab initio calculation, but this procedure can often lead to accuracy which could not otherwise be achieved. A typical procedure of this type, the G3 composite model developed by Pople and his co-workers,172 is illustrated in Fig. 6. Many-body methods and, in particular, MP2 theory are often used in validating such models. During the reporting period, a typical application is provided by the work of Rogers et al.,81 who calculated enthalpies of hydrogenation, isomerization and formation of extended linear polyacetylenes using the G3(MP2) model.173 Rogers et al. examined the conjugation enthalpies of linear polyynes with three, four, and five alternant triple bonds. Molecules as large as 1,3,5,7,9-decapentayne were studied. Space does not allow anything more than a brief overview of the many publications in which MP2 theory is used to investigate basis set truncation errors in molecular calculations during the period covered by this report.

9. Summary and future directions The present article continues our biennial survey of ‘‘Many-body Perturbation Theory and Its Application to the Molecular Structure Problem’’ covering the reporting period assigned to this volume: June 2005 to May 2007. Many-body perturbation theory in its lowest order form, which is often designated MP2, continues to be the most widely used of the ab initio approaches to the molecular electronic structure problem which go beyond an independent particle model and take account of the effects of electron correlation. The main focus of the present review has been on some of the emerging fields in which MP2 calculations are being carried out. Obviously, within the limited space available it has not possible to cover all of the fields of application. Some selectivity has been necessary, but the choices made do provide a ‘‘snapshot’’ of the range of contemporary applications of chemical modelling using many-body perturbation theory. Chem. Modell., 2008, 5, 208–248 | 243 This journal is

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Applications have been reported in an ever increasing range of research areas during the reporting period. Given the ‘many-body’ nature of quantum chemical problems, many-body perturbation theory will continue to play a central role in the analysis and study of molecular structure.

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Experiment and theory in the determination of molecular hyperpolarizabilities in solution; pNA and MNA in dioxane David Pugh DOI: 10.1039/b609719g

1. Introduction The computation of molecular polarizabilities, hyperpolarizabilities, magnetizabilities and hypermagnetizabilities attracts a significant part of the research effort in quantum chemistry. High level ab initio and density functional calculations have progressed to the point where it is reasonable to expect realistic results to be obtained for isolated organic molecules containing, say, up to about 100 atoms. The internal consistency of these methods can be assessed by comparing the results obtained by different techniques, and in many cases it is possible at least to find a reasonable measure of agreement between the values obtained for a particular parameter as calculated by the more fundamental quantum chemistry approaches. The objective of these studies is to be able to explain and predict experimental properties and to utilize the predictive capability to help control useful processes and to assist in the design of new materials with particular required properties. One of the main incentives for calculations of molecular hyperpolarizabilites has been the desirability of having high hyperpolarizability materials available for optoelectronic applications. The relationship between the computations and experimentally determined parameters is therefore all-important. While gas phase work on the hyperpolarizability of small molecules has been relatively free of problems concerned with the definitions of measured quantities and their formal relationship to computed quantities, the same cannot be said about solution studies of rather larger organic species. It is the latter that possess the very large nonlinear response functions that are of greatest interest. The prototype system for such studies has been 4-nitroaniline (pNA) and this review is mainly concerned with the relation between the measurements, in vacuo and in solution, of the hyperpolarizabilities of pNA and the closely related molecule, MNA (2-methyl, 4-nitroaniline) to ab initio and DFT calculations of these quantities. The first problem is concerned with the definitions of the quantities calculated and measured. It has only gradually emerged over the last decades that there was considerable ambiguity about the definition of some of the reported parameters. In 1992 Willetts et al.1 (referred to as WRBS) gave a detailed account of the various conventions that have been applied by different authors in defining the molecular response functions, but more recently Reiss2 has suggested that there are still inconsistencies in the way that the experimentally measured, macroscopic, response functions are reported. A second difficulty arises from the circumstance that almost all the effects of experimental or technological interest involve the frequency-dependent response functions. Frequently the theoretical values are calculated using a very high level static field calculation which is then adjusted for the frequency dependence using a lower level (often time-dependent Hartree-Fock) method. The semi-empirical approach, which was often closely connected with the identification of significantly large effects in organic molecules, was usually calibrated to experimental electronic excitation energies, which are related to resonances in the response functions. These Department of Pure and Applied Chemistry, University of Strathclyde, Thomas Graham Building, 295 Cathedral Street, Glasgow, UK G1 1XL


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methods could therefore be said to be adapted to identifying particularly strong resonant effects, but weak on producing quantitatively accurate results. The ab initio methods are now expected to achieve quantitative accuracy for the static response but are still arguably unreliable when it comes to some of the frequency-dependent phenomena. An even more serious impediment to obtaining a quantitative correspondence of theory and experiment is a consequence of measurements on the molecules of greatest interest necessarily being made in solutions or condensed phases. Internal field corrections are usually applied in a formal way using standard continuum theories (Lorentz-Lorenz, Clausius-Mossotti, Onsager).3 In the design of optimized crystal structures these same theories have often been applied, apparently successfully, but with even less theoretical justification.4 Recently simulation studies of the solutions have been favoured, using a mixture of ab initio or DFT methods for the solute molecule and a semi-empirical or molecular mechanics method for a surrounding cluster of solvent.5,6 A more rigorous approach to the internal field corrections in crystals has been pursued by Munn and collaborators7–10 and in combination with ab initio calculations, by Munn, Papadopoulos, Reiss et al.11 Finally, it is to be expected of an all-embracing theory that it will reflect the ubiquitous nature of the polarizability–hyperpolarizability formulation. A very wide range of coherent and incoherent optical and spectroscopic effects are analysed in terms of polarizabilities or hyperpolarizabilities that differ only in the type of frequency dependence. For all these phenomena the response for a given set of input and output frequencies is determined formally through the polarizabilities by a perturbation theory expansion over the molecular eigenstates. In the development of solution studies of organic molecules and pNA in particular the perturbation theory expansion over excited states has figured prominently. Even if the hyperpolarizabilities calculated by this method are only of use in providing a general comparison of different structures and semi-quantitative results they are of interest for another reason. Very often experimental studies have been accompanied by an attempt to explain the results obtained in terms of the SOS theory. An examination of the multiplicative constants in the perturbation theory expansion at least gives an indication of how the parameters measured experimentally are being defined by the authors and may be of help in resolving some of the doubts that exist over these definitions. The problem of establishing the values of clearly determined molecular parameters for pNA and MNA in solution and interpreting their values through fundamental non-empirical calculations has received a great deal of attention in the period extending from the 1970s to the present. There have been a number of recent studies, which have often included substantial reviews, and re-interpretations of earlier works. The aim of the present review is to describe the background to the present studies and identify the origins of some of the current controversies. In particular it seems to be essential to attempt to distinguish between the macroscopic measurements and the values of the first molecular hyperpolarizability deduced from them, which are often presented as an ‘experimental’ quantity, but are strongly dependent on the nature of the intervening theory. The most commonly studied problem is the determination of the first hyperpolarizability, b(2o;o,o) for second harmonic generation for solutions of pNA or MNA in dioxane solution. Views on the nature of dioxane at a microscopic level have changed over the period of these investigations. In earlier NLO work it was assumed to be a ‘non-polar’ solvent in which the properties of the active solute would be close to that of the isolated molecule. More recently it is regarded as being decidedly ‘polar’ on a molecular scale with an effective dielectric constant of 6. Much of the present article is concerned with EFISH measurements on these solutions. The plan of the article is as follows:—In section 2 the systematic definitions of the susceptibilities and (hyper)polarizabilities are described; section 3 briefly derives the standard SOS formulae; in section 4 the general theory of the field induced second 250 | Chem. Modell., 2008, 5, 249–278 This journal is

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harmonic generation experiment (EFISH), from which most information on the first and second order hyperpolarizabilities has been obtained, is derived; section 5 reviews ab initio and DFT calculations on the isolated pNA molecule; section 6 discusses the solitary gas phase experimental determination of Kaatz et al.12 and related deductions about the solution response based on hyper-Rayleigh scattering measurements; section 7 discusses the extension of the EFISH theory to two component solutions and reviews EFISH studies of pNA and MNA in dioxane; section 8 reviews recent work including computer simulation of the solution structure and section 9 briefly describes very recent related work that has not yet been incorporated into more detailed reviews.

2. General theory of the response to frequency-dependent electric fields 2.1 Macroscopic fields The derivation of formulae for the frequency-dependent nonlinear susceptibilities of nonlinear optics from the time-dependent response functions can be found in a number of sources, (Bloembergen,13 Ward and New,14 Butcher and Cotter,15 Flytzanis16). Here it is assumed that the susceptibilities can be expressed in terms of frequency-dependent quantities that connect individual (complex) Fourier components of the polarization with simple products of the Fourier components of the field. What then has to be shown is how the quantities measured in various experiments can be reduced to these simpler parameters. The applied field is expanded as, X E ¼ Eð0Þ þ Eðon Þ expðion tÞ ð2:1Þ n

and the polarisation in terms of the order of interaction, P = P(0) + P(1) + P(2) +

(2.2)

with the nth order term given by, PðnÞ ¼ PðnÞ ð0Þ þ

X

PðnÞ ðos Þ expðios tÞ

ð2:3Þ

s

In the above, quantities with no arguments are time-dependent, those with one argument, which is always a frequency, are time-independent Fourier coefficients. It is convenient to introduce one further type of symbol with two arguments, P(n)(t,os), where the quantity defined contains the full nth order time-dependent response at frequency os, which will contain both positive and negative complex frequency components. The composite quantities, P(n)(t,os), can always be expressed in terms of linear combinations of elementary complex polarizations, p(n)(t,os) = e0v(n)(os;o1,o2, . . . ,on)E(o1)E(o2) . . . E(on)exp(iost)

(2.4)

where a unique ordered product of the complex field components occurs on the right. The lower case symbol for the polarization has been introduced to indicate that this is an intermediate quantity related to the complex product field. It is nevertheless well defined since the complex product can be split into real and imaginary parts. All susceptibilities can be reduced to combinations of those defined in eqn (2.4) and the adoption of the susceptibilities defined through this equation has two advantages; firstly there is no possibility of introducing numerical factors associated with particular definitions of fields in terms of trigonometric functions (raised to the nth power), and secondly it can be shown15 that the susceptibilities defined in this way will not vary discontinuously when degeneracy occurs among the field frequencies. Chem. Modell., 2008, 5, 249–278 | 251 This journal is

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However the real fields in the experiment may be defined, the above susceptibilities refer to the response to a product of complex fields written as Eeiot. Any additional factors that occur in the definitions of the actual real fields and their products have to be accounted for explicitly and the response expressed in terms of the susceptibilities defined as above. There may, in principle, be contributions involving several different frequency-dependent susceptibilities in eqn (2.3)—for example if the input field contained components of frequencies (0,o,2o), when the output third order term in 3o would contain contributions from the combinations (0,o,2o) and (o,o,o)—but most experimental arrangements avoid such situations. Symmetry requirements on the w’s that result from the permutation of the order of application of the fields, from causality requirements and from the requirement that real fields must give rise to real polarizations (in the absence of line-width effects) ensure that the w’s are real, that, w(n) ijk . . . (os;o1,o2, . . . on) is invariant under permutations of the pairs, (i,o1), (j,o2), (k,o3), . . . (n) and that, w(n) ijk . . . (os;o1,o2, . . . on) = wijk . . . (os;o1,o2, . . . on)

(2.5)

As an example the case of the third order contribution to frequency doubling, where the input field consists of a static field plus a field at frequency, o, is considered as in the EFISH experiment. Both the fields are applied along the same axis E = E0 + Eo cos ot = E(0) + E(o)(eiot + eiot)

(2.6)

o

where, E = 2E(o) and a superscript notation has been used to indicate that a field amplitude is not defined in the standard form of eqn (2.1). The third order polarization at the frequency 2o is P(3)(t,2o) = 3e0[w(3)(2o;o,o,0)E(o)2E(0)e2iot + w(3)(2o;o,o,0)E(o)2E(0)e2iot]

(2.7)

which, using the symmetry properties of the susceptibilities, reduces to, P(3)(t,2o) = 32e0w(3)((2o;o,o,0)(Eo)2E0 cos 2ot

(2.8)

2.2 The molecular response The induced molecular dipole replaces the polarization and the local field, E/, acting on the molecule is introduced in place of the macroscopic field. There are two conventions in use for defining the hyperpolarizability series; one is the exact analogue of the macroscopic method (B convention), the other uses a Taylor series expansion (T convention) where a factor (1/n!) is introduced into nth order terms. The notation introduced by WRBS1 as been used. For a noncentrosymmetric molecule subjected to an internal field, E 0 (t) = E 0 o cos ot the second order induced dipole at 2o is given in the two conventions by, dm(t,2o) = 12bB(2o;o,o)(E 0 o)2 cos 2ot = 14bT(2o;o,o)(E 0 o)2 cos 2ot

(2.9)

where the hyperpolarizabilities are defined in a manner exactly analogous to the macroscopic susceptibilities except for the inclusion of a factor (1/n!) in the T convention terms. From eqn (2.9) bT = 2bB

(2.10)

Many of the discussions of the hyperpolarizability series and the interpretation of EFISH and other measurements carried out in the nonlinear optics 252 | Chem. Modell., 2008, 5, 249–278 This journal is

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physics community use the B convention. Phenomenological conventions defined in an ad hoc fashion for particular types of experiment have been extensively used particularly in reporting EFISH experiments. These will be discussed in later sections. Quantum Chemists and other workers in molecular theory have adhered to the T convention for defining the hyperpolarizabilities and almost all ab initio and density functional calculations have used it. The most suitable candidate for a standardised system therefore appears to be a combination of the power series expansion for the macroscopic susceptibilities and the Taylor series convention for the hyperpolarizabilities.

3. The sum over states method In the studies of the nonlinearities of organic molecules the sum over states method has played a leading part, to some extent in default of any other practicable approach. The earlier experimental EFISH studies have often been linked to analysis of the results by the semi-empirical SOS method. An examination of the exact SOS formulae used to interpret the data can provide confirmatory evidence of the convention that has been adopted. The standard SOS procedure, for the electronic contribution to the first hyperpolarizability for frequency doubling, is summarized below. Writing the perturbing field in the general form, X E¼ Eðof Þeiof t ð3:1Þ f 0

and the perturbing hamiltonian as, H = er E where r ¼

P

ra is the sum over the

a

position vectors of the electrons, leads to formulae for the first and second order corrections to the wave function, C(t) = C(0) + C(1) + C(2) + e X X 0 hnjr Ef j0i iðof o0 Þt e jni Cð1Þ ¼ h f n on0 þ of Cð2Þ ¼

e 2 X X 0 h

f ;g n;m

hnjr Ef jmihmjr Eg j0i eiðog þof o0 Þt jni ðom0 þ og Þðon0 þ og þ of Þ

(3.2) ð3:3Þ

ð3:4Þ

In deriving these formulae terms from the lower limit of the time integral from N to t can be omitted provided there are no secular terms corresponding to vanishing denominators.17,18 The prime on the summations over the molecular eigenfunctions implies that the ground state is omitted, which requires that the co-ordinate origin is taken at the electronic charge centroid of the molecule.18,19 The electric dipole is given by, l = ehC(0) + C(1) + C(2)|r|C(0) + C(1) + C(2)i

(3.5)

For a perturbing field, E ¼ E o cos ot ¼

E o iot ðe þ eiot Þ 2

ð3:6Þ

the induced dipole is given in terms of hyperpolarizabilities defined in the T convention as, o o Ej Ek 1 2iot 2iot dmi ð2oÞ ¼ fbijk ð2o; o; oÞe ð3:7Þ þ bijk ð2o; o; oÞe g 2 2 2 Chem. Modell., 2008, 5, 249–278 | 253 This journal is

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where the tensor summation convention is implied, and comparing this expression with the formulae derived from the perturbation theory expression leads to, bTijk ð2o; o; oÞ ¼ bTijk ð2o; o; oÞ ( ) rj0n rinm rkm0 2e3 X 0 ri0n rjnm rkm0 rk0n rjnm rim0 þ þ ¼ 2 h n;m ðom0 þ oÞðon0 þ 2oÞ ðom0 oÞðon0 2oÞ ðom0 þ oÞðon0 oÞ ð3:8Þ If the matrix elements are evaluated in an arbitrary co-ordinate system then, rnm ) (hn|r|mi h0|r|0idnm).

(3.9)

In the two state model, where the entire effect is attributed to the contribution of one pp* transition polarized along the direction (z) of the ground state dipole, the hyperpolarizability becomes, bTzzz ð2o; o; oÞ ¼

6e3 o2n0 jrn0 j2 rnn 2 2 ðon0 o2 Þðo2n0 4o2 Þ h

ð3:10Þ

This can be rewritten in terms of the oscillator strength of the transition and the change in the total electric dipole moment between the ground and excited state:bTzzz ð2o; o; oÞ ¼

6e2 on0 fn0 Dm 2 hm ðon0 o2 Þðo2n0 4o2 Þ

ð3:11Þ

The two state formula of eqns (3.10) and (3.11) have frequently been used in attempts to deduce frequency-dependent results from static field calculations, or conversely to deduce the static hyperpolarizability from measurements at optical frequencies. For any convention (C) defining the hyperpolarizability, the two-state model gives, bC zzz ð2o; o; oÞ ¼

nC e3 o2n0 : jrn0 j2 rnn 2 2 2 ðon0 o Þðo2n0 4o2 Þ h

ð3:12Þ

Values of nC are included in Table 1.

4. General theory of the EFISH experiment The input field is defined as, E ¼ E0 þ Eo cos ot ¼ E0 þ

Eo iot ðe þ eiot Þ 2

ð4:1Þ

Table 1 EFISH hyperpolarizability conventions WRBS label

nC

T

6

B

3

B*

1

X

3/2

g 0 = G/Nf 20fof2o bT 1 T Z 4 g þ 5kT bB 3 B Z 2 g þ 15kT bB 3 B Z þ 5kT 2 g bX Z gX þ 5kT

The notation, T, B, B*, X was introduced by WRBS. G is as defined in eqn (4.16) and nC is the constant in the two-state formula (3.12).


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In terms of the elementary susceptibilities, if the medium is isotropic in the absence of the fields so that 2nd order terms are zero, the part of the polarization that varies at the second harmonic frequency is (see eqn (2.8)), P(3) = 32e0v(3)(2o;o,o,0):E0(Eo)2 cos(2ot)

(4.2)

This quantity is the source of the second harmonic and is determined from its intensity and the macroscopic optical parameters. If the intensity of the optical input is also measured and the static field strength known then the susceptibility in the equation can be calculated. In practice the intensity of the SHG is measured relative to a known standard that for solution work has usually been quartz, occasionally lithium iodate. In the gas phase a calculated value for an inert gas has been used. The macroscopic third order susceptibility has to be related to the response functions for the active molecule in the solution. In the long-established procedure where the molecule is considered to be at the centre of a virtual cavity in a continuous medium the fields acting on the molecule, denoted by primed quantities are, at any frequency, O, g 0 E ðOÞ ¼ EO þ PO ; ð4:3Þ e0 where g is a geometric factor depending on assumptions about the cavity enclosing the molecule. For the static and o-field, the linear relationship between polarization and input field can be rewritten as, g 0 E ðOÞ ¼ EðOÞ þ e0 wð1Þ ðO; OÞ EðOÞ ¼ ½1 þ gwð1Þ ðO; OÞEðOÞ ¼ fO EðOÞ ð4:4Þ e0 where, fO = 1 + gw(1)(O), is the internal field factor. There is also an internal field at 2o given by, g 0 E ð2oÞ ¼ Pð2oÞ ð4:5Þ e0 [There has been some discussion recently about the inclusion of the 2o field; Wortmann and Bishop20 suggested it should be omitted, but Munn et al.21 have demonstrated that the term is present in an exact crystal lattice calculation, confirming the validity of the procedure used in the continuum model from the early days of nonlinear optics.] The molecular response is described in terms of polarizabilites and hyperpolarizabilities by equations analogous to the macroscopic equations. In this case the field is, E0 = E00 + E0o cos ot + E02o cos 2ot

(4.6)

and the definitions of the standard hyperpolarizabilities follow as in the macroscopic case except that an additional factor of 1/n! is included with each nth order term:dl2o=a(2o;2o):E02o+12 12b(2o;o,o):(E0o)2 + 16 32c(2o;o,o,0):E00(E0o)2 (4.7) The effect of the three internal fields, at 0, o and 2o, on one molecule is next treated, and after angular averaging the connection between the microscopic and macroscopic response functions established. The polarization of the dipole distribution by re-orientation of the molecules is induced by the static field. Introducing the Boltzmann factor and normalizing the distribution gives, for the number of molecules per unit volume with dipole moment inclined at an angle, y, to the fields, expðxÞ mE 0 cos y nðyÞ ¼ n sinh x ; where x ¼ z kT 2 x

ð4:8Þ


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and n is the number density of solute. Expanding and discarding terms nonlinear in the static field, mEz0 cos y nðyÞ ¼ n 1 þ ¼ nw ð4:9Þ kT When this distribution function is included the rotational averaging of the dipole moment induced in the molecule, 1 0o 2 dm(2o,t) = [azz(2o;o,o)E 0 2o z + 4bzzz(2o;o,o,0)(E z ) 00 0o 2 + gzzzz(2o;o,o,0)E z (E z ) ]cos2ot

(4.10)

2o

leads to a polarization, P cos(2ot) where, g P2o ¼ e0 n½hwazz ð2o; 2oÞi P2o þ e0 1 fhwbzzz ð2o; o; oÞifo2 ðEzo Þ2 þ hwgzzzz ð2o; o; o; 0Þif0 fo2 Ez0 ðEzo Þ2 g 4

ð4:11Þ

The rotational averages for the odd order terms in a and g come from the first 1 term in w and are a = 13aii and g ¼ 15 ð2giijj þ gijji Þ which is the scalar part of the tensor. For b the calculation is shown explicitly, since again there can be some ambiguity. The non-zero contribution to b comes from the second term in w. mEz0 hcos ybzzz ð2o; o; oÞi kT 0 mE ¼ z hazZ azI azJ azK ibIJK ð2o; o; oÞ kT mEz0 ½hcos4 yibZZZ þ hcos2 y sin2 y cos2 fifbZXX þ 2bXZX g ¼ kT þ hcos2 y sin2 y sin2 fi þ fbZYY þ 2bYZY g mE 0 1 1 bZZZZ þ fbZXX þ bZYY þ 2bXZX þ 2bYZY g ¼ z 15 kT 5 0 mE ¼ z bZ 5kT ð4:12Þ

hwbzzz ð2o; o; oÞi ¼

where, bZ = 13(bZII + 2bIZI) is the Z-component (the component in the direction of the molecular dipole) of the vector part of the b-tensor. Alternatively the rotational average, again obtained from the second term in w, can be written in terms of hbzi, the average component of the vector part of b in the direction of the applied field. Transforming the molecular quantity, bZ and averaging gives, bz = cos y bZ and, from the second term in w, b// = hbzi = 35bZ

(4.13)

so that, hwbi ¼

mEz0 b 3kT ==

ð4:14Þ

Eqn (4.11) becomes, ð1 nað2o; 2oÞÞP2o ¼

o 1n m bZ ð2o; o; oÞ þ gð2o; o; o; 0Þ f0 fo2 Ez0 ðEzo Þ2 4 5kT


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The factor multiplying the polarization on the right is easily shown to be equal to (1/f2o), and, using eqn (2.8), 3 ð2o; o; o; 0ÞE 0 ðE o Þ2 Pz ð2oÞ ¼ e0 wð3Þ 2 h zzzz i n m T ¼ bz ð2o; o; oÞ þ gT ð2o; o; o; 0Þ f0 fo2 f2o E 0 ðE o Þ2 4 5kT ¼ng0T f0 fo2 f2o E 0 ðE o Þ2

ð4:15Þ

If the objective is to express both the macroscopic and molecular quantities involved in the analysis of the EFISH experiment in terms of the fundamental definitions of section 2, then eqn (4.15) can be regarded as the standard form of the EFISH equation (see for example Reiss2). In eqn (4.15) the superscript has been re-inserted to emphasise that the equation relates a susceptibility defined by the standard method of section (2) to hyperpolarizabilities in the T convention. The hyperpolarizabilities are obtained by contraction of the tensors, bTijk(2o;o,o) and gTijk(2o;o,o,0) defined in the standard form of section 2. In discussing experimental and theoretical results in this paper we have tried to reduce all reported quantities to the form given in these equations, so that susceptibilities are related to eqn (2.6) and hyperpolarizabilities to eqn (2.9). The experimental studies have, however, used other conventions. Instead of eqn (4.15) the equation that often occurs is written, Pz(2o) = GE0(Eo)2 = ng 0 f0f2of2oE0(Eo)2 G = ng 0 f0f2of2o

(4.16)

where G is a macroscopic response function defined specifically for the EFISH experiment and g 0 is an effective second hyperpolarizability, in some cases also constructed specifically for the EFISH experiment. WRBS have enumerated the different versions of g 0 that are may have been used by different authors. These are given in Table 1. The T convention is as in eqn (4.15) and the B convention is obtained when the factors of 1/n! are omitted in the hyperpolarizability series. The fourth line can be interpreted as defining an empirical convention, tailored for the EFISH experiment, where the rotational averaging factors have been retained but the front factor, derived from the general definitions of susceptibilities and hyperpolarizabilities, has been omitted. In the B* convention an additional factor of 2/3 has appeared in the definition of g 0 in comparison with the X convention. The most plausible explanation of the origin of the B* convention is that the X convention has been adapted to the case where the macroscopic quantity being employed is the standard susceptibility, w(3)(o;o,o,0) rather than the ad hoc G. In interpreting the results of the experimental work the first priority is to identify clearly the macroscopic quantity reported. The various forms of analysis used to extract the hyperpolarizability value can be applied to any such macroscopic measurement and provided the latter is clearly defined there should be no difficulty in choosing the correct formula from Table 1, to obtain the hyperpolarizability in any desired convention. In section (7), therefore, an effort is made to isolate the parameters describing the macroscopic observation from the subsequent interpretation in terms of hyperpolarizabilities.

5. Ab initio and DFT calculations of the pNA b tensor Table 2 lists the results of calculations of the static first hyperpolarizability. As the perturbation theory expansion shows, for o = 0, Kleinman symmetry22 holds exactly and the components of the tensor are invariant under permutations of the three co-ordinate indices so that bZ as defined in eqn (4.12) reduces to, bZ = bZZZ + bZXX + bZYY

(5.1)


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Table 2 Ab initio and DFT calculations of the electronic part of b(0;0,0) for pNA Reference

Method

Soscun (2006)25

RHF MP2 B3LYP B3LYP MP2 CC2

Reiss (2005)5

bTZZZ

bTZXX

bTZYY

bTZ

Pol D95V(p*,d)

1057 2109 1822 2186 1646 1818

186 64.8 141.1 162.5 50 ?

39.8 61.8 72 70.4 50 ?

831 1982 1609 1953 1546 ?

Basis

opt.6-31G(d*,p)

Salek (2005)26

CCSD B3LYP B3LYP B3LYP

aug-cc-PVDZ cc-pVDZ aug-cc-PVDZ Sadlej

1919 1353 1765 1782

121.5 157.4 141.2 127.3

61.3 9.25 67.1 72.9

1736 1186 1557 1582

Luo (1993)24

RHF MCSCF RHF MP2 RHF MP2

2z + pol. funs 2z + pol. funs 3s2p2d/2s1p 3s2p2d/2s1p 3s2p2d/2s 3s2p2d/2s

1181 1372

208 15.0

25.4 15.0

1269 2058

191 78

48.0 46

947 1373 1271 2019 1030 1934

Sim (1993)23

All values are in atomic units. The Z-axis is in the direction of the ground state dipole and the X-axis in the plane of the benzene ring.

where Z is in the direction of the ground state dipole. The terms on the right are the only distinct non-zero elements of the tensor at zero frequency. Where the three elements are given in Table 2 the bZ column can be completed by a simple addition. It is perhaps useful to note that in some of the original papers the tensor components are given in the T convention (used here) but the figure quoted for bZ is often half that given in Table 2; the explanation being that the authors adhere to the Taylor series convention while performing molecular quantum chemistry calculations but revert to the B convention for quantities related to SHG. It is clear that in all cases the major contribution, by a factor between 5 and 10, is the diagonal component along the dipole, bZZZ, in agreement with the simple donor/ acceptor interpretation. The entry in the ‘method’ column refers to the level at which the actual hyperpolarizability calculation has been made—the geometry of the molecule has often been determined at a different level. In these cases alternative structure determinations have had little effect, but the question will be addressed below in connection with solution studies. Four Hartree-Fock level results appear and these are consistently lower, by substantial percentages, than any of the results obtained from methods that attempt to include the effects of electron correlation. The values given by Sim et al.23 are about 10–20% higher than those of Luo et al.24 and Soscun et al.,25 but the approximate HF value, taking the average is defined to about 15% as (1020 150). In all cases the finite field method has been used. Sim et al.23 discuss the consistent inclusion of contributions from higher order terms; Soscun et al.25 employ a more recent formulation of the finite difference equations. The variation in the results is almost certainly due to the difference in basis sets. The correlated results, taking account only of the highest level values from each source, vary from about 1300 au to just over 2000 au. for bTZ and the most recent ab initio calculations of Salek et al.26 and Reiss et al.5 give respectively 1736 au and 1546 au. Fig. 1 and Table 3 refer to the frequency dependence of bSHG obtained from the Z ab initio methods. Only those that attempt to include electron correlation effects are included and the DFT methods, where the density functional has often been adjusted by reference to 258 | Chem. Modell., 2008, 5, 249–278 This journal is

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Fig. 1 Frequency dependence of bTZ/au from ab initio calculation. r: Sim et al.;23 J: Salek et al.;26 K: Reiss et al.;5 &: Luo et al.24

time-dependent ab initio studies are omitted. The aim in this section is to try to define what the ab initio predictions of the gas phase hyperpolarizability have established. DFT methods can be extended to much more complex systems, but in the absence of clear experimental guidance, can only be considered to be as reliable as the ab initio methods from which the functionals have been derived. The CCSD results of Salek et al.26 (2005) and the MCSCF results of Luo et al.24 (1993) compute the frequency dependence as an intrinsic part of the correlated calculation. The work of Sim et al.23 (1993) and Reiss et al.5 (2005) takes the static value obtained at the MP2 level and scales it using the RPA method to get the frequency dependence. Luo et al.24 (1993) also report the result of an RHF/RPA calculation where the frequency dependence is the natural extension of the RHF method. The plotted points are at the four readily available laser frequencies that have been used in almost all experimental work. The most popular of these has been the YAG frequency corresponding to 1.17 eV or 1064 nm. At this frequency the spread of results ranges from about 1550 to 2600 au. If only the two fully frequencydependent correlated calculations are considered the range is from about 1700 to 2600 au. Salek et al., using the CCSD method find that as the frequency is increased from zero to 1.17 eV, bSHG increases from 1736 to 2667 au and Luo et al., using Z MCSCF, from 1373 to1898 au.

Table 3 Gas Phase ab initio calculations of bTZ(2o;o,o) Reference

l/nm

Method 24

Luo (1993) Sim (1993)23 Reiss5 Reiss5 Salek26 Luo (1993)24

RHF + RPA MP2 + RPA RPA Scaled MP2 CCSD MCSCF

0

1907

1064

968

833

947 1934

1042 2222

1505

1676

1546 1736 1373

1908 1509

1315 2779 1126 2091 2667 1898

3202 2171

2415

All values are in atomic units.


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The various authors make claims for the accuracy of their methods—in particular relating to the quality of the basis sets and the accuracy of the geometry that has been used—but it is difficult to believe that the theoretical value of bSHG (gas, Z electronic, 1.17 eV) has been settled to better than about 500 au. Provided a fixed geometry is assumed (whatever it may be) the quantity that is being calculated is clearly defined, but it excludes vibrational effects and the effects of intermolecular interactions. It has usually been assumed for frequency doubling, where two optical frequency fields have to interact, that the vibrational effects will be small and in the gas phase the internal field factors do not differ greatly from unity. Even if they are by no means the whole story, the quantities, bSHG (gas, electronic, 0 eV) and Z bSHG (gas, electronic, 1.17 eV), as calculated by reliable ab initio methods, ought to Z provide the foundation for a sound theoretical understanding of the large hyperpolarizability in pNA and similar molecules. A firmly established figure for the 1.17 eV value, accurate to about 100 au would be extremely valuable in helping to distinguish between the alternative interpretations of the experimental data.

6. Gas phase measurement Smaller molecules and inert gas atoms have been extensively studied using EFISH in the gas phase (see for example Miller and Ward,27 Ward and Miller,28,29 Shelton30). Shelton and Rice31 provide a comprehensive list of gaseous EFISH measurements on small molecules up to 1994. The only such result reported for molecules with donor/ acceptor substitution on a benzene ring appears to be that obtained by Kaatz et al.12 for pNA in 1998. In this experiment a gas mixture containing 0.075 mole fraction of pNA was used to obtain an EFISH measurement at 1064 nm at one temperature. The gT(2o;o,o,0) of eqn (4.15) was estimated from a THG experiment and taken as the intercept on a two point plot of g 0 versus 1/T. The value of bZ was calculated from the slope. The linearity of such plots has been confirmed in the work on smaller molecules. The gas phase method differs from that used for solutions in that the extrapolation to infinite dilution is not made since the molecular density in the gas is very much smaller. Also the internal field factors are close to unity. It is usually possible to make measurements over a sufficiently wide range of temperatures to obtain the quantity (mb/k) from the plot of G versus 1/T. In the case of pNA the value of the dipole was chosen as 6.87 D. The absolute calibration of these results is based on an absolute value of hgi obtained from an EFISH measurement on N2 which, in turn, has been calibrated against an ab initio calculation of the quantity for He. The latter is expected to have achieved a high degree of accuracy. The absolute value obtained for bZ is therefore independent of any solid state standard. The results are summarized below. Gas Phase EFISH Determination of bZ for pNA at 1064 nm mb G ¼ g þ z ; m ¼ 6:87D; bz ¼ 1787au 5kT It is reasonably certain that the above value does refer to bTz. The calibration is with respect to values measured in the gas phase or calculated where adherence to the standard conventions has been the norm. Kaatz and Shelton have also made Hyper-Rayleigh scattering measurements of a number of solvents and solutions32,33 including pNA in dioxane. The absolute values have been obtained by comparison with the hyper-Rayleigh scattering from carbon tetrachloride. An absolute value for CCl4 in the gas phase had been obtained34 by utilizing the N2 result as in the EFISH gas measurement described above. The same value was then taken to apply to liquid CCl4. Since these measurements were made an effective hyper-Rayleigh hyperpolarizability for liquid CCl4 has been obtained via measurements of acetonitrile as an intermediate in the gas phase, calibrated with respect to N2 and then directly with the liquid CCl4.35 If the more recent estimates 260 | Chem. Modell., 2008, 5, 249–278 This journal is

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are adopted, the value of 2470 au obtained from the HRS in dioxane by Kaatz and Shelton must be multiplied by a factor of 1.88 to give 4644 au at 1064 nm for the equivalent of the EFISH parameter, bTz. This result is compared with EFISH dioxane solution measurements in section 7.

7. Solution EFISH studies of pNA and MNA in dioxane The high first hyperpolarizability of pNA was first identified in EFISH measurements in the 1970s. The experimental method itself was developed by Levine and Bethea,36,37 Oudar,38 Chemla39 and others and the inherent difficulties of the procedure and the devices used to overcome them are well documented in the some of the papers. Measurements have been made on a large range of molecules (see for example Cheng et al.40,41 where full account of the method is given and Zyss and collaborators,4 where the main objective has been to characterise compounds suitable for growth in the form of nonlinear optically useful crystals.) The solvents used were often highly polar (acetone, DMSO) and where the nonlinearity was weak (as for example in nitrobenzene and aniline) neat liquids were used. A clear solutedependent signal was, however, obtained for pNA in dilute solutions. The quantitative study on which the interpretation of a great deal of subsequent work has depended was made by Teng and Garito42,43 (1983) on solutions of pNA and MNA(2-methyl-4-nitroaniline) in dioxane. They employed a careful extrapolation procedure, previously developed by Singer and Garito44 to obtain results at infinite dilution (the method parallels the Debye method used in the measurements of dipoles in solution) and made measurements at the four non-zero frequencies listed in Table 2. 7.1 Analysis of the macroscopic measurements Experimental EFISH results have often been reported in terms of a quantity, G, as defined eqn (4.16) in terms of the real field amplitudes, E = E0 + Eo cos ot

(7.1)

P2o(t) = P2o cos(2ot) = GE0(Eo)2 cos(2ot) P2o = GE0(Eo)2

(7.2)

through the equations,

There is usually no ambiguity about the definitions of the field strengths, which are as in eqn (7.1), and it follows that in terms of the standard susceptibility definitions of eqn (2.8), G = 32e0w(3)(2o;o,o,0)

(7.3)

[The inclusion or omission of the permittivity of free space, e0 in these equations does not affect the discussion of the integer or simple fractional numerical factors, although it does change the units in which G or w(3) are reported (see Appendix).] Optical theory then relates the 2o polarization to the observed second harmonic intensity. Assuming that G has been correctly determined in terms of the fields as defined in (7.1) it has then to be expressed in terms of the molecular parameters. The standard version of the susceptibility is immediately obtained from G through eqn (4.15), and from that equation, using the standard field factor theory for a single component system, 1 T bTZ m G ¼ Nf0 fo2 f2o g þ ð7:4Þ 4 5kT or, for other conventions, G = Nf0f2of2og 0

(7.5)


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where the appropriate g 0 can be found from Table 1. Within the limitations of the field-factor theory uncertainty arises only if there is doubt as to the definition of the macroscopic parameter and if the data for the field factors, dipole moment and temperature are not available. If this data is provided one can choose whichever convention is preferred and insert the corresponding formula on the right of eqn (7.5). Reiss2 has questioned whether the definitions of the macroscopic electric fields used in the experiments have always been correctly identified. A more general definition of the field would be, E = E0 + mEo cos ot

(7.6)

where m = 2 occurs if the field is defined as the sum of complex exponential functions. Then, introducing the symbol, Gm, and using the T convention,

1 bT m g þ Z E 0 ðmE o Þ2 P2o ¼ Gm E 0 ðE o Þ2 ¼ Nf0 fo2 f2o 4 5kT ð7:7Þ

T b m ¼ m2 Nf0 fo2 f2o g þ Z E 0 ðE o Þ2 5kT and for the case of m = 2,

bT m G2 ¼ Nf0 fo2 f2o g þ Z 5kT

ð7:8Þ

where, G2 would be the measured value if the field were defined with m = 2 in eqn (7.6). Reiss has introduced a notation for combining the four microscopic variants of WRBS with field definitions defined by the ‘m’ value. It might be preferable to keep the macroscopic adjustments separate from the molecular definitions. The question of the definition of the fields is related to the nature of the EFISH measurement. The essential observation is of the ratio between the intensity of second harmonic light emitted from the solution to that emitted by quartz standard. However the o-field is defined the ratio will be related to G by an equation (written schematically) of the kind, !1=2 SHG ISOL GEo Eo2 ¼G ð7:9Þ SHG d11 Eo2 IQ where G is a factor that depends on the optical arrangement and depends on refractive indices (and through them on coherence lengths). The optical fields cancel out in eqn (7.9), but the definition of the field is still relevant since it occurs implicitly in the definition of d11 for quartz. The definition of d11 involves (again schematically) a relation of the type, (ISHG ) = G 0 d11E2o Q 0

(7.10)

where the optical factor, G , the SHG intensity and the field strength must be determined absolutely. If the definition of the field in this experiment is the same as the standard definition (E = Eo cos ot) used in deriving the molecular expression on the right of eqn (4.15) then it will always be correct to take m = 1. The above discussion may, however, be over-simplified in that the signal obtained from the SHG cell also includes a contribution from the glass windows. Glass is isotropic and the glass signal arises from a term, GGE0E2o. The definition of the quantity GG must also be consistent with the field definition. In the cases discussed below, however, there seems to be no doubt that the appropriate value of m is one.


c


The theory thus far derived has dealt with one active species. Eqn (7.5) can be extended to a two component solute/solvent system as, GL = N0F0g00 + N1F1g01

(7.11)

2 2o FI = f 0I (f o I ) fI

(7.12)

where,

The internal field factors are now in general to be adjusted to take account of the differing environments of the molecules of the two species. In particular the reaction field introduced in the Onsager field will be different, even in very dilute solution, in the cavity surrounding the polar solute as compared to that of a non-polar solvent. The Onsager and Lorenz factors for the EFISH equations are, respectively, Onsager: fIO ¼

2 eððnO I Þ þ 2Þ

2e þ

2 ðnO I Þ

Lorenz: fIO ¼

2 ðnO I Þ þ2 3

ð7:13Þ

The reaction field that is responsible for the modification of the Lorenz factors to the Onsager form is present only if the dipole of the molecule in the cavity has become partially oriented in the direction of the electric field and may reasonably be assumed to be absent at the optical frequencies. At these frequencies it is also usually assumed that the refractive index may be taken as that of the solvent for both species. For the static field, however, the Onsager factor should, in principle be chosen and the effective refractive index is related to the response through the polarizability of the molecule in the cavity. For the solvent this gives rise to the solvent refractive index, but for the solute the refractive index is a partial quantity reflecting the contribution to the solution refractive index of a solute molecule in a dilute solution. Measurements of refractive index as a function of concentration and specific volume are necessary to determine it. In many EFISH studies this procedure has not been followed and less experimentally demanding approximations have been employed. Full accounts of the theory are to be found in the work of Singer and Garito,44 Cheng et al.40 and Meredith et al.45 The simplest treatment, which might be regarded as a reasonable approximation when the solute makes a contribution to the nonlinearity that is much larger than that of the solvent, is to neglect the first term in eqn (7.11) and assume that the refractive indices and dielectric constant occurring in the field factors are simply the measured values for the solution as a whole. The composition of the solution is defined in terms of the weight fraction of the solute and to obtain the NI values the measured density is required. The unapproximated eqn (7.11) becomes,

ð1 wÞ w 0 0 ð7:14Þ F0 g0 þ F1 g1 GL ¼ NA r M0 M1 which, on introducing the above approximations, reduces to, w 0 GL ¼ NA r f ð0Þf ðoÞ2 f ð2oÞg1 M1

ð7:15Þ

where the field factors are obtained from the measured solution properties. If it is further assumed that the variation of the field factors and the density with composition has a negligible effect, then the value of g01 can be calculated from the slope of a plot of GL versus w. Putting in the standard formulae for the field factors (Onsager factor for the static field and Lorenz factors for the optical fields) gives, 27M1 ð2e þ nðoÞ2 Þ @GL 0 ð7:16Þ g1 ¼ NA r eðnðoÞ2 þ 2Þ3 ðnð2oÞ2 þ 2Þ @w 0 Variations of this equation appear in which slightly different approximations for the field factors have been used, but these variations seem to have very little effect since Chem. Modell., 2008, 5, 249–278 | 263 This journal is

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the difference between the refractive indices of the fundamental and second harmonic is very small and the static dielectric constant is nearly equal to the square of the refractive index. The approximations used in deriving eqn (7.16) do, however, disregard some possibly important effects. The extrapolation to zero concentration, which is necessary to obtain reliable single molecule properties, needs to be treated more carefully as in the conventional Debye extrapolation procedure used in the determination of dipole moments in solution. The extension of the extrapolation to the nonlinear case has been treated in some detail by Singer and Garito.44 Their paper is concerned specifically with liquid mixtures where the introduction of separate refractive indices for the two components is an easily defined procedure. The interpretation of the parameters introduced for a solute is more questionable and in the application of the equation to the pNA/dioxane system43 these parameters are not explicitly given. Most other solution determinations have used simplified versions of their equation, neglecting terms that are thought to be small but retaining one or two larger terms that can give rise to appreciable corrections in eqn (7.14). The equation used by Teng and Garito43 is derived by differentiating eqn (7.14) with respect to w, when, substituting the solvent nonlinearity, G0 ¼ NA r0

F0 0 g M0 0

where appropriate, leads to,

0 @GL 1 @r 1 @F0 F1 g1 ¼ NA r0 þG0 1 r @w 0 F0 @w 0 @w 0 M1

ð7:17Þ

ð7:18Þ

If the term in the derivative of the field factor were negligible the expression on the left of this equation would be defined completely in terms of macroscopic measurable quantities. The specifics of the chosen cavity model enter the field factor derivative where Lorentz-Lorenz and Onsager factors may be mixed. The most commonly used procedure is to employ Onsager for the static field and Lorentz factors for the optical fields. For FI (i = 0,1), !2 ! 2 e nI þ 2 nðoÞ2 þ2 nð2oÞ2 þ 2 FI ¼ ð7:19Þ 3 3 n2I þ 2e where the optical factors, in which there will be no contribution from a reaction field, can be written in terms of the solution refractive index, while the low frequency factor has to be expressed in terms of a quantity which defines the reaction field for a central molecule of definite polarizability and dipole moment. The quantity, n0 can be determined experimentally from the pure solvent, but n1 is a construct from the polarizability of a solvated solute molecule isolated from other solute molecules as described above. FI changes with w through the variations of the two solution optical refractive indices and the dielectric constant. The formulae can be simplified, without appreciable loss of accuracy, by ignoring the frequency dependence of the solution refractive index, giving, 2 3 eðnI þ 2Þ n2 þ 2 FI ¼ ð7:20Þ 3 n2I þ 2e Using this formula for F0 and F1 in eqn (7.18) results in the equation employed by Teng and Garito43 to convert the macroscopic nonlinearity to a molecular quantity:

27M1 ðn21 þ 2e0 Þ @GL 1 @r 1 @F0 0 ¼ g1 þG 1 0 r0 @w 0 F0 @w 0 NA r0 e0 ðn20 þ 2Þ3 ðn21 þ 2Þ @w 0 ð7:21Þ 264 | Chem. Modell., 2008, 5, 249–278 This journal is

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where the field factor derivative is given explicitly by, 2

1 @F0 3 @n 1 2 @e ¼ þ F0 @w 0 n20 þ 2 @w 0 e0 2e0 þ n20 @w 0

ð7:22Þ

Unfortunately Teng and Garito do not report the details of the application of the formula, no values being given for n1 or for the partial derivatives. [This is in contrast to the earlier paper of Singer and Garito on liquid mixture calculations using the same formula where full details are provided.] It is therefore difficult to assess to what extent the smaller terms, not involving (@GL/@w)0 make significant contributions. The solvent quantity, G0 is rather less than 1% of the derivative, (@GL/@w)0, which at least suggests that the other terms may not be important. If the specific nature of the reaction field is ignored the sometimes unavailable factor n1 is not required. In some of the work discussed below the simplified versions of the formula have been employed. The first term in the curly brackets in eqn (7.18) is usually available and included. A simplified version avoiding the use of n1 is obtained if Lorenz type internal field factors are used for all three frequencies so that, !2 ! 3 eþ2 nðoÞ2 þ 2 nð2oÞ2 þ 2 e þ 2 n2 þ 2 ffi F0 ¼ F1 ¼ ð7:23Þ 3 3 3 3 3 Bosshard et al.46 adopt the approximation on the extreme right of eqn (7.23) and neglect the variations of the density and refractive index, leading directly from eqn (7.21) to the formula,

81M1 @GL 1 @e 0 g1 ¼ ð7:24Þ þG 1 0 e0 þ 2 @w 0 @w 0 NA ðe0 þ 2Þðn20 þ 2Þ3 r0 They provide measured data for MNA at 1064 nm for the dielectric constant derivative. Eqn (7.24) further reduces to eqn (7.25) if the dielectric constant variation is also neglected: 81M1 @GL 0 g1 ¼ ð7:25Þ þG 0 @w 0 NA ðe0 þ 2Þðn20 þ 2Þ3 r0 and this version appears to have been used by Burland et al.47 in one of the treatments of their EFISH data on MNA in dioxane. Eqns (7.16), (7.24) and (7.25) can be applied easily to available measured macroscopic quantities and have been used in many of the reported analyses of EFISH data. There remains the question of whether the neglect of the other terms in eqn (7.21) is justified and whether a proper assessment of the parameter, n1 is required. Whatever the validity of these formulae and the underlying cavity field theories it is very desirable that, before applying them, the definition and consistency of the macroscopic quantities as measured by different groups should be assessed. Much of the discussion in the literature deals directly with the final values of the hyperpolarizability presented in the various experimental papers. In calculating these values different versions of the above procedures may have been employed and, in particular, different values of the molecular dipole moment inserted into g0 to extract the b value. It is relatively easy to identify how the microscopic parameter has been obtained and which molecular convention is being used provided the identity of the macroscopic quantity is clearly established. The symbol, G0L (and its derivative, (@G 0 L/@w)0) is introduced here to denote provisionally a reported macroscopic nonlinearity before assessing its precise definition. The unprimed symbols are defined in accordance with eqn (4.16). The most troublesome ambiguity in the Chem. Modell., 2008, 5, 249–278 | 265 This journal is

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Table 4 Measured EFISH nonlinearities for pNA and MNA at 1064 nm

TG TG Bur(1) Bos P(1) P(2)

pNA MNA MNA MNA pNA pNA

Ref. q1 q1 q1 q1/ Li+ Li+

A

B

G0 0 (@G 0 /@w)0 0.053 0.1 10 0.3 0.053 0.1 9.4 0.3 0.039 10.8 0.046 9.5 2 — 15.7 — 15.7

G0 0 (@G 0 /@w)0 0.032 0.06 6.0 0.18 0.032 0.06 5.7 0.18 0.023 6.5 0.18 0.035 7.1 1.5 9.05 4.9

Adjustment factor 0.6 0.6 0.6 0.6 (0.5/0.4) 0.98/1.7 0.5624/1.80

G 0 0 and (@G/L/@w)0 are in units of 1012 esu2cm4. q1 is the older quartz standard, d11 = 1.2 109 esu; q1/ is referred to a quartz standard with d11 = 0.96 109 esu; Li+ is the older LiIO3 standard. These are the unmodified calibration standards in the original papers and correspond to the values in columns, A. The entries in columns B have been adjusted to the calibration standards that are currently thought to be correct. These are for quartz, d11 = 0.72 109 esu; and require an adjustment of the LiIO3 value by a factor of (0.98/1.7). The results of Paley et al.48 have also been further modified for reasons discussed in the text. TG: Teng and Garito;43 Bur: Burland et al.;47 Bos: Bosshard et al.;46 P: Paley et al.48

interpretation of the data arises from uncertainty as to whether the quantity employed is, G 0 L = GL = 32e0w(3)(2o;o,o) or G 0 L = e0w(3)(2o;o,o)

(7.26)

All the experimental studies described below have, at least in the initial analysis of the experimental results, related the macroscopic measured quantity to a molecular quantity defined by the formula referred to as the X convention in Table 1. The molecular quantity so extracted will then be in the X convention, provided the macroscopic quantity is defined by the first of the equalities in eqn (7.26); if the macroscopic quantity is defined by the second equality, then the molecular quantity will be in the B* convention. In the following section an attempt is made to identify and compare the macroscopic quantities, (@G0L/@w)0 reported by various authors. The values of G 0 0 are also usually reported and are useful in assessing consistency between different measurements. (Ancillary linear quantities, in particular (@e/@w)0 are not subject to conflicting definitions and when available are considered generally reliable.) In the subsequent section the molecular quantities evaluated by the equations given in this section are examined. The results to be considered are summarized in Table 4. In some cases their interpretation is not straightforward and is discussed in the following paragraphs. The values for the pure solvent nonlinearity and G00 in columns A are as far as possible taken directly from the original paper. In the Teng and Garito paper these values are stated explicitly. The paper includes an equation, which identifies G0 with G as defined in eqn (4.16) and the formula used for the SOS calculations also described is consistent with definition of bX z. Burland et al.47 refer to the nonlinearity as w(3)(2o;o,o) and their introduction of this quantity in the earlier part of their paper would lead to the conclusion that it is the standard susceptibility as defined in eqn (2.4), in which case the bz that they derive from it would be in the B* convention. However, the symbol, G, subsequently appears and it is stated that w(3)(2o;o,o) now has to be identified with G. A nearly linear plot of w(3)(2o;o,o) versus w is shown from which the value of (@G 0 L/@w)0 in column A has been obtained. An alternative method of analysis of the macroscopic data for G0L, which does not involve an explicit calculation of the derivative, is discussed below. 266 | Chem. Modell., 2008, 5, 249–278 This journal is

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The value attributed in column A to Bosshard et al.46 is quoted in their paper and has required only a conversion of the units. Their definitions make it clear that the quantity concerned is intended to be GL, so that their hyperpolarizability is in the X convention. Paley et al.48 do not give results of the macroscopic quantity directly, but it is made clear that their bX Z has been derived through a formula of the type given in eqn (7.25) with pure Lorenz factors. They also give the dipole moment used in the calculation so that it is possible to reconstruct the (@GL/@w)0 value for comparison with the other measurements. In columns B the results of columns A have been adjusted by factors required to relate them to a uniform standard based on that currently preferred for the reference standards. In comparing ab initio and DFT calculations with experiment most authors have adopted a revised value for the quartz d11 coefficient which is reduced by a factor of 0.6 from that used in earlier work. The results of Teng and Garito and Burland et al. in column B have simply been altered by this factor. Bosshard et al. give an unusual value for quartz which is (0.4/0.5) times the older (q1) figure. This factor has been removed before converting to the new quartz standard. There is a more significant problem in relation to the work of Paley et al. The primary standard used in these measurements has been Lithium Iodate and again a revised determination of the standard has led to a large change—by a factor of (0.98/ 1.7). There is also a more serious question that arises from the use of an intermediate standard—a 1% solution of MNA in dioxane. After this solution has been calibrated against LiIO3 all subsequent measurements were referred to it, the nonlinearity of the solution being reported as 3.13 1013 esu., converting to 1.80 1013 esu with the newer LiIO3 standard. Bosshard et al. have also used this method (although with quartz as the primary standard) and their equivalent figure after conversion of their units, multiplication by 0.6(0.5/0/4) and conversion to the new quartz standard is 0.5625 1013 esu. Similar figures can be derived from the work of Burland et al. and Teng and Garito by applying the equation, ! 0 @G0 0 GL ðw ¼ 0:01Þ ¼ G0 þ 0:01 @w 0

The near linearity of, for example, the results of Burland et al. justify this procedure. There is, therefore, a large and unaccountable difference between the absolute values of Paley et al. and other groups for the 0.01% solution. If these values were consistent with each other it would demonstrate the equivalence of the quartz and lithium iodate standards but the large discrepancy leaves the issue in doubt. In the second set of adjusted results in column B for Paley et al., the values have been recalibrated using 0.5625 1013 esu as the reference for the 1% MNA solution. In the two cases the results of Paley et al. are, respectively higher and lower than the other sets of results which are in much better mutual agreement. Focussing on the first four lines of the Table 4, all of which are calibrated with the quartz standard, and can be adjusted to the same value for this standard, it is apparent that there is reasonably good agreement between the three determinations of the macroscopic nonlinearity of MNA. This consistency, taken in conjunction with the equations as written in the papers, leads to the conclusion that, with a reasonable degree of certainty, the provisionally defined quantity, G 0 L can be identified with the GL of eqn (4.16). Having made this assumption, it follows that any differences in reported bz values that are not proportional to the small percentage differences in the GL values must be the result of variations in the method employed to convert macroscopic to molecular parameters. This question is examined in the next section. The alternative interpretation, that some of the results should be taken as being in the B* convention, is also discussed. Chem. Modell., 2008, 5, 249–278 | 267 This journal is

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Table 5 mbTZ: interpretation A

1 2 3 4 5 6 7 8 9 10

Ref.

Mol.

Eqn (7.24)

TG TG Bur(1) Bur(2) Boss Stah Wort Pal(1) Pal(2) KS

pNA MNA M NA MNA MNA pNA pNA pNA pNA pNA

G,X G,X G,X G,X G,X G,X G,X G,X G,X

eqn (7.25)

3.48 3.63 4.22

3.68 3.85 4.38

4.55

4.79 (3.17) 5.52 2.99

Published

Adjusted

2.91 2.88 4.40 3.87 4.82 3.17 3.28 5.52 2.99 3.25

2.91 2.88 3.28 (3.87) 3.61 2.50 (2.46) 4.36 2.36 (3.25)

The product mbTZ in units of au.Debye/104. Where the macroscopic data is available from Table 4 the product has been calculated from eqns (7.24) and (7.25) using the additional data from Table 7. The direct third order term, gT, has been neglected. The published value obtained from m and bTz as given in the papers is given in the seventh column. In rows 1 and 2 a dipole of 6.2D has been assumed (see text). In the final column the published values that have been obtained from simplified equations such as (7.24) and (7.25) have been reduced by the ratio of the ‘published’/’eqn (7.25)’ results of Teng and Garito. Values in brackets are unaltered, except in the case of Wortmann et al. where the published result has been multiplied by the missing 2o internal field factor. TG: Teng and Garito;43 Bur: Burland et al.;47 Boss: Bosshard et al.:46 Stah: Stahlein et al.:49 Wort: Wortman et al.;50 Pal: Paley et al.;48 KS Kaatz and Shelton,33 Hyper Rayleigh measurement. The bTz values are in atomic units and the value in the last column has been adjusted, where appropriate, by the same factors used for the macroscopic quantities in Table 4.

7.2 Calculation of the molecular bz from the experimental nonlinearities The formulae for effecting the conversion have been discussed in section (7.1). Data is not completely available to apply the general formula of Teng and Garito, but the simplified versions can readily be applied to the macroscopic measurements described in the previous section. A number of different values of the dipole moment have been adopted in extracting bTZ from g 0 . In all cases other than in refs. 42 and 43 the values of m and a form of b are given and it is possible to reconstruct the quantity mbZ. In comparing the interpretations of different studies this product, expressed in units of (au.Debye), has been used. The third order hyperpolarizability term, g of Table 6 mbTZ: interpretation B

1 2 3 4 5 6 7 8 9 10

Ref.

Mol.

TG TG Bur(1) Bur(2) Boss Stah Wort Pal(1) Pal(2) KS

pNA MNA M NA MNA MNA pNA pNA pNA pNA pNA

Published G,B* G,B* G,X G,X G,X G,B* G,X G,X G,X B*

4.37 (5.52) 4.32 (5.78) 4.40 3.87 4.82 4.75 3.28 (2.50) 5.52 2.99 4.87

The product mbTZ in units of au.Debye/104. The last column gives the mbTZ product calculated from the published values with the assumptions of the previous column. See Table 5 for key to references.


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Table 7 MPNA = 138, MMNA = 152 v0 =0.967 cm3 g1 o 1064 nm e0 = 2.209 F = f 0(f o)2f 2o f0 ¼ @e

no = 1.41253,

e0 ðn2o þ2Þ , n2o þ2e0

@w 0 ¼

fo ¼

n2o þ2 3 ,

n2o = 1.422

f 2o ¼

n22o þ2 3

42, for MNA46 also assumed for pNA.

eqn (4.15) is known to be negligible. Table 5 includes results computed using eqns (7.24) and (7.25) which can be compared with values obtained from the published data. The refractive index, dielectric constant and other data required to perform the calculations are given in Table 7. Table 6 gives an alternative interpretation of the published data. In Table 5, starting with the adjusted results of column B of Table 4, the two formulae have been applied and resulting values converted to atomic units and multiplied by a factor of 4, so that the values quoted (as in the paper of Reiss2) are for the quantity, bTZ(2o;o,o). Also included is the value reported in the original paper, again adjusted by the factors used to convert column A to column B in Table 4 and to bTZ(2o;o,o) in atomic units. The table also includes pNA values reported by Stahelin et al.49 and Wortmann et al.50 (for which it has not been possible to analyse the published macroscopic data as above), the result obtained by the alternative method of Burland et al.47 and the hyper-Rayleigh value of Kaatz et al.12 The values obtained from eqn (7.24), which includes the correction for the concentration dependence of the dielectric constant (from ref. 46), are lowered by between 5 and 10% compared to the uncorrected values. The more frequently used approximation appears to be that of eqn (7.25). The fundamental study of Teng and Garito43 is the only one that treats both pNA and MNA. Referring to the first two lines of Table 5 it can be seen that both the calculated and published values for the two molecules are very similar. The values calculated from eqn (7.25) are substantially larger than the published values calculated on the assumption that the dipole moment has been taken as 6.2D. [Although not stated in the publications this assumption has been made universally in discussions of these papers. It is clear from the better documented study of Singer and Garito that the dipole moment was determined from measurements of the dielectric constant in solution extrapolated to infinite dilution and it is known from other studies that this procedure leads to a value of 6.2D.] For pNA the ratio of the value given in the paper to the value obtained from the widely used uncorrected approximation of eqn (7.25) is rp = 0.79, for MNA it is rM = 0.74. It is possible that this discrepancy is the source of the assumption made by some authors that the reported bZ value referred to the B* convention, in which case the ratio would have been 0.67, the difference being attributed to the uncertain values of the correction terms used in applying eqn (7.21). The alternative, and probably more plausible explanation, is that the modified field factors and additional correction terms in eqn (7.21) which was used in ref. 43, account for the differences. If this is so then Teng and Garito’s results can be assigned to the (G, X) convention; and if it is accepted that the full equation produces a more accurate version of the microscopic response function than the approximate equations, then there is a case for adjusting other results that have been obtained through the approximate equations by the ratios, rp or rM. This method has been used, where appropriate, to produce the quantities in the final column of Table 5. As previously stated there is some ambiguity in the definition of the macroscopic quantity used in the work of Burland et al., but if this is taken to be w(3) the discrepancy with the results of ref. 43, assumed to be in the (G, X) convention, would Chem. Modell., 2008, 5, 249–278 | 269 This journal is

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amount to a factor of almost 2. In Table 5 the results of Burland et al. have been assigned to the (G, X) convention. The hyperpolarizability, calculated from eqn (7.25) and the data in Table 4, reproduce the published value obtained by the first method employed in the paper and this value has been adjusted by the factor rM in the final column. In ref. 47 a second method of obtaining the hyperpolarizability based on a numerical fit over the whole range of concentrations, rather than on an extrapolation to zero concentration, has also been reported. There is no justification for applying the above correction factor in this case and the value in the final column is unchanged. It is possible that this second method, which gives a lower hyperpolarizability, implicitly includes some of the additional corrections of eqn (7.21). Bosshard et al.46 work in the (G, X) convention. From Table 4 their value can be recalculated and adjusted by the factor, rM. The interpretation of the hyperpolarizability reported by Stahelin et al.49 is more difficult. Their publication is closely associated with that of Burland et al.47 and it might be assumed that they would employ the same convention. The issue again turns on their definition of a quantity referred to as w(3). Making the same assumptions as in the case of Burland et al. would lead to the (G, X) convention and values calculated via eqn (7.25) to be adjusted as before. This interpretation has been used in Table 5. Some of the authors of this publication have, however, in other work concluded that the Teng and Garito hyperpolarizabilities are in the B* convention and they compare their value with that of Teng and Garito. The close agreement found is rather illusory since Stahelin et al. use a dipole moment of 7.0. There is nevertheless a strong possibility that their quoted value is intended to refer to the B* convention. This alternative view is adopted in Table 6. The molecular quantities derived by Wortmann et al.50 are obtained through a very detailed analysis that includes, although in a different formalism, all the smaller terms treated in the full equation of Teng and Garito. It is reasonably clear that his values of the hyperpolarizability refer to the X convention. The adjustment by rp is not required, but division by the missing field factor, f2o [see section 3, and ref. 21] is necessary. The last three rows of the table exhibit the EFISH results of Paley et al.,48 the second version, corrected as described above probably being the more acceptable, and the Hyper-Rayleigh value calculated from the dipole moment and hyperpolarizability given in ref. 12. In Table 6 an alternative analysis is presented. The essential assumptions are that the hyperpolarizabilities reported by Teng and Garito and by Stahelin et al. and possibly even by Wortmann et al. and Kaatz et al. relate to the B* convention. The figure in brackets for the Teng and Garito results has been calculated from eqn (7.25), assuming that G/ = w(3). To maintain any reasonable degree of consistency in the results it still has to be assumed that refs. 46 and 47 have used the X convention. Since most of these results have been calculated with the approximate formulae they should be compared with the bracketed values for the Teng and Garito results. The agreement is then poor. If the general bias towards an interpretation in terms of the B* convention reflected in Table 6 has any validity then the average values of the product would be substantially higher than predicted from the results of Table 5. From Tables 5 and 6 it is possible to make an assessment of an average result. From rows 1, 6, 7, 9 and 10 of Table 5, for the case of pNA, and rows 2, 3 and 5 for MNA, pNA: mbTZ = 2.63 0.2 au Debye

MNA: mbTZ = 3.26 0.2 au Debye

where the second value of Burland et al.46 has been omitted on the grounds that the extrapolation method with the adjustment to the final column is more accurate. From Table 6 the uncertainties are greater but it can reasonably concluded that from both pNA and MNA the values of the product are in the range (4 to 5) au.Debye. There are clearly a number of uncertainties that make it impossible to 270 | Chem. Modell., 2008, 5, 249–278 This journal is

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Table 8 Estimates of bTZ for pNA and MNA in dioxane

pNA pNA pNA pNA pNA pNA MNA MNA MNA MNA

A A KS(X) (gas) B B KS(B*) (gas) A A B B

mbTZ/Debye.au/104

m/Debye

bTZ/au

2.63 2.63

6.2 7.0 6.87 6.2 7.0 6.87 6.2 7.4 6.2 7.4

4242 3757 1781 7256 6429 2671 5258 4405 7258 6081

4.5 4.5 3.26 3.26 4.5 4.5

The figure of 4.5 for the mbTZ product from interpretation B is an arbitrary value within the group of higher values found if the results of Pal (2) and Wort are omitted. See Table 5 for key to references.

reach completely reliable conclusions, but a consideration of the internal consistency of the analyses make it seem probable that the version given in Table 5 is the more likely to be correct. This is in broad agreement with the views put forward by Reiss for pNA. The inclusion of the MNA data draws attention to the major variations that arise when different procedures are used to convert the macroscopic measured nonlinearities (even when unambiguously defined) to molecular quantities. To extract the hyperpolarizability from the product data a value for the dipole moment has to be adopted. Again there is considerable variation between the values used by different groups. It has been argued that the dipole moment of the molecule in vacuo should be used, all other effects presumed to be subsumed into the cavity field formalism. If the cavity method were strictly accurate the analysis from the dielectric measurements should lead to the dipole moment of the isolated molecule. Gaseous dipole moments for pNA and MNA have not been reliably measured in the gas phase. Calculations tend to give higher values (B47 Debye) for both molecules, whereas the solution extrapolation for pNA in dioxane gives 6.2 Debye. In the case of pNA Teng and Garito (almost certainly!) used the value 6.2 Debye while all other authors have adopted higher values near to 7 Debye. For MNA it can again be assumed that the lower solution value was used by Teng and Garito while Burland et al. and Bosshard et al. have used a value of 7.4 Debye. In Table 8. The results obtained from the above averages of Table 5 are given for one low and one high value of the dipole and compared with the gas phase value from Kaatz et al.12 for pNA.

8. Theoretical approaches to the calculation of the EFISH nonlinearity of pNA in solution There is reasonable agreement between the bTZ values obtained from the gas-phase measurement of Kaatz et al.12 and the higher level theoretical calculations on the isolated molecule. If effects occurring in solution were completely accounted for by the introduction of the standard internal field factors in eqn (4.15) then the bTZ values derived from the solution measurements should also be in agreement with the gas phase values. It is clear from a perusal of the above tables that bTZ extracted from solution measurements is always substantially greater than the value established for the isolated molecule. This is true irrespective of whether the experimental data is interpreted as pertaining to X or to the B* conventions. A more detailed analysis of the solution behaviour is therefore required and, since applications exploiting the high nonlinearities possessed by pNA and similar molecules will inevitably use condensed phases, such an analysis should be of practical importance. Chem. Modell., 2008, 5, 249–278 | 271 This journal is

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In assessing recent work it might be useful to classify models used for calculating the response functions of liquids or solutions in general terms as follows:A: Continuum Models (I) Lorentz-Lorenz, Onsager internal field factors (QM(g), Continuum) (II) Self-Consistent Reaction Field (SCRF) Models (QM/Continuum) B: Discrete Models (III) Discrete SCRF Models (QM/MM/Continuum) In case (I), the theory that has been used to describe the EFISH analysis in earlier sections, the isolated molecule calculation provides the bZ value and the internal field factors adjust the fields to allow for the polarization on the cavity surface. The effect of reaction fields due to the additional polarization on the cavity walls induced by the permanent and induced dipoles on the central molecule is implicitly included in the low frequency Onsager field factor through the dielectric constant values. The choice of the high frequency dielectric constant (eN) in this formulation is rather ill defined and no account is taken of changes in the static or dynamic polarizabilities of the molecule as a result of the surrounding fields. In (II) the reaction field of the dipole is included in the molecular Hamiltonian, so that the QM calculation, at whatever level, is modified to give a new molecular wavefunction for one molecule at the centre of a cavity. This calculation can be carried out in the absence of an applied macroscopic field and would give the unperturbed properties (dipole moment, energy states etc.) of a solvated molecule. The macroscopic field has then usually been applied in a finite field calculation of the hyperpolarizability. One source of uncertainty in this procedure arises from the fact that when the reaction field is introduced into the hamiltonian it appears in a specific form, 1 2ðe 1Þ l a3 2e þ 1

ð8:1Þ

where e is the dielectric constant and l the dipole moment, which is very sensitive to the choice of the cavity radius, a. Method (III) attempts to include a discrete molecular description of the solvent structure around a central solute molecule. The solute molecule is described, again, through a QM calculation while the spatial distribution, charge distribution, polarizabilities etc. of the adjacent part of the solvent is represented by a parametrized molecular mechanics (MM) model. The parametrization may be achieved through high level calculations on the isolated solvent molecules. To relate this type of calculation to the macroscopic response it is still necessary to include long range effects through the continuum model. The basis of the procedure followed is justified by the theory that underlies the derivations of the Lorentz and Onsager approximations. In its simplest form a sphere is defined around a solute molecule and the calculation split into two parts. The sphere is a ‘virtual’ cavity. It is not implied that the material in the sphere has been removed. In two cases, a simple cubic array of dipoles and a homogeneous isotropic continuum, it can be shown that for cavities that are large on a molecular scale but much smaller than the macroscopic dimensions, the field arising from the molecules in the cavity tends to zero. The material outside the cavity modifies the macroscopic field through the usual field factors. There are slight variations in the methods used by different authors, but typically (following Jensen et al.51 and Wortmann and Bishop20) the field at the solute molecule is written, EINT = EMAC + EPOL ECAV POL + EDIS

(8.2)

where the first two terms on the right are the macroscopic field and the continuum polarization field that gives rise to the usual field factors. The part of the macroscopic polarization field that due to the material in the cavity is then removed and 272 | Chem. Modell., 2008, 5, 249–278 This journal is

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replaced by the discrete contributions, EDIS of the molecules in the cavity. It is possible to write, PERM EDIS = EIND DIS + EDIS

(8.3)

where the first term is the field due to the molecules in the cavity which has been induced by the macroscopic field, while the second term is the average field produced by these molecules in the absence of a macroscopic field. The separation is not always convenient. The macroscopic polarisation can always be written as, P = NhmINDi

(8.4)

where hmINDi is an average induced dipole contribution from a solute molecule. The polarization can now be expanded in several different ways. For example, if the expansion is in terms of powers of the first two terms on the right of eqn (8.2) which together comprise the effective fields as defined by the conventional field factors then effective polarizabilities and hyperpolarizabilities will be defined which incorporate the effects of the discrete terms that have been omitted from the fields in the power expansion. If the expansion is in terms of all contributions except EPERM DISC then the polarizabilities and hyperpolarizabilities so defined will represent effective properties of a solvated molecule. Since pNA and most of the chromophores of interest have large dipole moments an important feature of the continuum models is the introduction of the reaction field. The pNA molecule at the centre of the cavity in the continuum induces a polarization on the surface of the cavity, which produces the reaction field acting on the central molecule. This reaction field changes the dipole moment of the pNA molecule via the linear polarizability. A self consistent procedure is required in which the effects of the reaction field and also the effects of the applied macroscopic fields modified by the internal field factors are included in a self-consistent determination of the molecular response within a specified quantum mechanical model. Willetts and Rice52 attempted to develop a continuum model (type II) to calculate the EFISH response of liquid acetonitrile. Their results were compared with accurate calculations and measurements of the hyperpolarizability of the isolated molecule and measured solution for two different cavity radii. One ‘natural’ choice of cavity radius is based on the van der Waals radius of the compound and with this value of a the discrepancy between the gas and solution measurements is not resolved. A rather smaller value of a can be selected to give much better agreement. The arbitrariness of the choice of cavity radius places a serious limitation on the usefulness of the continuum theories and, in recent years, there has been a considerable effort to develop hyperpolarizability calculations based on discrete molecular simulations of liquid and solution structure. Several groups have developed models of type III based on a combination of quantum chemical calculations (QM) at various levels for a central model with the effect of the surrounding medium represented by an ensemble of molecules treated by molecular mechanics (MM). Mikkelsen et al.53 have employed a continuum reaction field response model using reference wave functions at SCF and MCSCF levels. The basis set selection is developed from work on RPA calculations on pNA by Agren et al.54 A planar C2v geometry has been employed, adapted from the crystallographically determined structure where deviations from planarity are small.55,56 In the MCSCF work a full p-orbital space is included with complete correlation constituting a CASSCF p-orbital calculation. The electronic excitation energies, transition moments and ground and CT dipole moments have been calculated in this approximation. These results are applied to a subsequent analysis of the two-state model. Effective solute hyperpolarizabilities are obtained over a range of continuum dielectric constants including those of the usual solvents. Chem. Modell., 2008, 5, 249–278 | 273 This journal is

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Jensen et al. use a discrete QM/MM model in which the quantum mechanical part of the calculation employs a DFT method. Using procedures such as those outlined above they calculate vacuum, solute and effective values for pNA at CCSD/B3LYP level. Reiss et al.5,6 have recently carried out an extensive theoretical study of the linear and nonlinear properties of pNA in the gas phase, of three solvents, cyclohexane (CH), 1,4-dioxane (DI) and tetrahydrofuran (THF) and of solutions of pNA in these solvents. The aim of the work is to provide a treatment in which the liquid phase molecules are treated discretely via a statistical simulation of the liquid and solution structures. The general strategy for attaining this goal can be summarized as follows:Firstly, the geometry of the pNA molecule in the gas and solution phases has been closely scrutinized using, for the solution, several self-consistent reaction field formalisms, the solute molecule being described at reasonably high levels of ab initio and DFT theory with averaging over conformations of the molecule. The question of the structure of pNA in gas, liquid and crystal phases has been previously addressed by several authors and the main features, confirmed and extended by this study, are that in the gas phase the C–NH2 group is pyramidal while under the influence of local fields in solutions or crystals the arrangement tends to become more planar. The crystal structure has been determined experimentally and is planar but with the C–NH2 plane twisted slightly with respect to the plane of the molecule. The computations in this work establish that there is a similar effect in a local field, represented by a self consistent reaction field, provided a rather high dielectric constant (B7) is used. Only THF of the three solvents has such a large permittivity, the observed (static) value for DI being 2.2. However the authors draw attention to ‘the dioxane effect’ where, probably because of high quadrupole interactions, this solvent behaves like a more polar solvent on a microscopic scale, as is demonstrated by high solvatochromic shifts. Torsional rotations of the amino group are also treated. The electronic structures of the three solvent molecules have been examined at the same level and the dipole and quadrupole moments evaluated so that the dipole, dipole gradient and quadrupole fields can be included in the discrete simulation. Calculations of the dipole moment of the isolated pNA molecule have also been made for use in the EFISH formula. Calculations of the vibrational contributions to the static polarizability and hyperpolarizability have also been attempted. As far as the EFISH experiment is concerned, which depends on the square of an optical frequency field, it is assumed that there will be no direct contribution to b(2o,o,o) from the vibrations although the static contribution is comparable with the static electronic contribution to b(0;0,0). An indirect vibrational effect through the linear polarizability of the solvent molecules is more important. Calculations of the vibrational effects in pNA cannot be carried out reliably even for the static case since the second term in the perturbation theory is much larger then the first and there is no evidence of convergence. The preliminary calculations are essentially of the properties of single molecules in the self-consistent continuum reaction field, taking account of the effect of the field on the structure of the single molecule. The data is then used in the discrete simulation, where the effective parameters (a,b,g) of the solute pNA molecule averaged over solution conformations are obtained. In this stage of the computation the averaged geometries from the continuum studies are used. The response function calculated in the simulation is the molecular response function of the solvated molecule. When the finite field procedure is applied the polarization of the solvent molecules by the macroscopic field is not included and this (substantial) effect has still to be allowed for when relating the results to the 274 | Chem. Modell., 2008, 5, 249–278 This journal is

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measured macroscopic susceptibility by including the usual Onsager-Lorentz-Lorenz factors. This feature might be considered a systematic weakness of the procedure in comparison with that of Jensen et al., where the macroscopic susceptibility is calculated directly from a finite field that interacts with both solvent and solute. The level of description in the latter calculation is much lower than that in the Reiss study, where it would not be computationally feasible at present to use the Jensen procedure. However, in the Reiss study, despite the detailed simulation of the solution for part of the calculation, the relation of the results to experiment still depends on the validity of the conventional internal field factors. The computed quantity is therefore,

mb g ¼ gel þ ; ð8:5Þ 5kT related to the standard macroscopic susceptibility by, 3 (3) 2wzzzz(2o;o,o,0)

= 14Nf0f2of2og,

(8.6)

Since Reiss et al. consistently use the T convention. In discussing the work of the above groups it is desirable to separate the comparison of their theoretical results from the comparison of theoretical results with experiment. As far as the second comparison is concerned a simplified version of the extensive discussion and comparison of models presented in the paper would be that, when all the enhancements associated with non-uniform fields and quadrupole effects have been included, there is reasonable agreement between the effective g 0 or first hyperpolarizability extracted from the analysis and the lower of the two estimates described in section 7. In other words, the agreement depends on identifying the results of Teng and Garito and Stahelin et al. as being correctly deduced in the X convention. In comparing solution calculations, however, there are still differences in the choice of the quantity that is used to characterise the response of the solution. A distinction is made between the property of a solvated pNA molecule and the effective hyperpolarizability of pNA in solution. In the former case the modifications of the pNA molecule have been calculated in the solution without any applied electric field present. In the second case the applied electric field has been present while the structure of the pNA and its immediate surroundings have been optimized. In both cases the calculated molecular quantity has to be used with internal field factors to recover the macroscopic response—but the choice of field factors may be different in the two cases.

9. Recent work on pNA/MNA The development of the discrete solution simulation methods and their application to other systems (the pNA/MNA/dioxane solutions have been used here as the most exhaustively studied exemplar, but similar problems are inherent in all studies of organic molecules of a comparable or greater size in solution) is well described in recent publications which can be located, in particular, via the recent series by Reiss and collaborators. Here attention is drawn to a few more recent publications that are relevant to the subject. Whitten et al.57 have critically re-examined the molecular dipole moment of MNA in the crystal. MNA, as opposed to pNA, crystallizes in a noncentrosymmetric group where the molecules are all closely aligned, and it would be expected that there would be a very large internal dipolar field acting on each molecule arising from the dipoles of its neighbours. The additional induced dipole in the ground state of the molecule under these circumstances would, if known, give an indication of an upper limit to the changes that might be expected Chem. Modell., 2008, 5, 249–278 | 275 This journal is

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in solution. An earlier experimental charge density analysis58 had concluded that the ground state dipole was enhanced by a factor of almost 3, to 23 Debye due to a combination of strong intermolecular interactions and crystal field effects. Whitten et al.57 have examined X-ray and neutron diffraction data carefully measured at 100 K and supplemented their analysis with ab initio Hartree-Fock calculations to re-assess the problem and have concluded that the degree of enhancement is only of the order of 30–40%. Their analysis includes anisotropic modelling of the motion of hydrogen atoms, integral use of periodic ab initio calculations and improved data quality, all of which represent advances over the earlier studies. Yang et al.59 have made an extensive study of the effects of basis sets in a TDDFT-SOS approach to the calculation of the hyperpolarizability of pNA and a squairaine molecule. They find that the method is not very sensitive to the choice of basis set. The convergence behaviour and efficiencies of various functionals are discussed. Calculations of the excited state energies of pNA in vacuo and in solution— through a polarized continuum model—have been reported using a TDDFT model by Scalmani et al.60 [See also Faustino et al.,61 who have attempted to relate calculations of the first hyperpolarizability of excited states of pNA to experimental hyper-Rayleigh scattering data].

Appendix I. Conversion of units 1.

Hyperpolarizabilities

Four systems occur: 2 n (1) ESU: dm(n)/esu.cm = a(n) esu(E/esu.cm ) 1 n (2) SI(1): dm(n)/Cm = a(n) (E/Vm ) SI(1) 1 n (3) SI(2): dm(n)/Cm = e0a(n) esu(E/Vm ) (n) (n) 2 n (4) AU: dm /e.a0 = aau (E/ea0 ) where, a(n) is the nth order polarizability [(n 1)th order hyperpolarizability]. The omission of the (1/n!) factors does not affect the conversion of units. The AU equation has been written in the form suitable for conversion to the esu system. If the velocity of light is written, c = B 108 ms1, where B = 2.99792458, then using, (Coulomb/esu) = (B 109)1, e0 = (4pB2 109)1, leads to, ðnÞ

aSIð1Þ =CV n mnþ1 ¼ ðnÞ

aSIð2Þ =mnþ2 V 1n ¼

10ð4nþ11Þ ðnÞ aesu =esu1n cm2nþ1 Bnþ1 4p 10ð4nþ2Þ ðnÞ aesu =esu1n cm2nþ1 Bn1

n1 a(n) (a0/cm)(2n+1)a(n) AU = (e/esu) esu,

where e = 4.8029750 1010 esu, a0 = 5.29167 109 cm. The numerical equivalents of 1 esu unit are given in the table for n = 1,2,3:

a b g

SI(1) 1.1126501 (16) 3.7224011 (21) 1.2379901 (25)

SI(2) 1.2566371 (5) 4.1916900 (10) 1.3981973 (14)


c


(AU) 6.74872532(24) 1.15756813(32) 1.98550683(39)

2.

Susceptibilities

There are only two systems in use: 2 n (1) ESU: P(n)/esu.cm2 = w(n) ESU(E/esu.cm ) (n) 2 (n) 1 n (2) SI: P /Cm = e0wSI (E/Vm ) ðnÞ

wSI =CV n mn2 ¼

4p ðnÞ w =esu1n cm2n2 104ðn1Þ Bn1 ESU

The numerical equivalents of 1 esu are given in the table for n = 1,2,3: w(1) 4p

3.

w(2) 4.1916900 (4)

w(3) 1.39811973 (8)

Dipole moments

1 Debye Unit = 1018esu cm = 3.3356410 1030 Cm.

References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

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34 P. Kaatz and D. P. Shelton, Mol. Phys., 1996, 88, 683. 35 R. D. Pyatt and D. P. Shelton, J. Chem. Phys., 2001, 114, 9538. 36 B. F. Levine and C. G. Bethea, J. Chem. Phys., 1976, 65, 2429; B. F. Levine and C. G. Bethea, J. Chem. Phys., 1978, 69, 5240. 37 B. F. Levine, Chem. Phys. Lett., 1977, 37, 516. 38 J. L. Oudar, J. Chem. Phys., 1977, 67, 446. 39 J. L Oudar and D. S. Chemla, Opt. Commun., 1975, 13, 10. 40 L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken and S. R. Marder, J. Phys. Chem., 1991, 95, 10631. 41 L.-T. Cheng, W. Tam, S. R. Marder, A. E. Stiegman, G. Rikken and C. W. Spangler, J. Phys. Chem., 1991, 95, 10643. 42 C. C. Teng and A. F. Garito, Phys. Rev. Letters, 1983, 50, 350. 43 C. C. Teng and A. F. Garito, Phys. Rev. B, 1983, 28, 6766. 44 K. D. Singer and A. F. Garito, J. Chem. Phys., 1981, 75, 3572. 45 G. R. Meredith and B. Buchalter, J. Chem. Phys., 1983, 78, 1938. 46 Ch. Bosshard, G. Knopfle, P. Pretre and P. Gunter, J. Appl. Phys., 1992, 71, 1594. 47 D. M. Burland, C. A. Walsh, F. Kajzar and C. Sentein, J. Opt. Soc. Am. B, 1991, 8, 2269. 48 M. S. Paley, J. M. Harris, H. Looser, J. C. Baumert, G. C. Bjorklund, D. Jundt and R. J. Tweig, J. Org. Chem., 1989, 54, 3774. 49 M. Stahelin, C. R. Moylan, D. M. Burland, A. Willetts, J. E. Rice, D. P. Shelton and E. A. Donley, J. Chem. Phys., 1993, 98, 5595. 50 R. Wortmann, P. Kramer, C. Glania, S. Lebus and N. Detzer, Chem. Phys., 1993, 173, 99. 51 L. Jensen and P. van Duijnen, J. Chem. Phys., 2005, 123, 074307. 52 A. Willets and J. E. Rice, J. Chem. Phys., 1993, 99, 426. 53 K. V. Mikkelsen, Y. Luo, H. Agren and P. Jorgensen, J. Chem. Phys., 1994, 100, 8240. 54 H. Agren, O. Vahtras, H. Koch, P. Jorgensen and T. Helgaker, J. Chem. Phys., 993, 98, 6417. 55 F. Bertinelli, P. Palmieri, A. Brillante and C. Talliani, Chem. Phys., 1997, 25, 335. 56 S. P. Karna, P. N. Prasad and M. Dupuis, J. Chem. Phys., 1991, 94, 1171. 57 A. E. Whitten, P. Turner, W. T. Klooster, R. O. Piltz and M. A. Spackman, J. Phys. Chem. A, 2006, 110, 8763. 58 S. T. Howard, M. B. Hursthouse, C. W. Lehmann, P. R. Mallinson and C. S. Frampton, J. Chem. Phys., 1992, 97, 5616. 59 G. C. Yang, S. Q. Shi, W. Guan, L. Fang and Z. M. Su, J. Mol. Struct. (Theochem), 2006, 773, 9. 60 G. Scalmani, M. J. Frisch, B. Mennucci, J. Tomasi, R. Cammi and V. Barone, J. Chem. Phys., 2006, 124, 094107. 61 W. M. Faustino and D. V. Petrov, Chem. Phys. Lett., 2002, 365, 170.


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The floating spherical Gaussian orbital (FSGO) methodw A. H. Pakiari DOI: 10.1039/b801727c

Introduction This review is dedicated to my PhD supervisor Professor J. W. Linnett, to Professor N. C. Handy who taught me Configuration Interaction (CI) during my postdoctoral period, and to Professor M. J. S. Dewar, who taught me theoretical physical organic chemistry. The review includes the most important articles on the FSGO method dating from 1967, when the original idea of the FSGO method was introduced by A. A. Frost1 until the last FSGO paper to appear in the literature in 2004. Therefore, this is a review of 37 years of FSGO. One difficulty is that authors have applied FSGO with other methods such as Monte Carlo and the phrase ‘‘FSGO’’ has not been used in the title or abstract of their articles. Such articles cannot be found by searching databases. The date of finishing this review is July 2007, which is also the date of the final search for finding FSGO articles in databases. What we are trying to achieve in this article is to give a review of the subject for those who are interested, in such a way that they can start their research. Obviously, one is not able to go into excessive detail, but references cited in this review can help such people to peruse the matter. The references are given in such a way as to make them succinct due to the limitation of the length of the review article, but hopefully in an effective manner. Therefore, you may find the name of some authors in the article but no mention of them in the references. For example, Frost used Huzinaga’s pseudopotential, but reference to Huzinaga is not given in our reference list. Presumably, those people who are interested in this particular reference can read Frost’s paper, and find it. This review consists of four parts. Part I includes the original theory of FSGO and its characteristics. Part II describes various developments of the FSGO original theory; for example, describing molecular orbitals by multi-Gaussian expansions, open shell methodology, multi-determinant calculations, how to describe lone pairs, how to describe p-systems, force constants and pseudopotential etc. Part III contains a collection of useful papers, which may be used by readers for different purposes; this part is called ‘Application of FSGO’. Part IV contains other methods which FSGO has been applied to such as FSGO fragments, natural orbitals, Monte Carlo, DFT, perturbation theory etc. Obviously, it is not possible to cover all papers. It may be that some important papers have been neglected, partly due to the space limitation of review article and partly due to ignorance on the part of author, and I apologize for any major omission. However, the authors named in the review have many publications in this subject; perhaps their names form good keywords for more FSGO articles, for those readers who are interested. Frost last reviewed this subject in 1977.2 Chemists are always interested to see where the electrons are in molecule, which electrons from what valence shell of which atom participate in different part of molecules and why, and what is the of role of each electron in determining molecular properties. These are difficult questions to be answered, but such considerations enable us to study chemical reactivity, which is what chemistry all about. Perhaps, Chemistry Department, College of Sciences, Shiraz University, Shiraz 71454, Iran. E-mail: [email protected]; [email protected] { I have to acknowledge Dr A. Mohajeri faculty member of Chemistry Dept. Shiraz University for editing and scientifically correction of this manuscript. [email protected]


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the motivation for introducing the Floating Spherical Gaussian Orbital (FSGO) method was the above considerations. Computer programming and working with the FSGO method is simple, and you will find in the Appendix various useful expressions for evaluation of the electronic energy.

1- Part I: 1-A

Original theory

The FSGO method, which was introduced by Frost1 in 1967, is briefly as follows. The spherical Gaussian, which is the simplest Gaussian function, and looks like the simplest hydrogenic 1s orbital, is used as a constructing function for the basis set, ~i) = (2ai/p)3/4 exp[ai(~ ~i)2] Gi(~ rR rR

(1-A-1)

~i are the variational The orbital exponent ai and the location of the orbital center R parameters. These Gaussians are positioned in the molecule in the same way as in Lewis’s concept of valence theory. A single Slater determinant wave function for the non-orthogonal basis set of doubly occupied orbitals, GI is constructed, and the total electronic energy is given by the well-known formula X XX hijh1 jiiþ ð2hiijh2 jjji hijjh2 jijiÞ ð1-A-2Þ Eel ¼ 2 i

j

i

Here, h1 is the operator for kinetic and electron nuclear attraction energies, and h2 is the two electron operator (1/rij). Since the Gaussian orbitals in eqn (1-A-1) form a non-orthogonal basis set, the following transformation can be used in order to convert it to an orthogonal basis set; X ji ¼ ðS1=2 Þin Gn ð1-A-3Þ n

where S is the overlap matrix. The total electronic energy for a single-determinantal wave function (Slater determinant) for describing one spherical Gaussian per molecular orbital is given by: X X Eel ¼ 2 hij Tjk þ Rijkl ½2Tij Tkl Til Tjk ð1-A-4Þ j;k

k;l;p;q

where hij = hi|h1|ji, Rijkl = hij|h2|kli. T = S1 is the inverse of the overlap matrix. The total energy is obtained by adding the inter-nuclear repulsion energy. E ¼ Eel þ

N X Zm Zn mon

ð1-A-5Þ

rmn

where rmn and Z are inter-atomic distance and nuclear charge, respectively. The dipole moment can be written as follows, X X X hmi ¼ Rn Zn 2 Ri Tii 4 Rij Tij Sij ð1-A-6Þ n

i

j;k

where Rn, Ri and Rij are locations of the nuclei, the orbital centers, and the overlap centers, respectively. The force on each nucleus due to the electron distribution can also be calculated,

fn ¼ 2Zn

n X j;k

Tjk qn;jk

ðRn Rjk Þ s3


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ð1-A-7Þ

where s = |Rn Rjk|, and qn,jk = 8(2/p)1/2Sjk(s/pjk)3F1[2(s/pjk)2]

(1-A-8)

F1(t) = (1/2t)[Fo(t)et]

(1-A-9)

where Fo is the error-type function, which is required for energy integrals and finally pjk=f2ik Sik

(1-A-10)

Technically, the nuclear coordinates and orbital parameters, which consist of the Cartesian coordinates of the orbital center and its exponent, are all variational parameters and have to be optimized until the viral theorem is satisfied as a necessary condition (at least the total kinetic energy is equal and opposite to the total energy). For a sufficient condition, it is required that the obtained electronic and geometrical structure must make sense chemically and experimentally. The above formulas have been programmed in FORTRAN by Frost3 and Liu et al.,4 and there are integral forms for kinetic energy, electron-nuclear attraction and electron-repulsion with a modified routine for the error function eqn (1-A-9) in that paper. We have also provided for above integral forms using the ordinary error function (ERF available in the most FORTRAN compilers), in the Appendix. Frost1 presented the results for LiH by single Gaussian FSGO (SG-FSGO) in his first paper; they are 6.57269 au (in error by 18.6%), 6.56 Debyes (error 11.6%), 3.2260 au (error 7.0%) for the total energy, dipole moment and bond length, respectively. 1-B Characteristics of the SG-FSGO Frost applied the SG-FSGO method to one- and two-electron-pair systems in part II,5 first row of atom hybrids and hydrocarbons in part III,6 hydrocarbons in Part IV7 and diatomic molecules of first-row and second-row atoms,8 where he obtained about 1.7% to 6% error in bond length for hydrocarbons. In addition, the total energies evaluated by SG-FSGO have about 15–17% error. The percentage errors for bond length and bond angle for a molecule with lone pairs such as H2O are 14.5% and 8.0%, respectively. Despite these shortcomings SG-FSGO is rather good in molecule without lone pairs such as hydrocarbons; it cannot however predict correct properties for molecules with lone pairs. For example, a SG-FSGO calculation on F2 with six lone pairs shows dissociation.9 There are several major shortcomings in the original FSGO method (i) Lack of cusp conditions, which can be cured by using at least six concentric Gaussians at nuclei. (ii) Lack of suitable description of lone pair orbitals, which will be discussed later in section 2-H. (iii) Lack of general and suitable orbital description for s-bond. (iv) p-Bond will be given a rather general description in section 2-I. (v) A large number of parameters have to be optimized, which problem has not yet been solved. (vi) Description molecular orbital FSGO needs experience and a lot of chemical intuition, which is managed automatically in the SCF procedure by symmetry adapted concepts. Description of molecular orbitals is important in FSGO, it is used to employ description of orbitals instead of basis set. The results of chemical properties have low percentage of error when the description of MO orbitals is relevant. Therefore, all attempts are to choose a better description of MO orbitals. To get a better orbital description, the multi Gaussians orbital description section 2-A or using open shell and multi-determinantal wave function section 2-C and 2-D may help. 1-B-ii

Localized nature of FSGO wave-function

A localization procedure similar to that of Edmiston and Ruedenberg10 has been proposed by Suthers and Linnett11 for FSGO wave functions. The two electron Chem. Modell., 2008, 5, 279–311 | 281 This journal is

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integrals part in eqn (1-A-2) can be written as follows; ER = D + C 0 X 0 X D¼ ½iijii Diagonal terms

(1-B-1) ð1-B-2Þ

i

C0 ¼ 2

XX

X0 ¼ 2

½iijjj; i 6¼ j off diagonal Coulomb terms

ð1-B-3Þ

½ijjji; i 6¼ j off diagonal exchange terms

ð1-B-4Þ

j

i

XX j

i

The equivalent expressions in the non-orthogonal basis set can be written as: X XXXX ½mnjpq ðT1=2 Þim ðT1=2 Þin ðT1=2 Þip ðT1=2 Þiq ð1-B-5Þ D¼ m

C0 ¼ 2

n

p

q

XXXX m

n

p

i

½mnjpq

XX

q

ðT1=2 Þim ðT1=2 Þin ðT1=2 Þjp ðT1=2 Þjq ; i 6¼ j

j

i

ð1-B-6Þ X0 ¼

XXXX m

n

p

½mnjpq

XX

q

i

ðT1=2 Þim ðT1=2 Þjn ðT1=2 Þjp ðT1=2 Þiq ; i 6¼ j ð1-B-7Þ

j

Now, the set of localized orbitals can be obtained by the procedure of Lennard-Jones and Pople, or by the Reudenberg method. Sum of self repulsion terms D will be maximized by this procedure. This implies the simultaneous minimization of C0 and X0 .

2- Part II, Development of original theory 2-A

Multi-Gaussian FSGO

In 1969 Frost12 also developed the double Gaussian FSGO model (DG-FSGO) and improved his previous results. The constructing function is now a linear combination of two spherical Gaussians, ~i)2] + C2 exp[a2(~ ~i)2] Fi = C1 exp[a1(~ rR rR

(2-A-1)

where the Ci are linear coefficients. As shown in Table 1, the improvements for DG-FSGO in comparison with SG-FSGO, are about 10.514.0% in energy, 1.0–4.0% in bond length and 4.05.8% in inter-bond angle. The percentage of error will increase with increasing the number of lone pairs in molecule. A general form of linear combination of spherical Gaussians has been used for describing each molecular orbital.13 The multi-Gaussian function Ith molecular orbital is defined as X ~j Þ w I ¼ NI cj Gj ð~ rR ð2-A-2Þ j

where cj is the coefficient of Gaussian, and NI is the normalization factor which can be evaluated as; !1=2 X ci cj Sij ð2-A-3Þ NI ¼ i;j

The linear coefficients of the multi-Gaussian ci are obtained by direct optimization 282 | Chem. Modell., 2008, 5, 279–311 This journal is

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Table 1 The comparison of results of single and double Gaussian FSGO with Hartree-Fock and experimental results SG-FSGO

LiH CH4 NH3 H2O HF

Calc. error% Calc. Error% Calc error% Calc. error% Calc. error%

DG-FSGO

Energy

BL

Angle

Energy

6.572 17.7 33.992 14.7 47.568 15.0 64.288 15.3 84.571 16.0

1.712 7.3 1.115 2.0 1.011 0.06 0.881 7.9 0.784 14.5

—

7.6905 1.620 3.6 1.6 38.4851 1.038 3.5 5.0 53.81409 0.950 3 6.0 72.7921 0.861 4.1 10.0 94.571 0.794 5.5 13.4

— 87.6 17.8 88.4 15.4 —

BL

Experimental Angle Energya

BL

Angle

—

1.595 — 1.093 — 1.012 — 0.957 — 0.917 —

—

— 91.9 13.8 94.5 9.6 —

7.9851 — 39.8660 — 55.9748 — 75.9224 — 100.058 —

— 106.6 — 104.5 — —

a Hartree Fock energy, total energy is negative, BL is bond length in angstrom, and angle is in degree.

similar to other parameters. We use lower case letter i, j, k, l for representation of single Gaussian orbital description, and upper case letter I, J, K, L for representation of multi-Gaussian orbital description. To obtain total electronic energy of multiGaussian orbital description, it must be considered in eqn (2-A-4) that W = D1, and the lower case letters should be changed to upper case one. X X Eel ¼ 2 hIJ WIJ þ RIJKL ½2WIJ WKL WIL WJK ð2-A-4Þ I;J

I;J;K;L

where hIJ and RIJKL were explained for eqn (1-A-4). The MG-FSGO can be used in three ways. First, each Gaussian can only be used in one specific orbital description similar to eqn (1-A-4), which Frost used for single and double Gaussian calculations on LiH. This type is called Localized FSGO. The second type of application is that each Gaussian can be used in different combinations (orbital description) with identical exponent, position and coefficient. This method was applied to ethylene, vinyl lithium, diazene and formaldehyde.13 This type is called Semi-delocalized FSGO. Third, each Gaussian with identical exponent and position but different coefficient can make contributions in different orbital descriptions; therefore, it has more flexibility in our basis. This type of application can be called delocalized MG-FSGO (see section 2-G). This is similar to the SCF procedure. This method was applied to cyclobutadiene.13 2-B Floating ellipsoidal Gaussian orbitals (FEGO) It was thought in the early 1970s that it might be possible to improve the results of FSGO by using Floating Elliptic Gaussian Orbitals, FEGO. In a short article in 1974, Frost14 presented disappointing results for FEGO, which are for LiH; a total energy of 6.57277 au (compared with 6.57270 au for SG-FSGO) and the bond length is 3.212 au (compared with 3.226 au by SG-FSGO). The result for NH3 was also very disappointing, namely 1.96 au for the bond length, and an angle of 91.11. Therefore, there were no further articles about FEGO in the literature. 2-C

Open-shell wave-function

There are two types of open shell wave-function in the FSGO method. I- The first type of open-shell is called restricted open-shell, denoted RO-FSGO, and it was introduced by Blustin and Linnett.15 There are Chem. Modell., 2008, 5, 279–311 | 283 This journal is

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n-electrons in doubly occupied orbitals, and there is an extra orbital fm that is singly occupied, n.fm| C = det|f1f1 . . . fnf

(2-C-1)

One and two electron electronic energies are given, respectively by " # n n n X X X 2 E1 ¼ 2 hij Tij þ Am hmm 2 him Sjm Tij þ hij Uij i;j¼1

"

n X

E2 ¼

i;j¼1

Rijkl ð2Tij Tkl Tik Tjl Þ þ A2m

n X

ð2Rijmm Rimjm ÞTij

i;j¼1

i;j;k;l¼1

2

ð2-C-2Þ

i;j¼1

n X

Rijml Smk ð2Tij Tkl Tik Tjl Þ þ

i;j;k;l¼1

n X

#

ð2-C-3Þ

Rijkl ð2Uij Tkl Uik Tjl Þ

i;j;k;l¼1

P P in which Uij ¼ nk;l¼1 Tik Tjl Skm Slm , A2m ¼ 1=ð1 ni;j¼1 Tij Sim Sjm Þ. The unpaired electron density is, " # n n X X 2 rz ðrÞ ¼ Am pmm ðr Rmm Þ 2 Tij Sim pjm ðr Rjm Þ þ Uij pij ðr Rij Þ ð2-C-4Þ i;j¼1

i;j¼1

The dipole moment is expressed as, for closed shell part, X

mc ¼

rn Z n 2

And for open shell part is, " mo ¼

Rin Tii 4

n X

i¼1

nuclei

A2m

n X

Rm 2

n X

Rim Sim Sjm Tij þ 4

i;j¼1

ð2-C-5Þ

Rij Sij Tij

i4j

n X

# ð2-C-6Þ

Rij Sij Uij

i;j¼1

They applied their method to radicals such as HeH, LiH+, H4+, He2+ as a p-type lobe function, constructed from two equivalent spherical Gaussians, fm = Nm(fa fb), Nm = 1/(2 2hfa|fbi)1/2. II- The second type of open-shell wave-function is called unrestricted FSGO, or UFSGO, which was introduced by Pakiari and Linnett.16 When there are p electrons with a-spin and q electrons with b-spin, then, the wave-function is, C = det|f1f2 . . . fpfp+1fp+2 . . . fp+q|

(2-C-7)

The electronic energy is, Eel ¼

pþq X i;j¼1 i4j

hij 3

2

pþq p X 16 X þ 6 Rijkl Tij Tkl Rijkl Til Tjk 4 2 i;j;k;l¼1 i4j;k4l

i;j;k;l¼1 i4j;k4l

q X i;j;k;l¼pþ1 i4j;k4l

7 Rijkl Til Tjk 7 5

ð2-C-8Þ

The factor half is due to avoiding the evaluation of the same integral twice, since each integral must be evaluated once for single occupied orbital. This can be taken into account in computer programming in order to save computational time. It should be noted that the electronic energy in both types can easily be extended to


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cover the MG-FSGO formalism. Obviously, these types of wave functions (2-C-1) and (2-C-7) are not eigenfunction of S2. 2-D

FSGO calculation with many determinantal wave function

It is possible to achieve an eigenfunction of S2 or some type of configuration interaction with nonorthogonal basis sets17 by using a combination of Slater determinants. The element Hmn between two determinants of the electronic energy matrix is; X X 1 Hmn ¼ Tij hmn þ ð2-D-1Þ ðwim wjm jwkn wln ÞðTjl Tik Tjk Til Þ D mn ij ijkl ioj; k 6¼ l where Dmn is the determinant formed from the elements of the overlap matrix between the two determinants m and n, and the elements of T-matrix belong to the inverse of Dmn. 2-E First and second derivative of FSGO energy integrals There are difficulties for optimization in FSGO methods, when using subroutines such as the Harwell Library VA04A, since it is a direct search optimization procedure. Therefore, the optimization is very time consuming. If we want to use optimization routine using gradient or second derivative, first and second derivatives are needed. Numerical first and second derivatives (finite difference method) have large error. Analytical derivatives help to speed up the optimization. Additionally, those derivatives are vital for force constant and frequency calculations. The formulation for analytical first18 and second derivatives19 has been provided by Pakiari and Ofadeh. These derivatives are with respect to exponent, and Cartesian coordinates. Since first and second derivative by finite difference contains large error specially the second one, this makes large error in calculation of force constant and optimization. Therefore first and second analytical derivative of FSGO energy integrals, which can also used in SCF procedure, can quite help for more accurate force constant calculation and also fast optimization. A typical formula for the analytical first derivative of overlap integrals Sij with respect to coordinates of orbitals is, dSij ai aj drij dSij ¼ 2 rij Sij if k ¼ i ¼ j then ¼0 dqk ai þ aj dqij dqk Where q represents the Cartesian coordinates of orbitals x, y, z, and 8 drij drij xi xj þ yi yj þ zi zj < k ¼ i ¼ j then dqk ¼ 0 ¼m if k ¼ i then m ¼ 1 : dqk rij k ¼ j then m ¼ 1

ð2-E-1Þ

ð2-E-2Þ

For analytical first derivative of overlap integral Sij with respect to exponent ak, dSij 3ðai aj Þ a2n Sij ¼ ð1Þm Sij r2ij 4ak ðai þ ak Þ dak ðai þ aj Þ2

ð2-E-3Þ

8 ij < k ¼ i ¼ j then dS dak ¼ 0 if k ¼ i then m ¼ 1 and n ¼ j : k ¼ j then m ¼ 2 and n ¼ i

ð2-E-4Þ

Typical second analytical derivatives of overlap integral Sij are as follows. Chem. Modell., 2008, 5, 279–311 | 285 This journal is

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The second derivative with respect to exponents ak1 and ak2, if k1 = k2 = i, " # @ 2 Sij 3 ða2i a2j Þ þ 2ai aj ai aj @Sij ¼ Sij þ 4 @ak2 @ak1 ai ðai þ aj Þ @ak2 a2i ðai þ aj Þ2 ð2-E-5Þ 2 2 2aj aj 2 2 @Sij þ Sij rij rij ðai þ aj Þ3 ðai þ aj Þ2 @ak2 if k1 = i and k2 = j, @ 2 Sij 3 2ai aj ¼ Sij Sij r2ij @ak2 @ak1 2ðai þ aj Þ2 ðai þ aj Þ3 3 ai aj @Sij ai aj @Sij r2 4 ai ðai þ aj Þ @ak2 ai ðai þ aj Þ2 ij @ak2

ð2-E-6Þ

The second derivative of overlap integral with respect to ak1 and qk2, a2j @ 2 Sij 3 ai aj @Sij @rij 2 @Sij ¼ 2S r þ r ij ij ij 4 ai ðai þ aj Þ @qk2 ðai þ aj Þ2 @qk2 @ak1 @qk2 @qk2 The second derivative of overlap integral with respect to qk1 and qk2, @ 2 Sij 2ai aj @Sij @rij @rij @rij @ 2 rij rij ¼ þ Sij þ Sij rij @qk2 @qk1 ðai þ aj Þ @qk2 @qk1 @qk2 @qk1 @qk2 @qk1

ð2-E-7Þ

ð2-E-8Þ

where @ 2 rij @rij @rij 1 ¼ ð1Þm 3 @qk2 @qk1 @qk2 @qk1 rij

ð2-E-9Þ

8 < k2 ¼ i then m ¼ 2 k1 ¼ i and or : k2 ¼ j then m ¼ 1

ð2-E-10Þ

If

2-F Reconciliation of FSGO and SCF methods There have been some arguments that FSGO cannot provide total energy with the same accuracy as SCF results. If the calculation is carried out for instance for optimized LiH with a single Gaussian (one Gaussian for k-shell and one Gaussian for bonding positioned at inter-atomic distance) by FSGO method, and then taking the optimized results as input for a standard SCF procedure, we will obtained almost the same results as FSGO. But, when you repeat the calculation with the multiGaussian FSGO for LiH, the results of SCF is more stable than FSGO. The reason is that orbital description of any MO in FSGO is truncated with respect to SCF molecular orbital. Now, if we put the same linear combination as SCF for orbital description of FSGO method, we will obtain the identical results from both methods SCF and FSGO. In other words, FSGO molecular orbitals are a truncated linear combination of basis set of SCF method.20 Therefore, it is possible to reach to Hartree-Fock limit. For example for LiH energy 7.9867349 au with bond length 3.035 au obtained by FSGO, and 7.987315 au with bond length 3.034 au by Cade21 obtained by SCF method. 286 | Chem. Modell., 2008, 5, 279–311 This journal is

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2-G

Delocalized FSGO

The argument in sections 2-A and 2-F means that we let any orbitals in basis set contribute to all molecular orbital in molecule, which depend on their symmetry similar to SCF, the results of FSGO and SCF are identical. This way of thinking can be called delocalized orbital in FSGO method. Delocalized and localized FSGO have been demonstrated for cyclobutadiene for p system.12 2-H

Lone pair description

There is a shortcoming in FSGO for predicting inter-bond angle and bond length for molecules with lone pair such as H2O and NH3 molecules6 or F2 molecule.8 The angles and bond lengths are quite below the experimental expectation, as mentioned in section 1-B. Tan and Linnett22,23 tried to spread the lone pair orbital of NH3 and CH2 around the atom having lone pair (delocalized idea such as SCF), and they have obtained rather more correct results as in Table 2. Since this idea is difficult to generalize for two or three lone pairs on one atom, another later development is to include an axial lone pair.24 The last improvements for lone pair description were the linear combination of 1s + single p-orbital and 2s + double p-orbital.25 2-I

p-Description

The description of p-bonds is another important component of chemical valence, which chemists must be concerned about. Frost7 describes s- and p-bonds with one Gaussian for each side of plane of ethylene (top and bottom), and the results are given in Table 3. The model is called banana type, which was introduced by Pauling and Slater. This type description for p-bond makes it indistinguishable from s-bond, and Hoffman and Woodward rule can not be applied to it. Pakiari and Keshavarz26 proposed a s–p model of p-bond description, which has been introduced by Hu¨ckel. In this model, Gaussian orbitals for describing a s-bond are positioned on the interatomic vector, and a p-bond is made of two p-type Gaussians (positive and negative lobes), whose centers are optimized very near to nuclei. The above procedures are called the localization p-system. Let us see how delocalized p-orbital description works. Delocalized FSGO (section 2-A and 2-G) have been used for p-system of cyclobutadiene in both square Table 2 Different results for lone pair descriptions in FSGO calculations Lone pair CH2

NH3

H2O

F2

Axial Axial Delocalized Axial 1p Axial 2p 1s + 1p 2s + 2p

SG-FSGO BL (%) Angle (%) BL (%) Angle (%) BL (%) Angle (%) BL (%)

2.17b (3.0) 91.5b (11.0) 1.91a (0.0) 87.9a (18.0)

2.14b (2.0) 103.5b (0.01) 1.90a (0.0) 106.3a (0.0) — —

Dissociates unstable —

2.054 (2.2) 102.05 (0.34) 1.834 (4.1) 105.27 (1.34) 1.667 (7.9) 104.15 (0.33) —

2.135 (1.67) 102.3 (0.14) 1.872 (2.2) 107.9 (1.11) 1.724 (4.65) 104.22 (0.27) —

2.101 (0.05) 100.23 (2.11) 1.909 (0.21) 96.47 (9.59) 1.808 (0.0) 95.82 (8.31) 2.681 (0.4)

2.124 (1.14) 102.26 (0.14) 1.893 (1.04) 106.74 (0.04) 1.723 (4.70) 104.31 (0.18) —

Tan and Linnett,22 b Tan and Linnett.23 The rest of results are references.24,25b-(Angles in deg. and bond length, BL, in au). a


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Table 3 The results of bond length and bond angle for ethylene by different methods (au) Method

C–C BL

Error%

C–H BL

Error%

HCC BA

Error%

Banana model SG-FSGO7 Banana model DG-FSGOa s–p model FSGO26 s–p model FSGO13 MP2/6-31Ga13 QCISD/6-31Ga13 Experiment

2.554 2.500 2.555 2.547 2.524 2.527 2.527

1.31 0.83 1.35 1.03 0.12 0.24 —

2.081 1.945 2.072 2.028 2.051 2.056 2.053

1.86 4.8 1.42 0.73 0.39 0.64 —

120.7 120.6 120.5 121.8 121.7 121.8 121.3

0.49 0.58 0.66 0.43 0.33 0.45 —

a DG-FSGO stands for double Gaussian FSGO calculation. BL is for bond length, and BA is for bond angle.

and rectangular geometry.13 The total energy for localized description for square and rectangular geometries are 151.81688 au and 151.84776 au, respectively, while for delocalized form it is 151.8524 au. 2-J Force constant The first calculation of force constants in the FSGO method was done by Easterfield and Linnett.27 They studied BH4, CH4 and NH+4, and their results are shown in Table 4. Harmonic force fields for several polyatomic hydrides of 1st-row atoms were also calculated by Semkow28 with the FSGO method. Some of their results are in Table 5. Pakiari and Noorizadeh29 have suggested an ab initio determination of a rather accurate force constant for given FSGO wavefunctions for LiH, BH, H2 and H2O. In this procedure, an nth degree polynomial must be fitted to different calculated potential points from ab initio wave function. Brief results are collected in Table 6. Solimannejad and Pakiari developed ab initio analytical force constant for FSGO, and applied their theory to H2, LiH and BH; the results are 5.936 (3.21%), 1.208 (17.73%) and 3.168 (4.48%) mdyn A˚1, respectively.30 Ab initio analytic cubic force constants have also been formulated by Solimannejad and Pakiari31 in FSGO and applied to H2O and CH2. 2-K

Orbital energy

Orbital energy is well-known in SCF procedure, which is ei in FCi = eiSCi. Pakiari and Solimannejad have obtained an expression for e in FSGO method.32 One must consider multi-Gaussian (section 2-A) and open shell FSGO (section 2-C). Table 4 The experimental and calculated force constant by FSGO27 BH4 Force constant F11/mdyn F22/mdyn F33/mdyn F44/mdyn F34/mdyn

A˚ A˚1 A˚1 A˚1 A˚1 1

NH3+

CH4

Calc.

Expt.

Calc.

Expt.

Calc.

Expt.

3.80 0.36 2.58 0.44 0.27

3.05 0.29 2.71 0.29 0.15

8.15 0.71 8.67 0.66 0.38

5.84 0.49 5.38 0.46 0.21

12.45 1.21 16.66 0.80 0.10

5.49 0.60 5.13 0.45 0.06


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Table 5 The experimental and calculated force constant by FSGO27 BeH2

BH3

CH4

H2O

Force constant

Calc.

HF

Calc.

Expt.

—

Calc.

Expt. R

Calc.

R (au) a Frr Frr 0 Fa Fg Fr

2.52 — — 0.037 0.171 — 3.24

2.55 — — 0.04 0.13 — 2.23

2.344 — — 0.018 0.250 0.356 4.369

2.19 — — — 0.087 0.10 3.38

R — F11 F33 F22 F34 F44

2.133

2.065 — 5.842 5.383 0.638 0.270 0.458

1.802 1.809 105.36 104.52 — — 1.493 0.100 1.185 0.762 0.105 0.245 14.387 8.456

6.133 6.784 — — —

— — — — Fra —

Expt.

Table 6 Calculated force constant29 (unite is mdyne/A˚) LiH

Method Best in literature FSGO G98w Expt.

Best in literature FSGO G98w Expt.

eai

BH

H2

ke

Error%

ke

Error%

ke

Error%

1.066 — 1.061 1.061 1.026 H2O krr 8.484 7.983 9.762 8.454

3.89 — 3.41 3.41 —

2.680 — 3.344 3.358 3.032

11.6 — 4.12 10.31 —

5.907 6.240 6.189 6.283 5.751

2.20 7.97 7.52 8.71 —

Error% 0.22 5.56 15.52 —

Kyy 0.762 1.304 2.783 0.761

Error% 0.13 70.13 267.0 —

X

"

X

ci cj ðia jja ÞWaIJ " # XX XX a a a a þ NI NJ NK NL ci cj ck cl ði j jk l Þ

¼ NI

NJ

ij

J

J

K;L

IJ

ðWaIJ WaKL WaIL WaJK Þ " # XX XX a a b b NJ NK NL ci cj ck cl ði j jk l Þ WaIJ WbKL þ NI J

K;L

IJ

ð2-K-1Þ

KL

ð2-K-2Þ

KL

We will now present some of the results in Table 7. 2-L Pseudopotential on FSGO The first attempts for using pseudopotential concepts within the FSGO model have been made by Barthelat and Durand33,34,35 between 1972 and 1976; they took X WR;l ðrÞPl ð2-L-1Þ WR ¼ l

where Pl is the projection operator over the lth sub-space of spherical harmonic. The pseudopotential is a semi-local form, and functions WR,l(r) = Bl/r2

(2-L-2)


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Table 7 Ionization potential by FSGO and SCF in comparison with experimental values. (unit is electron volt) H

He

Li

Be

B

C

N

O

F

Ne

H2

Expt. 13.6 24.6 5.4 9.3 8.3 11.3 14.5 13.6 17.4 21.6 15.4 FSGO 13.6 24.9 5.6 8.4 8.2 11.5 14.6 13.6 17.9 22.5 16.19 Error% 0.0 1.25 3.63 9.7 1.3 1.88 0.29 0.16 2.51 4.54 4.78 SCF 13.6 24.9 5.3 8.4 8.6 11.9 15.5 14.2 18.3 22.9 16.2 Error% 0.1 1.40 1.1 9.7 3.6 5.6 6.26 3.39 5.30 6.12 1.0

N2

C2H4

15.6 10.5 17.0 10.7 1.99 1.9 17.0 10.4 9.0 0.5

Linnett et al.36 used the pseudopotential equation as, HMwn = Enwn

(2-L-3)

where 1 HM ¼ r2 þ VM and VM ¼ VHP þ VP þ VR 2

ð2-L-4Þ

VR is the pseudopotential, VP is polarization potential and VM is modified by VPP. X Vpp ¼ Z=r þ Bl Pl =r2 ð2-L-5Þ l

The results are in Table 8. Frost et al.37 have tested the multi-Gaussian Goddard form of pseudopotential formula, nv X X 2 Vðri Þ ¼ þ Bl;j rnl;j eai;j ri Pl ð2-L-6Þ ri j l and the Coffey form of the pseudopotential, nv nc ari X þ e þ ðen es ÞPc Vðri Þ ¼ ri ri i

ð2-L-7Þ

and Bonitacic and Huzinaga’s form of pseudopotential, X nv X nv 2 þ Aj eaj ri þ Bc Pc Vðri Þ ¼ ri r i c j

ð2-L-8Þ

where nc is the number of core electrons, en and ec are the one-electron valence and core level energies, and Pc is, Pc(ryj) = |Rc(r)Ylm(yjihRc(r)Ylm(yj)|

(2-L-9)

Table 8 Some of the results of reference36

Total energy X–H Bond length

H–X–H Bond angle

PS FSGO PS PSGO Expt. PS FSGO Expt.

LiH

CH4

NH3

H2O

HF

6.502 6.572 3.326 3.226 3.015 — — —

33.587 33.992 2.461 2.107 2.065 — — —

50.908 47.568 2.975 1.910 1.912 120.0 87.6 106.7

67.786 64.288 2.999 1.666 1.809 180.0 88.4 104.5

86.837 84.571 3.090 1.482 1.733 — — —

Energy and bond length are in atomic unit, angle in degree.


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Table 9 The results obtained by Frost38

C–H Bond length

C–C Bond length

H–C–H Bond angle

PS PSGO Expt. PS FSGO Expt. PS FSGO Expt.

CH4

C2H6 (Stag.)

C2H4

C2H2

2.113 2.107 2.065 — — — — — —

2.119 2.117 2.065 2.843 2.836 2.899 107.97 108.2 109.1

2.075 2.081 2.052 2.556 2.554 2.527 118.8 118.7 117.3

2.032 2.039 2.001 2.289 2.296 2.277 — — —

C3H6 2.114 2.108 — 2.888 2.897 2.854 113.8 114.4 —

The results of Frost et al. are compared with the results of some different pseudopotential for CH4 in this article. Their conclusion is that the ordinary pseudopotential cannot give satisfactory results. Their recommendation is that it must develop a specific pseudopotential for FSGO. Frost et al.38 proposed a new pseudopotential and used it for hydrocarbons such as C2H6, C2H4, C2H2 and C3H6 (cyclopropane). The proposed model potential is, Vmodel = Vp(r) + Vsp(r)Pl=0

(2-L-10)

where Pl is the angular momentum projection operator, nc ar2 ðe2s e1s Þ br2 e and Vsp ¼ e rn r

Vp ¼

ð2-L-11Þ

nc is the number of core electrons, e1s and e2s are orbital energies and finally a, b and n are parameters. The results are presented in Table 9. Frost et al. compare FSGO, Hartree-Fock-Roothaan and pseudopotential calculations for the lithium dimmer.39 The HF procedure consists of a pseudopotential Fock operator, NX v þNc

0 FPS i ¼ h ðiÞ þ

ð2Jij Kij Þ þ Vp ðiÞ

ð2-L-12Þ

j6¼i¼Nc þ1

h 0 (i) = h(i) + Nc/|ri| where Nv and Nc are valence and core electrons, respectively, and, X VL ðrÞjLMihLMj Vp ðrÞ ¼

(2-L-13)

ð2-L-14Þ

L;M

VL ðrÞ ¼

X

Aa;b ðLÞra exp½aab ðLÞr2

ð2-L-15Þ

a;b

The function VL(r) is evaluated from atomic ground state in such a way that FPF and F have identical eigenvalues for that reference atom. Ffi = eifi; FPSfi = eifi

(2-L-16)

while for FSGO, we have, VPFSGO ¼

X

VFSGO;L ðrÞjLMihLMj

ð2-L-17Þ

L;M


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Table 10 Some results of reference40

X–H Bond length

D3d

Si2H6

Ge2H6

SiH3GeH3 (s)

PS

2.817 — 2.797 — 4.373 4.384 107.2 — 109.8 —

2.923 — 2.886 — 4.516 4.554 108.3 — 109.8 —

2.817 2.923 2.806 2.889 4.342 4.454 109.1 108.0 108.2 109.3

Expt. X–X BL H–X–H Bond angle

PS Expt. PS Expt.

X can be, Si or Ge and s stands for staggered.

where ( VFSGO;L ¼

2

2r e3:13r 2 2 e1:7r 2r e3:13r þ 2:2653 r2

L1 L¼0

ð2-L-18Þ

The FSGO pseudopotential model has also been used by Topiol et al. for Group IV hydrides,40 for geometrical properties of molecules such as X2H6, CH3–XH3, (X = C, Si and Ge) and also for GeH3–GeH3–GeH3 and SiH3–GeH3. The core potential is, VPS ¼ VLmax ðrÞ

Lmax X1

VlLmax ðrÞ

jYlm ihYlm j

ð2-L-19Þ

ml

l¼0

VLmax ðrÞ; VlLmax ðrÞ ¼

X

X

Ai rni eai r

2

ð2-L-20Þ

i

where Ylm is the usual spherical harmonic. Some of their results are in Table 10. Ab initio studies, using the FSGO-pseudopotential method, of the electronic structure in Group IV cluster molecules M(LH3)4; M, L = C, Si, Ge.41 The pseudopotential used is, Vpp ¼

max L max X X LX

a

jLMaihLMaj½VL ðra Þ VLmax þ1 ðra Þ þ VLmax þ1 ðra Þ

ð2-L-21Þ

L¼0 M¼L

N. K. Ray et al.42 have published some articles on pseudopotentials. They have used the Schwartz and Switaliski model of core potential, V¼

Zc expðgr2 Þ þ ½As or Ap r r

ð2-L-22Þ

where Zc is the nuclear charge minus the number of core electrons. As and Ap are the potential parameters for s and p state of electrons of atom under consideration. They used uncontracted basis set 11s or 7p. They obtained their A and g values when the experimental ionization potential had been reached for that species. Some of the results are in Table 11. The s-type pseudopotential had been used for compounds with two electrons in a valence shell43 Vm = Zr + As exp[gr2]/r and some of their results are given in Table 12. 292 | Chem. Modell., 2008, 5, 279–311 This journal is

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(2-L-23)

Table 11 Ray’s results of pseudopotential42

Total energy

PS PSGO PS FSGO Expt. PS FSGO Expt.

Bond length

Bond angle

LiH

CH4

NH3

H2O

HF

6.536 6.573 3.63 3.23 3.02 — — —

33.990 33.992 2.13 2.11 2.10 — — —

48.628 47.568 1.84 1.91 1.91 83.9 87.6 106.6

65.683 64.288 1.63 1.67 1.81 85.3 88.4 104.5

86.938 84.571 1.46 1.48 1.73 — — —

Energy and bond length are in atomic units, and angles in degrees.

Table 12 Ray’s results of pseudopotential43 Species

LiH NaH KH LiK Li2 K2

Valence energy

0.5906 0.5755 0.5380 0.3409 0.3816 0.3020

Bond length, (au)

Force constant (mdynes/A˚)

PS

Expt.

PS

Expt.

3.63 3.81 4.59 6.18 5.65 6.98

3.06 3.57 4.24 6.22 5.35 7.41

0.82 0.69 0.42 0.15 0.23 0.11

1.03 0.79 0.56 0.15 0.26 0.11

Energy and bond length are in atomic units.

Ray et al.44,45 changed their pseudopotential to the below form, V¼

Zc expðg r2 Þ þ Aav r r

ð2-L-24Þ

and they used an effective Aav, which is a function of both As and Ap, Aav = k(NsAs + NpAp)/(Ns + Np)

(2-L-25)

They used this pseudopotential for evaluation of total energy. Some of the results are in Tables 13 and 14. They have applied eqns (2-L-24) and (2-L-25) to CH4, SiH4 and GeH4 for bond length and average polarizabilities,46 and the results are in Table 15. They have further used the same pseudopotential (Aav = As) in eqn (2-L-24) for open shell systems, and some of their results47 in Table 16. Mehandru and Ray48 Table 13 Ray’s results of pseudopotential44

Total energy au Bond length au

Bond angle degree

PS PSGO PS FSGO Expt. PS FSGO Expt.

NaH

PH3

H2S

HCl

138.208 138.199 3.33 3.76 3.57 — — —

295.490 394.778 2.71 2.67 2.68 106.2 90.9 93.2

344.646 343.866 2.44 2.46 2.52 108.9 87.7 92.2

398.374 347.569 2.28 2.27 2.41 — — —

Energy and bond length are in atomic units, and angles in degrees.


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Table 14 Ray’s results of pseudopotential45 Species LiH LiCl NaF NaCl

Force constant (mdynes/A˚)

Bond length (au) PS

FSGO

Expt.

PS

Expt.

3.95 4.40 4.25 4.50

2.72 4.23 3.48 4.42

2.89 3.72 3.72 4.46

2.37 1.81 1.46 1.87

1.92 1.50 1.31 1.19

Table 15 The results of polarizability46

Bond length (au)

PS FSGO Expt. PS FSGO

Average polarizability

CH4

SiH4

GeH4

2.14 2.11 2.10 29.6 24.4

2.76 2.80 2.76 58.5 44.0

2.82 — 2.89 58.7 —

Polarizablities in 1025 Cm3.

Table 16 Results of open shell calculation47 Species

+

LiH NaH+ Li2+ Na2+ LiNa+

Valence energy in (au)

0.4261 0.4255 0.2277 0.2168 0.2224

Bond length (au)

Dissoication energy (De)

PS

Expt.

PS

Expt.

5.10 5.84 6.34 6.70 6.52

4.5 5.4 5.8 6.7 6.5

0.05 0.03 0.86 0.79 0.72

0.09 0.03 1.22 0.97 0.92

have further used the following model potential, V0m ¼

2 Zm m expðgm r Þ þ ½Am s or Ap r r

ð2-L-26Þ

where Zm (effective nuclear charge) is the nuclear charge minus the number of core m electrons. Am s and Ap are the same as before belong to atom m. They have done calculations for C2H6, Si2H6 and Ge2H6 geometries and internal rotation barriers. They also used the model potential in eqn (2-L-22) for calculation of equilibrium geometries of some homonuclear diatomics,49 such as Li2, Be2, B2, C2, N2, O2, F2, P2 and Cl2. Molecular anisotropy in some diatoms such as H2, Li2, C2, N2, O2 (singlet), F2, HF and CO have been evaluated with same model potential.50 A nonlocal pseudopotential in the FSGO model: Vc ðiÞ ¼

Zc þ ðkc =r2c Þjcihcj rci

ð2-L-27Þ

Where Zc is the core charge, kc and rc are pseudopotential parameters, and study of some organometallic systems,51 such as LiCH3, HBeCH3, CH3–Be–CH3, BeCH3 and LiCCH. Gaspar and Gaspar52,53 have used the model of core potential, Vmod ðrÞ ¼

X Z a Na

C

a

ra

þ

lc þ1 XX a

Vl ðra ÞPal

l¼0


c


ð2-L-28Þ

Table 17 Gaspar’s results of first paper53 Geometry

BeH2 BH3 CH4 C2H6

Be–H B–H C–H C–H C–C H–C–H

Valence energy

Calculated

Exp.

Calculated

Exp.

2.5055 2.2399 2.0596 2.0624 2.7386 107.371

2.54 2.2431 2.0617 2.0825 2.9026 109.31

2.03260 4.16068 7.12590 13.25928

2.19390 4.56349 8.10769 15.03376

Energy and bond length are in atomic units.

where Za is number of the nuclear at the position Ra, NaC is the number of electrons in the core a, Pal are the angular momentum projection operator with the center of Ra, ra = |r Ra|. The analytical form of the radial part is, X Vl ðrÞ ¼ Ail expðail r2 Þ ð2-L-29Þ i

We have resented the results in Table 17. They have published three articles on pseudopotential fragments: 1- Ab initio molecular fragment calculations with pseudopotentials: hydrocarbon calculations of double-zeta quality.54 2- Ab initio molecular fragment calculations with pseudopotentials.55 3- Ab initio molecular fragment calculations with pseudopotentials. Some fragments containing nitrogen and oxygen.56 They have used pseudopotentialHamiltonian, ! Nn Nn n X X X 1 1 ps ps H ¼ DðiÞ þ Vk ðiÞ þ ð2-L-27Þ 2 r i4j ij i¼1 k¼1 Vps k is the pseudopotential associated with core of atom k, Vps k ¼

Z X Ai expðai r2 ÞPl þ r i

ð2-L-28Þ

where Z is the net charge core containing electrons and nucleus of atom Ai and ai. The results have been presented in Table 18. They applied the same pseudopotential as before in the paper with title ‘‘ab initio molecular calculations with pseudopotentials: calculations of double-zeta quality on ethylene, acetylene, and water’’, but a linear combination of FSGO has been used in the following articles.57 A description of C–C bond for double and triple bond has been given in the pseudopotential-FSGO method. They have used model of lone pairs for water. It would be interesting to check the result CC bond length in ethylene and acetylene, and O–H bond length and H–O–H angle for water. The results have been collected in Table 19. They have additional two publications, which have the following titles ‘‘FSGO study of the Gauss-type pseudopotential for the Lithium and Lithium(1-) and its isoelectronic series,58 ab initio molecular fragment calculations with pseudopotentials, calculation on Li2, LiH and BeH2,59 and ‘‘Ab initio molecular fragment calculations with pseudopotentials: model peptide studies’’.60 Bakhshi has published two articles on this subject with titles: 1- Average electric polarizabilities and magnetic susceptibilities of hydrocarbons, alcohols, ethers, aldehydes, ketones, amines and nitriles.61 2- Double-zeta pseudopotential study of methyllithium, methylberyllium and dimethylberyllium.62 Chem. Modell., 2008, 5, 279–311 | 295 This journal is

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Table 18 Results of pseudopotential fragment methods. (au) Geometry

Total valence energy

Ref. 54 C2H6 C2H4 C3H8 Ref. 55 CH3OH CH3NH2 HONH2 N2H2 Ref. 56 HCOOH HCHO HCONH2

Expt.

Calc.

Dev.%

Expt.

Calc.

Dev.%

2.90 2.47 112.41

2.64 2.56 110.61

9.0 3.6

15.0338 13.8265 21.9361

13.2754 12.3504 19.4158

11.7 10.7 11.5

C–O C–N N–O N–N

2.68 2.80 2.76 2.78

2.73 2.78 2.75 2.79

1.83 0.7 0.25 0.52

24.1466 18.6456 27.7901 22.2730

21.1994 16.3843 24.3031 19.4787

12.2 12.1 12.6 12.5

CQO CQO CQO

2.35 2.29 2.36

2.30 2.34 2.32

2.07 2.48 1.67

39.0289 22.9797 33.5268

34.6840 20.6493 29.8568

11.13 10.14 10.95

C–C C–C C–C–C angle

Table 19 Results in reference.57 (unit is au)

C2H4

C2H2

H2O

CQC C–H HCH angle Total energy CRC CH Total energy O–H H–O–H Total energy

Expt.

Calc.

FSGO

2.47 2.02 110.01 76.830 2.28 2.00 78.639 1.81 104.41 76.444

2.02 2.34 106.81 76.830 2.34 1.92 76.369 1.70 103.91 74.204

2.55 2.08 118.71 65.835 2.30 2.04 64.678 1.67 88.41 64.288

There is another publication on this subject, ‘‘A simple nonlocal FSGO pseudopotential for first-row atoms’’.63

3- Part III, Application of FSGO method In addition to all above references, which are contained many applications of FSGO method, we are presenting some of other applications of the FSGO method in the following.

3-A

Atomic calculations:

The calculation with minimum Gaussian per orbital has been done by using different orbital different spin (DODS) and double quartet theory64 for first row of periodic table. The trend of ionization potential of this calculation is correct, but the error begins to increase when p-type orbital must be used for B to Ne. Suthers and Linnett65 repeated these calculations by using p-orbitals (made of two Gaussians in opposite sign positioned at different side of nuclear centre) with a multi-Gaussian description of orbitals. The percentage of errors has been reduced to 4% and 1% for double and triple Gaussian, respectively. The former calculations have been repeated using the DODS concept and double quartet theory with multi-Gaussian description per orbitals,66 and the same order of error as Suthers’s have been obtained. 296 | Chem. Modell., 2008, 5, 279–311 This journal is

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3-B Solid state: There are some articles on solid state FSGO which are as follows: 1-Gaussian orbital calculations of solids. Crystalline lithium hydride.67 2-Ab initio Gaussian orbital calculation of the (100) surface of crystalline lithium hydride.68 3-Quantum mechanical ab initio calculations of the surface structure of crystalline lithium hydride.69 4Absolute calculation of the adsorption of a hydrogen molecule on the (100) surface of crystalline lithium hydride.70 3-C

Spectroscopy:

3-C-A Chemical shift. Frost and Nelson71 have employed the FSGO model to predict the ESCA chemical shifts for CH4, C2H6, C2H4, C2H2, C3H8, C3H6, isoC4H10, and CMe4. They have obtained rather good results. 3-C-B Photoelectron spectroscopy. Cox et al.72 employed the FSGO method to evaluate orbital energy for photoelectron spectroscopy, their conclusion is that the method is unreliable for prediction of correct values. O’Donnell et al.73 carried out a calculation for the uracil ground electronic state by the ab initio FSGO fragment method (see section 4-A). They obtained properties such as ionization potentials, permanent dipole moment, the electronic charge distributions of the HOMO and LUMO, the bond populations and bond orders. They also studied the dipole binding energies of the three H bonding sites used for base pairing in RNA. They found a linear relationship between the calculated orbital energies and one-electron ionization potentials measured by UV photoelectron spectroscopy. 3-C-C Spectroscopic constants. Linnett and Semkow74 used the SG-FSGO method and its extension to contracted basis sets of Gaussians to describe how each electron pair can be used to calculate the rotational-vibrational constants, and bond lengths of MH (M = Li, B, N, and F). Their results are near the results corresponding to the HF limit for those properties associated with vibronic motion. 3-C-D Band structure and XPS. Andre et al.75 compared experimental XPS and theoretical calculations by HF and FSGO, their recommendation is that it is able to provide computationally cheap results by FSGO in comparison with more complete ab initio ones. 3-D

Cluster calculations:

A series of calculations have been reported by I. Tamassy-Lentei and J. Szaniszlo over the period 1973 until 2003, as follows; 1- Alkaline clusters: a- KnHm, KnLim and KnNam cluster,76 b- KnHm+, KnLim+ and KnHm+ cluster,77 c- NanHm anionic sodium–hydrogen clusters,78 dNanLim + cationic sodium–lithium alkali clusters,79 e- theoretical investigation of neutral, anionic and cationic states of Na microclusters,80 f- stabilities and structures of small NanHm+ cationic and NanHm anionic sodium–hydrogen clusters,81 g- anionic states of small Li clusters,82 h- several superalkali molecules,83 h- Ab initio theoretical study of the properties of the Li(H2)n alkaline anionic clusters (n = 1, 2, 3).84 2- Cation affinities: a- hydrogen-, lithium-, sodium- and potassium of the XY (X, Y = H, Li, Na, K) alkali dimmers.85 3- Stability: a- stability of small anionic lithium–hydrogen systems,86 b- the stability of beryllium oxide, hydroryberyllium(1+), and aquaberyllium(2+) (BeO, BeOH+, and BeOH+2) systems,87 c- Ab initio theoretical study of Chem. Modell., 2008, 5, 279–311 | 297 This journal is

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the structures and stabilities of the lithium beryllium hydride (LiBeH3, Li2BeH4 and LiBe2H5) molecules.88 4- Ab initio theoretical investigation of the properties of lithium compounds with hydrogen, oxygen, and nitrogen (Li2O, Li2OH+, Li3O+ and Li2N, Li2NH, Li2NH2+, Li3N, Li3NH+, Li4N+).89 5- Theoretical description of the hydrogen cyanide–hydrogen isocyanide and lithium cyanide–lithium isocyanide isomerization using the FSGO method.90 6- Theoretical determination of the dissociation energy for the noble gas hydride ions HeH+, NeH+ and ArH+ by the FSGO method.91 7- Polymerization of the beryllium hydride molecule. Ground state of the beryllium hydride (BeH2, Be2H4, and Be3H6) systems have been studies by the FSGO.92 8- Calculation of the proton affinity of several small molecules by the FSGO (Proton affinities were calculated of H, H2, He, Be, F, HF, O2, HO, H2O, N3, NH2, NH3, CO, CN, H2CO, S2, HS and H2S by using the ESGO method).93 9- Interaction of atoms and ions with two electrons: Ground state of the lithium ion-helium, lithium ion-lithium ion complexes at small inter-nuclear distances.94 10- Theoretical study of the lithium hydride dimmers using the FSGO method.95 11- FSGO study of the interaction potentials between the lithium ion and hydrogen, and the helium and hydrogen systems.96 12- Theoretical investigation of the Na2BeH4, Na2MgH4 complex hydrides and the Na2BH4+, Na2AlH4+ isoelectronic ions.97 3-D There is a series of calculations by S. A. Cruz and J. Soullard from 1991 to 2001, concerning: 1- Pressure effects on the electronic and structural properties of molecules.98 2- Calculation of molecular integrals for systems confined by hard spherical walls: use of the single-center expansion of floating spherical Gaussians.99 3- Bond stopping cross sections for protons incident on molecular targets within the OLPA/FSGO implementation of the kinetic theory.100 4- Chemical bond effects on the low-energy electronic stopping power of protons: use of molecular fragments.101 5- Chemical bond effects on the low energy electronic stopping power theory.102

3-E Polymers J. M. Andre et al. have several papers on polymer such as: 1- A Floating Spherical Gaussian Orbital (FSGO) model for polymers: calculation of X-ray diffraction structure factors.103 They have evaluated electron densities and related X-ray structure factors for polyethylene. 2- An extended formalism and computational procedure for saturated and conjugated systems.104 A general formula was employed to compute electronic properties of stereoregular polymers by using FSGO fragment approach (see section 4-A). 3- On the calculation of long-range coulombic contributions to the direct space LCAO CO matrix elements of model polymers.105 They applied the FSGO model for an infinite chain of LiH molecules. 4- An FSGO-CO study of the long-range effects on calculated conformational stability and on one-electron levels of polyethylene.106 Two conformations of polyethylene were considered. 5- A floating spherical Gaussian orbital model for polymers.107,108 6- J. L. Bredas published Ab initio SCF-FSGO study of the structure of an isolated poly (sulfur nitride), (SN)x, chain.109 D. R. Armstrong et al. have a paper on polymers ‘‘The electronic structure of polymers by FSGO’’.110 298 | Chem. Modell., 2008, 5, 279–311 This journal is

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3-F Electric field, polarizalbities and magnetic susceptibilities FSGO double Gaussian calculation has been employed to obtain electric field gradients near the deuteron for CH3D, H2CCHD, and HCCD. The obtained results are rather accurate.111 The FSGO method has been applied for transferability of bond, core and lone-pair and to obtain average electric polarizabilities and magnetic susceptibilities of some alcohols, ethers and amines. The calculated results agree with available experimental data.112 Andre et al.113 have carried out a calculation by FSGO Hartree Fock (RHF, UHF, PHF and EHF procedures) to examine the total energy, potential curves and the dissociation of H2 in external electric fields of different strength. The FSGO method is used by Gruendler114 for properties of H, He, He+, H2, H2+, HeH+ and H3+ in a strong uniform electric field to evaluate polarizabilities including their field strength dependence. Electric-field-variant (EFV) AOs basis with FSGO method have been used115 for calculation of molecular polarizabilities from electric-field-variant atomic orbitals, and applied to the hydrogen molecule and to the alkane series CnH2n+2, n = 1, 2, 3, 4, 5, and 6. Further information is given in the reference.61

4- Part IV, Using the FSGO concept in other methods 4-A

FSGO fragment

This method is actually an SCF procedure with FSGO basis set, and is designed for large molecule (containing r200 electrons, and r75 nuclei) calculations using transformability of bond functions with four variational parameters (exponent and Cartesian coordinates) in order to reduce the number of parameters to be optimized. This method is ab initio and was introduced in 1969 by Christoffersen and Maggiora.116 FSGO fragment was also reviewed by Christoffersen117 in 1972, and he has many references in this type of work in this review. The procedure is as follows, (1-) Perform FSGO calculation for suitable fragments, i.e. X and Y, and obtaining bond lengths and bond angles. (2-) Bring the fragments together with desired distance and angles. Any extra hydrogen must be removed from fragments X–H and Y–H with their electrons, to make an X–Y bond. (3-) Now, it is possible to get an optimum molecular description for the molecule by the SCF procedure, and each molecular orbital can be written as ci ¼

fragments NA X X A¼1

A cA ji Gj

ð4-A-1Þ

j¼1

Let us have an example for calculation C2H6 from CH4 fragments. We have the following basis set, A B B {GA 1 , . . . ,G5 ,G1 , . . . ,G5 }

ci ¼

NA AX and B X F¼1

cFji GFj

(4-A-2) ð4-A-3Þ

j¼1

Therefore, the Hartree Fock equation can be expressed as FC = SCE

(4-A-4)

where A B FAB rs ¼ hGr jhjGs i þ

Q X Q X NC X ND X C¼1 D¼1 t¼1

A B C D PCD tu fhGr Gs jGt Gu i

u

1 GB jGC GD ig hGA 2 r u t s

ð4-A-5Þ


c


Table 20 The comparison of results of Bacsky and Linnett with other calculations. (au)118,120 He

H2

Bacsky, Linnett Scwartz Roothaan —

2.87852 2.87903 2.86168 —

H3

Bacsky, Linnett 1.1699 Bacsky, Linnett Exact 1.17447 STO with 176 Configurations. SCF 1.13363 CI — — Correlated wavefunction

1.65170 1.65207 1.64993 1.6621

where PCD tu is charge and bond order matrix; PCD tu ¼ 2

occ X

D cC tj cuj

ð4-A-6Þ

j¼1

4-B Natural orbitals In 1972, Bacsky and Linnett published a series of paper which they named ‘Mixed Basis Functions in Molecular Quantum Mechanics’.118,119,120 They used a CI-type calculation as follows: a- Evaluation of the basis integrals, which are the overlap, kinetic energy, electron–nuclear attraction energy and electron–electron repulsion energy in a mixed set of 1s Slater and Gaussian functions b- Construction of orthogonal symmetry orbitals. c- Transformation of above evaluated integrals in terms of atomic or bond function into molecular integrals. d- The CI wave function is constructed by an orthonormal set of configurations, and obtaining the lowest eigenvalue by Nesbet’s procedure. e- The set natural orbital is constructed. Typical results are collected in Table 20. 4-C

Monte carlo FSGO

The Monte Carlo method, which is based on statistical concepts, has been used for a long time in quantum chemistry. It is based on random sampling and averaging instead of integration. This subject was reviewed by W. A. Lester, Jr and R. N. Barnett in 1990. Although Anderson has suggested that Quantum Monte Carlo QMC began in 1975, there is some evidences of Monte Carlo in quantum chemistry long time before this time. For example, you will find some 1934 references to the work of D. H. Weinstein and others in Allen’s paper.121 As far as we know a number of different techniques have been used such as QMC, Variation QMC (VMC), Diffusion QMC (DQMC or DMC), Green’s function QMC (GQMC) and least squares QMC (LSQMC). FSGO has been employed by two techniques of Monte Carlo; DQMC and LSQMC. In 1972 T. L. Allen used Monte Carlo for FSGO method by least squares solution of the Schro¨dinger equation for many electron atoms and molecules. The least squares solution of the Schro¨dinger equation was introduced by D. H. Weinstein in 1934, and developed by others. Let us define the local energy, ei, for our system of interest as Hji = eifi ) ei = Hji/ji (ji a 0)

(4-C-1)

Local energies at points xi (I = 1, 2, . . . , N) are calculated. Each ei is weighted by j2i and suitable volume elements Dni. The average energy will approach the expectation value of the energy as N increases. 300 | Chem. Modell., 2008, 5, 279–311 This journal is

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N P

hwe i ¼ i¼1N P

i¼1

N P

R ji Hji Dn i jHjdn ¼ i¼1N ) R 2 P j dn j2i Dn i j2i Dn i

j2i Dn i ei

ð4-C-2Þ

i¼1

The selected points have to be obtained by careful sampling using a modified Monte Carlo technique. The FSGO function is chosen for the electron density, because it can be optimized to get the best result. The Gaussian distribution wq with standard deviation s is taken, wq = (1/2ps2)1/2 exp[q2/2s2]

(4-C-3)

in such a way that for two electrons, we can write, g = wx1wy2, . . . wz2 = (1/2ps2) exp [(r21 r21)/2s2]

(4-C-4)

where s = r/2, and r is orbital radius. The result of this calculation for electronic energy of Helium with six parameters is 2.9062 au, while the results obtained by very accurate procedure by E. A. Hylleraas122 is 2.90324 au. Another Monte Carlo approach for molecular electronic structure through the SCF procedure has been developed since 1986 by D. M. Ceperley and B. Alder.123 The diffusion quantum Monte Carlo (DQMC) technique in one of the different concepts is used to transform the Schro¨dinger equation into imaginary time to obtain a reaction-diffusion equation, and then solve it by random sampling. There are two fundamental problems in DQMC; time step-size and node problems. Finite time step-size of propagating solution introduces errors in treating the continuous time evolution of the Schro¨dinger equation. The nodal surface of the unknown exact wave function makes good sampling difficult. Shyn-Yi Leu and Chung-Yuan Mou124 in 1994 developed a new algorithm for DQMC using FSGO orbitals jG = Naeax

2

(4-C-5)

instead of Slater-type orbitals, as guiding functions for molecular systems to determine the electronic energies of the ground states of atomic and molecular system within the Born Oppenheimer approximation. This type of uG in DQMC procedure prevents also the divergence of the Coulombic potential. Since FSGO has a good flexibility for both the positions and exponents of orbitals, it can be a better guiding function, and additionally reduce some unnecessary nodes in the guiding wave function. Using FSGO in DQMC helps to evaluate gradient easier, therefore, make easier to calculate derivatives of QMC energy. Let us go very briefly to the mathematics of DQMC-FSGO and then to the results. Let us have a transformed imaginary time operator, which determines the evolution of distribution of psips, ˜ T˜ = jGet(HE)jG1 = et(HE)

(4-C-6)

where H˜ is the transformed operator, H˜ jGHjG1=12 p2 + ipG + EL(x)

(4-C-7)

where H is original Hamiltonian of the desired system, t is time step in simulation, and the local energy is EL = jGHjG1

(4-C-8)

The drift velocity field is generated by, G = r ln|jG|

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Table 21 Ground state of H2, LiH and HF molecules by DQMC/FSGO. (au)

Experiment Hartree-Fock limit STO-3G FSGO-EN FSGO (1G) FSGO (2G) Reynolds Subramaniam

H2

LiH

HF

1.17447 1.1336 1.1745 — 1.1745 1.1742 1.1745 —

8.0702 7.9873 8.0593 8.06904 — — 8.059 8.0699

100.4534 100.0703 — 100.450 — — — 100.433

In DQMC the corresponding imaginary time Schro¨dinger eqn (4-C-6) appears for the branching process as, @f 1 2 ¼ r f rðGfÞ ðEL ET Þf; f ¼ jG co @t 2

ð4-C-10Þ

where co is the exact wave function, and ET is the reference energy adjusted in the simulation process. f represents the distribution of psip in a random walk simulation. (EL ET)f in eqn (4-C-10) is called reaction term and can be simulated by the random process of the birth and death. The first two terms in eqn (4-C-10) give the effect of the reference Fokker-Planck operator in the Brownian motion theory, r (12 r G),

(4-C-11)

and drift velocity eqn (4-C-9). For solution of eqn (4-C-10), one must split the operator approximation to eqn (4-C-6) into reaction term and random walk parts, 2

T˜L = et(p +ipG)et(ELE) For further splitting of diffusion operator, we have, 2

T˜L = et(p +ipG) = etKetD

(4-C-12)

where K 12p2, D ipG In the computational procedure, the Ornstein-Uhlenbeck process is applied to onedimensional random walks under a harmonic velocity field. The velocity field obtained by the Gaussian function eqn (4-C-5) is, Gx ¼

d ðax2 Þ ¼ 2ax dx

ð4-C-14Þ

and then Fokker-Planck equation is, @f 1 @ 2 @ f þ 2a ðf xÞ ¼ @t 2 @ 2 x @x

ð4-C-15Þ

Then, the iterative procedure given by a Markov chain will be applied, xnþ1 ¼ axn þ gn s; a ¼ e2at ; s2 ¼

1 a2 4a

ð4-C-16Þ

Let t denote the step size of time. xn represents the position of the nth generation as xo is the initial position, and gn is a Gaussian random number with variance one and average zero (Table 21). 302 | Chem. Modell., 2008, 5, 279–311 This journal is

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Table 22 Calculated Compton profile and Radial momentum Densities RMD (au)130 NaCl CP Q = 0.0 Q = 1.4 Q = 5.0 RMD P = 0.0 P = 1.0 P = 5.0

LiF

Free ion model 7.314 3.152 0.444 Free ion model 0.0 7.706 2.072

Calc. 7.308 3.155 0.444 Calc. 0.0 7.729 2.072

Expt. 8.130 2.979 0.453 — — — —

Free ion model 3.462 1.908 0.102 Free ion model 0.0 3.308 0.264

Calc. 3.450 1.914 0.102 Calc. 0.0 3.266 0.264

Expt. 3.848 1.951 0.113 — — — —

The same procedure can be applied in 3N dimensional space. Let us consider some results by Shyn-Yi Leu and Chung-Yuan Mou. In the literature, Lu, Shih-I have used DQMC/FSGO and published some articles between 2001 and 2004 with following titles: 1- A diffusion quantum Monte Carlo method based on floating spherical Gaussian orbitals and Gaussian geminals: Dipole moment of lithium hydride molecule.125 2- Ornstein-Uhlenbeck diffusion quantum Monte Carlo calculations on BH and HF with the floating spherical Gaussian orbitals and spherical Gaussian geminals.126 3- The accuracy of diffusion quantum Monte Carlo simulations in the determination of molecular equilibrium structures.127 4- Electron affinities with diffusion quantum Monte Carlo for C2 and BO molecules.128 4-D

Compton profile

Ray et al.129 first applied FSGO in Compton profile calculations. S. Bhargava and N. K. Ray130 used FSGO to calculate Compton Profiles (CP) for ionic system such as LiF, LiCl, NaF and NaCl. The method had been proposed by Epstein and Lipscomb. Radial momentum density for an s-type Gaussian I(p) is given by the relationship, R I(p) = |x(p)|2p2 sin ypdypdfpdp (4-D-1) where |x(p)|2 is the momentum density. Eqn (4-D-1) can also be written, XX IðpÞ ¼ 4p 2Tjk fj ðpÞfk ðpÞjo ðpRjk Þp2 j

ð4-D-2Þ

k

where Rjk is the distance between jth and kth Gaussians, jo(pRjk)=sin (pRjk)/pRjk, and Tjk=(S1)jk fj(p) (i.e. the Fourier transform) is also Gaussians and given by, fj(p) = (r2/2p)3/4 exp (p2r2j /4) Then, the CP is, JðqÞ ¼

1 2

Z1

IðpÞ dp p

ð4-D-3Þ

jqj

where q is the projection of the electron’s initial momentum upon the scattering vector. The results for NaCl are in Table 22. There are some other publications of this type in literature.51,62,131 Chem. Modell., 2008, 5, 279–311 | 303 This journal is

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4-E FSGO in perturbation theory Well-known procedures for the calculation of electron correlation energy involve using virtual Hartree-Fock orbitals to construct corresponding wavefunctions, since such methods computationally have a good convergence in many-body perturbation theory (MBPT). Although we know the virtual orbitals are not optimized in the SCF procedure. Alternatively, it is possible to transform the virtual orbitals to a number of functions. There are some techniques to do such transformation to natural orbitals, Brueckner orbitals and also the Davidson method. In 1984, Adamowicz and Bartlett132 used a linear combination simple spherical Gaussians (FSGOs) for constructing correlation orbitals. They performed SCF calculation with extended basis sets of M functions, which give resulting N-occupied (index a, b, . . . ) and M–N virtual orbitals ( index i, j, . . . ). Then, the second order perturbation energy can be expressed as, Eð2Þ ¼

XX i4j

jhabjjijij2 e þ ej ea eb a4b i

ð4-E-1Þ

The task is to optimize the local and non-linear parameters for another basis set, {f 0 }, which consists of M’ FSGOs to produce as accurately as possible the pair functions of Gaussian and second order energy of (1), where M 0 r M N. To start the procedure we have to choose arbitrary FSGOs and construct, f0 = Q|f 0 i, where Q = |aiha|. W define a new set of M 0 orthogonal excited orbitals a 0 via |a 0 i = |f0 i, where T is transformation matrix. In order to have optimized virtual orbitals, we write the Fock matrix F = Fo + Fv, and we have the following expression to Fv, which is similar to the well-known Roothaan equation, F 0 v C 0 v = D 0 v C 0 ve 0 v 0

0

(4-E-2) 0

Here F v = SCvev C*vS*v, D v = SCv C*vS, S = hf |fi and T = C*vS*v. Once we have T for a given choice of {f 0 p}, optimization of the variational parameters of FSGO for the {f 0 p} is carried out by repeating evaluation of E 0 (2). The results for LiH with 36 Gaussians for SCF and second order perturbation are 7.986415 and 0051030 au, respectively. In 1997, Pakiari and Mohammadi133 used the FSGO basis set for a perturbation variation Rayleigh Ritz (PV = RR) calculation. We used a matrix representation Schro¨dinger equation for the configuration interaction calculation, HC = SCE (4-E-3) 1 1 P P where Sij = hDi|Dji, Hij = hDi|H|Dji, C ¼ Cj lj , E ¼ Ej lj and H = H0 + lH1. j¼0

j¼0

The S-matrix is diagonal, and is a unit matrix for configuration interaction in application of quantum chemistry, therefore, it is called So from now on. Then, eqn (4-E-3) can be expanded with respect to power of l as follows, (HC)n = So (CE)n n = 0, 1, 2, . . . ,n

(4-E-4)

After evaluation of the zero order energy HoCo = SoCoEo and the first order energy w pp Epp then, the energy of the nth, (2n)th and (2n+1)th can be evaluated 1 = [Co H1Co] as diagonal elements of following matrices: y y Epp n ¼ Co H1 Cn1 Co So

n1 X

Cj Enj for n 2

ð4-E-5Þ

j¼1

" Epp 2n

¼

½Cyn1 ðH1

pp

So E1 ÞCn

n X n1 X

#pp Cyj So Ei C2nij

i¼2 j¼ni


c


ð4-E-6Þ

" Epp 2nþ1

¼

½Cyn ðH1

pp

So E1 ÞCn

n n1 X X

#pp Cyj So Ei C2nijþ1

ð4-E-7Þ

i¼2 j¼niþ1

In this method, we have selected H2 with FSGO basis set to evaluate total energy for only few doubly excited configurations, and obtained E = 1.14307404 au. In 2005, Pakiari and Mohajeri134 have constructed a Fock matrix with second and third order Moller-Plesset perturbation theory, and employing the FSGO basis set. The theory is called Møller–Plesset Goldstone Perturbational Hartree Fock or briefly MPGHF is presented in this paper. The MPGHF method is a procedure in which different orders of perturbation energies are taken into account for obtaining formulae involving the Fock operator, and replace the ordinary procedure of SCF for obtaining the total energy. This type of energy contains appropriate correlation contribution according to the Møller–Plesset perturbation theory and using the Goldstone diagram approach. Since this procedure is variational, therefore it will correct MP approaches that lack variance. This approach will also ease the theory of configuration interaction (CI) by looking at the problem as a single particle approach. A typical Fock operator with second order perturbation F(S) mn is in the following; HF FðSÞ mn ¼ Fmn þ

N=2 X vit X

D1 abrs

X

rs

ab

ðPoccvit Þ2 ðmnjslÞ½2ðmnjlsÞ ðmsjlnÞ ls

ð4-E-8Þ

ls

There are twelve third-order perturbation diagrams, consequently twelve Fock operators containing third-order correction, F(T) mn , in the Goldstone diagram, we write only the first one F(T1) here. mn ðT1Þ ¼ FHF Fmn mn þ

N=2 X vit X ab

D1 abrs

X

rs

ðPoccvit Þ2 ðmnjslÞ½2ðmnjlsÞ ðmsjlnÞ ls

ð4-E-9Þ

ls

A test of theory and programming has been carried out for some atoms and molecules in this research. 4-F FSGO in density functional theory (FSGO-DFT) Like HF method, there is no contribution of correlation terms in FSGO. A research has been done by Pakiari and Mohajeri135 to introduce density functional theory (DFT), which has been designed for FSGO. Their principal objective was to apply a combination of energy functionals to the FSGO densities. The functionals used are separated into exchange and correlation parts. For the exchange part the the Becke exchange that includes gradient correction was used while for the correlation part the Lee, Yang and Parr gradient-corrected functional has been applied. In this procedure first a traditional FSGO calculation is carried out for each species, so the fully optimized basis set is chosen. Then the density function is calculated using the following equation. X r¼2 wj w k Tjk ð4-F-1Þ j;k

To develop the FSGO-DFT, the FSGO exchange energy is removed from the FSGO and EBx are total energy. The local density and the Becke exchange functionals ELDA x calculated using eqns (4-F-2) and (4-F-3), respectively. ELDA x

¼ Cx

XZ s

rs4=3 dr;

3 1=3 Cx ¼ ð3=2Þ 4p

ð4-F-2Þ


c


EBx ¼ ELDA b x

XZ

r4=3 s

s

x2s dr ð1 þ 6bxs sinh1 xs Þ

ð4-F-3Þ

where b is a constant and xs = |rrs|/r4/3 s . The parameter b has been determined by a least squares fit to exact atomic Hartree-Fock data giving a value b = 0.0042 au. To estimate the LYP correlation energy ELYP , the functional is computed using c eqns (4-F-2) and (4-F-3) for closed shell and open shell systems, respectively. The LYP correlation energy for a closed shell is given by, Z 1 1 1 2 2=3 5=3 cr1=3 Ec ¼ a dr r þ br C r 2t þ þ r e t r F w w 9 18 1 þ dr1=3 ð4-F-4Þ for an open shell system it is, Z n h gðrÞ 8=3 5=3 2=3 r þ 2br 2 CF ra8=3 þ 22=3 CF rb rtw Ec ¼ a 1 þ dr1=3 1 1 1=3 dr þ ðra taw þ rb tbw Þ þ ðra r2 ra þ rb r2 rb Þ ecr 9 18

ð5-F-5Þ

where the coefficients a = 0.04918, b = 0.132, c = 0.2533, d = 0.349 and CF ¼ 3 2 2=3 and other parameters in the LYP equation are given by 10 ð3p Þ tw ¼

1 jrrðrÞj 1 2 r r 8 rðrÞ 8 "

gðrÞ ¼ 2 1

r2a ðrÞ þ r2b ðrÞ

ð5-F-6Þ #

r2 ðrÞ

ð4-F-7Þ

which give EBLYP . Finally the total The total correction is the sum of EBx and ELYP c xc BLYP energy is obtained by adding the Exc to the FSGO energy. The most complicated part of density functional researches is the selection of simple, accurate and fast method for numerical integration over different functionals. For numerical integration it is best to use a spherical polar coordinate system (r,y,j) for each Gaussian, so the volume integral for a given function is a familiar expression: I¼

Z2pZpZ1

fðr; y; jÞr2 sin y dr dy dj

ð4-F-8Þ

0 0 0

Since our functionals are made of spherical Gaussians, they are independent of angular coordinates (y, j) the integral of the angular part becomes constant, Z2pZp

sin ydydj ¼ 4p

ð4-F-9Þ

0 0

Therefore we have the following integral for our different functionals I ¼ 4p

Z1

f½rðrÞr2 dr

ð4-F-10Þ

0

For the radial part the Gauss-Legendre method was used for numerical integration of different functionals using 40 radial points. The procedure has been programmed using FORTRAN90. This procedure was tested for four atoms and four ionic systems. 306 | Chem. Modell., 2008, 5, 279–311 This journal is

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Table 23 Total energy and bond length of LiH. (units are au) Arthur

Method

Basis set

Total energy

Bond length

Perez Pakiari18 G98w G98w Cade19

SCF FSGO SCF SCF HF-limit

1s GTO, 1s STO FSGO 6-31G** 6-311G** —

7.98283 7.98673 7.98134 7.98579 7.98731

3.040 3.035 3.236 3.191 3.034

It is worth pointing out that the DFT energies obtained through HF are not variational with respect to the Schro¨dinger Hamiltonian, and hence are not guaranteed to lie above the exact values, and the results are sometimes a little bit lower than exact values. In DFT-FSGO the optimization of all variational parameters (exponents and coefficients of different Gaussians) are performed and the energies, which are almost above the exact values clearly demonstrate that, this calculation is variationally stable during optimization. This type of calculation is basis set dependent, and if a large basis set is used, although the result of exchange will be roughly the same as for a weak basis set, and the total energy will substantially improve. 4-G

Mixed atomic basis set: GTO and STO in SCF method

As previously mentioned in section 4-2, Linnett and Bacsky in 1972 used a mixed basis set to improve the results of total energy. In 1999 J. E. Perez et al.136 has applied 1s Gaussian and 1s Slater types in an SCF calculation on LiH in order to improve the results. Table 23 is the presentation for comparison the different results. 4-H Fourier space Hartree-Fock (FS-RHF) with using distributed basis set of s-type Gaussian function (DSGF) I. Flamant137 et al. successfully applied the FGSO basis set in Fourier Space Restricted Hartree Fock (FS-RHF) in a study of identification of conformational signatures in valence band of polyethylene. In 1998, they used a distributed basis set of s-type Gaussian function (DSGF) in FS-RHF.138 The method briefly is to use RHF-Bloch states jn(k,r), which are doubly occupied up to the Fermi energy EF and orthonormalized. k and n the wave number and the band index, respectively. X Cpn ðkÞbp ðk; rÞ ð4-H-1Þ jn ðk; rÞ ¼ p

where bp ðk; rÞ ¼ N1=2

1 X

expði2pmkÞwm p ðrÞ

ð4-H-2Þ

m¼1

where p is the label atomic function. Then, the one electron energy eigenvalues En(k) are the solution of the following equation, X X Fpq ðkÞCqn ðkÞ ¼ En ðkÞ Spq ðkÞCqn ðkÞ ð4-H-3Þ q

q

where Spq is overlap matrix elements, Spq = hbp(k,r)|bq(k,r)i

(4-H-4)


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and Fock matrix elements are, Fpq(k) = Tpq(k)+Vpq(k) + Jpq(k) + Xpq(k)

(4-H-5)

where Tpq(k) kinetic, Vpq(k) electron–nuclear attraction, Jpq(k) electron–electron repulsion, and Xpq(k) is exchange term.

Appendix We have mentioned before there is another type formula for FSGO integrals, which has published by Frost.3 They are different from the following integrals by electron– nuclear attraction and electron–electron repulsion integrals, because they have used the modified error function. Different integrals of evaluation of electronic energy are given in the following: The overlap integral, Sij, is, " #3=4 4ai aj ai aj 2 ðAp-1Þ exp r Sij ¼ ai þ aj ij ðai þ aj Þ2 where a is exponent of Gaussian, and rij is, rij = [(xi xj)2 + (yi yj)2 + (zi zj)2]1/2

(Ap-2)

Kinetic energy, KE, is, ðKEÞij ¼ Sij

ai aj ai aj 2 32 rij ai þ aj ai þ aj

Electron nuclear attraction, ENA, is, X 1 Zn erf½ðai þ aj Þ1=2 rijn ðENAÞij ¼ Sij r ijn n

ðAp-3Þ

ðAp-4Þ

where " rijn ¼

XNn

ai xi þ aj xj ai þ aj

2 #1=2 ai yi þ aj yj 2 ai zi þ aj zj 2 þ YNn þ ZNn ai þ aj ai þ aj ðAp-5Þ

where XN, YN and ZN are Cartesian coordinates of nucleus n, erf is ordinary error function. If rijn = 0, then X 1 Zn pffiffiffi ðai þ aj Þ1=2 ðAp-6Þ ðENAÞij ¼ Sij p n The electron–electron repulsion integral EER(ijkl) is, ( ) 1 ðai þ aj Þðak þ al Þ 1=2 ðEERÞijkl ¼ Sij Skl erf rijkl rijkl ai þ aj þ ak þ al

ðAp-7Þ

where,

rijkl

" ai xi þ aj xj ak xk þ al xl 2 ai yi þ aj yj ak yk þ al yl 2 ¼ þ ai þ aj ak þ al ai þ aj ak þ al 2 #1=2 ai zi þ aj zj ak zk þ al zl þ ai þ aj ak þ al


c


ðAp-8Þ

If rijkl = 0 then, 1 ðai þ aj Þðak þ al Þ 1=2 ðEERÞijkl ¼ Sij Skl pffiffiffi p ai þ aj þ ak þ al

ðAp-9Þ

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Advances in valence bond theory Peter B. Karadakov DOI: 10.1039/b608774b

1. Introduction Since the early days of quantum mechanics, there have been two distinct approaches to the description of molecular electronic structure: Molecular orbital (MO) theory and the valence bond (VB) method. In the minds of many theoretical chemists, the competition between these two approaches is still far from over. Testimony to this is the interesting discussion between Hoffmann, Shaik and Hiberty entitled ‘A Conversation on VB vs. MO Theory: A Never-Ending Rivalry?’1 which highlights the most important past and present encounters between the two schools of thought and creates the impression that VB theory continues to be an equally capable adversary in the MO-VB contest. However, a repetition of the pre-ab initio period, when Linus Pauling’s ideas made VB theory the more influential approach, is highly unlikely. While VB theory will certainly remain a source of highly visual qualitative interpretations of chemical bonding and reactivity which are arguably more versatile than their MO counterparts, from a quantitative viewpoint, with the proliferation of ab initio approaches to molecular electronic structure VB has been gradually relegated to a somewhat backstage role. The main reasons for this are the much higher computational costs associated with VB calculations, and the fact that VB wavefunctions often have to be ‘hand-tailored’ to a particular problem which makes them difficult to use by non-specialists and may introduce an undesirable strong dependence of the outcome of a calculation on the construction of the wavefunction. The rapid developments in computer technology within the last two decades have enabled a number of previously unfeasible VB calculations. In turn, this has stimulated a series of theoretical developments and brought about a resurgence of interest in VB theory, which some authors see as a VB ‘renaissance’. Evidence to this is provided by the appearance of several books on VB theory and its applications: the first expert textbook on ab initio VB theory, Gallup’s ‘Valence Bond Methods— Theory and Applications’,2 another recent textbook on the same topic, ‘A Chemist’s Guide to Valence Bond Theory’ by Shaik and Hiberty,3 ‘Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective’ by Weinhold and Landis4 which examines classical VB concepts from the viewpoint of the natural bond orbital (NBO) approach, a volume of the Elsevier Theoretical and Computational Chemistry series, dedicated to VB theory.5 While VB-based methods are far from catching up with the quantitative character of the most advanced MO approaches, it has become relatively straightforward to calculate close VB analogues of the completeactive-space self-consistent field (CASSCF) construction in MO theory. These provide VB-style explanations of electronic structure and reactivity that stem from ab initio wavefunctions of a reasonable quality. VB calculations of this type have become an important tool in arsenal of the theoretical chemist and their easy-tointerpret results are sufficient to ensure the survival of VB ideas well into the 21st century. The aim of this review is to present a summary and critical analysis of the more important recent work in the VB field. It is subdivided into three sections, the first of which includes a detailed comparison between the molecular orbital (MO) and valence bond (VB) approaches, while Sections 2 and 3 deal with methodological developments in the VB area, and applications of VB theory, respectively. Department of Chemistry, University of York, Heslington, York, UK YO10 5DD


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2. Comparison of the MO and VB approaches Initially, the MO and VB approaches were developed with different aims in view: the primary task of MO theory was to explain the electronic spectra of molecules, while the VB method was concerned mainly with the problems of bonding and valency. This is directly reflected in the construction of the wavefunctions used in the most well-known examples of the two approaches, the Hartree-Fock (HF) method and classical (Heitler-London6) VB theory. The HF wavefunction takes the form of a single Slater determinant, constructed of spin-orbitals, the spatial parts of which are molecular orbitals (MOs). Each MO is a linear combination of atomic orbitals (LCAOs), contributed by all atoms in the molecule. The wavefunction in classical VB theory is a linear combination of covalent and ionic configurations (or structures), each of which can be represented as an antisymmetrised product of a string of atomic orbitals (AOs) and a spin eigenfunction. The covalent structures recreate the different ways in which the electrons in the AOs on the atoms in the molecule can be engaged in bonding or lone pairs. An ionic structure contains one or more doubly-occupied AOs. Each of the structures within the classical VB wavefunction can be expanded in terms of several Slater determinants constructed from atomic spin orbitals. At the ab initio level of theory, a single Slater determinant built from MOs offers considerable computational advantages in comparison with a multideterminant expression involving determinants built from AOs. Without any loss in generality, the MOs within a Slater determinant can be assumed to be orthogonal, whereas in classical VB theory, one can orthogonalise the AO basis without changing the wavefunction only in the full-VB limit, that is, for a VB wavefunction that includes all possible covalent and ionic structures. The matrix element of the Hamiltonian between two identical Slater determinants constructed from orthogonal orbitals involves a single and a double sum over the one-electron and Coulomb and exchange integrals, respectively (see Table 1). If one wishes to improve the HF wavefunction through a multideterminant expansion employing orthogonal MOs, such as, say, configuration interaction (CI), the only additional non-zero Hamiltonian matrix elements are those between Slater determinants that differ by one and two orbitals. The expressions for these involve a single sum over two-electron integrals and just two two-electron integrals, respectively. In contrast, the Hamiltonian matrix element between any pair of determinants of non-orthogonal AOs from a classical VB wavefunction incorporates a double sum over one-electron integrals, and a quadruple sum over two-electron integrals (see Table 1). Given the fact that, to match the quality of even a single-determinant HF wavefunction, one would need to construct a classical VB wavefunction involving a large number of covalent and ionic structures, it is not difficult to understand the current almost complete absence of interest in ab initio implementations of Heitler-London-style approaches. However, Table 1 Hamiltonian matrix elements between Slater determinants constructed from orthogonal and non-orthogonal orbitals. hˆ stands for the one-electron operator from the nonrelativistic Hamiltonian Hˆ, the two-electron integrals are denoted as hij|kli = hi(1)j(2)|r1 12 | k(1)l(2)i, Fai is obtained from F = |f1f2 fN| by replacing orbital fi with orbital fa ( a 4 N), finally, Duv(i|j) and Duv(ij|kl) are the first and second-order cofactors of the overlap determinant D = hFu|Fvi7 Orthogonal orbitals P ^ P ^ þ 12 ðhijjiji hijjjiiÞ hFjHjFi ¼ hijhjji i i;j ^ þ P ðhajjiji hajjjiiÞ ^ ¼ hajhjii hFai jHjFi j

ˆ hFab ij |H|Fi = hab|iji hab|jii

Non-orthogonal orbitals P ^ ^ v i ¼ Duv ðijjÞhijhjji hFu jHjF i;j P þ 12 Duv ðijjklÞhijjkli i;j;k;l

ˆ hFabc ijk |H|Fi = 0


c


it should be emphasised that the frequently mentioned ‘N! problem’ which suggests that the computational effort associated with non-orthogonal CI scales as N!, where N is the number of electrons, is nothing more than a myth. The LU factorisation that can be used to calculate the determinant of an N N matrix scales as N3 and, as Lo¨wdin’s has shown in his 1955 paper7 which contains the derivation of the expressions for the Hamiltonian and overlap matrix elements between Slater determinants built from non-orthogonal orbitals, the Hamiltonian matrix element can be calculated very efficiently using the inverse of the orbital overlap matrix. Lo¨wdin’s ideas have been transformed into very efficient algorithms by van Lenthe and co-workers (see e.g. ref. 8 and references therein). The introduction of ionic structures for neutral species in VB theory goes against chemical intuition, yet without these a classical VB wavefunction may even fail to match the quality of its HF counterpart. One example is provided by the simplest classical VB description of benzene in terms of covalent structures only (for the 6 p-electrons, within a single-z basis, these include the two well-known Kekule´ structures and three Dewar structures featuring p-bonds between para-positioned carbons) the energy of which has been demonstrated to be 0.0728 Hartree above the corresponding HF solution.9 Although the p-space full-VB wavefunction for benzene improves on the HF energy by 0.0792 Hartree, the analysis of its composition leads to the counter-intuitive result that the combined weight of the chemically-significant covalent structures is just 8.16%, while the rest of the wavefunction is made by the 170 singly, doubly and triply ionic structures.9 In contrast to MO approaches, having more than one basis function on an atomic centre is a major problem for classical VB theory. For example, if in the above-mentioned p-only VB description of benzene we decide to switch f rom a single-z to a double-z basis, the number of covalent structures increases from 5 to 26 5 = 320 and, according to Weyl’s dimension formula which gives the number of linearly independent configurations for N electrons distributed between M orbitals, 2S þ 1 Mþ1 Mþ1 ð2:1Þ DðM; N; SÞ ¼ N=2 S M þ 1 N=2 þ S þ 1 the overall number of covalent and ionic structures goes up from D(6,6,0) = 175 to D(6,12,0) = 15730. In fact, classical VB theory has never ventured beyond double-z basis sets, and even these have been used to treat directly no more than six electrons. One solution to this basis set problem in more recent classical VB-style approaches, such as some types of VBSCF (VB self-consistent-field) wavefunctions10 and the BOVB (‘breathing’ orbital VB) method,11,12 is to use variational hybrid atomic orbitals (HAOs) expanded in terms of the basis functions on a single centre only. While all classical VB-style approaches, even those using variational HAOs, can be formulated only in terms of localised atom-centred basis sets, MO theory can use basis sets of any type, including plane wave basis sets. There are no restrictions on the type of basis that can be used in modern VB (GVB or SC) calculations, and it would be interesting to see what is the size of the plane wave basis that would be required in order to reproduce the usually well-localised orbitals observed when working within atom-centred basis sets. The number of configurations required to construct a VB wavefunction of reasonable quality is considerably smaller in approaches implementing the Coulson-Fischer idea13 to employ orbitals delocalised over more than one atom. Two of the more widely used modern VB methods, spin-coupled (SC) theory14 and the generalised VB (GVB) approach,15 make full use of this idea by using orbitals constructed as LCAOs, just as in MO theory. Both of these methods do not include any ionic structures in the wavefunction for a neutral system. Other implementations 314 | Chem. Modell., 2008, 5, 312–349 This journal is

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include the ‘breathing’ orbitals suggested by Hiberty et al.11 which are expanded in terms of the basis functions on a single centre or on a specific molecular fragment, and the Linnett-type orbitals used by Amovilli et al.16 which span the basis functions on an atom and one of its nearest neighbours. In fact, the substitution of AOs by orbitals expanded in terms of selected basis functions is the only way in which VB theory can accommodate the extended basis sets required for obtaining results comparable to those of MO theory. The fact that, as a rule, the orbitals appearing in modern VB wavefunctions are non-orthogonal makes them markedly different from standard MOs, despite the use of similar expansion techniques. At the moment, orbital non-orthogonality, which is an inherent feature of VB theory remains the single most important performance-limiting factor in modern VB techniques. While the most common variant of the GVB approach dispenses with most of the nonorthogonality in an ad hoc manner, by imposing strong-orthogonality (SO) restrictions on orbital overlaps, this is at the expense of a significant decrease in the flexibility of the wavefunction. The usually well-localised nature of the orbitals appearing in VB wavefunction makes spatial symmetry more difficult to use than in the MO case. In MO theory, symmetry can be introduced and utilised at the orbital level: Each delocalised MO can be constructed as a symmetry-adapted linear combination (SALC) of basis functions, which is straightforward to implement in program code and can be exploited to achieve substantial computational savings. As a rule, the individual localised orbitals from VB wavefunctions are not symmetry-adapted, but transform into one another under the symmetry operations of the molecular point group. The use of symmetry of this type normally requires prior knowledge of the orbital shapes and positions and is very difficult to handle without human intervention. Performance is not the only area in which VB methods lose out to their MO counterparts. MO theory offers clear ways of introducing systematic improvements to a reference HF or complete-active-space self-consistent-field (CASSCF) wavefunction through Møller-Plesset many-body perturbation theory of various orders (MPn), coupled-cluster (CC) constructions, second-order perturbation theory based on a CASSCF reference (CASPT2), CI and multi-reference CI (MRCI), etc. Against this hierarchy of MO wavefunctions, the VB arsenal looks markedly deficient. The quantitative characteristics of GVB and SC wavefunctions place them in-between the HF and CASSCF MO constructions. Examples of post-GVB wavefunctions are provided by GVB with restricted configuration interaction (GVB-RCI),17 GVBLMP2 (a localised MP2 treatment based on a GVB reference)18 and GVB-RCC (GVB restricted coupled cluster),19,20 while the SC wavefunction can be improved through non-orthogonal CI, known as SCVB.21 Many of the remaining VB wavefunctions in current use are ‘hand-tailored’ to specific molecular systems and, in a direct ab initio comparison of say, calculated ground-state energies, would fare worse than HF, the simplest MO approach. At the moment, analytical gradients are available for a limited number of VB wavefunctions, GVB with perfect pairing (PP) and SO restrictions (GVB-PP-SO, within GAUSSIAN22), SC theory (thanks to the CASVB23,24 module in MOLPRO25), and for VBSCF wavefunctions10 (within GAMESS-UK26). The forte of VB theory is that its results are very easy to interpret using wellknown qualitative concepts such as Lewis structures, hybridisation and resonance. Electron pairs associated with bonding interactions and lone pairs are immediately obvious, and so are unpaired electrons, resonance is embedded in the wavefunction, bond breaking and formation can be modelled in a very straightforward manner by allowing the electrons in the AOs to be engaged in different bonding pairs within different configurations. All of this is particularly useful when studying and trying to understand the sometimes profound changes in the electronic structure of a reacting system occurring during a chemical reaction.


c


In order to produce qualitative information about bonding patterns and lone pairs, MO theory has to resort to orbital localisation procedures or more complicated interpretation techniques such as, for example, Bader’s Atoms in Molecules (AIM) approach.27 However, orbital localisation procedures can be applied only to wavefunctions which remain invariant with respect to non-singular linear transformations of the orbitals, i.e. single-configuration and complete-active-space (CAS) MO wavefunctions and even for these, the use of orthogonal localised orbitals is not sufficient to reproduce all of the chemically-important information included in modern VB wavefunctions. Evidence of the superiority of the interpretational facilities offered by VB wavefunctions is provided by the continuing efforts to transform CAS wavefunctions to VB-style form.23,24,28–30 An interesting discussion of the qualitative aspects of the VB vs. MO rivalry has been published by Shaik and Hiberty.31 These authors examined four perceived fundamental ‘flaws’ of classical VB theory, due to which it has often been considered to be inferior to MO theory,32,33 that (a) VB theory predicts that O2 has a singlet ground state given by the Lewis structure OQO, as opposed to MO theory which correctly shows that the molecule is paramagnetic; (b) VB theory wrongly predicts that cyclobutadiene is stabilised by resonance just as benzene while MO theory shows correctly that cyclobutadiene has zero resonance energy and should be rectangular, with alternating C–C bond lengths; (c) in contrast to MO theory, VB theory fails to classify ions such as C3H3+ and C3H3, C5H3+ and C5H5, C7H7+ and C7H7, etc., as aromatic or antiaromatic; (d) in contrast to MO theory, VB theory fails to predict that the photoelectron spectrum of CH4 exhibits two different ionisation peaks. The analysis of the origins of these ‘flaws’ showed that two of these, (a) and (d), should be looked upon as ‘myths’ of uncertain origins, while (b) and (c) are the result of the misuse of simple resonance theory that just counts resonance structures. As Shaik and Hiberty have demonstrated, in each of the four cases the proper use of relatively simple qualitative VB theory leads to correct predictions which are just as convincing as those made by MO theory.

3. Developments in VB methodology A careful look at the methodological work in the VB area in recent years reveals that a very significant part of it is taken up by developments in a ‘horizontal’ dimension which can be either re-formulations of ideas already present in VB methods that incorporates spin in explicit form within spin-free approaches, or improvements to the computational algorithms for calculating well-known VB-style wavefunctions. As a consequence, after outlining well-established VB methods such as GVB, SC and BOVB, the current review focuses on recent developments in VB methodology which have essentially new features. The only spin-free VB approach which is discussed in greater detail is the VBCI method.34–36 The main reason for this is that it is not difficult to show that any spin-free VB construction has an exact spin-containing equivalent (see e.g. ref. 37) and the permanents involving orbital overlaps that arise in spin-free approaches are computationally more demanding than the corresponding determinants in spin-containing methods. The most general N-electron VB wavefunction that incorporates spin in explicit form can be written down as X ^ K YN Þ C¼ CSKk AðF ð3:1Þ SM;k K;k

^ is the antisymmetriser where A ^ ¼ ðN!Þ1 A

X

eP P

P


c


(the summation runs over all N! permutations of the electron coordinates), FK stands for the orbital product FK ¼

N Y

cKm

ð3:2Þ

m¼1

and YN SM;k is an N-electron spin function with total spin S and z-projection of the total spin M. The main differences between this VB wavefunction and a CI expansion in terms of configuration state functions (CSFs) in MO theory are that at least some, if not all of the orbitals cKm are non-orthogonal, as a result of which the VB wavefunction represents a non-orthogonal CI expansion, and that VB theory prefers to use spin functions from the Rumer spin basis38 rather than the Kotani spin functions39 which are the usual choice in MO theory. It is difficult to draw a strict dividing line between VB and MO theory as methods from both camps often employ very similar approximations, for example, the GVB and SC approaches use orbitals in MO form, while the CAS and restricted-active-space state-interaction approaches40 (CASSI and RASSI) solve non-orthogonal CI problems. The number of the linearly independent N-electron spin functions corresponding to a total spin S is given by the expression ð2S þ 1ÞN! N N N ð3:3Þ fS ¼ N N ¼ 1 S S 1 N þ S þ 1 ! 12 N S ! 2 2 2 The Rumer spin basis represents a set of fN S linearly independent spin functions, in which N2S electrons (m1, ,mN2S) form singlet pairs, and the remaining 2S electrons (mN2S+1, ,mN) are assigned spins a (in the resulting spin function M = S): R

1 1 YN SM;k ¼ pffiffiffi ½aðm1 Þbðm2 Þ aðm2 Þbðm1 Þ . . . pffiffiffi ½aðmN2S1 ÞbðmN2S Þ 2 2 aðmN2S ÞbðmN2S1 ÞaðmN2Sþ1 Þ . . . aðmN Þ

ð3:4Þ

but not all VB approaches make use of all of these. A frequently used extended label that uniquely defines a Rumer spin function is provided by the list of its singlet pairs: k (m1m2, m3m4, . . . ,mN2S1–mN2S)

(3.5)

fN S

linearly independent Rumer spin functions can be obtained through the A set of diagrammatic technique suggested by Rumer38 and extended to nonsinglet states by Simonetta et al.41 In this approach, each Rumer spin eigenfunction is associated with an extended Rumer diagram. The diagrams for an N electron system with spin quantum number S are drawn as follows: N + 1 points are placed so as to form a convex polygon (usually, a regular polygon). The first N points are labelled, say clockwise, with the numbers from 1 to N, and the last point becomes the pole (P). 1 2 N S lines are then drawn between pairs of the first N points, additional lines are drawn between the remaining 2S points and the pole; none of the lines are allowed to intersect. It can be shown41 that the number of distinct extended Rumer diagrams which can be drawn in this way is fN S , and the associated Rumer spin eigenfunctions, in which the singlet pairs contain electrons with numbers connected by the first 1 2 N S lines, while the remaining electrons (with numbers connected to the pole) are placed within a spin functions, are linearly independent. The extended Rumer diagrams and the associated Rumer spin functions for N = 5, S ¼ 12 ( f15 ¼ 5 ) are 2 shown in Fig. 1. ^ KYN If all orbitals cKm are different, AF defines a covalent structure. In ionic SM;k structures, there are one or more pairs of identical orbitals (i.e. one or more orbitals are doubly-occupied). In all modern VB approaches the orbitals cKm are approximated by expansions in subsets A, B, etc., of a finite AO basis for the whole Chem. Modell., 2008, 5, 312–349 | 317 This journal is

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molecule {wp|p = 1,2, . . . ,M} as cKm ¼

X

ð3:6Þ

cKm;p wp

p2A

A subset may involve all basis functions on an atom or on a molecular fragment, all basis functions on atom or on a molecular fragment of a particular symmetry, all basis functions on an atom and its nearest neighbours (in this case some of the subsets may overlap), or may consist of all available basis functions. 3.1 The GVB-PP-SO and SC methods Two popular modern VB approaches, GVB-PP-SO15 and SC,14 use fully-variational wavefunctions including a single orbital product. In the GVB-PP-SO wavefunction this single orbital product is combined with a single ‘perfect-pairing’ (PP) spin function which in the Rumer spin basis corresponds to (1–2, 34, . . . ,(N2S1)(N–2S)): 201 1 !3 N 2NS Y Y ^ 4@ CGVBPPSO ¼ A ð3:7Þ c2m1 c2m ðab baÞA cn a 5 m¼1

n¼N2Sþ1

The orbitals in the GVB-PP-SO wavefunction are expanded in the AO basis for the whole molecule {wp|p = 1, 2, . . . ,M}, and are required to be ‘strongly-orthogonal’ (SO), i.e. the only non-orthogonal orbitals are those, the spins of which are coupled to singlets by the spin function: c1 and c2, c3 and c4, etc. It is straightforward to eliminate non-orthogonality altogether by replacing the orbitals c2m1 and c2m within each singlet pair by a couple of orthogonal GVB natural orbitals defined as cmg = (c2m1 + c2m) [2(1 + hc2m1|c2mi]1/2, cmu = (c2m1 c2m) [2(1 hc2m1|c2mi]1/2

(3.8)

This makes GVB-PP-SO the easiest to implement and most computationally efficient modern VB approach. It has been incorporated within several popular ab initio packages, including GAUSSIAN22 (which can calculate GVB-PP-SO energies, analytical gradients and numerical Hessians), GAMESS-US42 (capable of computing GVB-PP-SO energies, analytical gradients and Hessians) and Jaguar43 which can not only evaluate single-point energies and perform geometry optimisations with GVB-PP-SO wavefunctions, but also do calculations of these types with two postGVB constructions, GVB-RCI17 and GVB-LMP2,18 both of which use a GVB-PPSO reference. At the moment, Jaguar is the only ab initio package that implements pseudospectral methods for calculating two-electron integrals.44,45 This allows all methods implemented in Jaguar, including GVB, GVB-RCI and GVB-LMP2 to scale more favourably than M3 (where M is the basis set size). In Jaguar, the computational effort associated with GVB-PP-SO calculations is not prohibitively larger than that for HF calculations, which makes the advantages of GVB-based approaches, for example, correct bond dissociation, accessible even for larger systems. Jaguar is also capable of performing GVB-DFT calculations, in which the electronic density for a molecule is calculated using a GVB-PP-SO wavefunction, and then this density is used to carry out a density function theory (DFT) energy evaluation. The GVB-DFT combination was first suggested by Kraka46 who applied an approach of this type, GVB-LSDC (GVB followed by a DFT calculation utilising the local density correlation functional) to the evaluation of homolytic bond dissociation energies for a set of 27 small molecules, achieving excellent agreement with experiment. The main disadvantage of the GVB-PP-SO wavefunction is that, due to the use of a single spin function, it cannot handle molecules for which resonance is important. For example, a GVB-PP-SO calculation on benzene produces a 318 | Chem. Modell., 2008, 5, 312–349 This journal is

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Fig. 1 Extended Rumer diagrams and Rumer spin functions for N = 5, S ¼ 12.

wavefunction of D3h symmetry suggesting alternating single and double carbon–carbon bonds. Similar problems occur if an attempt is made to use the GVB-PP-SO wavefunction to describe a reaction in which the singlet orbital pairs in the reactant(s) and product(s) are different, for example, the Diels-Alder reaction. The SC wavefunction which was first proposed by Gerratt and Lipscomb,47 followed shortly by Ladner and Goddard who introduced an identical construction under the full-GVB name,48 also uses a single orbital product but, in contrast to the GVB-PP-SO wavefunction, it makes full use of the spin space: " ! # N Y N ^ CSC ¼ A ð3:9Þ cm Y SM

m¼1

where YN SM is a general N-electron spin function which is expanded in terms of all fN S linearly independent spin functions from an N-electron spin basis N

YN SM

¼

fS X

CSk YN SM;k

ð3:10Þ

k¼1

SC theory does not assume any orthogonality between the orbitals cm which, just as in the GVB-PP-SO case, are expanded in the AO basis for the whole molecule {wp|p = 1, 2, . . . ,M}. The use of the full spin space and the absence of orthogonality requirements allow the SC wavefunction to accommodate resonance which is particularly easy to identify if YN SM is expressed within the Rumer spin basis. In addition to the Rumer spin basis, the SC approach makes use of the Kotani spin basis, as well as of the less common Serber spin basis.49,50 When analysing the nature of the overall spin function YN SM in the SC wavefunction (3.9), it is often convenient to switch between different spin bases. The transformations between the representations of YN SM in the Kotani, Rumer and Serber spin bases can be carried out in a straightforward manner with the use of a specialised code for symbolic generation and manipulation of spin eigenfunctions (SPINS, see ref. 51). The Kotani spin eigenfunction KYN SM can be generated in two possible ways: (i) by addition of the spin of the Nth electron to that of the N1 electron Kotani spin eigenfunction corresponding to S 12 K

1=2 K N1 YN YS1;M1;k0 aðNÞ SM;k ¼½ðS þ MÞ 2

2

bðNÞð2SÞ1=2 þ ðS MÞ1=2 K YN1 S1;Mþ1;k0 2

ð3:11Þ

2

(ii) by subtraction of the spin of the Nth electron from that of the N1 electron Kotani spin eigenfunction corresponding to S þ 12 K

1=2 K N1 YN YSþ1;M1;k0 aðNÞ SM;k ¼ ½ðS M þ 1Þ 2

2

bðNÞð2S þ 2Þ1=2 þ ðS þ M þ 1Þ1=2 K YN1 Sþ1;Mþ1;k0 2

ð3:12Þ

2


c


Fig. 2

Kotani branching diagram.

A Kotani spin eigenfunction is fully defined by the sequence of partial resultant spins obtained after combining the spins of the first 1, 2, . . . ,N1 electrons: k (S1S2 . . . SN1)

(3.13)

The consecutive formation of Kotani spin eigenfunctions can be illustrated through the Kotani branching diagram (see Fig. 2). Each circle in this diagram corresponds to an allowed value of the total spin S for a given number of electrons N and contains the corresponding fN S [see eqn (3.3)], while the upwards and downwards pointing arrows correspond to the addition and subtraction eqn (3.11) and (3.12), respectively. The Serber spin eigenfunctions SYN SM;k are assembled through a process similar to that used to obtain their Kotani counterparts, but from the two-electron singlet and triplet spin eigenfunctions Y200 = 21/2 (abba), Y210 = 21/2 (ab + ba), Y211 = aa, Y21,1 = bb (and, if the number of electrons is odd, the spin function for the last electron). Just as in the case of the Kotani spin functions, the intermediate values of the total spin can be used in order to introduce a compact definition of a Serber spin function: 8 ð. . . ðð s12 s34 ÞS4 ; s56 ÞS6 ; . . . SN2 ; sN1;N Þ for N even > > > |fflffl{zfflffl} < k

N=21

ðð. . . ðð s12 s34 ÞS4 ; s56 ÞS6 ; . . . SN3 ; sN2;N1 ÞSN1 Þ for N odd; > > > : |fflfflffl{zfflfflffl}

ð3:14Þ

ðN1Þ=2

where sm1,m may take the values 0 or 1 depending on the singlet or triplet coupling of the spins of electrons m and m 1. In the odd-electron case, it is not necessary to include the spin of the last electron to be added to SN1 as it is always 12. A Serber branching diagram (see Fig. 3) can be constructed similarly to the Kotani branching diagram (see Fig. 2). A two-electron triplet state can be added to (N2,S) in one of three possible ways: These are denoted by an arrow pointing upwards from (N2,S) to (N,S + 1), an arrow pointing downwards from (N2,S) to (N,S1) and a horizontal arrow pointing from (N2,S) to (N,S), respectively (the arrows from the last two cases can be drawn only if S 4 0 at N2). The addition of a twoelectron singlet spin state to (N2,S) is marked by a dashed horizontal arrow pointing from (N2,S) to (N,S). The odd-electron spin states are connected to their even-electron precursors by upwards and downwards pointing arrows, as in the 320 | Chem. Modell., 2008, 5, 312–349 This journal is

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Fig. 3 Serber branching diagram.

Kotani branching diagram (Fig. 2), however, on the Serber branching diagram these states have no successors. For larger molecules, it is usual to employ GVB-PP-SO and SC wavefunctions with core-valence separation which leads to a significant reduction of the required computational effort. The SC wavefunction with n doubly-occupied and N active electrons (overall number of electrons: 2n + N) takes the form52 " ! # ! n N Y Y N ^ ð3:15Þ CSM ¼ A ji aji b cm Y SM

i¼1

m¼1

The most convenient and widely accessible way of calculating fully-variational SC wavefunctions is associated with the development of a strategy, known as CASVB.23,53– 55 CASVB makes use of the fact that a CAS wavefunction, just as any full-CI construction, is invariant not only to a unitary, but also to a general non-singular linear transformation of the active orbitals. This property can be exploited in order to transform it to an alternative equivalent representation dominated by a small number of configurations. The transformation of the full-CI space induced by a non-unitary transformations of orbital space can be carried out exactly by means of the efficient computational schemes developed by Thorsteinsson and Cooper.24 One straightforward application of the CASVB approach is associated with the generation of representations of a CAS wavefunction dominated by a single or multi-configuration modern VB component. If we decompose a CAS wavefunction CCAS into a VB component CVB and its orthogonal complement within CCAS, Y> VB, according to the recipe CCAS = SVB CVB + (1S2VB)1/2 Y> VB

(3.16)

the contribution of CVB to CCAS can be maximised by maximising the overlaprelated quantity SVB SVB ¼

hCCAS jCVB i hCVB jCVB i1=2

ð3:17Þ

This procedure is relatively inexpensive computationally and, with reasonable choices for the form of CVB, it is fairly robust. An obvious alternative is to minimise the energy expectation value EVB ¼

^ VB i hCVB jHC hCVB jCVB i

ð3:18Þ

The minimisation of EVB is more expensive than the maximisation of SVB, because it requires evaluation of the Hamiltonian matrix element hCVB|Hˆ|CVBi and its Chem. Modell., 2008, 5, 312–349 | 321 This journal is

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derivatives, but this may be achieved by adapting the efficient procedures already available in various CASSCF codes. It turns out, however, that if one employs the same VB ansatz within each of these two approaches, they produce very similar CVB which, in practice, makes the maximisation of SVB the preferred procedure. Both optimisations utilise reliable Newton-Raphson-based techniques involving first and second derivatives. The CASVB strategy for the fully-variational optimisation of modern VB wavefunctions relies on a linked two-step iterative strategy. Just as in CASSCF, this starts with choosing active and inactive spaces, and an appropriate form for CVB. The first ‘non-orthogonal’ step involves the minimisation of EVB using the CASVB algorithms, whereas the second ‘orthogonal step’ involves orbital optimisation using standard CASSCF procedures. When starting from a converged CASSCF wavefunction, convergence to a final VB wavefunction which has a high overlap with the CASSCF one (say, the SC wavefunction) can involve a very small number of iterations. This often makes the CASVB calculation of SC wavefunctions somewhat cheaper than the older traditional direct SC optimisation procedures.52,56 The full CASVB module is incorporated in MOLPRO,25 and has more recently been implemented within another popular ab initio package, MOLCAS.57 It is usual to benchmark the performance of a SC wavefunction with N active orbitals against its ‘N in N’ CASSCF counterpart. In the case of benzene, the amount of ‘non-dynamic’ ‘6 in 6’ p space CASSCF correlation energy recovered by a SC wavefunction with six orbitals is 89.6%.58 If SC theory is used to study the evolution of the electronic structure of a reacting system along a reaction path, the SC wavefunction accounts, as a rule, for more than 90% of the CASSCF correlation energy. In the case of the Diels-Alder reaction, the percentages are 92.9% at the transition structure (TS), and 95.4% and 95.8% at the intrinsic reaction coordinate (IRC) points at 0.6 amu1/2bohr and 0.6 amu1/2bohr, respectively.59 This shows that in the majority of cases the single orbital product SC wavefunction can provide a reasonably close approximation to its CASSCF counterpart even without the inclusion of ionic structures. The SC wavefunction may include smaller amounts of the CASSCF correlation energy for molecules, in which the active orbitals are crammed together within a small volume of space. For example, for ethene (H2CQCH2) a SC wavefunction with four active ‘bent-bond’ orbitals recovers 76.5% of the ‘4 in 4’ CASSCF correlation energy (the corresponding percentage for a two-pair GVB-PP-SO construction is 57.1%), while for ethyne (HCRCH) a SC wavefunction with six active ‘bent-bond’ orbitals recovers only 66.3% of the ‘6 in 6’ CASSCF correlation energy (which is still better than the 48.5% achieved by a three-pair GVB-PP-SO construction).60 A distinct advantage of the GVB-PP-SO and SC approaches over other VB methods is that they employ wavefunctions which require minimal amounts of ‘hand-tailoring’. In most cases, the GVB-PP-SO and SC wavefunction-related input data is similar to that for a CASSCF calculation and involves choosing a suitable basis set, the numbers of core and active orbitals, and specifying initial guesses for these orbitals. Just as MO-based approaches, such as HF and CASSCF, GVB-PP-SO and SC calculations can converge to wavefunctions corresponding to different local minima on the energy hypersurface, some of which may not exhibit the full symmetry of the nuclear framework. This happens more often in the SC case, due to the increased variational flexibility of the wavefunction. Choosing appropriate initial guesses for the core and active orbitals is not always sufficient to ensure convergence to the lowest local minimum, or to a solution of the required symmetry. The SC codes described in ref. 52 utilise a second-order constrained minimisation procedure which makes it straightforward to introduce simple orbital symmetry constraints cpm zpmqn(R)cqn = 0


c


(3.19)

where cpm is the coefficient of basis function wp in orbital cm and zpmqn(R) is a number associated with a symmetry operation R of the symmetry group of the molecule which can take the values 0, 1. These simple symmetry constraints can help enforce s-p separation in conjugated systems and achieve convergence to ‘bent-bond’ solutions in systems with multiple bonds.60 Finding symmetry-adapted solutions for molecules that contain a symmetry axis of an odd order (say, C3 or C5) may require more general symmetry constraints of the form X cpm kpmqn ðRÞcqn ¼ 0 ð3:20Þ q;n

where the values of kpmqn(R) are not restricted to 0, 1. Symmetry constraints of this type have been implemented in more recent versions of the SC codes described in ref. 52. While the use of symmetry constraints allows avoiding broken-symmetry solutions in the majority of cases, it is important to emphasise that, as discussed in detail in ref. 37, VB wavefunctions that incorporate a single orbital product can be prevented from achieving the full symmetry of the problem by construction. In such situations, the only way to obtain a symmetry-adapted wavefunction is to use an appropriate symmetry projection operator which, when applied to a single orbital product VB wavefunction, produces a VB construction involving multiple orbital products. One potential pitfall in SC calculations is associated with the fact that all SC orbitals cm in eqn (3.9) and eqn (3.15) are non-orthogonal and there is no mechanism that would prevent three or more orbitals from trying to become the same during the variational optimisation of the wavefunction. This is in contrast to standard MO theory and the GVB-PP-SO approach where the orthogonality requirements allow no more than two spatial orbitals to be identical. When three or more SC orbital become very similar, the antisymmetry requirement makes the norm of the SC wavefunction a very small number which leads to a numerical instability in the optimisation procedure and a convergence failure. This situation which can be aptly termed ‘self-annihilation’ of the SC wavefunction is observed for some O-containing systems, for example, formaldehyde (H2CQO), in which the attempt to describe the carbon–oxygen double bond with a SC wavefunction with four active orbitals succeeds only if one imposes s-p separation through the orbital symmetry constraints (20). In an unconstrained calculation all four active orbitals ‘collapse’ onto the oxygen atom and the wavefunction optimisation fails to converge. In order to prevent the ‘self-annihilation’ of the SC wavefunction it would be necessary to find a way of ensuring that the matrix of SC orbital overlaps hcm|cni remains non-singular throughout the wavefunction optimisation procedure. A requirement of this type (equivalent to an inequality constraint) is far from straightforward to implement in code and although the CASVB strategy appears to be more robust with respect to the SC wavefunction ‘self-annihilation’ than direct codes,52,56 the problem is still unresolved. The SCVB approach21 which is a non-orthogonal CI expansion based on a SC reference introduces a set of Fock-like one-electron operators, one for each occupied SC orbital. The eigenvalues of these Fock-like operators resemble orbital energies, and each occupied SC orbital represents the eigenfunction with the lowest eigenvalue of its Fock-like operator. The remaining eigenfunctions of each Fock-like operator form ‘stacks’ of virtual orbitals which can be used to construct additional configurations for the non-orthogonal CI expansion. It is usual to consider only ‘vertical’ excitations, in which an occupied SC orbital is replaced by virtuals from its ‘stack’ only. As a rule, the orbitals from each ‘stack’ are localised within the same region of space which makes all excitations reasonably local. Perhaps the most successful application of this SCVB strategy to date is the extremely thorough study of all singlet and triplet valence excited states, as well as the n = 3, 4 singlet and triplet Chem. Modell., 2008, 5, 312–349 | 323 This journal is

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Rydberg states of benzene below the first ionisation potential at 9.25 eV carried out by da Silva et al.61 The use of the SCVB approach allowed the authors to classify the valence excited states as covalent or ionic in a straightforward manner, and it was shown that covalent states were well described using the usual assumption of s-p separation. The errors in the computed transition energies to the ionic states were observed to be much larger, an indication that these states require additional s-p correlation for their proper description. The use of a suitable s core, derived from a calculation on the C6H6+ cation, produced very good descriptions of the Rydberg states. This work showed that compact and easy-to-interpret SCVB constructions are capable of describing excited states with numerical accuracy comparable to that of much larger HF or CASSCF-based CI expansions. The SCVB approach has two main drawbacks, both of which are associated with the definition and use of the Fock-like operators. Firstly, in contrast to HF orbital energies, the eigenvalues of the Fock-like operators in the SC case do not provide easily identifiable contributions to the total electronic energy and have no physical interpretation, as there is no SC analogue of Koopmans’ theorem. Secondly, when taken together, the ‘stacks’ comprise an overcomplete set of orbitals which may lead to linear dependence problems in larger SCVB expansions. A simpler idea which still has not been exploited fully would be to construct a nonorthogonal CI on top of a SC wavefunction using the orbitals from the orthogonal complement to the SC orbitals (and the core orbitals, if any). One way of generating ‘canonical’ virtual orbitals which have orbital energies similar to the HF ones was suggested in ref. 52, where it was shown that the core orbitals in the SC wavefunction (3.15) can be chosen as eigenfunctions of a projected Fock-like operator, PˆvFˆPˆvj = eiji

(3.21)

Pˆv in this equation is the projection operator in the orthogonal complement to the SC orbital space, P^v ¼ 1

N X

~ j jcm ihc m

ð3:22Þ

m¼1

The definition of Pˆv involves the dual orbitals c~m of the SC orbitals cm, ~ ¼ c m

N X

cn ðD1 Þnm ;

Dmn ¼ hcm jcn i

ð3:23Þ

n¼1

The Fock-like operator Fˆ incorporates contributions from the core and valence orbitals, F^ ¼ h^ þ

n N X X ð2 Jî K^ i Þ þ 14D1 DðmjnÞð4 J^mn K^ mn K^ nm Þ

ð3:24Þ

m;n¼1

i¼1

In this equation Jî and Kî denote the standard Coulomb and exchange operators involving core orbital ji, D and D(m|n) stand for the normalisation integral for the SC wavefunction and the elements of the first-order density matrix in the space of the SC orbitals, and Jˆmn and Kˆmn are generalised Coulomb and exchange operators with matrix elements hwp|Jˆmn|wqi = hwpcm|wqcni, hwp|Kˆmn|wqi = hwpcm|cnwqi. At least N out of the M eigenvalues of the projected Fock-like operator PˆvFˆPˆv must be equal to zero: these correspond to eigenvectors spanning the SC orbital space. The remaining MN eigenvalues resemble HF orbital energies so closely that it is even possible to prove an analogue of Koopmans’ theorem demonstrating that the ei values represent approximations to the corresponding ionisation potentials and electron affinities. Another possibility is to subject the orbitals from the orthogonal complement of the occupied orbitals to a localisation procedure which would help with the interpretation of the non-orthogonal CI construction in VB terms. 324 | Chem. Modell., 2008, 5, 312–349 This journal is

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Similarly to any other truncated CI, a limited SCVB expansion is not sizeconsistent but this drawback is not particularly serious for calculations on small molecules. Another alternative to the SCVB approach is provided by the SCVB* treatment,62–64 in which one generates one or more ‘optimal’ virtual orbitals for each occupied SC orbital and then employs these virtuals to construct a non-orthogonal CI expansion. The simplest case involves the calculation of one ‘optimal’ virtual orbital c+ m for each SC orbital cm. If the SC wavefunction (3.9) is augmented with all vertical double excitations into these virtuals, X Cmn Cmn ð3:25Þ CD ¼ C0 CSC þ m4n

where Cmn is the configuration obtained by replacing orbitals cm and cn in CSC by + c+ m and cn , respectively, the lowest eigenvalue of the secular problem associated with CD can be approximated by means of an expression which resembles the MP2 energy " # X ðH0;mn E0 S0;mn Þ2 1 ð2Þ ED ¼ H00 ð3:26Þ S00 Hmn;mn E0 Smn;mn m4n in which H00 = hCSC|Hˆ|CSCi, S00 = hCSC|CSCi, E0 = H00/S00, H0,mn = hCSC|Hˆ|Cmni, etc. It should be mentioned that eqn (3.26) is slightly more general than the expression reported in refs. 62–64 which is valid only if the reference SC wavefunction is normalised, i.e. for S00 = 1. The ‘optimal’ virtual orbitals c+ m are calculated through minimisation of E(2) D . The final SCVB* wavefunction is constructed as a non-orthogonal CI expansion including all single and double excitations cm - c+ m : X X Cm Cm þ Cmn Cmn ð3:27Þ CSCVB ¼ C0 CSC þ m

m4n

A SCVB* wavefunction incorporating just 25 structures has been shown to produce highly accurate results for the He LiH van der Waals system,62 with marked improvements over a SCVB treatment making use of a much larger number of structures. The multi-reference SCVB* (MR-SCVB*) approach64,65 represents an extension of the idea behind SCVB* which makes use of more than one reference wavefunction. The starting point, just as in the SCVB* case, is the calculation of the SC wavefunction for the lowest state of a given symmetry which is followed by the determination of single-configuration SC approximations to one or more of the lowest excited states. These are obtained through the standard SC optimisation procedure while making sure that either all active orbitals in the excited state SC wavefunction are orthogonal to one orbital from the ground state SC wavefunction (an N:1 scheme), or one excited state orbital is orthogonal to all ground state orbitals (an 1:N scheme). The next step in the MR-SCVB* approach is the generation of SCVB*-style virtual orbitals for all single-configuration SC ground and excited-state wavefunctions. The MR-SCVB* wavefunction incorporating all single and double excitations arising from the reference set {CK} can be written down as ! X X X CMRSCVB ¼ CK CK þ CKmK CKmK þ CKmK n K CKmK nK ð3:28Þ mK

K

mK 4n K

where CKmK arises from CK through the single excitation cmK - c+ mK, etc. The MR-SCVB* approach has been used to obtain near-full-CI quality but still easy-to-interpret results for the first excited state of the LiH2+ cation.65 Thorsteinsson and Cooper have developed a CASVB methodology for describing low-lying excited states66 which makes use of two different ideas. The simpler one of Chem. Modell., 2008, 5, 312–349 | 325 This journal is

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these is to look for VB-style representations of the CASSCF wavefunctions for different electronic states utilising overlap criteria similar to that in eqn (3.17). A more sophisticated approach is to use a second-order energy optimisation procedure which seeks a saddle point at which the second-derivative matrix (Hessian) has i1 negative eigenvalues in order to find an approximation to the wavefunction of state number i of a given symmetry. Test applications to the 21Ag state of trans-1,3butadiene, the c˜1A1 state of methylene and the first singlet excited state of benzene show that this methodology represents an interesting alternative to the more traditional SCVB approach. The separate optimisations of states of the same symmetry should result in non-orthogonal wavefunctions, but this can be remedied through an additional non-orthogonal CI calculation, just as in the case of the CASSI and RASSI approaches. 3.2 VB wavefunctions incorporating multiple orbital products The variety of VB wavefunctions incorporating multiple orbital products is enormous and next to impossible to arrange within a formal classification scheme as most of these do not emerge as the result of more or less systematic attempts to improve a given reference as common in MO theory, but as specific constructions aimed to achieve an economic and easy-to-interpret description of a particular problem. The generalised multi-configuration SC (GMCSC) and optimised basis set-GMCSC (OBS-GMCSC) methods developed by Penotti67,68 can be viewed as multi-configuration extensions of the SC approach. The GMCSC wavefunction can be written down as CGMCSC ¼

L X

CS;K CSM;K

ð3:29Þ

K¼1

where L is the number of SC configurations included in the wavefunction and CSM;K is a single SC configuration defined analogously to the SC wavefunction from eqn (3.9), " ! # N Y N ^ cm Y CSM;K ¼ A ð3:30Þ K

SM;K

m¼1

where N

YN SM;K

¼

fS X

K CSk YN SM;k

ð3:31Þ

k¼1

It is not difficult to establish that this definition of the GMCSC wavefunction is identical to that of the most general N-electron VB wavefunction that incorporates spin in explicit form [see eqn (3.1)]. Just as in SC theory, all orbitals cmK are nonorthogonal and approximated by MO expansions in a suitable AO basis. The orbitals and the coefficients CS;K and CK Sk are determined variationally, through minimisation of the energy expectation value corresponding to CGMCSC. The OBSGMCSC method goes one step further and adds the exponential factors in a Slatertype basis used for expanding the cmK to the list of variational parameters. Penotti has described68 a very thorough OBS-GMCSC study of the 1A 0 1 ground state of BH3 which starts with the determination of a SC wavefunction with eight active orbitals, followed by the calculation of OBS-GMCSC wavefunctions incorporating three, six, seven and eight SC configurations. The eight-configuration OBS-GMCSC wavefunction has been shown to yield a ground-state energy lower than that obtained with a large MO-based CI wavefunction including single, double, triple and quadruple excitations (CISDTQ) utilising a much larger fixed basis. 326 | Chem. Modell., 2008, 5, 312–349 This journal is

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The OBS-GMCSC wavefunction takes into account all degrees of variational freedom available within a non-orthogonal CI wavefunction involving nonorthogonal orbitals. In principle, it is not possible to define a more general non-orthogonal CI approach. When working, as standard, in a fixed basis, and if all structures are constructed from the same set of orbitals, the GMCSC wavefunction becomes equivalent to another non-orthogonal CI construction utilising variational orbitals, the VBSCF method suggested by Van Lenthe and BalintKurti.8,10,69,70 One of the problems associated with these VB-style approaches is that the numerical effort needed for the calculation of OBS-GMCSC, GMCSC and VBSCF wavefunctions is much higher than that required for orthogonal MO-based constructions of similar quality. A second, potentially more serious problem is that the choice of configurations and orbitals is usually made through chemical intuition, with guidance from classical VB ideas that can be traced back to the Lewis theory of valence. This can work reasonably well, if the configurations correspond to easily identifiable resonance structures such as those in the allyl radical or the cyclopentadienyl anion. Usually, in such cases the modern VB approaches relying on a single orbital product, GVB-PP-SO and SC, produce broken-symmetry solutions and the multi-configuration VB constructions provide a way of obtaining symmetry-adapted solutions. However the choice of configurations that should be included in a GMCSC or a VBSCF wavefunction in order to obtain a description of the ground state of a small molecule, say water, comparable in quality to that provided by a MO approach capable of achieving chemical accuracy, for example, CCSD(T) (CC with singles, doubles and perturbative corrections for connected triple excitations) is far from straightforward and cannot be decided upon through any other but an extremely time-consuming trial-and-error approach. The BOVB method suggested by Hiberty et al.11,12 employs a multi-configuration classical VB-style wavefunction. Each ‘breathing’ orbital is a variational HAO expanded just in terms of the basis functions on a single centre. The BOVB wavefunction includes all Lewis structures that are required to describe the expected bond-breaking and bond-formation processes in a system in VB terms. Each VB structure is constructed from a different set of orbitals in order to accommodate the changes in the environments experienced by orbitals within different covalent and ionic structures. The orbitals and structure coefficients are determined variationally, by minimising the energy expectation value corresponding to the BOVB wavefunction. When dealing with larger molecules BOVB makes use of a core of doublyoccupied orbitals, which can be localised or delocalised. The two different choices are treated as two different levels of BOVB theory, denoted by ‘L’ and ‘D’ respectively. It should be mentioned that if the doubly-occupied orbitals are determined variationally one would expect the difference between these two levels to disappear, although it is possible that the different initial guesses for the core orbitals in the ‘L’ and ‘D’ schemes may guide the optimisation procedure to different local minima. Other levels of BOVB theory are distinguished according to whether ionic electron pairs are described by doubly-occupied orbitals (‘D’) or pairs of singlyoccupied orbitals (‘S’). Using these symbols, BOVB wavefunctions can be classified as ‘L’, ‘SL’, ‘D’, or ‘SD’. A number of computational studies performed using the BOVB method (see e.g. ref. 12 and references therein) indicate that this method includes some of the dynamic correlation energy which is missing in approaches such as GVB-PP-SO and SC. This dynamic correlation energy comes from the fact that BOVB allows the orbitals in different structures to become different which, for a wavefunction with N active electrons and L structures leads to an ‘N electrons in LN orbitals’ construction. A BOVB wavefunction of this type can perform better than its GVB-PP-SO and SC counterparts which represent ‘N electrons in N orbitals’ constructions. In certain circumstances, it can even perform better than an ‘N in N’ CASSCF wavefunction. However, it is not fair to compare a BOVB wavefunction with N active electrons and L structures to an ‘N in N’ CASSCF wavefunction as its direct Chem. Modell., 2008, 5, 312–349 | 327 This journal is

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CASSCF equivalent would be an ‘N in LN’ CASSCF construction. An ‘N in LN’ CASSCF wavefunction will always perform better than a BOVB wavefunction with N active electrons and L structures, as it is not handicapped by the restrictions imposed on the basis functions that can contribute to each ‘breathing’ orbital. It would be interesting to perform the following numerical experiment, which should be within the capabilities of the VBSCF codes in GAMESS-UK:26 Starting with an optimised BOVB wavefunction, let us substitute the HAOs by MOs, using the HAOs as initial guesses for the MOs. A re-optimisation of the resulting wavefunction will lead to a lower ground-state energy, due to the increased variational flexibility of the orbitals, and it is also to be expected that if the system is a neutral molecule, the weights of the ionic structures will decrease considerably in comparison to those within the BOVB description. It is important to add that Penotti’s OBS-GMCSC results reported in ref. 68 include sizable dynamic correlation effects which emphasises the fact that the inclusion of dynamic correlation energy in BOVB is due to its multiconfiguration nature and is not associated with the use of ‘breathing’ orbitals. The VBCI method34–36 can be viewed as a MRCI extension of the VBSCF approach. This method, which has been developed as a spin-free approach, starts with the calculation of a VBSCF wavefunction. The orbitals used to construct the initial wavefunction are formed as linear combinations of AOs from different subsets (or ‘blocks’) as in eqn (3.6). The virtual orbitals needed for the additional VBCI configurations come from the orthogonal complements to the occupied orbitals for each subset from the original VBSCF wavefunction. The most convenient way of finding these virtual orbitals to diagonalise the representation of the projection onto the occupied space operator for each subset, X PA ¼ jcIm iðD1 ð3:32Þ A ÞIm;Jn hcJn j Im2A;Jn2A

in the basis functions making up the subset. D1 A denotes the inverse of the overlap matrix with elements hcIm|cJni hIm, Jn A Ai. The eigenvectors corresponding to zero eigenvalues provide the required virtual orbitals. Incidentally, the correct expression for the representation of PA in the basis functions from subset A, w PA = SACAD1 A CA SA

(3.33)

where SA is the matrix of overlaps between basis functions hwp|wqi(p,q A A) and CA is the matrix assembled from all columns of coefficients for occupied orbitals cIm(Im A A), differs from the expression reported in refs. 34 and 36 which omits the first factor, SA. Excited structures are obtained by substituting an occupied orbital expanded in a certain subset by a virtual orbital expanded in the same subset. The VBCIS, VBCISD, VBCISDT wavefunctions involve, as suggested by the acronyms, single, single and double, single, double and triple excitations, respectively. One conclusion following from a very thorough VBCI and BOVB study of the reaction barrier for the hydrogen exchange reaction35 is that the VBCISD results are usually sufficiently accurate and there is no need to include higher excitations. In ref. 36 Song et al. introduced an analogue of Davidson’s correction71 for VBCISD wavefunctions, which provides an estimate of the contribution of quadruple excitations that are products of double excitations to the correlation energy. This has been shown to alleviate, just as in the MO case, the size-consistency problem in calculations on small molecules.

4. Applications of VB theory 4.1 Modern VB descriptions of electronic reaction mechanisms The electronic structure rearrangements that take place as a chemical reaction progresses from reactants, through one or more transition states and, possibly, reaction intermediates, to products, are reflected by changes of the characteristics of 328 | Chem. Modell., 2008, 5, 312–349 This journal is

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individual orbitals and within the total electronic density of a suitable wavefunction describing the reacting system. If this wavefunction is based on a multiconfiguration ansatz, then the variation of the relative weights of individual configurations along the reaction pathway can also provide useful information about the reaction mechanism. Several very popular and successful interpretations of electronic reaction mechanisms employ orbital models, for example, the Woodward-Hoffmann rules,72,73 Fukui’s frontier orbital theory,74 the Dewar-Zimmerman treatment.75–77 One common feature of these models is that they are derived from very low-level theory which, as a rule, is the Hu¨ckel molecular orbital (HMO) approach or its extended (EHT) version. This implies a many-electron wavefunction in the form of a single Slater determinant. The qualitative features of the orbitals used in the HMO and EHT-based models do not change much at higher levels of theory if the wavefunction remains a single Slater determinant and are usually well-reproduced by ab initio HF calculations. However, now it is wellrecognised that the accurate description of the bond-breaking and bond-making processes that take place during a chemical reaction requires a more flexible and complicated ab initio wavefunction constructed at or beyond the CASSCF level of theory. A wavefunction of this type typically involves a large number of configurations built from orbitals, the most important of which, those in the active space, have fractional occupation numbers. As a rule, such a wavefunction is too complicated to allow qualitative interpretations similar, say, to the ideas of Woodward and Hoffmann. Orbital shapes can be considered to have physical significance only if each orbital is either singly or doubly-occupied or, at least, has an occupation number very close to one or two. In the context of a multiconfiguration wavefunction which involves several configurations with comparable weights, as it is often the case at or in the vicinity of a TS, the individual orbitals have little, if any physical relevance and should be regarded as no more than mathematical components of the wavefunction ansatz. The obvious conclusion is that the wavefunctions which are most appropriate for generating orbital-based interpretations of reaction pathways are those that incorporate a single product of orbitals. If we leave aside the HF wavefunction which is not sufficiently flexible and is prone, in its closed-shell form, to wavefunction instabilities, the choice of singleorbital-product wavefunctions is limited to spin-projected HF (spin-PHF) (see e.g. ref. 78 and references therein), half-projected HF (HPHF),79,80 GVB and SC wavefunctions. The first two of these approaches, though free of the wavefunction instabilities plaguing the closed-shell HF method, are no longer in wide use. The presence of a single spin eigenfunction of the perfect pairing type within the most common variant of the GVB wavefunction, GVB-PP-SO, very much restricts its ability to follow more complicated electronic structure rearrangements, for example those that take place during a pericyclic reaction passing through an aromatic TS. The SC wavefunction is much more versatile, as a consequence of the fact that it includes, by definition, all spin-coupling modes between the set of active orbitals which is sufficient to accommodate all possible bond-breaking and bond-making processes. An alternative treatment is offered by Bader’s AIM approach27 which can be used to extract potentially very useful information from the changes in the total electronic density of the reacting system. However, it is not straightforward to transcribe this information into, say, a reaction scheme that makes use of full and half-arrows which any chemist would find easy to understand. The modern VB description of the electronic mechanism of a chemical reaction based on SC theory involves two steps. Firstly, the transition structure and a sequence of geometries along the reaction path, also known as the IRC, in the directions of reactants and products, are calculated using an existing efficient implementation of a high-level MO method. In principle, the TS can be optimised Chem. Modell., 2008, 5, 312–349 | 329 This journal is

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at the SC level of theory using the CASVB module in MOLPRO,25 but it is usually more convenient to use the efficient geometry optimisation and IRC algorithms implemented in GAUSSIAN,22 in conjunction with a more conventional approach such as CASSCF, MP2 or DFT. This is followed by SC calculations at the geometries along the reaction path and a detailed analysis of the results of these calculations which involves examining the variations in the shapes of the SC orbitals and in the changes to the mode of spin coupling. This methodology has provided new, interesting and often unexpected insights into the electronic mechanisms of a number of reactions, including the Diels-Alder reaction between butadiene and ethene59 and the hetero-Diels-Alder reaction of acrolein and ethene,81 the electrocyclic isomerisation of cyclobutene to cis-butadiene,82 the electrocyclisation of hexatriene,83 the 1,3-dipolar additions of fulminic acid to ethyne,84,85 of diazomethane to ethene86 and of methyl azide to ethene,87 the electrophilic addition of hydrochloric acid to ethylene,88 SN2 identity reactions,89 the addition reactions of singlet dihalocarbenes with ethene,90 the Claisen Rearrangement of allyl vinyl ether,91 the [1s, 5s] hydrogen shift in (Z)-1,3pentadiene.92 The SC models of these reactions have been shown to provide a theoretical vindication of the popular homolytic and heterolytic reaction schemes, drawn using half-arrows (‘harpoons’) and full-arrows, respectively, that can be found in many organic chemistry textbooks, in a context which is slightly different from the standard textbook interpretation, but makes very good sense from a quantum-chemical viewpoint: The half-arrows indicate changes in the shapes of individual orbitals, accompanying the breaking of the bonds in which these orbitals participate in the reactants and their re-engagement in new bonds within the product, rather than movements of electrons; the full-arrows correspond to relocations of orbital, rather than electron pairs. An important feature of the SC models for the electronic mechanisms of pericyclic reactions is the possibility to identify aromatic transition states, by analogy with the well-known SC description of benzene.58,93,94

4.1.1 Diels-Alder reactions. The SC wavefunction used to describe the electronic mechanism of the Diels-Alder reaction between cis-butadiene and ethene59 involves six active orbitals which are sufficient to accommodate the four butadiene and two ethene p electrons participating in the bond-breaking and bond-formation processes. Each orbital turns out to be well-localised about one carbon atom and remains associated with the same carbon atom throughout the reaction, with the main changes being in the degree of spx character and in the extent and direction of the deformations of the orbitals. Initially, the p bonds in butadiene are formed by the symmetry-related pairs (c1, c2) and (c3, c4), while the pair (c5, c6) corresponds to the p bond in ethene (see the right-hand column of Fig. 4 which shows the symmetry-unique SC orbitals c1, c2 and c6 at a displacement of 0.6 amu1/2bohr from the TS towards the reactants). At the TS (middle column of Fig. 4), it is not difficult to notice the emerging distortions of c1 and c6 towards one another, as well as that of of c2 in the direction of its symmetry-related counterpart c3. At the same time, there is an obvious decrease in the overlap between c1 and c2. As shown in Fig. 5(a), in the vicinity of the TS the overlaps between all SC orbitals participating in breaking and forming bonds become very much the same. Continuing towards reactants (see the left-hand column of Fig. 4), orbitals c1 and c6 become much more sp3-like, and correspond to one of the two new s bonds. Similarly, the pair (c2, c3) depicts the new p bond. The changes in the shapes of the orbitals are accompanied by a re-coupling of the electron spins. For this reaction, it proves most convenient to express the total active-space spin function Y600 in the Rumer basis. As shown in Fig. 5(b), the two Kekule´-like functions (1–2, 3–4, 5–6) and (1–6, 2–3, 4–5) are dominant over the 330 | Chem. Modell., 2008, 5, 312–349 This journal is

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Fig. 4 Symmetry-unique spin-coupled orbitals for the Diels-Alder reaction. The remaining orbitals can be obtained from those shown through reflections in the sh plane passing through the middles of the ethene and central butadiene C–C bonds.

entire IRC segment considered, with one corresponding to the spin-coupling within reactants and the other to that within the product. These two Kekule´-like functions attain equal weight in the vicinity of the TS. At this stage the orbital overlaps, the mode of spin coupling and the estimated ‘resonance energy’59 are all strongly reminiscent of the SC description of benzene58,93,94 and so there are good reasons to argue in favour of an ‘aromatic’ transition state. In this way, the SC study of the electronic mechanism of the Diels-Alder reaction59 was able to provide the first evidence of an ‘aromatic’ TS coming directly from the analysis of a post-HF wavefunction. The changes in the SC orbitals and spin-coupling mode during the course of the reaction strongly suggest that the best schematic representation of the Diels-Alder reaction between cis-butadiene and ethene is through a homolytic mechanism, in which six half-arrows indicate the simultaneous breaking of the

Fig. 5 (a) overlap integrals and (b) spin-coupling weights for the Diels-Alder reaction.


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three p bonds on the reactants and formation of the three new bonds, two s and one p, in the product:

The SC description of the electronic mechanism of the hetero-Diels-Alder reaction of acrolein (H2CQCH–CHQO) and ethene81 was found to be very similar to that of the Diels-Alder reaction between cis-butadiene and ethene. Despite the fact that this concerted reaction is markedly asynchronous, with the breaking of the carbon– oxygen p bond, and the formation of the new carbon–oxygen s bond, ‘lagging behind’ somewhat the other bond-making and bond-breaking processes, this reaction can be described through a simplistic homolytic scheme similar to Scheme A, and it can be argued that the reacting system passes through a geometry, soon after the TS along the reaction path in the direction of the product, at which it can be considered to be significantly aromatic. 4.1.2 1,3-dipolar cycloadditions. The nature of the electronic rearrangements that take place during 1,3-dipolar cycloaddition (13DC) reactions is one interesting source of controversy amongst theoretical chemists. In the case of the 13DC reaction between fulminic acid and ethyne, both the SC84 and restricted HF (RHF)95,96 treatments support a heterolytic mechanism illustrated by Scheme B1 (see below) which involves the movement of electron pairs that are retained throughout the course of the reaction (the N to O arrow indicates that the corresponding N–O p bond is strongly polarised towards the oxygen atom). However, in the case of the 13DC reaction involving diazomethane and ethene, the SC analysis86 results in Scheme C, while RHF theory95,97,98 suggests an electron flow in the opposite direction, of the type illustrated by Scheme B2.

Nguyen et al.99 argued that the 13DC reaction between fulminic acid and ethyne should be described by Scheme B2. Use was made of two approaches, the first of which, CI-LMO-CAS, involves localisation of the ‘6 in 5’ CASSCF active-space orbitals and analysis of the weights of configurations expressed in terms of these localised orbitals. However, the two configurations responsible for the reaction mechanism do not dominate the CASSCF wavefunction, but have a combined weight that never exceeds ca. 44%. The ‘6 in 5’ active space requires one of the active orbitals to be doubly-occupied and this orbital localises about the more electronegative atoms, O and N, which explains the direction of the leftmost arrow in Scheme B2. The second approach which employed DFT-based reactivity descriptors for fulminic acid can be considered to have a limited predictive power because it examined an isolated molecule which was unable to ‘feel’ the influence of the other 332 | Chem. Modell., 2008, 5, 312–349 This journal is

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reactant and had a linear equilibrium geometry that was very different from its conformation near the TS. A qualitative classical VB analysis carried out by Harcourt100 led to the formulation of a ‘concerted biradical’ mechanism for 13DC reactions (see also ref. 101 and references therein). This combination of terms may sound unusual to those dealing with pericyclic reactions (as a rule, within the accepted terminology ‘biradical mechanism’ is synonymous with a non-concerted, stepwise mechanism involving a biradical intermediate), but in this instance ‘biradical’ was used to denote a homolytic electronic rearrangement. In another paper, Harcourt and Schulz102 discussed possible homolytic mechanisms for the 13DC reaction between fulminic acid and ethyne. These mechanisms are illustrated by Schemes B3 and B4. The dots on the two sides of the O–N bond in B3 represent singly-occupied px (ON) and py (ON) localised molecular orbitals (LMOs), while the thick and thin lines in B3 and B4 reflect Harcourt’s notation for normal and fractional electron-pair bonds, respectively. According to Harcourt and Schulz, the preferred scheme should be B3. The presence of mutually perpendicular LMOs of p symmetry in this scheme is associated with the fact that the analyses of the reaction mechanisms corresponding to B3 and B4 in ref. 102 were based on the electronic structure of HCNO at its linear gas phase equilibrium geometry and did not take into account the presence of the second reactant and the changes in the geometry of the reacting system along the reaction path. As a rule, the wavefunctions used by Harcourt and co-workers are molecule-specific, carefully handcrafted classical VB constructions which are defined within a minimal set of basis functions. Wavefunctions of this type cannot ordinarily be used to provide a consistent picture of electronic structure changes along a general reaction path, as such hand-tuning is likely to require different wavefunction constructions at different points along this path. Harcourt and Schulz criticised Scheme B1 (and, effectively, Scheme B2) stating that this scheme should involve charge transfer between the species. This is not correct: The 13DC reaction between fulminic acid and ethyne is a concerted, almost synchronous process and so all electron rearrangements depicted by Schemes B1 and B2 take place simultaneously, without any noticeable charge transfer between the reactants. Although the terms ‘heterolytic’ and ‘homolytic’ can be used to distinguish between mechanisms B1 and A, these do not imply the existence of any biradical or zwitterionic intermediates (see ref. 59). Sakata103 carried out a HF-level population analysis along the reaction path of the 13DC reaction between fulminic acid and ethyne. His results suggest a mechanism described by Scheme B5, which is very similar to B1 and in line with previous HFlevel results. Nguyen et al.104 initiated a discussion about the existing models for the electronic mechanism of the HCNO + C2H2 13DC reaction which was continued in refs. 85 and 101. In their opening comment, Nguyen and co-workers claimed that the contrast between the predictions coming from two very different VB treatments (compare Scheme B184 to B3 and B4102) was as indication that VB theory was unable to provide credible coherent results about the electronic structure rearrangements that take place during 13DC reactions. In fact, although the results of the existing SC calculations had suggested the heterolytic mechanisms B1 and C for the 13DC reactions between fulminic acid and ethyne, and diazomethane and ethene, respectively, no restriction within the wavefunction ansatz would prevent SC theory from predicting homolytic mechanisms for other 13DC reactions. This was confirmed by the SC analysis of the electronic mechanism of the 13DC reaction between methyl azide (CH3N3) and ethene87 which showed that this reaction should follow a homolytic mechanism. Let us briefly examine the most important features of the heterolytic and homolytic mechanisms proposed by SC theory for the 13DC reactions between diazomethane and ethene,86 and methyl azide and ethene.87 The changes in the shapes of the SC orbitals during the 13DC reactions between diazomethane and ethene are illustrated in Fig. 6. Chem. Modell., 2008, 5, 312–349 | 333 This journal is

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Fig. 6 Spin-coupled orbitals for the 13DC of diazomethane to ethene at ‘after’-TS, TS and ‘before’-TS geometries (RIRC = 1.2, 0, 1.2 amu1/2bohr). Contour plots of the orbitals in the plane of the heavy atoms; dotted lines correspond to negative values.

The six SC orbitals from the rightmost column in Fig. 6 fall into two distinct groups, each of which is associated with one of the two reactants. The orbital pair (c3, c4) is responsible for the ethene carbon–carbon p bond. The remaining four orbitals are localised on the diazomethane fragment and very much reproduce the well-known SC description of isolated CH2N2106,107 according to which the central nitrogen atom is ‘hypercoordinate’, taking part in more than four covalent bonds: an almost double bond to C and an almost triple bond to the terminal N. In an undistorted C2v-symmetry diazomethane molecule, the pair (c5, c6) would describe the p component of the carbon–nitrogen bond, while the pair (c1, c2) would correspond to one of the p components (the one that points out of the molecular plane) of the nitrogen-nitrogen multiple bond.


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Upon proceeding to the TS (see the middle column in Fig. 6), the most pronounced changes are observed in c3 and c4. Later on, these orbitals become involved in the forming carbon–carbon bond (see the leftmost column in Fig. 6). The carbons to be joined by this bond are separated by a shorter distance than the carbon and nitrogen atoms from the other forming bond throughout the IRC interval considered and, as a consequence, the major changes in the shapes of the orbitals involved in that other bond, c1 and c2, occur after the TS. The changes in the shapes of the remaining two orbitals, c5 and c6, are relatively minor and suggest that at the TS these orbitals still realise a partial p bond between the carbon and central nitrogen atoms of the CH2N2 moiety. After the TS, both orbitals shift in the direction of the nitrogens to form the lone pair on the central nitrogen. Throughout the part of the reaction path studied at the SC level of theory, the active-space spin-coupling pattern Y600 remains markedly dominated by its PP component, in which the spins of the strongly overlapping orbital pairs (c1, c2), (c3, c4), and (c5, c6) are singlet-coupled. As there is no evidence of significant resonance that could be associated with aromatic properties, it is safe to assume that the reacting system remains non-aromatic throughout the course of the reaction. This analysis clearly indicates that the orbital pairs (c1, c2), (c3, c4) and (c5, c6) are retained throughout the 13DC of diazomethane to ethene. The orbital pair shifts responsible for the bonding rearrangements associated with this reaction can be summarised through the homolytic Scheme C in which the leftmost, middle and rightmost arrows depict the shifts of (c5, c6), (c1, c2) and (c3, c4), respectively. Apart from some relatively minor differences in the ‘timelines’ of changes in the shapes of some of the orbitals along the reaction path, this mechanism is very similar to that of the 13DC between fulminic acid and ethyne.84 The SC descriptions of the electronic mechanisms of the 13DC between fulminic acid and ethyne, and between diazomethane and ethene involve bond rearrangements, achieved through the movement of singlet orbital pairs through space, during which at least one of the orbitals within a pair becomes completely detached from the atomic centre with which it was associated initially and ends up localised about another centre. The ability of the SC wavefunction to produce a description of this type follows from the fact that the SC orbitals are singly-occupied non-orthogonal MOs. A more orthodox VB approach, insisting on strict localisation of the orbitals such as, for example, BOVB would need a number of additional and, in this case, rather unphysical, ionic structures in order to compensate for the insufficient flexibility of the orbitals throughout the reaction path. The SC wavefunction used to describe the electronic mechanism of the 13DC reaction between methyl azide and ethene involves eight active orbitals.87 The changes in the shapes of the SC orbitals during this reaction are illustrated in Fig. 7. At the ‘before’-TS geometry (see the top group of orbitals in Fig. 7), the pair (c4, c5) corresponds to the ethene p bond, while the remaining six orbitals are all on the methyl azide fragment. The pair (c1, c6) is responsible for the nearly-p bond between the central N and the N connected to the methyl group, while the two nearly-p bonds between the central and terminal nitrogens are described by the orbital pairs (c2, c3) and (c7, c8). The ‘before TS’ active space spin-coupling pattern is strongly dominated by reactant-like Rumer spin functions, which couple to singlets the spins of orbitals residing on the same reactant only. Of these, the most important one is (1–6, 2–3, 4–5, 7–8), which is in line with the assignment of orbital pairs to bonding interactions. The shapes of the orbitals at the TS (see the middle group of orbitals in Fig. 7) are more product-like. Two new pairs, (c3, c4) and (c5, c6), reflect the formation of the two new bonds closing the ring, c1, and c2 are to become responsible for the lone pair on the central nitrogen in the product, while c7 and c8 now clearly form the almost p component of the NQN bond. The main contribution to the active space spin-coupling pattern now comes from the Rumer spin eigenfunction (1–2, 3–4, 5–6, 7–8) in which the spins of all pairs of active orbitals which are becoming involved in Chem. Modell., 2008, 5, 312–349 | 335 This journal is

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Fig. 7 Spin-coupled orbitals for the 13DC of methyl azide to ethene before the TS, at the TS, and after the TS. The ‘before’ and ‘after’-TS geometries correspond to IRC points 1.2 amu1/2bohr away from the TS.

bonds or in the central nitrogen lone pair in the product are coupled to singlets. Second in importance is the Rumer spin eigenfunction (1–2, 3–6, 4–5, 7–8) which, in contrast to the first Rumer spin eigenfunction, is reactant-based: It couples to singlets the spins of the active orbitals that are becoming engaged in the central nitrogen lone pair (c1 and c2), the orbitals that were involved in the ethene p bond (c4 and c5), the orbitals that initially described one of the components of the NRN bond in methyl azide and later, the almost p component of the NQN bond in the product (c7 and c8), and of the two orbitals which reside on different ends of the 1,3dipole: c3 and c6.


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As the reactants approach one another, the weight of Rumer spin function (1–2, 3–6, 4–5, 7–8) starts to increase, in parallel with the decrease of the weight of spin function (1–6, 2–3, 4–5, 7–8), and peaks near the TS. This finding, in combination with the observed changes in the shapes of the valence orbitals and their overlaps indicates that, on approaching the TS, the bonds realised by the orbital pairs (c1, c6) and (c2, c3) gradually weaken, and that this is accompanied by the formation of a lone pair on the central nitrogen, represented by the singlet-coupled orbitals c1 and c2. As the new bonds involving c3 and c6 are not yet fully developed, the active space spin function in the vicinity of the TS displays competition (or ‘resonance’) between two spin-coupling patterns: the product-like (1–2, 3–4, 5–6, 7–8) and the reactant-based (1–2, 3–6, 4–5, 7–8). These patterns differ only in the mode of coupling the spins of the four orbitals responsible for the two new bonds closing the ring: c3, c4, c5 and c6. The large separation and low overlap between orbitals c3 and c6 suggests that the presence of the less important pattern, (1–2, 3–6, 4–5, 7–8) is an indication that the reacting system attains, near the TS, some singlet biradical character. However, there is not even a hint of a benzene-like ‘resonance’ as observed near the transition states of the Diels-Alder reaction between butadiene and ethene59 and the electrocyclisation of hexatriene.83 The ‘after-TS’ SC picture (see the bottom group of orbitals in Fig. 7) shows a further development of the product-like features observed at the TS. The only particularly noticeable changes in the forms of the SC orbitals are some compacting of the lone pair orbitals c1 and c2. The active-space spin-coupling pattern is largely dominated by the product-like spin eigenfunction (1–2, 3–4, 5–6, 7–8), while the weight of the reactant-based singlet biradical pattern (1–2, 3–6, 4–5, 7–8) has become so low that it is safe to discard its contribution. Throughout the 13DC reaction of methyl azide with ethene, each SC orbital remains distinctly associated with a single atom while its form, overlap with other SC orbitals and participation in the active-space spin-coupling pattern adjust to accommodate the differences in the nature of the bonding in reactants and product. The bond-breaking and bond-formation processes realised in this way can be illustrated through the following homolytic scheme:

4.1.3 The [1s, 5s] hydrogen shift in (Z)-1,3-pentadiene. Various geometric, energetic and magnetic criteria strongly suggest that the TS of the [1,5]-H shift in (Z)-1,3pentadiene is aromatic.108,109 As a consequence, it would have been reasonable to expect that the SC description of the reacting system near the TS would show benzene-like features and very much repeat the picture observed for the prototypical Diels-Alder reaction (see section 1). In fact, the SC description of the TS of the [1,5]H shift in (Z)-1,3-pentadiene92 turned out to be markedly different: The active space was found to involve two strongly interacting three-orbital moieties, one of which (centred on the carbon opposite the shifting hydrogen) is almost identical to the three-orbital ‘antipair’ SC picture of the p active space in the allyl radical,110 while the second one (centred on the shifting hydrogen) is very similar in nature, but involves more s-like orbitals. However, in previous SC work ‘antipairs’ had often been associated with antiaromaticity and radical or diradical character (see e.g. the SC description of square cyclobutadiene in ref. 111). The detailed analysis carried out in ref. 92 showed that, on their own, ‘antipairs’ should not be considered as a sign of antiaromaticity or diradical character, as these can be observed even within Chem. Modell., 2008, 5, 312–349 | 337 This journal is

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Fig. 8 Spin-coupled orbitals for the [1,5]-H shift in (Z)-1,3-pentadiene.

an alternative p space SC wavefunction for benzene. Therefore, ‘antipairs’ can be expected to feature within the SC descriptions of reactions which follow heterolytic mechanisms (involving movements of orbital pairs) and pass through aromatic transition states; this was observed for the first time in the SC study of the [1,5]-H shift in (Z)-1,3-pentadiene. The [1,5]-H shift in (Z)-1,3-pentadiene is an example of a degenerate pericyclic process. As a consequence, the corresponding IRC is symmetric with respect to the TS and it was sufficient to examine the changes in the SC wavefunction in just one of the two possible directions starting at the TS. The shapes of the six SC orbitals at the reactant geometry (which coincides with that of the product), at two intermediate IRC points (0.1, amu1/2bohr and 0.6 amu1/2bohr), and at the TS are shown in Fig. 8. At the reactant/product geometry (see the leftmost column in Fig. 8), the six SC orbitals in are engaged in three well-defined bonds. The pair (c1, c2) is responsible 338 | Chem. Modell., 2008, 5, 312–349 This journal is

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for s bond attaching the hydrogen that migrates during the course of the reaction from the leftmost carbon (C1) to the rightmost carbon (C5), while the remaining two pairs, (c3, c4) and (c5, c6), describe the diene p bonds. With the progress of the reaction towards the TS, both orbitals from the pair (c1, c2) develop distortions towards C5. Orbital c1 remains always associated with the migrating H atom while, by the end of the reaction, its partner c2 shifts from C1 onto C5. At the TS, c2 looks very much like the in-phase superposition of two sp3-like hybrids, based on C1 and C5, respectively. The changes within orbital pair (c3, c4) along the reaction path closely parallel those within (c1, c2). Here orbital c4 remains permanently attached to C3, while c3 relocates from C2 onto C4. At the TS c1, c2, c3 and c4, are fully symmetric with respect to the sh plane passing through the migrating H and C3. Away from the TS, the pair (c5, c6) is responsible for a p bond (between C4 and C5 in the reactant, or between C1 and C2 in the product). At the TS, c5 and c6 are semilocalised over two atomic centres and both orbitals are antisymmetric with respect to the sh plane. The analysis of the composition of the active-space spin-coupling pattern, together with evidence coming from orbital overlaps, clearly indicates that the orbital pairs (c1, c2), (c3, c4) and (c5, c6) are preserved all the way from the reactant, through the TS, to the product. The movements of these three orbital pairs during the [1,5]-H shift in (Z)-1,3pentadiene can be described using Scheme E

which suggests that the reaction proceeds through a heterolytic mechanism that is markedly different from that depicted in the textbook-style Scheme F.

Scheme E involves an unusually long-range relocation of orbital pair (c5, c6) which, at first glance, is difficult to accept. However, a closer examination of the more traditional Scheme F shows that the required movements of the orbital pairs responsible for the diene p bonds cannot be described by a SC wavefunction which reproduces the Cs symmetry of the TS. The shifts of orbital pair (c1, c2) and (c3, c4) shown in Scheme E are consistent with the Cs symmetry of the TS, but then the only option left to orbital pair (c5, c6) is to embark on a long-range journey across the ring. In order to achieve a better understanding of the electronic structure of the TS for the [1,5]-H shift in (Z)-1,3-pentadiene, the orbitals in the SC wavefunction were reordered as C600 = A [(core)c1c2c3c4c5c6Y600] = A [(core)c2c6c1c4c3c5Y600 0 ]

(4.1)

where 0

Y600 ¼

5 X

S

0

C0k S Y600;k

ð4:2Þ

k¼1

stands for the active-space spin function corresponding to the new ordering of the active orbitals, expressed in the Serber spin basis. This new ordering of the SC orbitals allowed the identification of two ‘antipairs’ , involving the orbitals c3 and Chem. Modell., 2008, 5, 312–349 | 339 This journal is

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Fig. 9

Symmetry-unique SC orbitals from the ‘antipair’ description of benzene.

c5, and c2 and c6, respectively. The spins of the orbitals from an ‘antipair’ are almost entirely triplet-coupled, which is not straightforward to notice within a spin basis other than the Serber one. The orbital shapes and coupling of spins within the second group of three orbitals in eqn (4.2), c4, c3 and c5, closely resemble the SC description of the p space of the allyl radical.110 The first three orbitals, c2, c6 and c1, are of predominantly s character but exhibit similar properties. This shows that at the TS the SC active space for the [1,5]-H shift in (Z)-1,3-pentadiene involves an allyl-like moiety and its s-space equivalent. Another example of a SC active space comprised of two allyl-like moieties involving ‘antipairs’ has been observed in the TS for the Cope rearrangement of 1,3,4,6-tetracyano-1,5-hexadiene.112 The SC wavefunction at the TS for the [1,5]-H shift in (Z)-1,3-pentadiene is very different from its counterpart in the case of the Diels-Alder reaction (see section 4.1.1), and from the well-known SC description of benzene, in both of which the SC orbitals are numbered in a clockwise fashion around the ring. However, as shown in ref. 92, if the orbitals in benzene are ordered as in eqn (4.2) and the active-space spin-coupling pattern is expressed in the Serber spin basis, it becomes very similar to that at the TS for the [1,5]-H shift in (Z)-1,3-pentadiene. An additional symmetry-constrained SC calculation on benzene produced an ‘antipair’ solution which is just about 1 mhartree above the well-known unconstrained solution with localised orbitals. The orbitals from this ‘antipair’ solution are shown in Fig. 9 (the orbitals from the standard solution are visually indistinguishable from c1). As the very small energy separation between the standard and ‘antipair’ SC solutions for benzene makes the latter a viable alternative description of the molecule, it can be argued that, by inference, the TS for the [1,5]-H shift in (Z)-1,3-pentadiene should be considered to be aromatic. The fact that it is possible to devise a SC wavefunction for benzene which involves ‘antipairs’ and is just 1 mhartree above the well-known standard SC wavefunction, indicates that the presence of ‘antipairs’ in the SC description should not be considered, on its own, as a reliable indication of antiaromatic, radical or diradical character. 4.1.4 SN2 identity reactions. In ref. 89 SC theory was applied to study the electronic rearrangements taking place during gas-phase SN2 identity reactions X + RX - XR + X where X = Cl and R is methyl, ethyl or t-butyl, and X = F and R is methyl. As the generally accepted notion is that these reactions involve four active electrons, two from the forming s bond and two from the breaking s bond, the SC calculations were performed with a wavefunction including four active orbitals. For all SN2 identity reactions studied it was observed that these four orbitals are engaged in two pairs, both of which are preserved during the course of the reaction. Initially, one of the orbitals pairs is on the incoming nucleophile and the other one is engaged in the R–X bond. At the TS, the bond-formation and bond-breaking processes are almost equally advanced and the pairs are responsible for the bonding interactions between R and the incoming and leaving nucleophiles (see Fig. 10). Beyond the TS, the pair that was initially on the 340 | Chem. Modell., 2008, 5, 312–349 This journal is

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Fig. 10 SC orbitals at the transition states of the SN2 identity reactions involving (a) CH3Cl and Cl, (b) CH3CH2Cl and Cl and (c) C(CH3)3Cl and Cl.

incoming nucleophile becomes engaged in the forming bond, while the other pair localises onto the leaving nucleophile. It is interesting to compare the shapes of the SC orbitals at the transition states of the reactions involving CH3Cl and Cl, CH3CH2Cl and Cl and C(CH3)3Cl and Cl (see Fig. 10). While the transition states for the CH3Cl + Cl and CH3CH2Cl + Cl reactions show well-defined Cl–C–Cl bonding interactions, slightly weakened for the second reaction, the SC model for the TS of the SN2 gas-phase identity reaction between C(CH3)3Cl and Cl can be described fairly accurately as involving a C(CH3)3+ carbocation ‘clamped’ between two Cl ions. This suggests a tendency for a switch from the SN2 to the SN1 mechanism. A mechanism of this type cannot be described within a gas-phase study, but it is obvious that in the presence of suitable solvent molecules the components of the TS would separate to form individual solvated C(CH3)3+ and Cl ions. Although the SC descriptions of the CH3Cl + Cl and CH3F + F gas-phase SN2 identity reactions were found to be qualitatively similar, a significantly larger extent of bond formation was observed at the TS for the fluorine case, with electronic rearrangements starting within a much earlier region of the reaction path. Another VB study of SN2 identity reactions has been reported by Song, Wu, Hiberty and Shaik.113 These authors applied the BOVB method on its own and within the VBPCM framework (VB coupled with a polarised continuum model)114 to the reactions X + CH3X - XCH3 + X where X = F, Cl, Br and I, in vacuum and in aqueous solution. The VB structure set was chosen so as to include all possible ways of distributing the four electrons of the incoming nucleophile X: and the H3C–X bond between the three fragments incoming X, CH3 and leaving X: X : CH3 X |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

X : : CH Xþ 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

X : CHþ : X 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

X CH3 : X |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

X : CH X 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Xþ : CH : X 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

1

4

2

5

3

6

Structures 1 and 4 can be identified as covalent and describe the electron pairing within the reactant and the product. The most plausible ionic structure is 3, which places a methyl carbocation between two halogen anions. Structure 5 implies a not so plausible combination of a singlet pair involving the unpaired electrons on the two halogens and a negative charge on the methyl fragment. Each of structures 2 and 6 also places a negative charge on the methyl fragment, together with a positive Chem. Modell., 2008, 5, 312–349 | 341 This journal is

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charge on one of the halogens. Structures 2, 5 and 6 contradict chemical intuition and it does not come as a surprise that their combined weight never exceeds 5.6% at a TS, or 11.7% at the ion-molecule complex (both numbers correspond to X = I in aqueous solution). If the orbitals had been allowed to extend over the basis set contributed by all atoms in the reacting system, it can be expected that the combined weight of structures 2, 5 and 6 would have become even lower. The activation barriers for all four reactions, calculated using the BOVB approach and its PCM version were found to compare well to their CCSD(T)-level counterparts, as well as to experimentally derived barriers in solution. 4.2 Aromaticity Aromaticity is one of a number of remarkably important but ‘fuzzy’ concepts in chemistry that does not lend itself easily to widely applicable clear-cut definitions. One of the more recent attempts to summarise what chemists with different backgrounds see in this term was made by Schleyer.115 Valence bond theory has been, since Kekule´’s times, a major source of qualitative and quantitative interpretations of aromaticity, and it is surprising that it has not been mentioned, even once, in Schleyer’s editorial. Of course, this omission does not diminish the importance that VB methods have had and still have in providing easy-to-understand ideas about aromatic behaviour which are often markedly superior to those coming from other theoretical sources. In order to justify this claim from a current perspective, in this section we look at the recent VB studies of the aromaticity of a series of ‘benzenelike’ D6h rings, and of clamped benzenes. 4.2.1 VB-style representations of ‘benzene-like’ rings. It is often intriguing to find out whether a certain six-electron cyclic p-electron system is aromatic or not. Similarity to the paragon of aromaticity, benzene, can be misleading: For example, ‘inorganic benzene’ or borazine (B3N3H6) is well-known to have little, if any, aromatic character.116 Three recent papers117–119 compare the aromaticity of benzene and the ‘benzene-like’ D6h ring systems Si6H6, B6, Al6, N6 and P6, using different VB-style approaches and ring-current calculations. Sakai’s paper117 makes use of his CiLC approach which closely overlaps with the CILMO-CAS approach suggested earlier by the same author and mentioned in section 4.1.2. In general, a CiLC calculation involves three steps: (i) an appropriate preliminary CASSCF calculation; (ii) localisation of the orbitals making up the active space using the Boys procedure;120 (iii) the LMOs obtained in this way are used to construct either a compact CI wavefunction in an attempt to approximate the CASSCF wavefunction using a smaller number of chemically important configurations, or a full CI wavefunction. If the initial CASSCF calculation is of the ‘N in N’ type, the Boys localisation of the active space provides N singly-occupied orthogonal LMOs and one could expect that the results of a subsequent CI calculation in terms of these LMOs would have some common features with those coming from a SC calculation utilising the same number of active orbitals. In fact, as shown in ref. 119, CiLC and SC theory produce very different results, and there are simple but theoretically-sound arguments that indicate that the results to be trusted are the SC ones. The calculations performed on the D6h ring systems C6H6, Si6H6, B6, Al6, N6 and P6 in ref. 119 were of three different types: A. Transformation of the CASSCF wavefunction to an alternative modern-VB representation dominated by a single orbital product ansatz identical to the SC one, using the EVB energy-based criterion [see eqn (3.18)], without any constraints on the overlaps between the active orbitals cm. B. The same as in A, except that all of the active orbitals cm are constrained to be mutually orthogonal. C. Fully variational determination of SC wave functions, imposing s-p separation, so as to enable proper comparisons with A and B. 342 | Chem. Modell., 2008, 5, 312–349 This journal is

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The optimised EVB energies obtained in type A calculations were found out to be reasonably close to their CASSCF counterparts, with energy differences of 7–8 mhartree for the rings based on first-row atoms and 4–5 mhartree for those based on second-row atoms. The weight of the single orbital product ansatz in the CASSCF wavefunctions was always observed to be above 98.5%. These numbers strongly suggest that type A calculations produce very accurate SC-style approximations to the corresponding CASSCF wavefunctions. In fact, the results of the full SC calculations (type C) are almost indistinguishable from the type A results. The orbitals obtained through the type A calculations are shown in columns (a) and (a 0 ) of Fig. 11. In all type A calculations the active-space spin function was found to be dominated by the two Kekule´-type Rumer spin functions, with weights between 38–40% each. All of this is consistent with the traditional VB notions of aromaticity. The mutually orthogonal orbitals produced by Type B calculations are shown in columns (b) and (b 0 ) of Fig. 11. A key difference from those obtained without such orthogonality constraints is a reduction in the deformations towards neighbouring atoms, which are replaced by small orthogonalisation tails on the adjacent centres. There is also a dramatic reduction in the weights of the single Kekule´-type spincoupling modes which are now between 17–18%, with higher net contributions coming from Dewar-type Rumer spin functions. The EVB energies obtained in type B calculations are considerably higher, by between 0.277–0.692 hartree than their type

Fig. 11 Symmetry-unique active orbitals for various D6h ring systems, generated by transforming the CASSCF wave function without constraints on the overlaps between the active orbitals [columns (a) and (a 0 )] and with mutual orthogonality imposed [columns (b) and (b 0 )].


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A counterparts. The weight of the single orbital product ansatz in the CASSCF wavefunctions falls down to between 23–36%, with the rest taken up by singly, doubly and triply ionic structures, of which the singly-ionic ones are dominant, with weights between 47–49%. The weights of doubly ionic structures are also significant, between 17–25%, and even triply ionic structures can have non-negligible contributions (over 2% in B6 and C6H6). Type B calculations, the results of which were demonstrated to be rather disappointing, can be viewed as an enhanced version of Sakai’s CiLC approach. The use of unoptimised Boys-localised orbitals in CiLC can lead only to even greater weights of the ionic structures. The relative aromacities of the various D6h rings studied in ref. 119 were estimated using two criteria, the first of which is the SC resonance energy, defined as the difference between the SC energy and that of a single Kekule´ structure constructed from the same orbitals, Eres = |ESCE1K|. The second criterion is the proportion Qres of the total active-space only electronic energy represented by Eres. Qres provides a better balanced criterion as a low active-space only electronic energy will lead to a low Eres value even if the system exhibits significant resonance. The results strongly suggest the following ordering of the different ring systems according to their degree of aromaticity: C6H6 B B6 4 N6 4 Al6 B Si6H6 4 P6 Sakai’s CiLC aromaticity criterion117 is based on the difference between two combined weights of covalent and singly ionic determinants in the CASSCF wavefunction constructed from LMOs which are associated with the two Kekule´ structures. Each of these weights involves all covalent determinants that place pairs of orbitals with different spins over the ‘double’ bonds in a Kekule´ structure, and the singly ionic determinants that place an electron pair over one of the atoms involved in a ‘double’ bond. A smaller difference is interpreted as an indication of a higher degree of aromaticity. As the combined weight of all covalent and singly ionic determinants makes up a smaller part of the full CASSCF wavefunction than the single orbital product VB components calculated through the type A and type C approaches in ref. 119, it should be expected that the predictions made using the CiLC aromaticity criterion are less reliable than those based on Eres and Qres. Sakai’s criterion predicts B6 to be more aromatic than benzene and Al6 to be more aromatic than N6: B6 4 C6H6 4 Al6 4 N6 B Si6H6 4 P6 A magnetic criterion for the aromaticity of these systems is provided by the contributions from the p bonds to the total nucleus-independent chemical shift (NICS) values calculated for a point 0.5 A˚ above the ring centre. The NICS(p) values reported by Schleyer and co-workers121 are C6H6 (16.8), N6 (15.9), P6 (14.7) and Si6H6 (14.1). All results discussed ref. 119 lead to the conclusion that C6H6 is more aromatic than N6, and that the D6h rings involving first-row atoms are more aromatic than their second-row analogues. On the other hand, all criteria in refs. 117 and 119 suggest that Si6H6 is more aromatic than P6, whereas the fairly close NICS(p) values suggest the opposite order. The C6H6, N6, P6 and Si6H6 (constrained) D6h rings have also been considered by Engelberts et al.118 who used a combination of VB calculations based on strictly atomic non-orthogonal orbitals, fully variational SC calculations, and the analysis of current-density maps in a study of various ‘inorganic benzenes’. All four of these homonuclear systems showed characteristics of aromaticity, similar to those described in ref. 119. C6H6 and N6 were reported to show strong ring currents, with those for the second row analogues being less than half the strength calculated for benzene. 344 | Chem. Modell., 2008, 5, 312–349 This journal is

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Fig. 12 Symmetry-unique SC orbitals for (a) tris(bicyclo[2.1.1]hexano)benzene and (b) tris(benzocyclobutadieno)benzene.

4.2.2 The aromaticity of clamped benzenes. Organic chemists have been interested for a long time in the properties of a non-aromatic isomer of benzene, 1,3,5-cyclohexatriene, or bond-fixed benzene. One way of obtaining this structural unit is by annealing suitable substituent rings (clamping groups) to a benzene molecule. Two examples are provided by tris(bicyclo[2.1.1]hexano)benzene [molecule (a), see Fig. 12(a)] which was synthesised by Siegel et al.,122 and tris(benzocyclobutadieno)benzene [molecule (b), see Fig. 12(b)] which was synthesised by Vollhardt and co-workers.123 Calculations of p current density maps through the ipsocentric approach (see e.g. ref. 124; also known as CTOCD-DZ, continuous transformation of origin of current density-diamagnetic zero) can be used to investigate the aromaticity of molecules containing benzene rings.125–127 Fowler et al.125 confirmed a diatropic ring current in the tris(bicyclo[2.1.1]hexano)benzene, but found no such current (and hence, by the ipsocentric definition, no aromaticity) in the tris(benzocyclobutadieno)benzene. Their results are consistent with the NICS values of 8.0 for the central ring of molecule (a)128 and 1.1 in molecule (b),129 where lower values are indicative of increased aromaticity and benzene has a NICS value of 9.7.130 While both the ipsocentric and NICS methodologies loosely define aromaticity as the occurrence of diatropic ring currents, in ref. 131 molecules (a) and (b) were examined from the VB (and chemically intuitive) viewpoint of resonance. It was shown that modern VB theory in its SC form strongly suggests that molecule (a) retains some of the resonance between electronic bonding structures observed in benzene,58,93,94 whilst the electronic structure of molecule (b) displays no signs of resonance and delocalisation of orbitals across the central ring. Additional insights provided by this investigation are that the aromaticity of molecule (a) is significantly reduced in comparison to benzene, and that the decrease of aromaticity caused by each set of clamping groups is more pronounced than that associated with an equivalent geometric distortion of a unsubstituted benzene ring. The analysis of molecules (a) and (b) in terms of magnetic properties via ipsocentric and NICS methodologies also suggests that molecule (a) retains aromatic character (in this case diatropic ring currents) whilst molecule (b) succeeds in generating a 1,3,5-cyclohexatriene moiety. As the definitions of aromaticity in terms of resonance and magnetic properties differ greatly, it is reassuring to observe that both suggest that molecule (a) is aromatic but (b) is not, which is by no means guaranteed given the insensitivity of NICS to small variations in benzene geometry.117 However, neither of the magnetic properties based methodologies observed the reduction of aromaticity in (a) when compared with benzene to anywhere near the same extent as in the case of SC theory. Chem. Modell., 2008, 5, 312–349 | 345 This journal is

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5. Concluding remarks As it has been demonstrated in this review, VB theory is continuing to provide intuitive, easy-to-visualise explanations of electronic structure and reactivity which in many cases are derived from ab initio wavefunctions of near-CASSCF quality. However, in quantitative terms current-day VB theory is just as far from catching up with MO theory as twenty years ago. The main reason for this is that the number of researchers working on VB-related methodological developments is relatively small and their efforts are not well-coordinated which can be explained by the fact that, on the whole, VB theory does not have the relatively well-organised hierarchical structure exhibited by several groups of popular MO-based approaches. ‘Reinventing the wheel’ is a relatively frequent phenomenon in the VB area, and this does not help progress. In addition to this, despite the significant progress in the VB area, MO theory has not been standing still, but has been developing at an even faster pace, which is wellillustrated by a recent review on coupled-cluster theory by Bartlett and Musiaz.132 An increasing number of experimental chemists are running and publishing the results of their own MO calculations. This almost never happens with VB codes. While this is partially due to the much wider availability of fast MO approaches, coupled with efficient geometry optimisers, the fact that in many cases the VB wavefunctions have to be constructed by hand and the process may involve decisions made on the basis of chemical intuition, can be a major turning-off point. An approach using a wavefunction ansatz pre-tailored for a specific problem can never be as convincing as an approach in which a general wavefunction is left to decide what form to adopt without human intervention. One major step in the right direction for VB theory would be to abandon the Heitler-London model and use Coulson-Fischer ideas throughout. Modern VB approaches which have already taken this step, such as GVB and SC, enjoy wider popularity and require very much the same level of man-made decisions as CASSCF. One important task in front of VB theory is to find ways of making better use of the SC wavefunction. The SC wavefunction is the most general construction which employs a single orbital product. In contrast to the most popular spin-adapted single orbital product wavefunction, the RHF one, the SC wavefunction, very much as its CASSCF counterpart, can be calculated at any point of the potential energy surface for any reaction, provided that the active space contains enough orbitals to describe all bondbreaking and bond-formation processes. This makes the SC wavefunction a very suitable reference for developing higher-level CI, many-body perturbation theory (MBPT) or CC approaches. Several successful post-SC methods have already been formulated, including SCVB,21 SCVB*,62–64 MR-SCVB*,64,65 GMCSC.67,68 However, the real challenge is to develop SC-based equivalents of size-consistent and size-extensive HF-based approaches such as MP2 and CCD. QM/MM approaches, in which the QM part is provided by, or includes a SC construction, could become a powerful tool for studying the electronic mechanisms of biologically-important reactions. Given the huge volume of work on the calculation of molecular properties in MO theory, it is surprising that there is so little effort in this direction within the VB area. The fact that the orbitals used in VB approaches are, as a rule, well-localised, suggests that these approaches would be very suitable for the calculation and interpretation of local properties, such as NMR shielding tensors and nuclear spin-spin coupling constants.133

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Numerical methods in chemistry T. E. Simos DOI: 10.1039/b608781g

A. Newton–Cotes formulae for the numerical integration of the Schro¨dinger equation 1. Introduction In recent years, the research area of construction of numerical integration methods for ordinary differential equations that preserve qualitative properties of the analytic solution was of great interest. Here we consider Hamilton’s equations of motion which are linear in position p and momentum q q_ = mp, p_ = mq,

(1)

where m is a constant scalar or matrix. It is well known that eqn (1) is a an important one in the field of molecular dynamics.3,5,115–117 In order to preserve the characteristics of the Hamiltonian system in the numerical solution it is necessary to use symplectic integrators. In the recent years work has been done mainly in the construction of one step symplectic integrators (see ref. 11). In their work Zhu et al.26 and Chiou and Wu4 constructed multistep symplectic integrators by writing open Newton–Cotes differential schemes as multilayer symplectic structures. The last decades much work has been done on exponential fitting and the numerical solution of periodic initial value problems (see refs. 1,2,6–114,118 and references therein). In this paper we follow the steps described below: We present closed Newton–Cotes differential methods as multilayer symplectic integrators. We apply the closed Newton–Cotes methods on the Hamiltonian system (1) and we obtain as a result that the Hamiltonian energy of the system remains almost constant as the integration proceeds. We develop trigonometrically-fitted methods. We note that the aim of this paper is to generate methods that can be used for non-linear differential equations as well as linear ones. The new method developed in this paper has derived from a corresponding classical method and has an extra property without sacrificing any existing properties from the classical one. The new property provides higher efficiency in all problems and especially in problems with oscillating solutions. There is no case that the new method will be less efficient than the corresponding classical since it has the same algebraic order and the extra property.

2. Basic theory on symplectic schemes and numerical methods The following basic theory on symplectic numerical schemes and symplectic matrices is based on that developed by Zhu et al.26 The proposed methods can be used for non-linear differential equations as well as linear ones. Dividing an interval [a, b] with N points we have x0 = a, xn = x0 + nh = b, n = 1,2, . . . N. Department of Computer Science and Technology, School of Sciences and Technology, University of Peloponnese, GR-22100 Tripolis, Greece


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(2)

We note that x is the independent variable and a and b in the equation for x0 (eqn (2)) are different than the a and b in eqn (3). The above division leads to the following discrete scheme: pnþ1 anþ1 bnþ1 p ¼ Mnþ1 n ; Mnþ1 ¼ ð3Þ qnþ1 qn cnþ1 dnþ1 Based on the above we can write the n-step approximation to the solution as: an bn an1 bn1 a1 b1 p0 pn p ¼ ¼ Mn Mn1 M1 0 : qn cn dn cn1 dn1 c1 d1 q0 q0 Defining S ¼ Mn Mn1 M1 ¼

An Cn

Bn Dn

the discrete transformation can be written as: p pn ¼S 0 : qn q0 A discrete scheme (3) is a symplectic scheme if the transformation matrix S is symplectic. A matrix A is symplectic if ATJA = J where 0 1 : J¼ 1 0 The product of symplectic matrices is also symplectic. Hence, if each matrix Mn is symplectic the transformation matrix S is symplectic. Consequently, the discrete scheme (2) is symplectic if each matrix Mn is symplectic.

3. Trigonometrically-fitted closed Newton–Cotes differential methods 3.1 General closed Newton–Cotes formulae The closed Newton–Cotes integral rules are given by: Zb

f ðxÞdx zh

k X

ti f ðxi Þ;

i¼0

a

where h¼

ba ; xi ¼ a þ ih; i ¼ 0; 1; 2; . . . ; N: N

The coefficient z as well as the weights ti are given in Table 1. From Table 1 it is easy to see that the coefficients ti are symmetric, i.e. we have the following relation: k ti ¼ tki ; i ¼ 0; 1; . . . ; : 2 Closed Newton–Cotes differential methods were produced from the integral rules. For Table 1 we have the following differential methods: h k ¼ 1 ynþ1 yn ¼ ðfnþ1 þ fn Þ; 2 h k ¼ 2 ynþ1 yn1 ¼ ðfn1 þ 4fn þ fnþ1 Þ; 3 Chem. Modell., 2008, 5, 350–487 | 351 This journal is

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Table 1 Closed Newton–Cotes integral rules k

z

t0

t1

t2

t3

0 1 2 3 4 5 6

1 1/2 1/3 3/8 2/45 5/288 1/140

1 1 1 1 7 19 41

1 4 3 32 75 216

1 3 12 50 27

1 32 50 272

k ¼ 3 ynþ1 yn2 ¼

k ¼ 4 ynþ2 yn2 ¼

k ¼ 5 ynþ2 yn3 ¼

7 75 27

19 216

41

3h ðfn2 þ 3fn1 þ 3fn þ fnþ1 Þ; 8

2h ð7fn2 þ 32fn1 þ 12fn þ 32fnþ1 þ 7fnþ1 Þ; 45

5h ð19fn3 þ 75fn2 þ 50fn1 þ 50fn þ 75fnþ1 þ 19fnþ2 Þ; 288

k ¼ 6 ynþ3 yn3 h ð41fn3 þ 216fn2 þ 27fn1 þ 272fn þ 27fnþ1 þ 216fnþ2 þ 41fnþ3 Þ: ¼ 140 In the present review we will investigate the cases k = 2(2)6 and we will produce trigonometrically-fitted differential methods. 3.2 Trigonometrically-fitted closed Newton–Cotes differential method 3.2.1 First family of methods (case k = 6) 3.2.1.1 First method of the family. We requirie the differential scheme (see for details in ref. 105): yn+3 yn3 = h(a0fn3 + a1fn2 + a2fn1 + a3fn + a4fn+1 + a5fn+2 + a6fn+3),

(4)

to be accurate for the following set of functions (we note that fi = yi 0 , i = n 3(1)n + 3): {1,x,x2,x3,x4,x5,cos(wx),sin(wx)} Then, the following set of equations must hold: a0 + a1 + a2 + a3 + a4 + a5 + a6 = 6, 6a0 4a1 2a2 + 2a4 + 4a5 + 6a6 = 0, 27a0 + 12a1 + 3a2 + 3a4 + 12a5 + 27a6 = 54, 32a1 + 4a4 + 32a5 + 108a6 4a2 108a0 = 0, 80a1 + 5a4 + 80a5 + 405a6 + 5a2 + 405a0 = 486, uh sin(uh)(a0 + a2 a4 + a6 + 2a1 cos(uh) 2a5 cos(uh) 4a6 cos(uh)2 + 4a0 cos(uh)2) = 0, 352 | Chem. Modell., 2008, 5, 350–487 This journal is

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(5)

uh (a1 + a3 a5 + a4 cos(uh) + 2a5 cos(uh)2 + 4a6 cos(uh)3 3a6 cos(uh) + a2 cos(uh) + 2a1 cos(uh)2 + 4a0 cos(uh)3 3 cos(uh)a0) = 2 sin(uh)(1 + 4 cos(uh)2), We note that the first, second, third, fourth and fifth equations are produced requiring the scheme (4) to be accurate for xj, j = 0(1)5, while the sixth and seventh equations are obtained requiring the algorithm (4) to be accurate for cos(wx), sin(wx). We note here that the requirement for the accurate integration of functions (5), leads the method to be accurate for all the problems with solution which has behavior of trigonometric functions. In order to solve the above equation and to obtain the coefficients we use the program mentioned in Appendix A. The Local Truncation Error for the above differential method is given by: L:T:E:ðhÞ ¼

9h9 ð9Þ ðy þ u2 yð7Þ n Þ: 1400 n

The LTE is obtained expanding the terms ynj and fnj, j = 1(1)3 in (4) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. The behaviour of the coefficients of the method is given in the following figures (Fig. 1–4): where a_{i} = ai, i = 0(1)3 and where a4 = a2, a5 = a1, a6 = a0.

Fig. 1 Behaviour of the coefficient a0.


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3.2.1.2 Second method of the family. We requirie the differential scheme (4) to be accurate for the following set of functions (see for details in ref. 107): {1,x,x2,x3,cos(wx),sin(wx),x cos(wx),x sin(wx)}

(6)

Then, the following set of equations must hold: a0 + a1 + a2 + a3 + a4 + a5 + a6 = 6 6a0 4a1 2a2 + 2a4 + 4a5 + 6a6 = 0 27a0 + 12a1 + 3a2 + 3a4 + 12a5 + 27a6 = 54 uh sin(uh)(a0 + a2 a4 + a6 + 2a1 cos(uh) 2a5 cos(uh) 4a6 cos(uh)2 + 4a0 cos(uh)2) = 0 2u sin(uh) (1 + 4 cos(uh)2) = uh(a1 + a3 a5 + 4a0 cos(uh)3 + 2a1 cos(uh)2 3 cos(uh)a0 3a6 cos(uh) + a4 cos(uh) + a2 cos(uh) + 4a6 cos(uh) + 2a5 cos(uh)2) 6 cos(uh)h( 3 + 4 cos(uh)2) = h(4a6 cos(uh)3 a1 + a3 a5 + 2a1 cos(uh)2 3 cos(uh)a0 + 2a5 cos(uh)2 3a6 cos(uh) 4 cos(uh)ha1u sin(uh) 4 cos(uh)ha5u sin(uh) 2 cos(uh)a5ux sin(uh) + 2 cos(uh)a1ux sin + a2 cos(uh) 12 sin(uh)ha0u cos(uh)2 + a4 cos(uh) + 4a0 cos(uh)3 + 3 sin(uh)ha0u sin(uh)a0ux + sin(uh)a2ux sin(uh)ha2u 354 | Chem. Modell., 2008, 5, 350–487 This journal is

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sin(uh)a4ux + sin(uh)a6ux + 3 sin(uh)ha6u sin(uh)ha4u + 4 sin(uh)a0ux cos(uh)2 4 sin(uh)a6ux cos(uh)2 12 sin(uh)ha6u cos(uh)2) 2 sin(uh)x(1+4 cos(uh)2) = h(sin(uh)a0 sin(uh)a2 9ha6u cos(uh)+a4 ux cos(uh) + cos(uh)ha4u + 2a1ux cos(uh)2 + 9 cos(uh)ha0u + cos(uh)a2ux + 2a5ux cos(uh)2 3 cos(uh)a6ux + 4ha5u cos(uh)2 4ha1u cos(uh)2 ha2u cos(uh) 3 cos(uh)a0ux 4 sin(uh)a0 cos(uh)2 + 4 sin(uh)a6 cos(uh)2 + 12ha6u cos(uh)3 sin(uh)a6 + sin(uh)a4 2 cos(uh)a1 sin(uh) + 4a6ux cos(uh)3 + 2 cos(uh)a5 sin(uh) + a3xu a1ux a5ux 2ha5u + 2ha1u 12ha0u cos(uh)3 + 4a0ux cos(uh)3) We note that the first and second equations are produced requiring the scheme (4) to be accurate for xj, j = 0(1)3, while the third, fourth, fifth and sixth and seventh equations are obtained requiring the algorithm (4) to be accurate for cos(wx), sin(wx), x cos(wx), x sin(wx). We mention here that the requirement for the accurate integration of functions (6), leads the method to be accurate for all the problems with solution which has behavior of trigonometric functions. Chem. Modell., 2008, 5, 350–487 | 355 This journal is

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In order to solve the above equation and to obtain the coefficients we use the program mentioned in Appendix B. The Local Truncation Error for the above differential method is given by: L:T:E:ðhÞ ¼

9h9 ð9Þ 4 ð5Þ ðy þ 2u2 yð7Þ n þ u yn Þ: 1400 n

The LTE is obtained expanding the terms ynj and fnj, j = 1(1)3 in (4) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. The behaviour of the coefficients of the method is given in the following figures (Fig. 5–8): where a[i] = ai, i = 0(1)3 and where a4 = a2, a5 = a1, a6 = a0. 3.2.2 Second family of methods (case k = 4) 3.2.1.1 First method of the family. We requirie the differential scheme (see for details in ref. 103): yn+2 yn2 = h(a0fn2 + a1fn1 + a2fn + a3fn+1 + a4fn+2),

(7)

to be accurate for the following set of functions (we note that fi = yi 0 , i = n 2(1)n + 2): {1,x,x2,x3,cos(wx),sin(wx)} 356 | Chem. Modell., 2008, 5, 350–487 This journal is

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(8)



Then, the following set of equations must hold: a0 + a1 + a2 + a3 + a4 = 4, 4a0 2a2 + 2a3 + 4a4 = 0, Chem. Modell., 2008, 5, 350–487 | 357 This journal is

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12a0 + 3a1 + 3a3 + 12a4 = 16, uh sin(uh)(a1 a3 2a4 cos(uh) + 2a0 cos(uh)) = 0, uh(a0 + a2 a4 + 2a0 cos(uh)2 + 2a4 cos(uh)2 + a3 cos(uh) + a1 cos(uh)) = 4 cos(uh) sin(uh), 358 | Chem. Modell., 2008, 5, 350–487 This journal is

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We note that the first, second and third equations are produced requiring the scheme (7) to be accurate for xj, j = 0(1)3, while the fourth and fifth equations are obtained requiring the algorithm (7) to be accurate for cos(wx), sin(wx). It is important to be noted here that the requirement for the accurate integration of functions (8), leads the method to be accurate for all the problems with solution which has behavior of trigonometric functions. In order to solve the above equation and in order to obtain the coefficients we use the program mentioned in Appendix C. The Local Truncation Error for the above differential method is given by:

L:T:E:ðhÞ ¼

8h7 ð7Þ ðy þ u2 yð5Þ n Þ: 945 n

The LTE is obtained expanding the terms ynj and fnj, j = 1(1)2 in (7) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. The behaviour of the coefficients of the method is given in the following figures (Fig. 9–11): where a_i = ai, i = 0(1)2 and where a3 = a1, a4 = a0. Chem. Modell., 2008, 5, 350–487 | 359 This journal is

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3.2.1.2 Second method of the family. We requirie the differential scheme (7) to be accurate for the following set of functions (we note that fi = yi 0 , i = n 1, n, n + 1) (see for details in ref. 101): {1,x,cos(wx),sin(wx),x cos(wx),x sin(wx)}

(9)

Then, the following set of equations must hold: a0 + a1 + a2 + a3 + a4 = 4, uh sin(uh)(a1 a3 2a4 cos(uh) + 2a0 cos(uh)) = 0, uh(a0 + a2 a4 + 2a0 cos(uh)2 + 2a4 cos(uh)2 + a3 cos(uh) + a1 cos(uh)) = 4 cos(uh) sin(uh), h(a0 + a2 + a4 4a0uh cos(uh) sin(uh) 4a4uh cos(uh) sin(uh) a1uh sin(uh) + a1ux sin(uh) + 2a0 cos(uh)2 + a1 cos(uh) + a3 cos(uh) + 2a4 cos(uh)2 a3ux sin(uh) a3uh sin(uh) 2a4ux cos(uh) sin(uh) + 2a0ux cos(uh) sin(uh)) = 4h (1 + 2 cos(uh)2) h(a3 sin(uh) a1 sin(uh) 4a0uh cos(uh)2 + 2a0ux cos(uh)2 + a1ux cos(uh) + 4a4uh cos(uh)2 + 2a4ux cos(uh)2 + a3ux cos(uh) + a3uh cos(uh) + 2a4 cos(uh)sin(uh) a1uh 360 | Chem. Modell., 2008, 5, 350–487 This journal is

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cos(uh) 2a0 cos(uh)sin(uh) a0ux + a2ux a4ux 2a4uh + 2a0uh) = 4x cos(uh)sin(uh) We note that the first equation is produced requiring the scheme (7) to be accurate for xj, j = 0,1, while the second, third, fourth and fifth equations are obtained requiring the algorithm (7) to be accurate for cos(wx), sin(wx), x cos(wx), x sin(wx). We note here that the requirement for the accurate integration of functions (9), leads the method to be accurate for all the problems with solution which has behavior of trigonometric functions. In order to solve the above equation and in order to obtain the coefficients we use the program mentioned in Appendix D. The Local Truncation Error for the above differential method is given by: L:T:E:ðhÞ ¼

8h7 ð7Þ 4 ð3Þ ðy þ 2u2 yð5Þ n þ u yn Þ: 945 n

The LTE is obtained expanding the terms ynj and fnj, j = 1(1)2 in (7) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. Chem. Modell., 2008, 5, 350–487 | 361 This journal is

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The behaviour of the coefficients of the method is given in the following figures (Fig. 12–14): where a_i = ai, i = 0(1)2 and where a3 = a1, a4 = a0. 3.2.3 Third family of methods (case k = 2) 3.2.3.1 First method of the family. We requirie the differential scheme (see for details in ref. 104): yn+1 yn1 = h(a0fn1 + a1fn + a2fn+1),

(10)

to be accurate for the following set of functions (we note that fi = yi 0 , i = n 1, n, n + 1): {1,x,cos(wx),sin(wx)} Then, the following set of equations must hold: a0 + a1 + a2 = 2, uh sin(uh)(a0 a2) = 0, uh(a1 + a0 cos(uh) + a2 cos(uh)) = 2 sin(uh), 362 | Chem. Modell., 2008, 5, 350–487 This journal is

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(11)


We note that the first equation is produced requiring the scheme (10) to be accurate for xj, j = 0,1, while the second and third equations are obtained requiring the algorithm (10) to be accurate for cos(wx), sin(wx). We mention here that the requirement for the accurate integration of functions (11), leads the method to be accurate for all the problems with solution which has behavior of trigonometric functions. In order to solve the above equation and in order to obtain the coefficients we use the program mentioned in Appendix E. The Local Truncation Error for the above differential method is given by:

L:T:E:ðhÞ ¼

h5 ð5Þ ðy þ u2 yð3Þ n Þ: 90 n

The LTE is obtained expanding the terms ynj and fnj, j = 0,1 in (10) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. The behaviour of the coefficients of the method is given in the following figures (Fig. 15 and 16): where a_{i} = ai, i = 0,1 and where a2 = a0. Chem. Modell., 2008, 5, 350–487 | 363 This journal is

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3.2.1.2 Second method of the family. We requirie the differential scheme (11) to be accurate for the following set of functions (we note that fi = yi 0 , i = n 1, n, n + 1) (see for details in ref. 106): {cos(wx),sin(wx),x cos(wx),x sin(wx)}

(12)

Then, the following set of equations must hold: uh sin(uh)(a0 a2) = 0, uh((a0 + a2)cos(uh) + a1) = 2 sin(uh), h(a0 cos(uh) a2 cos(uh) a0ux sin(uh) + a2uh sin(uh) a1 a2ux sin(uh) + a0uh sin(uh)) = 2h cos(uh) h(a0ux cos(uh) a2uh cos(uh) + a2ux cos(uh) + a0uh cos(uh) + a0 sin(uh) a2 sin(uh) + a1ux) = 2x sin(uh) We note that The equations are obtained requiring the algorithm (11) to be accurate for cos(wx), sin(wx), x cos(wx), x sin(wx). It is important to be mentioned here that the requirement for the accurate integration of functions (12), leads the method to be accurate for all the problems with solution which has behavior of trigonometric functions. In order to solve the above equation and in order to obtain the coefficients we use the program mentioned in Appendix F. The Local Truncation Error for the above differential method is given by: L:T:E:ðhÞ ¼

h5 ð5Þ 4 ð1Þ ðy þ 2u2 yð3Þ n þ u yn Þ: 90 n

The LTE is obtained expanding the terms ynj and fnj, j = 0,1 in (11) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. Chem. Modell., 2008, 5, 350–487 | 365 This journal is

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The behaviour of the coefficients of the method is given in the following figures (Fig. 17 and 18): where a_i = ai, i = 0,1 and where a2 = a0.

4. Error analysis In this section we will investigate theoretically the methods presented above and in this paper. The scope of this investigation is to find a quantitative estimation for the extent of the accuracy gain to be expected from the exponentially-fitted versions. Definition 1. A method is called classical if it has constant coefficients Remark 1. A trigonometrically-fitted method is not a classical one because it has coefficients which are dependent on the quantity u = wh, where w is the frequency of the problem and h is the step length of the integration. Consider the radial Schro¨dinger equation: y00 (x) = [l(l + 1)/x2 + V(x) k2]y(x) = f(x)y(x) 2

(13)

2

where f(x) = U(x) k and U(x) = l(l + 1)/x + V(x). We write f(x) in (1) in the form f(x) = g(x) + d

(14)

where g(x) = U(x) Uc = g, where Uc is the constant approximation of the potential and d = v2 = Uc k2. 366 | Chem. Modell., 2008, 5, 350–487 This journal is

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So, g(x) depends on the potential and the constant approximation of the potential while d shows the energy dependence. We will compare the following methods: The classical fourth order closed Newton–Cotes formulae (Method I) The classical sixth order closed Newton–Cotes formulae (Method II) The classical eighth order closed Newton–Cotes formulae (Method III) The closed Newton–Cotes formulae developed in ref. 104 (Method IV) The closed Newton–Cotes formulae developed in ref. 106 (Method V) The closed Newton–Cotes formulae developed in ref. 103 (Method VI) The closed Newton–Cotes formulae developed in ref. 101 (Method VII) The closed Newton–Cotes formulae developed in ref. 105 (Method VIII) The closed Newton–Cotes formulae developed in ref. 107 (Method IX) We now present the formulae of the Local Truncation Error (LTE) for the above methods. For the Method I is equal to: L:T:E:ðhÞMI ¼

h5 ð5Þ y 90 n

ð15Þ

8h7 ð7Þ y 945 n

ð16Þ

For the Method II is equal to: L:T:E:ðhÞMII ¼


c


For the Method III is equal to: 9h9 ð9Þ y 1400 n

ð17Þ

h5 ð5Þ ðy þ u2 yð3Þ n Þ 90 n

ð18Þ

h5 ð5Þ 4 ð1Þ ðy þ 2u2 yð3Þ n þ u yn Þ 90 n

ð19Þ

8h7 ð7Þ ðy þ u2 yð5Þ n Þ 945 n

ð20Þ

8h7 ð7Þ 4 ð3Þ ðy þ 2u2 yð5Þ n þ u yn Þ 945 n

ð21Þ

9h9 ð9Þ ðy þ u2 yð7Þ n Þ 1400 n

ð22Þ

9h9 ð9Þ 4 ð5Þ ðy þ 2u2 yð7Þ n þ u yn Þ 1400 n

ð23Þ

L:T:E:ðhÞMIII ¼ For the Method IV is equal to: L:T:E:ðhÞMIV ¼ For the Method V is equal to: L:T:E:ðhÞMV ¼ For the Method VI is equal to: L:T:E:ðhÞMVI ¼ For the Method VII is equal to: L:T:E:ðhÞMVII ¼ For the Method VIII is equal to: L:T:E:ðhÞMVIII ¼ For the Method IX is equal to: L:T:E:ðhÞMIX ¼

(7) (5) (3) We express, now, the derivatives y(9) n , yn , yn , yn in terms of eqn (13), i.e. d d ð3Þ gðxÞ yðxÞ þ ðgðxÞ þ dÞ yðxÞ ; yð2Þ ¼ f ðxÞyðxÞ; y ¼ n n dx dx

yð5Þ n ¼

2 2 d3 d d d d yðxÞ þ 2 gðxÞ gðxÞ yðxÞ þ 3 gðxÞ yðxÞ dx dx dx3 dx2 dx2 d d gðxÞ þ ðgðxÞ þ dÞ2 yðxÞ þ 2ðgðxÞ þ dÞyðxÞ dx dx

ð24Þ

etc. We note that g(n)(x) = U(n)(x) for the nth order derivative with respect to x. Introducing the expressions obtained in (24) into the Local Truncation Error of the methods mentioned above (see relations (15)–(23)), we obtain the following expressions (as polynomials of d) for Local Truncation Error of the methods. 1 d3 1 d2 d L:T:E:ðhÞMI ¼h5 gðxÞ yðxÞ gðxÞ yðxÞ 90 dx3 30 dx2 dx 2 d 2 d gðxÞ gðxÞ yðxÞ gðxÞ d yðxÞ 45 dx 45 dx 1 d 1 d 1 d 2 yðxÞ gðxÞ yðxÞ gðxÞd yðxÞ d 2 90 dx 45 dx 90 dx ð25Þ 368 | Chem. Modell., 2008, 5, 350–487 This journal is

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5 4 8 d 8 d d yðxÞ L:T:E:ðhÞMII ¼h7 gðxÞ yðxÞ gðxÞ 945 dx5 189 dx4 dx 3 3 88 d 88 d yðxÞ yðxÞ gðxÞ gðxÞ gðxÞ d 3 3 945 945 dx dx 2 2 8 d d 104 d d gðxÞ yðxÞ yðxÞ gðxÞ gðxÞ gðxÞ dx2 dx2 63 dx 945 dx 2 2 104 d d 16 d d yðxÞ gðxÞ yðxÞ gðxÞ d dx2 945 dx 189 dx dx 8 d 16 d yðxÞ gðxÞ gðxÞ2 yðxÞ gðxÞ gðxÞd 105 dx 105 dx 8 d 8 d yðxÞ gðxÞ d 2 yðxÞ gðxÞ3 105 dx 945 dx 8 d 8 d 8 d yðxÞ gðxÞ2 d yðxÞ gðxÞd 2 yðxÞ d 3 315 dx 315 dx 945 dx

ð26Þ

18 d ðgðxÞ þ dÞ3 yðxÞ gðxÞ L:T:E:ðhÞMIII ¼h9 175 dx 4 4 27 d d 9 d d gðxÞ yðxÞ ðgðxÞ þ dÞ yðxÞ gðxÞ gðxÞ 100 dx 35 dx dx4 dx4 5 3 99 d 9 d2 d ðgðxÞ þ dÞyðxÞ gðxÞ gðxÞ yðxÞ gðxÞ 5 2 3 700 25 dx dx dx 3 3 63 d d d 207 d 2 gðxÞ yðxÞ ðgðxÞ þ dÞ gðxÞ yðxÞ gðxÞ 100 dx dx 700 dx3 dx3 3 2 ð27Þ 333 d d 9 d ðgðxÞ þ dÞyðxÞ gðxÞ gðxÞ yðxÞ gðxÞ 350 dx 50 dx dx2 2 7 153 d d 9 d ðgðxÞ þ dÞ2 yðxÞ gðxÞ gðxÞ yðxÞ 2 7 700 dx 1400 dx dx 2 6 117 d d 9 d d ðgðxÞ þ dÞ yðxÞ gðxÞ yðxÞ gðxÞ 350 dx dx 200 dx6 dx ! 2 2 81 d d 9 d yðxÞ ðgðxÞ þ dÞ4 yðxÞ gðxÞ 200 dx2 dx 1400 dx

1 d3 1 d2 d gðxÞ yðxÞ gðxÞ L:T:E:ðhÞMIV ¼h5 yðxÞ 90 dx3 30 dx2 dx 2 d 1 d gðxÞ gðxÞ yðxÞ gðxÞ d yðxÞ 45 dx 30 dx 1 d 1 d 2 yðxÞ gðxÞ yðxÞ gðxÞd 90 dx 90 dx

ð28Þ

1 d 1 d3 1 d2 d L:T:E:ðhÞMV ¼h5 yðxÞ gðxÞ yðxÞ gðxÞ gðxÞ d yðxÞ 45 dx 90 dx3 30 dx2 dx 2 d 1 d yðxÞ gðxÞ gðxÞ yðxÞ gðxÞ2 45 dx 90 dx

ð29Þ Chem. Modell., 2008, 5, 350–487 | 369 This journal is

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5 4 8 d 8 d d yðxÞ L:T:E:ðhÞMVI ¼h7 gðxÞ yðxÞ gðxÞ 945 dx5 189 dx4 dx 3 3 88 d 16 d yðxÞ yðxÞ gðxÞ gðxÞ gðxÞ d 945 189 dx3 dx3 2 2 8 d d 104 d d gðxÞ gðxÞ gðxÞ gðxÞ yðxÞ yðxÞ dx2 dx2 63 dx 945 dx 2 2 16 d d 16 d d yðxÞ gðxÞ yðxÞ gðxÞ d 189 dx 189 dx dx dx2 8 d 16 d yðxÞ gðxÞ gðxÞ2 yðxÞ gðxÞ gðxÞd 105 dx 135 dx 8 d 8 d yðxÞ gðxÞ d 2 yðxÞ gðxÞ3 189 dx 945 dx 16 d 8 d yðxÞ gðxÞ2 d yðxÞ gðxÞd 2 945 dx 945 dx

ð30Þ

2 16 d 8 d d L:T:E:ðhÞMVII ¼h7 gðxÞ d yðxÞ gðxÞ d 2 yðxÞ dx2 945 dx 135 dx 3 16 d 8 d 8 d gðxÞ d yðxÞ gðxÞ gðxÞd yðxÞ yðxÞ gðxÞ2 d dx3 189 dx 105 945 dx 5 4 3 8 d 8 d d 88 d gðxÞ yðxÞ gðxÞ gðxÞ gðxÞ yðxÞ yðxÞ dx3 945 dx5 189 dx4 dx 945 2 2 8 d d 104 d d gðxÞ gðxÞ gðxÞ gðxÞ yðxÞ yðxÞ dx2 dx2 63 dx 945 dx ! 2 16 d d 8 d 8 d 2 3 gðxÞ yðxÞ yðxÞ gðxÞ gðxÞ yðxÞ gðxÞ 189 dx dx 105 dx 945 dx

ð31Þ 9 d4 d 99 d5 yðxÞ d L:T:E:ðhÞMVIII ¼h9 gðxÞ gðxÞ yðxÞgðxÞ 4 5 40 dx dx 700 dx 7 6 2 9 d 9 d d 81 d2 d yðxÞ yðxÞ gðxÞ yðxÞ gðxÞ gðxÞ 7 6 2 1400 dx 200 dx dx 200 dx dx 3 4 9 d 9 d 27 d d 3 yðxÞ gðxÞd gðxÞ yðxÞ gðxÞ yðxÞ gðxÞ 4 1400 dx 50 dx 100 dx dx 3 3 9 d2 d 63 d d d gðxÞ yðxÞ gðxÞ yðxÞ gðxÞ gðxÞ 25 dx2 100 dx dx dx3 dx3 2 27 d 27 d d 9 d 3 yðxÞ gðxÞ d gðxÞ yðxÞ d yðxÞ gðxÞ d 3 1400 dx 100 dx dx 200 dx 9 d4 d 351 d 27 d 2 yðxÞ gðxÞ yðxÞ gðxÞ gðxÞ d yðxÞ gðxÞ gðxÞd 2 gðxÞ 4 35 dx dx 1400 dx 140 dx 2 18 d 27 d 117 d d 3 2 2 yðxÞ gðxÞ gðxÞ yðxÞ gðxÞ d gðxÞ yðxÞ gðxÞ 175 dx 1400 dx 350 dx dx 3 9 d 27 d5 207 d 4 yðxÞ gðxÞ yðxÞ gðxÞ yðxÞd gðxÞ gðxÞ2 1400 dx 200 dx5 700 dx3 2 2 3 9 d 153 d d 27 d d yðxÞ yðxÞ yðxÞ gðxÞ d 2 gðxÞ gðxÞ2 gðxÞ d 2 3 2 2 40 700 dx 200 dx dx dx dx 3 2 729 d 333 d d yðxÞ gðxÞ yðxÞ gðxÞ gðxÞd gðxÞ gðxÞ 1400 350 dx dx3 dx2 2 2 171 d d 99 d d gðxÞ yðxÞ yðxÞ gðxÞ d gðxÞ gðxÞd 200 dx 280 dx dx2 dx2

ð32Þ 370 | Chem. Modell., 2008, 5, 350–487 This journal is

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27 d4 d 99 d5 yðxÞ d L:T:E:ðhÞMIX ¼h9 gðxÞ gðxÞ yðxÞgðxÞ 4 5 140 dx dx 700 dx 7 6 2 9 d 9 d d 81 d2 d yðxÞ yðxÞ gðxÞ yðxÞ gðxÞ gðxÞ 7 6 2 1400 dx 200 dx dx 200 dx dx 3 4 3 9 d 27 d d 9 d2 d gðxÞ yðxÞ gðxÞ yðxÞ gðxÞ gðxÞ yðxÞ gðxÞ 4 2 3 50 dx 100 dx 25 dx dx dx 3 63 d d d 9 d 3 gðxÞ yðxÞ yðxÞ gðxÞ d gðxÞ 100 dx dx 700 dx dx3 2 36 d d 9 d 9 d4 d gðxÞ yðxÞ d yðxÞ gðxÞ d 3 yðxÞ gðxÞ gðxÞ 4 175 dx dx 700 dx 35 dx dx 27 d 18 d 18 d yðxÞ gðxÞ gðxÞ2 d yðxÞ gðxÞ gðxÞd 2 yðxÞ gðxÞ gðxÞ3 140 dx 175 dx 175 dx 2 9 d 117 d d 9 d 2 2 yðxÞ gðxÞ d gðxÞ yðxÞ gðxÞ yðxÞ gðxÞ4 1400 dx 350 dx dx 1400 dx 3 3 9 d5 207 d 9 d 2 yðxÞ yðxÞ gðxÞ yðxÞd gðxÞ gðxÞ gðxÞ d2 70 dx5 700 56 dx3 dx3 2 2 153 d d 99 d d yðxÞ yðxÞ gðxÞ gðxÞ2 gðxÞ d 2 700 dx 1400 dx dx2 dx2 3 2 9 d 333 d d gðxÞ yðxÞ yðxÞ gðxÞ gðxÞd gðxÞ gðxÞ 3 2 20 350 dx dx dx 2 2 531 d d 27 d d gðxÞ yðxÞ yðxÞ gðxÞ d gðxÞ gðxÞd 2 2 700 dx 100 dx dx dx

ð33Þ The leading terms (in d) of the above expressions are given by: 1 d L:T:E:ðhÞMI ¼ h5 d 2 yðxÞ 90 dx

ð34Þ

8 d L:T:E:ðhÞMII ¼ h7 d 3 yðxÞ 945 dx

ð35Þ

L:T:E:ðhÞMIII

9 d yðxÞ ¼h d 1400 dx 9 4

ð36Þ

1 d 1 d gðxÞ yðxÞ yðxÞ gðxÞ L:T:E:ðhÞMIV ¼ h5 d 30 dx 90 dx

ð37Þ

1 d L:T:E:ðhÞMV ¼ h5 d gðxÞ yðxÞ 45 dx

ð38Þ

8 d 8 d L:T:E:ðhÞMVI ¼ h7 d 2 gðxÞ yðxÞ yðxÞ gðxÞ 189 dx 945 dx

ð39Þ

16 d gðxÞ yðxÞ L:T:E:ðhÞMVII ¼ h7 d 2 945 dx

ð40Þ

9 d 9 d L:T:E:ðhÞMVIII ¼ h9 d 3 gðxÞ yðxÞ yðxÞ gðxÞ 200 dx 1400 dx 9 d gðxÞ yðxÞ L:T:E:ðhÞMVIII ¼ h9 d 3 700 dx

ð41Þ

ð42Þ


c


From the above equations we have the following theorem: Theorem 1. For the Closed Newton–Cotes formulae studied in this paper we have: In the trigonometrically-fitted methods (Methods MIV, MV, MVI, MVII, MVIII and MIX) the error increases with lower power of d than the corresponding classical ones. In the trigonometrically-fitted method MIV and MV, the error increases as the first power of d while in the classical one (Method MI) the error increases as the second power of d. In the trigonometrically-fitted method MVI and MVII, the error increases as the second power of d while in the classical one (Method MII) the error increases as the third power of d. In the trigonometrically-fitted method MVIII and MIX, the error increases as the third power of d while in the classical one (Method MIII) the error increases as the fourth power of d.

5. Closed Newton–Cotes can be expressed as symplectic integrators The details for this development can be found at refs. 2, 3, 101–107. Theorem 2 (See refs. 101–107). A discrete scheme of the form b a qn b a qnþ1 ¼ pnþ1 pn a b a b is symplectic. Proof. We rewrite (10) as 1 qnþ1 b a b ¼ pnþ1 a b a

a b

qn pn

ð43Þ

Define M¼

b a a b

1

b a a b

¼

1 b2 þ a2

b2 a2 2ab

2ab b a2

2

and it can easily be verified that MTJM = J thus the matrix M is symplectic. 5.1 Family of methods (case k = 2) In ref. 2 Zhu et al. have proved the symplectic structure of the well-known secondorder differential scheme (SOD), yn+1 yn1 = 2hfn

(44)

The above method has been obtained by the simplest Open Newton–Cotes integral rule. Based on the paper Chiou et al.3 the closed Newton–Cotes differential schemes can be presented as multilayer symplectic structures. Application of the Newton–Cotes differential formula for n = 2 to the linear Hamiltonian system (1) gives qn+1 qn1 = s(a0pn1 + a1pn + a2pn+1) pn+1 pn1 = s(a0qn1 + a1qn + a2qn+1) where s = mh, where m is defined in (1).


c


(44a)

From (44) we have that: qn+1 qn1 = 2spn pn+1 pn1 = 2sqn Substituting pn and qn into (44a) we obtain the following formula in matrix form (see for details refs. 101–107): 1 a21 1 a21 qnþ1 qn1 saa01 sa0a1 ¼ sa0 1 2 pnþ1 sa0 1 2 pn1 which is a discrete scheme of the form (43) and hence it is symplectic. We note here than in ref. 3 Chiou et al. have re-written Open Newton–Cotes differential schemes as multilayer symplectic structures based on (44). 5.2 Family of methods (case k = 4) As in the previous paragraph we base our study on ref. 27 where Zhu et al. have proved the symplectic structure of the well-known second-order differential scheme (SOD), yn+1 yn1 = 2hfn yn+2 yn2 = 4hfn

(45)

Application of the Newton–Cotes differential formula for n = 2 to the linear Hamiltonian system (1) gives qn+2 qn2 = s(a0pn2 + a1pn1 + a2pn + a3pn+1 + a4pn+2) pn+2 pn2 = s(a0qn2 + a1qn1 + a2qn + a3qn+1 + a4qn+2)

(46)

where s = mh, where m is defined in (1). From (45) we have that: qn+2 qn2 = 4spn pn+2 pn2 = 4sqn

(47)

qn+1 qn1 = 2spn pn+1 pn1 = 2sqn

(48)

qn+12 qn12 = spn pn+12 pn12 = sqn

(49)

Substitution of the approximation which is based on (49) for (m + 1)-step to (47) gives: qn+1 qn1 = (qn + spn+12) + (qn spn12) = 2qn + s(pn+12 pn12) = (2 s2)qn

(50)

Similarly we have: pn+1 pn1 = (pn sqn+12) + (pn + sqn12) = 2pn s(qn+12 qn12) = (2 s2)pn

(51)

Substituting (50) and (51) into (46), considering a0 = a4, a1 = a3 and using (47) we obtain the following formula in matrix form (see for details refs. 101–107): qnþ2 qn2 TðsÞ sa0 TðsÞ sa0 ¼ sa0 TðsÞ pnþ2 sa0 TðsÞ pn2 Chem. Modell., 2008, 5, 350–487 | 373 This journal is

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where TðsÞ ¼ 1

a1 ð2 s2 Þ þ a2 4

ð52Þ

which is a discrete scheme of the form (43) and hence it is symplectic.

5.3 Family of methods (case k = 6) In ref. 27 Zhu et al. have proved the symplectic structure of the well-known secondorder differential scheme (SOD), yn+1 yn1 = 2hfn, yn+2 yn2 = 4hfn, yn+3 yn3 = 6hfn.

(53)

The above methods have been produced by the well known Open Newton–Cotes integral formulae. Based on the paper Chiou et al.3 the closed Newton–Cotes differential schemes can be written as multilayer symplectic structures. Application of the Newton–Cotes differential formula for n = 3 to the linear Hamiltonian system (1) gives qn+3 qn3 = s(a0pn3 + a1pn2 + a2pn1 + a3pn + a4pn+1 + a5pn+2 + a6pn+3), pn+3 qn3 = s(a0qn3 + a1qn2 + a2qn1 + a3qn + a4qn+1 + a5qn+2 + a6qn+3)

(54)

where s = mh, where m is defined in (1). From (53) we have that: qn+3 qn3 = 6spn, pn+3 pn3 = 6sqn

(55)

qn+2 qn2 = 4spn, pn+2 pn2 = 4sqn

(56)

qn+1 qn1 = 2spn, pn+1 pn1 = 2sqn

(57)

qn+3/2 qn3/2 = 3spn, pn+3/2 pn3/2 = 3sqn

(58)

qn+1/2 qn1/2 = spn, pn+1/2 pn1/2 = sqn.

(59)

Considering the approximation based on the first formula of (59) for (n + 1)-step gives (taking into account the second formula of (59)): qn+1 qn1 = (qn + spn+1/2) + (qn spn1/2) = 2qn + s(pn+1/2 pn1/2) = (2 s2)qn.

(60)

Similarly we have: pn+1 + pn1 = (pn + sqn+1/2) (pn + sqn1/2) = 2pn + s(qn+1/2 qn1/2) = (2 s2)pn.

(61)

Considering the approximation based on the first formula of (59) for (n + 2)-step gives (taking into account the second formula of (58) and (61)): qn+2 + qn2 = (qn+1 + spn+3/2) + (qn1 spn3/2) = qn+1 + qn1 +s(pn+3/2 pn3/2) = (2 s2)qn 3s2qn = 2(1 2s2)qn.

(62)

Similarly we have: pn+2 + pn2 = (pn+1 + sqn+3/2) + (pn1 + spn3/2) = pn+1 + pn1 +s(qn+3/2 qn3/2) = (2 s2)qn 3s2qn = 2(1 2s2)qn. 374 | Chem. Modell., 2008, 5, 350–487 This journal is

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(63)

Substituting (60)–(63) into (54), considering that a0 = a6, a1 = a5 and a2 = a4 and using (55) we obtain the following formula in matrix form (see for details refs. 101–107): qnþ3 qn3 TðsÞ sa0 TðsÞ sa0 ¼ ; sa0 TðsÞ pnþ3 sa0 TðsÞ pn3 where TðsÞ ¼ 1

2a1 ð1 2s2 Þ þ a2 ð2 s2 Þ þ a3 6

which is a discrete scheme of the form (43) and hence it is symplectic.

6. Numerical example In this section we present some numerical results to examine the efficiency of the new introduced methods. Consider the numerical integration of the Schro¨dinger equation: y00 (x) = [l(l + 1)/x2 + V(x) k2]y(x).

(64)

using the well-known Woods–Saxon potential (see refs. 12, 26, 30–32) which is given by VðxÞ ¼ Vw ðxÞ ¼

u0 u0 z ð1 þ zÞ ½að1 þ zÞ2

ð65Þ

with z = exp[(x R0)/a], u0 = 50, a = 0.6 and R0 = 7.0. In Fig. 18 we give a graph of this potential. In the case of negative eigenenergies (i.e. when E A [50,0]) we have the well-known bound-states problem while in the case of positive eigenenergies (i.e. when E A [0,1000]) we have the well-known resonance problem (see refs. 12, 26, 30–32, 40). Many problems in chemistry, physics, physical chemistry, chemical physics, electronics etc., are expressed by eqn (26) (see refs. 49–52) (Fig. 19). 6.1 Resonance problem In the asymptotic region the eqn (64) effectively reduces to lðl þ 1Þ y00 ðxÞ þ k2 yðxÞ ¼ 0; x2

ð66Þ

for x greater than some value X. The above equation has linearly independent solutions kxjl(kx) and kxnl(kx), where jl(kx), nl(kx) are the spherical Bessel and Neumann functions, respectively. Thus the solution of eqn (64) has the asymptotic form (when x-N) y(x) C Akxjl(kx) Bnl(kx) C D[sin(kx pl/2) + tan dl cos(kx pl/2)]

(67)

where dl is the phase shift which may be calculated from the formula tan dl ¼

yðx2 ÞSðx1 Þ yðx1 ÞSðx2 Þ yðx1 ÞCðx2 Þ yðx2 ÞCðx1 Þ

ð68Þ

for x1 and x2 distinct points on the asymptotic region (for which we have that x1 is the right hand end point of the interval of integration and x2 = x1 h, h is the stepsize) with S(x) = kxjl(kx) and C(x) = kxnl(kx). Since the problem is treated as an initial-value problem, one needs y0 and y1 before starting a four-step method. From the initial condition, y0 = 0. The values yi, i = 1,2, . . . are computed using the high order Runge-Kutta method of Prince and Chem. Modell., 2008, 5, 350–487 | 375 This journal is

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Fig. 19 The Woods–Saxon potential.

Dormand.99,100 With these starting values we evaluate at x1 of the asymptotic region the phase shift dl from the above relation. The Woods–Saxon potential. As a test for the efficiency of our methods we consider the case of the numerical solution of the Schro¨dinger eqn (64) with l = 0 in the well-known case where the potential V(r) is the Woods–Saxon one (65). One can investigate the problem considered here, following two procedures. The first procedure consists of finding the phase shift d(E) = dl for E A [1,1000]. The second procedure consists of finding those E, for E A [1,1000], at which d equals p/2. In our case we follow the first procedure i.e. we try to find the phase shifts for given energies. The obtained phase shift is then compared to the analytic value of p/2. The above problem is the so-called resonance problem when the positive eigenenergies lie under the potential barrier. We solve this problem, using the technique fully described in refs. 12, 26, 30–32. The boundary conditions for this problem are: y(0) = 0, pffiffiffiffi yðxÞ cos½ E x for large x: The domain of numerical integration is [0, 15]. For comparison purposes in our numerical illustration we use the following methods: The well known Numerov’s method (which is indicated as Method I) The Explicit Numerov-Type Method developed by Chawla and Rao96 (which is indicated as Method II) The P-stable Exponentially Fitted Method developed by Kalogiratou and Simos114 (which is indicated as Method III) 376 | Chem. Modell., 2008, 5, 350–487 This journal is

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Fig. 20 Errors Err max = max(log10|Ecalculated Eanalytical|) of the highest eigenenergy E3 = 989.701916 for several values of NFE.

The four-step method developed by P. Henrici118 (which is indicated as Method IV) The Closed Newton–Cotes Formula developed in ref. 104 (which is indicated as Method V) The Closed Newton–Cotes Formula developed in ref. 106 (which is indicated as Method VI) The Closed Newton–Cotes Formula developed in ref. 101 (which is indicated as Method VII) The Closed Newton–Cotes Formula developed in ref. 103 (which is indicated as Method VIII) The Closed Newton–Cotes Formula developed in ref. 105 (which is indicated as Method IX) The Closed Newton–Cotes Formula developed in ref. 107 (which is indicated as Method X) The numerical results obtained for the five methods, with several number of function evaluations (NFE), were compared with the analytic solution of the Woods–Saxon potential resonance problem, rounded to six decimal places. Fig. 20 show the errors Err = log10|Ecalculated Eanalytical| of the highest eigenenergy E3 = 989.701916 for several values of NFE (Fig. 21–23). 6.2 Conclusions We have presented a new approach for constructing efficient methods for the numerical solution of the Schro¨dinger type equations. From the numerical results we have the following remarks: The Explicit Numerov-Type Method developed by Chawla and Rao96 has better behavior than the well known Numerov’s method The P-stable Exponentially Fitted Method developed by Kalogiratou and Simos114 has better behavior than the explicit Numerov-type method with minimal phase-lag of Chawla et al.96 for small number of function evaluations. Chem. Modell., 2008, 5, 350–487 | 377 This journal is

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The four-step method developed by P. Henrici118 has better behavior than all the previous mentioned methods. The closed Newton–Cotes formulae case k = 6 have better behaviour (generally) than the closed Newton–Cotes formulae case k = 4. The closed Newton–Cotes formulae case k = 4 have better behaviour (generally) than the closed Newton– Cotes formulae case k = 2. The most effieicient method is the closed Newton–Cotes formulae case k = 6 which integrates exactly any linear combination of the functions: {1,x,x2,x3,cos(wx),sin(wx),x cos(wx),x sin(wx)}. All computations were carried out on a IBM PC-AT compatible 80 486 using double precision arithmetic with 16 significant digits accuracy (IEEE standard).

B. Stabilization of a multistep exponentially-fitted methods and their application to the Schro¨dinger equation In this part of the paper we present the study of the stabilization of the multistep exponentially-fitted methods. More specifically we present a family of singularly Pstable exponentially-fitted four-step methods and a family of six-step exponentiallyfitted methods for the numerical solution of the radial Schro¨dinger equation.

7. Introduction The one-dimensional Schro¨dinger equation can be written as: y00 (x) = [l(l + 1)/x2 + V(x) k2]y(x).

(69)

The above boundary value problem occurs frequently in theoretical physics and chemistry, material sciences, quantum mechanics and quantum chemistry, electronics etc. (see for example refs. 115–118). Chem. Modell., 2008, 5, 350–487 | 379 This journal is

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We give some definitions for: The function W(x) = l(l + 1)/x2 + V(x) is called the effective potential. This satisfies W(x) - 0 as x - N The quantity k2 is a real number denoting the energy The quantity l is a given integer representing the angular momentum V is a given function which denotes the potential. The boundary conditions are: y(0) = 0

(70)

and a second boundary condition, for large values of x, determined by physical considerations. The last decades a lot of research has been done on the development of numerical methods for the solution of the Schro¨dinger equation. The aim of this research is the development of fast and reliable methods for the solution of the Schro¨dinger equation and related problems (see for example refs. 12, 14, 18, 21, 24, 29, 32–34, 39, 40, 46, 47, 49, 50, 68, 69, 104, 109, 120, 121, 123–135). The methods for the numerical solution of the Schro¨dinger equation can be divided into two main categories: Methods with constant coefficients Methods with coefficients depending on the frequency of the problem.w In this chapter we will investigate methods of the second category. We will investigate the exponentially-fitted methods. More specifically we will present a family of exponentially-fitted methods of sixth algebraic order for the numerical solution of the radial Schro¨dinger equation. The developed method is also almost Pstable i.e. it has an interval of periodicity equal to (0,N) (in the case that the frequency of the exponential fitting is the same as the frequency of the scalar test equation). We will present also a family of six-step exponentially-fitted methods. We apply the new presented methods to the resonance problem. This is one of the most difficult problems arising from the radial Schro¨dinger equation. The above application shows the efficiency of the new obtained method.

8. The new trigonometrically-fitted family of four-step method 8.1 The first method of the family 8.1.1 Construction of the new method. We introduce the following family of methods for the numerical solution of the problem y00 = f(x,y): yn+2 + ayn+1 (2 + 2a)yn + ayn1 + yn2 = h2[b0(y00 n+2 + y00 n2) + b1(y00 n+1 + y00 n1) + b2y00 n]

(71)

We demand the above method (71) to be exact for any linear combination of the functions {1,x,x2,x3,x4,x5,exp(Ivx)}

(72)

where I¼

pffiffiffiffiffiffiffi 1

So, the following system of equations must hold: 2a cos(w) 2 + 2 cos(2w) 2a = 2w2 cos(2w)b0 2w2b1 cos(w) w2b2 4 + a = 2b0 + 2b1 + b2

(73) (74)

{ When using the trigonometrically-fittedpmethod for the solution of the radial Schro¨dinger ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi equation, the fitted frequency is equal to: jlðl þ 1Þ=x2 þ VðxÞ k2 j.


c


16 + a = 48b0 + 12b1

(75)

where w = vh. We study, now, that stability of the presented method. We apply the new method (71) to the scalar test equation: y00 = v2y.

(76)

We obtain the following difference equation: A(v,h)(yn+2 + yn2) + B(v,h)(yn+1 + yn1) + C(v,h)yn = 0

(77)

A(v,h) = 1 + v2h2b0, B(v,h) = a + v2h2b1,C(v,h) = 2 2a + v2h2b2.

(78)

where

The corresponding characteristic equation is given by: A(v,h)(l4 + 1) + B(v,h)(l3 + l) + C(v,h)l2 = 0

(79)

Definition 1. (See ref. 21) A symmetric four-step method with the characteristic equation given by (11) is said to have an interval of periodicity (0,w20) if, for all w A (0,w20), the roots zi, i = 1,2 satisfy z1,2 = eiy(vh), |zi| r 1, i = 3,4

(80)

where y(vh) is a real function of vh and w = vh. Definition 2. (See ref. 21) A method is called P-stable if its interval of periodicity is equal to (0,N). In order the new method to be P-stabley we require the characteristic eqn (79) to have the following roots: exp(Ivh), exp(Ivh), exp(Ivh), exp(Ivh)

(81)

In order (81) to be roots of the characteristic eqn (79), the following system of equations must hold: 4(1 + v2h2b0)cos(vh)2 2(a + v2h2b1)cos(vh) 4 2v2h2b0 2a + v2h2b2 = 0 (82) 4(1 + v2h2b0)cos(vh)2 + 2(a + v2h2b1)cos(vh) 4 2v2h2b0 2a + v2h2b2=0 (83) Solving the system of eqns (73)–(75) and (82) and (83) the values of the coefficients of the methods are obtained (see for more details ref. 108). In order to solve the above equation and in order to obtain the coefficients we use the program mentioned in Appendix G. The behaviour of the coefficients of the method is given in the following figures (Fig. 24–27): where b_i = bi, i = 0(1)2 and a_ = a. The local truncation error of this method is given by: LTE ¼

2h8 ð8Þ ðy þ v2 yð6Þ n Þ 945 n

ð84Þ

The LTE is obtained expanding the terms ynj and fnj, j = 0(1)2 in (71) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. The LTE of any method (71) which integrates exactly any linear combination of the functions (72) is given by: { We note here that the frequency of the exponential fitting is equal with the frequency of the stability analysis. } In the case that the frequency of the exponential fitting is the same as the frequency of the scalar test equation (which in the case of (8) has been obtained).


c


Fig. 24 Behaviour of the coefficient b0.

2 31a 2 ð6Þ LTE ¼ h8 þ ðyð8Þ n þ v yn Þ 945 60480

ð85Þ

Based on the analysis presented in ref. 108 we conclude that the error constant’s smallest value can be obtained for a = 0.z 8.1.2 Stability analysis. We apply the new method to the scalar test equation: y00 = p2y,

(86)

where p a v. We obtain the following difference equation: A(p,h)(yn+2 + yn2) + B(p,h)(yn+1 + yn1) + C(p,h)yn = 0

(87)

A(p,h) = 1 + p2h2b0, B(p,h) = a + p2h2b1,C(p,h) = 2 (1+a) + p2h2b2.

(88)

where

} For this value the classical method i.e. the method produced from (73)–(75) and (82) and (83) for w - 0 is completely unstable since the characteristic equation has no roots.


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The corresponding characteristic equation is given by: A(p,h)(l4 + 1) + B(p,h)(l3 + l) + C(p,h)l2 = 0

(89)

Theorem 3. (See ref. 94) A symmetric four-step method with the characteristic equation given by (89) is said to have an interval of periodicity (0,H20) if, for all H A (0,H20) the following relations are hold P1(H,w) Z 0,P2(H,w) Z 0,P3(H,w) Z 0 P2(H,w)2 4P1(H,w)P3(H,w) Z 0

(90)

where H = ph, w = vh and: P1(H,w) = 2A(H,w) 2B(H,w) + C(H,w) Z 0, P2(H,w) = 12A(H,w) 2C(H,w) Z 0, P3(H,w) = 2A(H,w) + 2B(H,w) + C(H,w) Z 0, N(H,w) = P2(H,w)2 4P1(H,w)P3(H,w) Z 0.

(91)


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Definition 2. A method is called singularly P-stable if its interval of periodicity is equal to (0,N) only when the frequency of the exponential fitting is the same as the frequency of the scalar test equation, i.e. H = w. Definition 3. A method is called singularly almost P-stable if its interval of periodicity is equal to (0,N) S8 only when the frequency of the exponential fitting is the same as the frequency of the scalar test equation, i.e. H = w. Definition 4. A method is called P-stable if its interval of periodicity is equal to (0,N). Definition 5. A method is called almost P-stable if its interval of periodicity is equal to (0,N) S.* Based on the above we present the w H plane (Fig. 28). We have the following remarks (for more details see at ref. 108): If the frequency of the exponential fitting is equal to the frequency of the scalar test equation (first diagonal of the w H plane) the method is singularly P-Stable. If the frequency of the exponential fitting is different from the frequency of the scalar test equation the method is not P-stable (i.e. there are areas in the Fig. 28 that are white and in which the conditions of P-stability are not satisfied). In the case the frequency of the exponential fitting is equal to the frequency of the scalar test equation, we have for the stability polynomials, also, the following figure (Fig. 29 and 29b). || Where S is a set of distinct points. * Where S is a set of distinct points.


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Fig. 27 Behaviour of the coefficient a.

Remark 1: In the case that p a v i.e. the frequency of the exponential fitting is different from the frequency of the scalar test equation, then the system of eqns (73)– (75) and (82) and (83)ww haven’t a solution. 8.2 The second method of the family 8.2.1 Construction of the new method. We introduce the family of methods presented in (71). We demand the above method (71) to be exact for any linear combination of the functions {1,x,x2,x3,exp(I,vx),x exp(I,vx)} where I¼

(92)

pffiffiffiffiffiffiffi 1:

So, the following system of equations must hold: 4 cos(vh)2+2a cos(vh)2a 4 = (4b0 cos(vh)2 + 2b1 cos(vh) + b2 2b0)h2v2 (93) {{ Of course in this case v a p.


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Fig. 28 w H plane for the presented method.

2 sin(vh)h(4 cos(vh) + a) = 2h2v(hb1v sin(vh) 4hb0v cos(vh) sin(vh) + 4b0 cos(vh)2 + 2b1 cos(vh) + b2 2b0) (94) 8h2 + 2ah2 = 2h2(2b0 + 2b1 +b2)

(95)

Based on the analysis of the previous paragraph 7.1, the system of eqns (82) and (83) must also hold. Solving the system of eqns (93)–(95) and (82) and (83) the values of the coefficients of the methods are obtained (see for more details ref. 109). In order to solve the above equation and in order to obtain the coefficients we use the program mentioned in Appendix H. The behaviour of the coefficients of the method is given in the following figuers (Fig. 30–33): where b_i = bi, i = 0(1)2 and a_ = a. The local truncation error of this method is given by: LTE ¼

2h8 ð8Þ 4 ð4Þ ðy þ 2v2 yð6Þ n þ v yn Þ: 945 n

ð96Þ

The LTE is obtained expanding the terms ynj and fnj, j = 0(1)2 in eqn (71) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method. 8.2.2 Stability analysis. We apply the new method to the scalar test eqn (86) where p a v. We obtain the difference eqn (87). The corresponding characteristic equation is given by (89). Based on the above we present the w H plane (Fig. 34). We have the following remarks (for more details see at ref. 109): 386 | Chem. Modell., 2008, 5, 350–487 This journal is

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Fig. 29 (a) Stability polynomials for the presented method in the case that the frequency of the exponential fitting is equal to the frequency of the scalar test equation. (b) Stability polynomials (polynomial N(H,w)) for the presented method in the case that the frequency of the exponential fitting is equal to the frequency of the scalar test equation.

If the frequency of the exponential fitting is equal to the frequency of the scalar test equation (first diagonal of the w H plane) the method is singularly P-Stable. If the frequency of the exponential fitting is different from the frequency of the scalar test equation the method is not P-stable (i.e. there are areas in the Fig. 34 that are white and in which the conditions of P-stability are not satisfied). Chem. Modell., 2008, 5, 350–487 | 387 This journal is

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In the case the frequency of the exponential fitting is equal to the frequency of the scalar test equation, we have for the stability polynomials, also, the following figures. Remark 2: In the case that p a v i.e. the frequency of the exponential fitting is different from the frequency of the scalar test equation, then the system of eqns (93)–(95) and (82) and (83)zz haven’t a solution. 8.3 The third method of the family 8.3.1 Construction of the new method. We introduce the family of methods presented in (71). We demand the above method (71) to be exact for any linear combination of the functions {1,x,exp(Ivx),x exp(Ivx),x2 exp(Ivx)}

(97)

where I¼

pffiffiffiffiffiffiffi 1:

So, the following system of equations must hold: 2a cos(vh) 2 + 2 cos(2vh) 2a = 2h2v2 cos(2vh)b0 2h2v2b1 cos(vh) h2v2b2 {{ Of course in this case v a p.


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(98)


2x 2xa + 2xa cos(vh) + 2x cos(2vh) = 2h2v2 cos(2vh)b0x 2h2v2b1x cos(vh) h2v2b2x 2

2

2ah sin(vh) + 4h sin(2vh) = 4h v cos(2vh)b0 + 2h vb2 4h v b0 sin(2vh) 2h3v2b1 sin(vh) + 4h2vb1 cos(vh) 2

2

2

2

2

(99)

3 2

(100)

2

2x a 2x +8h cos(2vh) + 2x cos(2vh) + 2x a cos(vh) + 2ah cos(vh) = 2h4v2b1 cos(vh) + 4h2b1 cos(vh) 8h4 cos(2vh)b0v2 8h3b1v sin(vh) 2h2b1v2x2 cos(vh) h2b2x2v2 2h2 cos(2vh)b0v2x2 + 4h2 cos(2vh)b0 + 2h2b2 16h3b0v sin(2vh) (101) 8xh sin(2vh) + 4axh sin(vh) = 4h3b1v2x sin(vh) + 8h2 cos(2vh)b0vx 8h3b0v2x sin(2vh) + 4h2b2xv + 8h2b1vx cos(vh)

(102)

Based on the analysis of the previous paragraph 7.1, the system of eqns (82) and (83) must also hold. Solving the system of eqns (99)–(102) and (82) and (83) the values of the coefficients of the methods are obtained (see for more details ref. 110). In order to solve the above equation and in order to obtain the coefficients we use the program mentioned in Appendix I. The behaviour of the coefficients of the method is given in the following figures (Fig. 35–38): where b_i = bi, i = 0(1)2 and a_ = a. Chem. Modell., 2008, 5, 350–487 | 389 This journal is

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The local truncation error of this method is given by:

L:T:E: ¼

2h8 ð8Þ 4 ð4Þ 6 ð2Þ ðy þ 3v2 yð6Þ n þ 3v yn þ v yn Þ 945 n

ð103Þ

The LTE is obtained expanding the terms ynj and fnj, j = 0(1)2 in (71) into Taylor series expansions and substituting the Taylor series expansions of the coefficients of the method.

8.3.2 Stability analysis. We apply the new method to the scalar test eqn (86) where p a v. We obtain the difference eqn (87). The corresponding characteristic equation is given by (89). Based on the above we present the w H plane (Fig. 39). We have the following remarks (for more details see at ref. 110): If the frequency of the exponential fitting is equal to the frequency of the scalar test equation (first diagonal of the w H plane) the method is singularly P-Stable. If the frequency of the exponential fitting is different from the frequency of the scalar test equation the method is not P-stable (i.e. there are areas in the Fig. 39 that are white and in which the conditions of P-stability are not satisfied). In the case the frequency of the exponential fitting is equal to the frequency of the scalar test equation, we have for the stability polynomials, also, the following figures. 390 | Chem. Modell., 2008, 5, 350–487 This journal is

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Remark 3: In the case that p a v i.e. the frequency of the exponential fitting is different from the frequency of the scalar test equation, then the system of eqns (98)– (102) and (82) and (83)yy haven’t a solution.

9. Numerical results—Conclusion In order to illustrate the efficiency of the new presented methods we apply them to the radial Schro¨dinger equation. In order to apply the presented methods to the Schro¨dinger equation the value of parameter v is needed. For every problem of the radial Schro¨dinger equation given by the parameter v is given by v¼

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jqðxÞj ¼ jVðxÞ Ej

ð104Þ

where V(x) is the potential and E is the energy. We use as potential the well known Woods–Saxon potential (65). For some well known potentials, such as the Woods–Saxon potential, the definition of parameter v is not given as a function of x but based on some critical }} Of course in this case v a p.


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Fig. 34 (a) Stability polynomials for the presented method in the case that the frequency of the exponential fitting is equal to the frequency of the scalar test equation.

points which have been defined from the study of the appropriate potential (see for details ref. 40). For the purpose of obtaining our numerical results it is appropriate to choose v as follows (see for details ref. 40): 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 þ E ; > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < p37:5 þ E; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25 þ E ; > > > 12:5 þ E ; > : pffiffiffiffi E;

for x 2 ½0; 6:5 2h; for x ¼ 6:5 h for x ¼ 6:5 for x ¼ 6:5 þ h for x 2 ½6:5 þ 2h; 15

9.1 Radial Schro¨dinger equation—the resonance problem Consider the numerical solution of the Schro¨dinger eqn (64) in the well-known case that the potential is the Woods–Saxon potential (65). In order to solve this problem numerically we need to approximate the true (infinite) interval of integration by a finite interval. For the purpose of our numerical illustration we take the domain of integration as x A [0,15]. We consider eqn (64) in a rather large domain of energies, i.e. E A [1,1000]. Following the analysis of section 6, we will solve the so-called resonance problem. This problem consists either of finding the phase-shift dl or finding those E, for 392 | Chem. Modell., 2008, 5, 350–487 This journal is

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E A [1,1000], at which dl ¼ p2. We actually solve the latter problem, known as the resonance problem when the positive eigenenergies lie under the potential barrier. The boundary conditions for this problem are: pffiffiffiffi yð0Þ ¼ 0; yðxÞ ¼ cosð E xÞ for large x:

ð105Þ

We compute the approximate positive eigenenergies of the Woods–Saxon resonance problem using: the Numerov’s method which is indicated as Method I the exponentially-fitted method of Numerov type developed by Raptis and Allison46 which is indicated as Method II the exponentially-fitted method of Numerov type developed by Ixaru and Rizea40 which is indicated as Method III the two-step P-stable exponentially-fitted method developed by Kalogiratou and Simos114 which is indicated as Method IV the two-step P-stable method obtained by Chawla122 which is indicated as Method V the four-step method mentioned in Henrici119 which is indicated as Method VI the exponentially-fitted four-step method developed by Raptis120 which is indicated as Method VII the P-stable four-step method developed by Wang136 which is indicated as Method VIII Chem. Modell., 2008, 5, 350–487 | 393 This journal is

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the P-stable trigonometrically-fitted four-step method developed by Simos108 which is indicated as Method IX the P-stable trigonometrically-fitted four-step method developed by Simos109 which is indicated as Method X the P-stable trigonometrically-fitted four-step method developed by Simos110 which is indicated as Method XI The computed eigenenergies are compared with exact ones. In Fig. 40–43 we present the maximum absolute error log10(Err) where Err = |Ecalculated Eaccurate|

(106)

of the eigenenergies E0, E1, E2 and E3, respectively, for several values of NFE = Number of function evaluations. 9.2 Conclusions In the present chapter we have presented a new family of exponentially-fitted fourstep methods for the numerical solution of the one-dimensional Schro¨dinger equation. For these methods we have examined the stability properties. The new methods satisfy the property of P-stability only in the case that the frequency of the exponential fitting is the same as the frequency of the scalar test equation (i.e. they are singularly P-stable methods). The new methods integrate also exactly every linear combination of the functions 394 | Chem. Modell., 2008, 5, 350–487 This journal is

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{1,x,x2,x3,x4,x5,exp(Ivx)}.

(107)

{1,x,x2,x3,exp(I,vx),x exp(I,vx)}

(108)

2

{1,x,exp(I,vx),x exp(I,vx),x exp(I,vx)}

(109)

In order to illustrated their efficiency, we have applied the new methods to the resonance problem of the one-dimensional Schro¨dinger equation. Based on the results presented above we have the following conclusions: For the set of methods Method I—Method V (category of 2-step methods) we have the following remarks: 1. The method of Ixaru and Rizea40 is the most efficient of the category of 2-step methods 2. The method of Kalogiratou and Simos114 is the next most efficient method 3. The Method of Raptis and Allison46 is more efficient than the Numerov’s method and the P-stable method of Chawla122 4. Finally the method of Chawla122 has similar efficiency with the Numerov’s method. For the set of methods Method VI—Method XI (category of 4-step methods) we have the following remarks: 1. The method of Simos110 is the most efficient one in the category of 4-step methods Chem. Modell., 2008, 5, 350–487 | 395 This journal is

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2. The method of Simos109 is the next efficient one in the category of 4-step methods 3. The method of Simos108 is the third most efficient one in the category of 4-step methods 4. The method of Wang136 is the fourth most efficient one in the category of 4-step methods 5. The method of Henrici119 is the fifth most efficient one in the category of 4-step methods 6. Finally the method of Raptis120 is the sixth most efficient one in the category of 4-step methods From the results presented the most efficient methods are: The P-stable four-step trigonometrically-fitted method of Simos110 The P-stable four-step trigonometrically-fitted method of Simos109 The P-stable four-step trigonometrically-fitted method of Simos108 All computations were carried out on a IBM PC-AT compatible 80486 using double precision arithmetic with 16 significant digits accuracy (IEEE standard).

10. Comments on recent bibliography In ref. 137 the author presented a theoretical framework for a new type of phasefitted and amplification-fitted two-step hybrid methods. This type of methods have beem introduced by the same author in ref. 138. These type of methods is a simple modification of dissipative two-step hybrid methods since two free parameters are 396 | Chem. Modell., 2008, 5, 350–487 This journal is

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Fig. 39 w H plane for the presented method.


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Fig. 40 Comparison of the maximum errors Err max in the computation of the resonance E0 = 53.588872 using the Methods I–XI. The values of Err max have been obtained based on the NFE. The absence of values of Err max for some methods indicates that for these values of NFE = Number of Function Evaluations, the Err max is positive.


added in order the phase-lag and the amplification error to be eliminated. We notice here that this type of methods is useful only in the case in which a good estimate of the frequency of the problem is known in advance. The parameters are depended on the product of the estimated frequency and the stepsize. In the paper the author studies the algebraic order, the zero-stability, the linear stability and the phase properties. The numerical results are presented using sixth-order explicit phase-fitted and amplification-fitted two-step hybrid methods. In ref. 139 the authors present an explicit method for the numerical solution of the Schro¨dinger equation. The Schro¨dinger equation is first transformed into a Hamiltonian canonical equation and then the author obtained several methods up to the 398 | Chem. Modell., 2008, 5, 350–487 This journal is

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eighth order. They also construct new third and fourth order methods. The efficiency of the new methods is illustrated via several numerical experiments. In ref. 140 the author studies the enhancement of the accuracy of the line-based perturbation method via the introduction of the perturbation corrections. They effectively develop the first and the second order corrections. The authors present an error analysis in order to predict that the introduction of successive corrections substantially enhances the order of the method from four, (with zeroth order version), to six and ten (with the first and the second-order corrections). Also they remove the effect of the accuracy loss due to cancellation of terms when evaluating the perturbation corrections constructing alternative asymptotic formulae using the Chem. Modell., 2008, 5, 350–487 | 399 This journal is

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Maple manipulation package. Finally, they develop a procedure for choosing the step size in terms of the preset accuracy. In ref. 141 the author presents a new and accurate method for the numerical solution of the eigenvalue problem of second-order ordinary differential equation The shooting method is used. The method has three steps. The initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method. The initial-value problem is solved using new, highly accurate formulas of the linear multistep method. The eigenvalue is properly corrected at the matching point. The author examines with success the efficiency of the methods by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Po¨schl-Teller potential in quantum mechanics. In ref. 142 the authors are studied the Numerov-type ODE solvers for the numerical solution of second-order initial value problems. They present a powerful and efficient symbolic code in MATHEMATICA for the derivation of their order conditions and principal truncation error terms. They also present the relative tree theory for such order conditions along with the elements of combinatorial mathematics, partitions of integer numbers and computer algebra which are the basis of the implementation of the symbolic code. We must that one of the authors is an expert on this specific field. In ref. 143 the authors develop a third-order 3-stage diagonally implicit RungeKutta-Nystro¨m method embedded in fourth-order 4-stage for solving special second-order initial value problems. The obtained method has been developed in order to have minimal local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The authors also study the stability of the method. The new proposed method is illustrated via a set of test problems. In ref. 144 the author presents the construction of a non-standard explicit algorithm for initial-value problems. The order of the developed method is two and also is A-stable. The new proposed method is proven to be suitable for solving different kind of initial-value problems such as non-singular problems, singular problems, stiff problems and singularly perturbed problems. Some numerical experiments are considered in order to check the behaviour of the method when applied to a variety of initial-value problems. In ref. 145 the author presents a new explicit Numerov-type method. The development is based on a modification of a sixth-order explicit Numerov-type method recently produced by Tsitouras [Ch. Tsitouras, Explicit Numerov type methods with reduced number of stages, Comput. Math. Appl. 2003, 45, 37–42]. The author adds two free parameters in order nullify the phase-lag and the amplification error. We notice here that this type of methods is useful only in the case in which a good estimate of the frequency of the problem is known in advance. The parameters depend on the product of the estimated frequency and the stepsize. In ref. 146 the authors present a non-standard (nonlinear) two-step explicit Pstable method of fourth algebraic order and 12th phase-lag order for solving secondorder linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. In ref. 147 the authors apply Adomian decomposition method to develop an efficient algorithm of a special second-order ordinary initial value problems. The Adomian decomposition method is known that does not require discretization, so is computer time efficient. The authors are studied the Adomian decomposition method and the results obtained are compared with previously known results using the Quintic C2-spline integration methods. In ref. 148 the author obtain a new kind of trigonometrically fitted explicit Numerovtype method for the numerical integration of second-order initial value problems with 400 | Chem. Modell., 2008, 5, 350–487 This journal is

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oscillating or periodic solutions. This new method is based on the fifth-order method which is dispersion of order eight and dissipation of order five proposed in ref. 149. The authors in this paper require the method to be accurate for any linear combination of the trigonometric functions. For the finally constructed method the stability, phase-lag (dispersion) and dissipation was examined using the methodology introduced by Coleman and Ixaru (see ref. 150). The efficiency of the new proposed methos is examined via comparisons with the methods produced in refs. 149 and 151. In ref. 152 the author produces methods based on numerical differentiation some classes of special multistep methods. For these methods the regions of absolute stability are shown. Numerical efficiency of the methods is examined by application of some of Henrici119 and of some methods obtained in this paper of the same order to a second-order initial value problem. In ref. 153 the author studies a second-order differential equation whose solution is periodic with two frequencies has important applications in many scientific fields. This problem is a periodic stiff problem for most of the available linear multi-step methods. These phenomena are similar to studied for the popular Sto¨mer-Cowell class of linear multi-step methods for one-frequency problems. Based on the stability theory introduced by Lambert, ‘periodic stiffness’ appears in a two-frequency problem because the production of the step-length and the bigger angular frequency lies outside the interval of periodicity. It is known also that for a two-frequency problem, the error in the numerical solution obtained by a P-stable trigonometrically-fitted method with one frequency would be too high for practical applications even if a small step-length has been used. In this paper the author demonstrates that the interval of periodicity and the local truncation error of a linear multi-step method for a two-frequency problem can be greatly improved by a new introduced trigonometric-fitting methodology. A trigonometrically-fitted Numerov method with two frequencies is obtained and has been studied for P-stability with vanishing local truncation error for a two-frequency test problem. In ref. 154 the author presents an implicit hybrid two step method for the solution of second order initial value problem. The cost of the new obtained method is only six function evaluations per step and the algebraic order is eighth. The author studies the P-stability property and the conclusion is the new method satisfies this property requiring one stage less. In ref. 155 the authors study the implementation of high order implicit formulas (specially the Gauss methods) for solving second-order differential systems having high frequencies and small amplitudes superimposed. More specifically, they discuss the choice of an appropriate iterative scheme. In this investigation they have studied the predictors (initial guesses) and a variable order strategy to select the best predictor at each integration step is supplied. In ref. 156 the author studies the stability properties of a family of exponentially fitted Runge-Kutta-Nystro¨m methods. More specifically the author investigates the P-stability which is a very important property usually required for the numerical solution of stiff oscillatory second-order initial value problems. In this paper P-stable exponentially fitted Runge-Kutta-Nystro¨m methods with arbitrary high order are developed. The results of this paper are proved based on a symmetry argument. The authors in this paper157 present an explicit symplectic method for the numerical solution of the Schro¨dinger equation. A modified symplectic integrator with the trigonometrically fitted property which is based on this method is also produced. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential. In this paper158 the authors produced a class of non-linear explicit second-order methods for solving one-dimensional periodic initial value problems (IVPs). These methods are P-stable (based on the definition given by Lambert and Watson90) and they have phase-lag of high-order. The authors introduce a special vector operation such that the obtained methods to be extended to be vector-applicable directly. With this extension the produced methods can be applied to multi-dimensional problems. Chem. Modell., 2008, 5, 350–487 | 401 This journal is

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Since it is not sufficient to investigate the stability behavior of the vector form of the new methods based on the scalar test equation y00 = l2y, the authors have introduced a new test system and they have extended the definitions of the interval of periodicity and the phase-lag order to the new test system. Based on the above the stability property and the phase-lag of the vector-applicable methods is investigated. The summary of the investigation is: The methods are P-stable and of high phase-lag for one-frequency problems. For multi-frequency problems, the new methods are of only second phase-lag order and may have not a primary interval of periodicity. In this paper159 the authors present a new P-stable Obrechkoff four-step method. By introducing the idea of requirement the new methods to be stable for the scalar test equation y00 = o2y, the authors produce a trigonometric fitting P-stable Obrechkoff four-step method. Investigating the interval of periodicity of the Obrechkoff four-step methods one can see that the new method has an interval of periodicity equal to (0,N) while the previous methods had an interval of periodicity equal to (0,16). At the same time the method has kept the advantage in the accuracy and efficiency. The authors have tested the new methods to the well-known problems, (1) the test-equation; (2) Stiefel and Bettis problem; (3) Duffing equation without damping; and (4) Bessel equation. The numerical results show that the new developed method is more accurate than any previous method. One question which must be investigated in the future is the potential of the classical exponential fitting and the production of P-stable parameter dependent methods of the above form. In ref. 160 the author investigates the behaviour of the undamped Duffing oscillator. The author has remarked that with non-linearities, the frequency spectrum of an undamped Duffing oscillator should be composed of odd multiples of the driving frequency which can be interpreted as resonance driving terms. So, based on the above remark, it is expected that the frequency spectrum of the corresponding numerical approximation with high accurateness should contain nearly the same components. Based on this conclusion, the author consider that the key to develop a new numerical method with high stability, accuracy and efficiency for the Duffing equation, is the numerical approximation which is produced from the numerical method, to contain these Fourier components in order to calculate the amplitudes of these components in a more accurate and efficient way. The author based on the above constructed four types of Numerov methods: the first one is the traditional Numerov Method, which contains no Fourier component, the second one is a Numerov-type Method which contains only the first resonance term, the third one is a Numerov-type Method which contains the first two resonance terms, and the last one contains is a Numerov-type Method which contains the first three resonance terms In order to illustrate the efficiency of the new produced methods, the author applied them to the well-known undamped Duffing equation with Dooren’s parameters. The numerical results show that the Numerov method fitted with the Fourier components is much more stable, accurate and efficient than the one with no Fourier component. The accuracy of the fitted method with the first three Fourier components can attain 10 9 for a remarkable range of step sizes, which is much higher than the one of the traditional Numerov method, with eight orders for step size of p/2.011. In ref. 161 the authors have produced exponentially-fitted BDF-Runge-Kutta type formulas (of second-, third- and fourth-order). The good behaviour of the new produced methods for stiff problems is proved. The stability regions of the new proposed methods have been examined. It is proved that the plots of their absolute stability regions include the whole of the negative real axis. Finally the author propose several procedures to find the parameter of the new methods. 402 | Chem. Modell., 2008, 5, 350–487 This journal is

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In ref. 162 the numerical integration of Hamiltonian systems is investigated. Trigonometrically fitted symplectic partinioned Runge-Kutta methods of second, third and fourth orders are obtained. The methods are tested on the numerical integration of the harmonic oscillator, the two body problem and an orbital problem studied by Stiefel and Bettis. In ref. 163 a class of explicit modified Runge-Kutta-Nystro¨m (RKN) methods for the numerical solution of second-order IVPs with oscillatory solutions is produced. For this class of methods the symplecticness conditions and the exponential fitting conditions are investigated and derived. Explicit modified RKN integrators with two and three stages per step which have algebraic orders two and four, respectively, are obtained based on the above conditions. These new integrators are symplectic when they are applied to Hamiltonian problems. The new proposed methods integrate also exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(lt),exp(lt)}, l A C. In this paper also the stability properties of the new proposed methods is analysed and generalized periodicity regions for the linear scalar test equation are obtained. In ref. 164 new and efficient trigonometrically-fitted adapted Runge-KuttaNystro¨m methods for the numerical solution of perturbed oscillators are obtained. These methods combine the benefits of trigonometrically-fitted methods with adapted Runge-Kutta-Nystro¨m methods. The necessary and sufficient order conditions for these new methods are produced based on the linear-operator theory. In ref. 165 a symplectic explicit RKN method for Hamiltonian systems with periodical solutions is obtained. The characteristics of this new method are: algebraic order three phase-lag order six cost of three function evaluations per step. In ref. 166 a kind of trigonometrically fitted explicit two-step hybrid method which has algebraic order six is obtained. The new method has the following characteristics: is zero dissipative, is phase fitted, and is almost P-stable. In ref. 167 the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge-Kutta (EFRK) methods is investigated. The sufficient conditions on symplecticity of EFRK methods are presented. A family of symplectic EFRK two-stage methods with order four has been produced. This new method includes the symplectic EFRK method proposed by Van de Vyver168 and a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. In ref. 169 the extension of the algorithm of Scheifele170 for the construction of a fourth-order hybrid method without explicit first derivatives is investigated. In ref. 171 new Runge-Kutta-Nystro¨m methods for the numerical solution of periodic initial value problems are obtained. These methods are able to integrate exactly the harmonic oscillator. In this paper the analysis of the production of an embedded 5(3) RKN pair with four stages is presented. In ref. 172 a new embedded pair of explicit Runge-Kutta-Nystro¨m (RKN) methods adapted to the numerical solution of general perturbed oscillators is produced. This pair is based on the methods developed by Franco.173 These methods can be used for general problems. It is proved that the embedded methods have algebraic order 4 and 3. In ref. 174 a new embedded pair of explicit exponentially fitted Runge-KuttaNystro¨m methods is developed. The methods integrate exactly any linear combinations of the functions from the set {exp(mt),exp(mt)} (m A R or m A iR). The new methods have the following characteristics: it has four stages and algebraic orders five and three. Chem. Modell., 2008, 5, 350–487 | 403 This journal is

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Appendix A 4 restart; 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n–1]: = exp(v*(xh)); 4 f[n–1]: = diff(y[n1],x$1); 4 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n–2]: = exp(v*(x–2*h)); 4 f[n–2]: = diff(y[n–2],x$1); 4 y[n+3]: = exp(v*(x+3*h)); 4 f[n+3]: = diff(y[n+3],x$1); 4 y[n–3]: = exp(v*(x–3*h)); 4 f[n–3]: = diff(y[n–3],x$1); 4 4 4 final: = y[n+3]–y[n–3] = h*(a[0]*f[n–3]+a[1]*f[n–2]+a[2]*f[n–1]+a[3]*f[n]+ a[4]*f[n+1]+a[5]*f[n+2]+a[6]*f[n+3]); final : = e(v(x+3h)) e(v(x3h)) = h(a0ve(v(x3h)) + a1ve(v(x2h)) + a2ve(v(xh)) + a3ve(vx) + a4ve(v(x+h)) + a5ve(v(x+2h)) + a6ve(v(x+3h))) 4 final: = combine(final/exp(v*x)); final : = e(vx)(e(v(x+3h)) e(v(x3h))) = e(vx)h(a0ve(v(x3h)) + a1ve(v(x2h))) + a2ve(v(xh)) + a3ve(vx)) + a4ve(v(x+h)) + a5ve(v(x+2h)) + a6ve(v(x+3h))) 4 final: = expand(final); final : ¼ ðeðvhÞ Þ3 ¼

ha0 v ðeðvhÞ Þ3

þ

1 ðeðvhÞ Þ3 ha1 v ðeðvhÞ Þ2

þ

ha2 v þ ha3 v þ ha4 veðvhÞ þ ha5 vðeðvhÞ Þ2 þ ha6 vðeðvhÞ Þ3 eðvhÞ

4 final: = simplify(convert(final,trig)); final : = 2(16 cosh(vh)6 + 16 cosh(vh)5 sinh(vh) 24 cosh(vh)4 16 cosh(vh)3 sinh(vh) + 9 cosh(vh)2 + 3 cosh(vh)sinh(vh) 1)/(cosh(vh) + sinh(vh))3 = vh(a0 a2 + a4 a6 + 16a5 cosh(vh)4 sinh(vh) + 8a4 cosh(vh)3 sinh(vh) 32a6 cosh(vh)3 sinh(vh) + 32a6 cosh(vh)5 sinh(vh) 4a4 cosh(vh) sinh(vh) + 2a2 cosh(vh) sinh(vh) + 4a3 cosh(vh)2 sinh(vh) + a1 cosh(vh) + a1 sinh(vh) + 2a2 cosh(vh)2 + 4a3 cosh(vh)3 + 8a4 cosh(vh)4 + 16a5 cosh(vh)5 + 32a6 cosh(vh)6 12a5 sinh(vh) cosh(vh)2 + 6a6 cosh(vh) sinh(vh) + 18a6 cosh(vh)2 + 5a5 cosh(vh) + a5 sinh(vh) a3 sinh(vh) 20a5 cosh(vh)3 8a4 cosh(vh)2 3a3 cosh(vh) 48a6 cosh(vh)4)/(cosh(vh) + sinh(vh))3 4 4 4 y[n]: = exp(–v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n–1]: = exp(–v*(x–h)); 4 f[n–1]: = diff(y[n–1],x$1); 4 y[n+2]: = exp(–v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 404 | Chem. Modell., 2008, 5, 350–487 This journal is

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4 y[n–2]: = exp(–v*(x–2*h)); 4 f[n–2]: = diff(y[n–2],x$1); 4 y[n+3]: = exp(–v*(x+3*h)); 4 f[n+3]: = diff(y[n+3],x$1); 4 y[n–3]: = exp(–v*(x–3*h)); 4 f[n–3]: = diff(y[n–3],x$1); 4 4 4 final1: = y[n+3]–y[n–3] = h*(a[0]*f[n–3]+a[1]*f[n–2]+a[2]*f[n–1]+a[3]*f[n]+ a[4]*f[n+1]+a[5]*f[n+2]+a[6]*f[n+3]); final1 : = e(v(x+3h)) e(v(x3h)) =h (a0 ve(v(x3h)) a1 ve(v(x2h)) a2 ve(v(xh)) a3 ve(vx) a4 ve(v(x+h)) a5 ve(v(x+2h)) a6 ve(v(x+3h))) 4 final1: = combine(final1/exp(v*x)); final1 : = (e(v(x+3h)) e((x+3h)v))e(vx) = h (a0 ve((x+3h)v) a1 ve(v(x2h)) a2 ve(v(xh)) a3 ve(vx) a4 ve(v(x+h)) a5 ve(v(x+2h)) a6 ve(v(x+3h)))e(vx) 4 final1: = expand(final1); final1 :¼

1 ðeðvhÞ Þ3

ðeðvhÞ Þ3 ¼ ha0 vðeðvhÞ Þ3 ha1 vðeðvhÞ Þ2 ha2 veðvhÞ

ha3 v

ha4 v ha5 v ha6 v eðvhÞ ðeðvhÞ Þ2 ðeðvhÞ Þ3

4 final1: = simplify(convert(final1,trig)); final1 : = 2 (16 cosh(vh)6 + 16 cosh(vh)5 sinh(vh) 24 cosh(vh)4 16 cosh(vh)3 sinh(vh) + 9 cosh(vh)2 + 3 cosh(vh) sinh(vh) 1)/(cosh(vh)) sinh(vh))3 = vh (a0 + a2 a4 + a6 + 6a0 cosh(vh) sinh(vh) + 2a4 cosh(vh)2 sinh(vh) 4a2 cosh(vh) sinh(vh) + 4a3 cosh(vh)2 sinh(vh) + 5a1 cosh(vh) + a1 sinh(vh) 8a2 cosh(vh)2 + 4a3 cosh(vh)3 + a5 cosh(vh) + a5 sinh(vh) a3 sinh(vh) + 2a4 cosh(vh)2 3a3 cosh(vh) + 8a2 cosh(vh)3 sinh(vh) + 16a1 cosh(vh)4 sinh(vh) + 32a0 cosh(vh)5 sinh(vh) + 32a0 cosh(vh)6 + 16a1 cosh(vh)5 + 8a2 cosh(vh)4 48a0 cosh(vh)4 + 18a0 cosh(vh)2 20a1 cosh(vh)3 12a1 sinh(vh) cosh(vh)2 32a0 cosh(vh)3 sinh(vh))/ (cosh(vh) + sinh(vh))3 4 4 eq1: = final; eq1 : = 2 (16 cosh(vh)6 + 16 cosh(vh)5 sinh(vh) 24 cosh(vh)4 16 cosh(vh)3 sinh(vh) + 9 cosh(vh)2 + 3 cosh(vh) sinh(vh) 1)/(cosh(vh) + sinh(vh))3 = vh (a0 a2 + a4 a6 + 16a5 cosh(vh)4 sinh(vh) + 8a4 cosh(vh)3 sinh(vh) 32a6 cosh(vh)3 sinh(vh) + 32a6 cosh(vh)5 sinh(vh) 4a4 cosh(vh) sinh(vh) + 2a2 cosh(vh) sinh(vh) + 4a3 cosh(vh)2 sinh(vh) + a1 cosh(vh) + a1 sinh(vh) + 2a2 cosh(vh)2 + 4a3 cosh(vh)3 + 8a4 cosh(vh)4 + 16a5 cosh(vh)5 + 32a6 cosh(vh)6 12a5 sinh(vh) cosh(vh)2 + 6a6 cosh(vh) sinh(vh) + 18a6 cosh(vh)2 + 5a5 cosh(vh) + a5 sinh(vh) a3 sinh(vh) 20a5 cosh(vh)3 8a4 cosh(vh)2 3a3 cosh(vh) 48a6 cosh(vh)4)/(cosh(vh) + sinh(vh))3 4 eq2: = final1; eq2 : = 2 (16 cosh(vh)6 + 16 cosh(vh)5 sinh(vh) 24 cosh(vh)4 16 cosh(vh)3 sinh(vh) + 9 cosh(vh)2 + 3 cosh(vh) sinh(vh) 1)/(cosh(vh) + sin(vh))3 = vh (a0 + a2 a4 + a6 + 6a0 cosh(vh) sinh(vh) + 2a4 cosh(vh) sinh(vh) 4a2 cosh(vh) sinh(vh) + 4a3 cosh(vh)2 sinh(vh) + 5a1 cosh(vh) + a1 sinh(vh) 8a2 cosh(vh)2 + 4a3 cosh(vh)3 + a5 cosh(vh) + a5 sinh(vh) a3 sinh(vh) + 2a4 cosh(vh)2 3a3 cosh(vh) + 8a2 cosh(vh)3 sinh(vh) + 16a1 cosh(vh)4 sinh(vh) + 32a0 cosh(vh)5 sinh(vh) + 32a0 cosh(vh)6+16a1 cosh(vh)5+8a2 cosh(vh)4 48a0 cosh(vh)4+18a0 cosh(vh)2 20a1 cosh(vh)3 12a1 sinh(vh) cosh(vh)2 32a0 cosh(vh)3 sinh(vh))/ (cosh(vh) + sinh(vh))3 Chem. Modell., 2008, 5, 350–487 | 405 This journal is

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4 eq1i: = subs(v = I*v,eq1); eq1i : = 2(16 cosh(vhI)6 + 16 cosh(vhI)5 sinh(vhI) 24 cosh(vhI)4 16 cosh(vhI)3 sinh(vhI) + 9 cosh(vhI)2 + 3 cosh(vhI) sinh(vhI) 1)/(cosh(vhI) + sinh(vhI))3 = vh (a0 a2 + a4 a6 + 16a5 cosh(vhI)4 sinh(vhI) + 8a4 cosh(vhI)3 sinh(vhI) 32a6 cosh(vhI)3 sinh(vhI) + 32a6 cosh(vhI)5 sinh(vhI) 4a4 cosh(vhI) sinh(vhI) + 2a2 cosh(vhI) sinh(vhI) + 4a3 cosh(vhI)2 sinh(vhI) 12a5 sinh(vhI) cosh(vhI)2 + 6a6 cosh(vhI) sinh(vhI) + a1 cosh(vhI) + a1 sinh(vhI) + 2a2 cosh(vhI)2 + 4a3 cosh(vhI)3 + 8a4 cosh(vhI)4 + 16a5 cosh(vhI)5 + 32a6 cosh(vhI)6 + 18a6 cosh(vhI)2 + 5a5 cosh(vhI) + a5 sinh(vhI) a3 sinh(vhI) 20a5 cosh(vhI)3 8a4 cosh(vhI)2 3a3 cosh(vhI) 48a6 cosh(vhI)4)I/(cosh(vhI) + sinh(vhI))3 4 eq1il: = 2*(16*cosh(v*h*I)^6+16*cosh(v*h*I)^5*sinh(v*h*I) 24*cosh(v*h*I)^4 16*cosh(v*h*I)^3*sinh(v*h*I) + 9*cosh(v*h*I)^2+3*cosh(v*h*I)*sinh(v*h*I) 1)/(cosh(v*h*I) + sinh(v*h*I))^3; 4 eq1ir: = v*h*(a[0] a[2]+a[4] a[6] 32*a[6]*cosh(v*h*I)^3*sinh(v*h*I) 4*a[4]*cosh(v*h*I)*sinh(v*h*I) + 16*a[5]*cosh(v*h*I)^4*sinh(v*h*I) + 32*a[6]*cosh(v*h*I)^5*sinh(v*h*I) + 2*a[2]*cosh(v*h*I)*sinh(v*h*I) + 4*a[3]*cosh(v*h*I)^2*sinh(v*h*I) + 8*a[4]*cosh(v*h*I)^3*sinh(v*h*I) + 5*a[5]*cosh(v*h*I) + a[5]*sinh(v*h*I) + 8*a[4]*cosh(v*h*I)^4 +32*a[6]*cosh(v*h*I)^6+16*a[5]*cosh(v*h*I)^5+4*a[3]*cosh(v*h*I)^3+2 a[2]*cosh(v*h*I)^2+a[1]*cosh(v*h*I) + a[1]*sinh(v*h*I)–48*a[6]*cosh(v*h*I)^4–a[3]*sinh(v*h*I)–20*a[5]*cosh(v*h*I)^3–3*a[3]*cosh(v*h*I)– 8*a[4]*cosh(v*h*I)^2+18*a[6]*cosh(v*h*I)^2+6*a[6]*cosh(v*h*I)*sinh(v*h*I) –12*a[5]*sinh(v*h*I)*cosh(v*h*I)^2)/(cosh(v*h*I) + sinh(v*h*I))^3*I; 4 eq2i: = subs(v = I*v,eq2); eq2i : = 2(16 cosh(vhI)6 + 16 cosh(vhI)5 sinh (vhI) 24 cosh(vhI)4 16 cosh(vhI)3sinh(vhI) + 9 cosh(vhI)2 + 3 cosh(vhI)sinh(vhI) 1)/(cosh(vhI) + sinh(vhI))3 = Ivh(a0 + a2 + a6 + 16a1 cosh(vhI)4 sinh(vhI) 32a0 cosh(vhI)3 sinh(vhI) 12a1 sinh(vhI) cosh(vhI)2 + 2a4 cosh(vhI) sinh(vhI) 4a2 cosh(vhI) sinh(vhI) + 4a3 cosh(vhI)2 sinh(vhI) + 5a1 cosh(vhI) + a1 sinh(vhI) 8a2 cosh(vhI)2 + 4a3 cosh(vhI)3 + a5 cosh(vhI) + a5 sinh(vhI) a3 sinh(vhI) + 2a4 cosh(vhI)2 3a3 cosh(vhI) + 6a0 cosh(vhI) sinh(vhI) + 8a2 cosh(vhI)3 sinh(vhI) + 32a0 cosh(vhI)5 sinh(vhI) + 32a0 cosh(vhI)6 + 16a1 cosh(vhI)5 + 8a2 cosh(vhI)4 48a0 cosh(vhI)4 + 18a0 cosh(vhI)2 20a1 cosh(vhI)3)/(cosh(vhI) + sinh(vhI))3 4 eq2il: = 2*(16*cosh(v*h*I)^6+16*cosh(v*h*I)^5*sinh(v*h*I)24*cosh(v*h*I)^416*cosh(v*h*I)^3*sinh(v*h*I)+9*cosh(v*h*I)^2+3*cosh(v*h*I)*sinh(v*h*I)1)/(cosh(v*h*I) + sinh(v*h*I))^3; 4 eq2ir: = I*v*h*(20*a[1]*cosh(v*h*I)^3+16*a[1]*cosh(v*h*I)^5+32* a[0]*cosh(v*h*I)^6+8*a[2]*cosh(v*h*I)^4+18*a[0]*cosh(v*h*I)^2 48*a[0]* cosh(v*h*I)^4+8*a[2]*cosh(v*h*I)^3*sinh(v*h*I) + 16*a[1]*cosh(v*h*I)^4*sinh(v*h*I) + 32*a[0]*cosh(v*h*I)^5*sinh(v*h*I) + 6*a[0]*cosh(v*h*I)*sinh(v*h*I) 32*a[0]*cosh(v*h*I)^3*sinh(v*h*I) 12*a[1]*cosh(v*h*I)^2*sinh(v*h*I) + 4*a[3]*cosh(v*h*I)^2*sinh(v*h*I) + 2*a[4]*cosh(v*h*I)*sinh(v*h*I) 4*a[2]* cosh(v*h*I)*sinh(v*h*I) + 5*a[1]*cosh(v*h*I) + a[1]*sinh(v*h*I) 8*a[2]* cosh(v*h*I)^2+4*a[3]*cosh(v*h*I)^3 a[3]*sinh(v*h*I) + a[5]*sinh(v*h*I) + 2*a[4]*cosh(v*h*I)^2+a[5]*cosh(v*h*I) 3*a[3]*cosh(v*h*I)a[0]+a[2] a[4]+a[6])/(cosh(v*h*I) + sinh(v*h*I))^3; 4 4 eq1ilr: = simplify(evalc(Re(eq1il))); eq1ilr : = 0 4 eq1ili: = simplify(evalc(Im(eq1il))); eqlili : = 2 sin(vh) (4 cos(vh)2 1) 4 eq1irr: = simplify(evalc(Re(eq1ir))); 406 | Chem. Modell., 2008, 5, 350–487 This journal is

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eqlirr : = vh sin(vh) (a0 + a2 a4 + a6 4a6 cos(vh)2 2a5 cos(vh) + 2a1 cos(vh) + 4a0 cos(vh)2) 4 eq1iri: = simplify(evalc(Im(eq1ir))); eqliri : =vh(a1 + a3 a5 3a6 cos(vh)3 + 4a6 cos(vh)3 + a4 cos(vh) + 2a5 cos(vh)2 + a2 cos(vh) + 2a1 cos(vh)2 + 4a0 cos(vh)3 3 cos(vh) a0) 4 4 eq2ilr: = simplify(evalc(Re(eq2il))); eq2ilr : =0 4 eq2ili: = simplify(evalc(Im(eq2il))); eq2ili : =2 sin(vh) (4 cos(vh)2 1) 4 eq2irr: = simplify(evalc(Re(eq2ir))); eq2irr : =vh sin(vh) (a0 + a2 a4 + a6 4a6 cos(vh)2 2a5 cos(vh) + 2a1 cos(vh) + 4a0 cos(vh)2) 4 eq2iri: = simplify(evalc(Im(eq2ir))); eq2iri : =vh (a1 + a3 a5 3a6 cos(vh) + 4a6 cos(vh)3 + a4 cos(vh) + 2a5 cos(vh)2 + a2 cos(vh) + 2a1 cos(vh)2 + 4a0 cos(vh)3 3 cos(vh) a0) 4 eq1: = eq1irr = eq1ilr; eq1 : =vh sin(vh) (a0 + a2 a4 + a6 4a6 cos(vh)2 2a5 cos(vh) + 2a1 cos(vh) + 4a0 cos(vh)2) = 0 4 eq2: = eq1iri = eq1ili; eq2 : =vh (a1 + a3 a5 3a6 cos(vh) + 4a6 cos(vh)3 + a4 cos(vh) + 2a5 cos(vh)2 + a2 cos(vh) + 2a1 cos(vh)2 + 4a0 cos(vh)3 3 cos(vh) a0) = 2 sin(vh) (4 cos(vh)2 1) 4 eq3: = eq2irr = eq2ilr; eq3 : =vh sin(vh) (a0 + a2 a4 + a6 4a6 cos(vh)2 2a5 cos(vh) + 2a1 cos(vh) + 4a0 cos(vh)2) = 0 4 eq4: = eq2iri = eq2ili; eq4 : =vh (a1 + a3 a5 + 3a6 cos(vh) + 4a6 cos(vh)3 + a4 cos(vh) + 2a5 cos(vh)2 + a2 cos(vh) + 2a1 cos(vh)2 + 4a0 cos(vh)3 3 cos(vh)a0) = 2 sin(vh)(4 cos(vh)2 1) 4 simplify(eq1–eq3); 0=0 4 simplify(eq2+eq4); 0=0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

y[n]: = x^n; f[n]: = diff(y[n],x$1); y[n+1]: = (x+h)^n; f[n+1]: = diff(y[n+1],x$1); y[n–1]: = (x–h)^n; f[n–1]: = diff(y[n–1],x$1); y[n+2]: = (x+2*h)^n; f[n+2]: = diff(y[n+2],x$1); y[n–2]: = (x–2*h)^n; f[n–2]: = diff(y[n–2],x$1); y[n+3]: = (x+3*h)^n; f[n+3]: = diff(y[n+3],x$1); Chem. Modell., 2008, 5, 350–487 | 407 This journal is

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4 y[n–3]: = (x–3*h)^n; 4 f[n–3]: = diff(y[n–3],x$1); 4 4 final2: = y[n+3]–y[n–3] = h*(a[0]*f[n–3]+a[1]*f[n–2]+a[2]*f[n–1]+ a[3]*f[n]+a[4]*f[n+1]+a[5]*f[n+2]+a[6]*f[n+3]); a0 ðx 3hÞn n a1 ðx 2hÞn n final2 :¼ ðx þ 3hÞn ðx 3hÞn ¼ h þ x 3h x 2h a2 ðx hÞn n a3 xn n a4 ðx þ hÞn n a5 ðx þ 2hÞn n a6 ðx þ 3hÞn n þ þ þ þ þ xh x xþh x þ 2h x þ 3h 4 n: = 0; n : =0 4 eq5: = simplify(final2); eq5 : =0= 0 4 n: = 1; n : =1 4 eq5: = simplify(final2); eq5 : =6h = h (a0 + a1 + a2 + a3 + a4 + a5 + a6) 4 eq5: = simplify(eq5/h); eq5 : =6 = a0 + a1 + a2 + a3 + a4 + a5 + a6 4 eq5: = eq5; eq5 : =6 = a0 + a1 + a2 + a3 + a4 + a5 + a6 4 n: = 2; n : =2 4 eq6: = simplify(final2); eq6 : =12xh = 2h (a0 x + 3a0 h a1 x + 2a1 h a2 x + a2 h a3 x a4 x a4 h a5 x 2a5 h a6 x 3a6 h) 4 eq6: = simplify(eq6–2*eq5*x*h); eq6 : =0 = 6a0 h2 4a1 h2 2a2 h2 + 2a4 h2 + 4a5 h2 + 6a6 h2 4 eq6: = simplify(eq6/(h^2)); eq6 : =0 = 6a0 4a1 2a2 + 2a4 + 4a5 + 6a6 4 eq6: = eq6; eq6 : =0 = 6a0 4a1 2a2 + 2a4 + 4a5 + 6a6 4 4 n: = 3; n : =3 4 eq7: = simplify(final2); eq7 : =18x2h + 54h3 = 3h (a0 x2 6h a0 x + 9a0 h2 + a1 x2 4h a1 x + 4a1 h2 + a2 x2 2h a2 x + a2 h2 + a3 x2 + a4 x2 + 2h a4 x + a4 h2 + a5 x2 + 4h a5 x + 4a5 h2 + a6 x2 + 6h a6 x + 9a6 h2) 4 eq7: = simplify(eq7–3*eq5*x^2*h–3*x*h^2*eq6); eq7 : =54h3 = 27a0 h3 + 12a1 h3 + 3a2 h3 + 3a4 h3 + 12a5 h3 + 27a6 h3 4 eq7: = simplify(eq7/(h^3)); eq7 : =54 = 27a0 + 12a1 + 3a2 + 3a4 + 12a5 + 27a6 4 eq7: = eq7; eq7 : =54 = 27a0 + 12a1 + 3a2 + 3a4 + 12a5 + 27a6 408 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 4 n: = 4;

n : =4 4 eq8: = simplify(final2);

eq8 : =24x3h + 216xh3 = 4h(a4x3 a5x3 a6x3 a1x3 12h2a1x 3h2a2x 3h2a4x 12h2a5x + 9ha0x2 + 6ha1x2 + 3ha2x2 27h2 a0x a3x3 + 27a0h3 + 8a1h3 + a2h3 a4h3 8a5h3 27a6h3 27h2a6x 3ha4x2 6ha5x2 9ha6x2 a2x3 a0x3) 4 eq8: = simplify(eq8–4*eq5*x^3*h–6*x^2*h^2*eq6–4*x*h^3*eq7);

eq8 : =0 = 108a0h4 32a1h4 4a2h4 + 4a4h4 + 32a5h4 + 108a6h4 4 eq8: = simplify(eq8/(h^4));

eq8 : =0 = 108a0 32a1 4a2 + 4a4 + 32a5 + 108a6 4 eq8: = eq8;

eq8 : =0 = 108a0 32a1 4a2 + 4a4 + 32a5 + 108a6 4 4 n: = 5;

n : =5 4 eq9: = simplify(final2);

eq9 : =30x4h + 540x2h3 + 486h5 = 5h(a2x4 + a5x4 + a4x4 32h3a1x 4h3a2x + 4h3a4x + 32h3a5x + 54h2a0x2 + 24h2a1x2 + 6h2a2x2 108h3a0x + 108h3a6x + 6h2 a4x2 + 24h2a5x2 + 54h2a6x2 + 4ha4x3 + 8ha5x3 + 12ha6x3 8ha1x3 4ha2x3 12ha0x3 + a3x4 + a1x4 + a0x4 + 81a0h4 + 16a1h4 + a2h4 + a4h4 + 16a5h4 + 81a6h4) 4 eq9: = simplify(eq9–5*eq5*x^4*h–10*x^3*h^2*eq6–10*x^2*h^3*eq7– 5*x*h^4*eq8);

eq9 : =486h5 = 405a0h5 + 80a1h5 + 5a2h5 + 5a4h5 + 80a5h5 + 405a6h5 4 eq9: = simplify(eq9/(h^5));

eq9 : =486 = 405a0 + 80a1 + 5a2 + 5a4 + 80a5 + 405a6 4 eq9: = eq9;

eq9 : =486 = 405a0 + 80a1 + 5a2 + 5a4 + 80a5 + 405a6 4 4 4 4 Chem. Modell., 2008, 5, 350–487 | 409 This journal is

c


solut: = solve(eq1,eq2,eq5,eq6,eq7,eq8,eq9,{a[0],a[1],a[2],a[3],a[4],a[5],a[6]}); 3 solut :¼ a2 ¼ ð7vh cosðvhÞ 68vh þ 28vh cosðvhÞ sinðvhÞ2 20 þ 81 sinðvhÞ2 vh þ 75 sinðvhÞ 100 sinðvhÞ3 Þ=ðvhð4 cosðvhÞ þ 4 þ cosðvhÞ sinðvhÞ2 sinðvhÞ2 3 sinðvhÞ2 ÞÞ; a1 ¼

3 23vh cosðvhÞ þ 8vh þ 11vh cosðvhÞ sinðvhÞ2 þ 15 sinðvhÞ 20ðvhÞ3 ; 10 vhð4 cosðvhÞ þ 4 þ cosðvhÞ sinðvhÞ2 3 sinðvhÞ2 Þ

a0 ¼

1 21vh cosðvhÞ 36vh þ 15 sinðvhÞ 20 sinðvhÞ3 þ 33 sinðvhÞ2 vh ; 20 vhð4 cosðvhÞ þ 4 þ cosðvhÞ sinðvhÞ2 3 sinðvhÞ2 Þ

a6 ¼

1 21vh cosðvhÞ 36vh þ 15 sinðvhÞ 20 sinðvhÞ3 þ 33 sinðvhÞ2 vh ; 20 vhð4 cosðvhÞ þ 4 þ cosðvhÞ sinðvhÞ2 3 sinðvhÞ2 Þ

1 a3 ¼ ð51vh cosðvhÞ 24vh þ 39vh cosðvhÞ sinðvhÞ2 þ 48 sinðvhÞ2 vh 5 þ 75 sinðvhÞ 100 sinðvhÞ3 Þ=ðvhð4 cosðvhÞ þ 4 þ cosðvhÞ sinðvhÞ2 3 sinðvhÞ2 ÞÞ; 3 a4 ¼ ð7vh cosðvhÞ 68vh þ 28vh cosðvhÞ sinðvhÞ2 20 þ 81 sinðvhÞ2 vh þ 75 sinðvhÞ 100 sinðvhÞ3 Þ=ðvhð4 cosðvhÞ þ 4 þ cosðvhÞ sinðvhÞ2 3 sinðvhÞ2 ÞÞ; a5 ¼

3 23vh cosðvhÞ þ 8vh þ 11vh cosðvhÞ sinðvhÞ2 þ 15 sinðvhÞ 20 sinðvhÞ3 10 vhð4 cosðvhÞ þ 4 þ cosðvhÞ sinðvhÞ2 3 sinðvhÞ2 Þ

4 assign(solut); 4 h: = 1; h : =1 4 a[0]: = combine(a[0]); a0 :¼

42v cosðvÞ þ 39v 10 sinð3vÞ þ 33v cosð2vÞ 150v cosðvÞ þ 100v 10v cosð3vÞ þ 60v cosð2vÞ

4 a0t: = convert(taylor(a[0],v = 0,20),polynom); a0t :¼

41 9 2 1 171 79 þ v þ v4 v6 v8 140 1400 12320 56056000 269068800 2801 19009 v10 v12 187125120000 31287320064000

4 a[1]: = combine(a[1]); a1 :¼

243v cosðvÞ þ 96v 33v cosð3vÞ þ 60 sinð3vÞ 150v cosðvÞ þ 100v 10v cosð3vÞ þ 60v cosð2vÞ


54 27 2 3 4 513 79 2801 v v þ v6 þ v8 þ v10 35 700 6160 28028000 44844800 31187520000 19009 v12 þ 5214553344000

4 a[2]: = combine(a[2]); a2 :¼

165v þ 42v cosð3vÞ þ 243v cosð2vÞ 150 sinð3vÞ 150v cosðvÞ þ 100v 10v cosð3vÞ þ 60v cosð2vÞ


c


)


27 27 2 3 4 513 79 2801 þ v þ v v6 v8 v10 140 280 2464 11211200 17937920 12475008000 19009 v12 2085821337600

4 a[3]: = combine(a[3]); a3 :¼

165v cosðvÞ 39v cosð3vÞ 96v cosð2vÞ þ 100 sinð3vÞ 75v cosðvÞ þ 50v 5v cosð3vÞ þ 30v cosð2vÞ


68 9 2 1 4 171 79 2801 v v þ v6 þ v8 þ v10 35 70 616 2802800 13453440 9356256000 19009 þ v12 1564366003200

4 a[4]: = combine (a[4]); a4 :¼

165v þ 42v cosð3vÞ þ 243v cosð2vÞ 150 sinð3vÞ 150v cosðvÞ þ 100v 10v cosð3vÞ þ 60v cosð2vÞ


27 27 2 3 4 513 79 2801 þ v þ v v6 v8 v10 140 280 2464 11211200 17937920 12475008000 19009 v12 2085821337600

4 a[5]: = combine(a[5]); a5 :¼

243v cosðvÞ þ 96v 33v cosð3vÞ þ 60 sinð3vÞ 150v cosðvÞ þ 100v 10v cosð3vÞ þ 60v cosð2vÞ


54 27 2 3 4 513 79 2801 v v þ v6 þ v8 þ v10 35 700 6160 28028000 44844800 31187520000 19009 þ v12 5214553344000

4 a[6]: = combine(a[6]); a6 :¼

42v cosðvÞ þ 39v 10 sinð3vÞ þ 33v cosð2vÞ 150v cosðvÞ þ 100v 10v cosð3vÞ þ 60v cosð2vÞ


41 9 2 1 171 79 þ v þ v4 v6 v8 140 1400 12320 56056000 269068800 2801 19009 v10 v12 187125120000 31287320064000

4 4 4 simplify(a[0]a[6]); 0 4 simplify(a[1]a[5]); 0 4 simplify(a[2]a[4]); 0 Chem. Modell., 2008, 5, 350–487 | 411 This journal is

c


4 4 restart; 4 qnp3: = convert(taylor(q(x+3*h),h = 0,13),polynom); 4 qnp2: = convert(taylor(q(x+2*h),h = 0,13),polynom); 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(xh),h = 0,13),polynom); 4 qnm2: = convert(taylor(q(x2*h),h = 0,13),polynom); 4 qnm3: = convert(taylor(q(x3*h),h = 0,13),polynom); 4 snp3: = convert(taylor(diff(q(x+3*h),x$1),h = 0,13),polynom); 4 snp2: = convert(taylor(diff(q(x+2*h),x$1),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$1),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$1),h = 0,13),polynom); 4 snm2: = convert(taylor(diff(q(x2*h),x$1),h = 0,13),polynom); 4 snm3: = convert(taylor(diff(q(x3*h),x$1),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$1); 4 4 4 a[0]: = subs(v = v*h,41/140+9/1400*v^2+1/12320*v^4171/ 56056000*v^679/269068800*v^82801/187125120000*v^1019009/ 31287320064000*v^12); 4 a[1]: = subs(v = v*h,54/3527/700*v^23/6160*v^4+513/ 28028000*v^6+79/44844800*v^8+2801/31187520000*v^10+19009/ 5214553344000*v^12); 4 a[2]: = subs(v = v*h,27/140+27/280*v^2+3/2464*v^4513/ 11211200*v^679/17937920*v^82801/12475008000*v^1019009/ 2085821337600*v^12); 4 a[3]: = subs(v = v*h,68/359/70*v^21/616*v^4+171/2802800*v^6+79/ 13453440*v^82801/9356256000*v^10 +19009/1564366003200*v^12); 4 4 lte: = simplify(qnp3qnm3h*(a[0]*snm3+a[1]*snm2+a[2]*snm1 +a[3]*sn+a[2]*snp1+a[1]*snp2+a[0]*snp3)); 23327 1 ðDð13Þ ÞðqÞðxÞh13 v4 h15 ðDð11Þ ÞðqÞðxÞ 34496000 422400 57 171 þ v6 h17 ðDð11Þ ÞðqÞðxÞ þ v6 h13 ðDð7Þ ÞðqÞðxÞ 640640000 56056000 1 79 v4 h17 ðDð13Þ ÞðqÞðxÞ þ v8 h15 ðDð7Þ ÞðqÞðxÞ 5821200 269068800 79 79 þ v8 h17 ðDð9Þ ÞðqÞðxÞ þ v8 h21 ðDð13Þ ÞðqÞðxÞ 1076275200 127135008000 79 2801 v8 h19 ðDð11Þ ÞðqÞðxÞ þ v10 h23 ðDð13Þ ÞðqÞðxÞ þ 9225216000 88416619200000 2801 v10 h21 ðDð11Þ ÞðqÞðxÞ þ 6415718400000 2801 2801 þ v10 h19 ðDð9Þ ÞðqÞðxÞ þ v10 h17 ðDð7Þ ÞðqÞðxÞ 748500480000 187125120000 171 19009 v6 h15 ðDð9Þ ÞðqÞðxÞ þ v12 h19 ðDð7Þ ÞðqÞðxÞ þ 224224000 31287320064000 19009 þ v12 h23 ðDð11Þ ÞðqÞðxÞ 1072708116480000 19009 v12 h21 ðDð9Þ ÞðqÞðxÞ þ 125149280256000

lte :¼


c


19009 1 v12 h25 ðDð13Þ ÞðqÞðxÞ v4 h13 ðDð9Þ ÞðqÞðxÞ 14783258730240000 49280 1 1 v4 h11 ðDð7Þ ÞðqÞðxÞ v2 h15 ðDð13Þ ÞðqÞðxÞ 12320 73500 3 9 2 11 ð9Þ v2 h13 ðDð11Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 16000 5600 19 9 2 9 ð7Þ v6 h19 ðDð13Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ þ 294294000 1400 47 9 ðDð11Þ ÞðqÞðxÞh11 ðDð9Þ ÞðqÞðxÞh9 30800 1400 þ

4 coeff(lte,h,9);

9 2 ð7Þ 9 v ðD ÞðqÞðxÞ ðDð9Þ ÞðqÞðxÞ 1400 1400

Appendix B 4 restart; 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$1); 4 y[n+3]: = exp(v*(x+3*h)); 4 f[n+3]: = diff(y[n+3],x$1); 4 y[n3]: = exp(v*(x3*h)); 4 f[n3]: = diff(y[n3],x$1); 4 4 4 final: = y[n+3]y[n3] = h*(a[0]*f[n3]+a[1]*f[n2]+a[2]*f[n1] + a[3]*f[n]+a[4]*f[n+1]+a[5]*f[n+2]+a[6]*f[n+3]); final : =e(v(x+3h)) e(v(x3h)) = h(a0ve(v(x3h)) + a1ve(v(x2h)) + a2ve(v(xh)) + a3ve(vx) + a4ve(v(x+h)) + a5ve(v(x+2h)) + a6ve(v(x+3h))) 4 final: = combine(final/exp(v*x)); final : =e(vx)(e(v(x+3h)) e(v(x3h))) = e(vx) h(a0ve(v(x3h)) + a1ve(v(x2h)) + a2ve(v(xh)) + a3ve(vx) + a4ve(v(x+h)) + a5ve(v(x+2h)) + a6ve(v(x+3h))) 4 final: = expand(final); final :¼ðeðvhÞ Þ3

1 ðeðvhÞ Þ3

¼

ha0 v ðeðvhÞ Þ3

þ

ha1 v ðeðvhÞ Þ2

þ

ha2 v þ ha3 v þ ha4 veðvhÞ eðvhÞ

þ ha5 vðeðvhÞ Þ2 þ ha6 vðeðvhÞ Þ3 4 final: = simplify(convert(final,trig)); final : =2(16 cosh(vh)6 + 16 cosh(vh)5 sinh(vh) 24 cosh (vh)4 16 cosh(vh)3 sinh(vh) + 9 cosh(vh)2 + 3 cosh(vh) sinh(vh) 1)/(cosh(vh) + sinh(vh))3 = vh(6a6cosh(vh) sinh(vh) 32a6 cosh(vh)3 sinh(vh) + a0 a2 + a4 a6 + 32a6 cosh(vh)5 sinh(vh) 4a4 cosh(vh) sinh(vh) + 16a5 cosh(vh)4 sinh(vh) + 4a3 cosh(vh)2 sinh(vh) + 2a2 cosh(vh) sinh(vh) + a1 cosh(vh) + a1 sinh(vh) + 2a2 cosh(vh)2 + 4a3 Chem. Modell., 2008, 5, 350–487 | 413 This journal is

c


cosh(vh)3 + 8a4 cosh(vh)4 + 16a5 cosh(vh)5 + 32a6 cosh(vh)6 + 8a4 cosh(vh)3 sinh(vh) 8a4 cosh(vh)2 3a3 cosh(vh) a3 sinh(vh) 12a5 sinh(vh) cosh(vh)2 + a5 sinh(vh) + 5a5 cosh(vh) + 18a6 cosh(vh)2 20a5 cosh(vh)3 48a6 cosh(vh)4)/ (cosh(vh) + sinh(vh))3 4 4 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$1); 4 y[n+3]: = exp(v*(x+3*h)); 4 f[n+3]: = diff(y[n+3],x$1); 4 y[n3]: = exp(v*(x3*h)); 4 f[n3]: = diff(y[n3],x$1); 4 4 final1: = y[n+3]y[n3] = h*(a[0]*f[n3]+a[1]*f[n2]+a[2]*f[n1]+ a[3]*f[n]+a[4]*f[n+1]+a[5]*f[n+2]+a[6]*f[n+3]); 2

final1 : =e(v(x+3h)) e(v(x3h)) = h(a0ve(v(x3h)) a1ve(v(xh)) a2ve(v(xh)) a3ve(vx) a4ve(v(x+h)) a5ve(v(x+2h)) a6ve(v(x+3h))) 4 final1: = combine(final1/exp(v*x)); final1 : =(e((x3h)v) e(v(x3h)))e(vx) =h (a0ve(v(x3h)) a1ve(v(x2h)) a2ve(v(xh)) a3ve(vx) a4ve(v(x+h)) a5ve(v(x+2h)) a6ve((x3h)v))e(vx) 4 final1: = expand(final1); final1 :¼

1 ðeðvhÞ Þ3

ðeðvhÞ Þ3 ¼ ha0 vðeðvhÞ Þ3 ha1 vðeðvhÞ Þ2 ha2 vðeðvhÞ Þ

ha3 v

ha4 v ha5 v ha6 v eðvhÞ ðeðvhÞ Þ2 ðeðvhÞ Þ3

4 final1: = simplify(convert(final1,trig)); final1 : = 2(16 cosh(vh)6 + 16 cosh(vh)5 sinh(vh) 24 cosh (vh)4 16 cosh(vh)3sinh(vh) + 9 cosh(vh)2 + 3 cosh(vh) sinh(vh) 1)/(cosh(vh) + sinh(vh))3 = vh (a0 + a2 a4 + a6 + 2a4 cosh(vh) sinh(vh) 12a1 sinh(vh) cosh(vh)2 32a0cosh(vh)3 sinh(vh) + 6a0 cosh(vh) sinh(vh) + 4a3 cosh(vh)2 sinh(vh) 4a2 cosh(vh) sinh(vh) + 5a1 cosh(vh) + a1 sinh(vh) 8a2 cosh(vh)2 + 4a3 cosh(vh)3 + 32a0 cosh(vh)5 sinh(vh) + 16a1 cosh(vh)4 sinh(vh) + 2a4 cosh(vh)2 3a3 cosh(vh) a3 sinh(vh) + 8a2 cosh(vh)3 sinh(vh) + 32a0 cosh(vh)6 + 16a1 cosh(vh)5 + 8a2 cosh(vh)4 + a5 sinh(vh) + a5 cosh(vh) 48a0 cosh(vh)4 + 18a0 cosh(vh)2 20a1 cosh(vh)3)/(cosh(vh) + sinh(vh))3 4 4 eq1: = final; eq1 : =2(16 cosh(vh)6 + 16 cosh(vh)5 sinh(vh) 24 cosh (vh)4 16 cosh(vh)3 sinh(vh) + 9 cosh(vh)2 + 3 cosh(vh) sinh(vh) 1)/(cosh(vh) + sinh(vh))3 = vh (6a6 cosh(vh) sinh(vh) 32a6cosh(vh)3 sinh(vh) + a0 a2 + a4 a6 + 32a6 cosh(vh)5 sinh(vh) 4a4 cosh(vh) sinh(vh) + 16a5cosh(vh)4 sinh(vh) + 4a3 cosh(vh)2 sinh(vh) + 2a2 cosh(vh) sinh(vh) + a1 cosh(vh) + a1 sinh(vh) + 2a2 cosh(vh) + 4a3 cosh(vh)3 + 8a4 cosh(vh)4 + 16a5 cosh(vh)5 + 32a6 cosh(vh)6 + 8a4 cosh(vh)3 sinh(vh) 8a4


c


cosh(vh)2 3a3 cosh(vh) a3 sinh(vh) 12a5 sinh(vh) cosh(vh)2 + a5 sinh(vh) + 5a2 cosh(vh) + 18a6 cosh(vh)2 20a5 cosh(vh)3 48a6 cosh(vh)4)/(cosh(vh) + sinh(vh))3 4 eq2: = final1; eq2 : =2(16 cosh(vh)6 + 16 cosh(vh)5 sinh(vh) 24 cosh(vh)4 16 cosh(vh)3 sinh(vh) + 9 cosh(vh)2 + 3 cosh(vh) sinh(vh) 1)/(cosh(vh) + sinh(vh))3 = vh (a0 + a2 a4 + a6 + 2a4 cosh(vh) sinh(vh) 12a1 sinh(vh) cosh(vh)2 32a0 cosh(vh)3 sinh(vh) + 6a0 cosh(vh) sinh(vh) + 4a3 cosh(vh)2 sinh(vh) 4a2 cosh(vh) sinh(vh) + 5a1 cosh(vh) + a1sinh(vh) 8a2 cosh(vh)2 + 4a3 cosh(vh)3 + 32a0 cosh(vh)5 + sinh(vh) + 16a1 cosh(vh)4 sinh(vh) + 2a4 cosh(vh)2 3a3 cosh(vh) + a3 sinh(vh) + 8a2 cosh(vh)3 sinh(vh) + 32a0 cosh(vh)6 + 16a1 cosh(vh)5 + 8a2 cosh(vh)4 + a5 sinh(vh) + a5 cosh(vh) 48a0 cosh(vh)4 + 18a0 cosh(vh)2 20a1 cosh(vh)3)/ (cosh(vh) + sinh(vh))3 4 eq1i: = subs(v = I*v,eq1); 4 eq2i: = subs(v = I*v,eq2); 4 4 eq1f: = simplify(evalc(Re(eq1i))); eq1f : =0 = vh sin(vh) (a0 + a2 a4 + a6 + 2a1 cos(vh) 2a5 cos(vh) 4a6 cos(vh)2 + 4a0 cos(vh)2) 4 eq2f: = simplify(evalc(Im(eq1i))); eq2f : =2 sin(vh) (1 + 4 cos(vh)2) = vh (a1 + a3 a5 + 4a0 cos(vh)3 + 2a1 cos(vh)2 3 cos(vh) a0 3a6 cos(vh) 3a6 cos(vh) + a4 cos(vh) + a2 cos(vh) + 4a6 cos(vh)3 + 2a5 cos(vh)2) 4 simplify(evalc(Re(eq2i))); 0 = vh sin(vh) (a0 + a2 a4 + a6 + 2a1 cos(vh) 2a5 cos(vh) 4a6 cos(vh)2 + 4a0 cos(vh)2) 4 simplify(evalc(Im(eq2i))); 2 sin(vh) (1 + 4 cos(vh)2) = vh (a1 + a3 a5 + 4a0 cos(vh)3 + 2a1 cos(vh)2 3 cos(vh) a0 3a6 cos(vh) + a4 cos(vh) + a2 cos(vh) + 4a6 cos(vh)3 + 2a5 cos(vh)2) 4 4 4 y[n]: = x*exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = (x+h)*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = (xh)*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = (x+2*h)*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = (x2*h)*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$1); 4 y[n+3]: = (x+3*h)*exp(v*(x+3*h)); 4 f[n+3]: = diff(y[n+3],x$1); 4 y[n3]: = (x3*h)*exp(v*(x3*h)); 4 f[n3]: = diff(y[n3],x$1); 4 4 4 final2: = y[n+3]y[n3] = h*(a[0]*f[n3]+a[1]*f[n2]+a[2]*f[n1]+ a[3]*f[n]+a[4]*f[n+1]+a[5]*f[n+2]+a[6]*f[n+3]); final2 : =(x + 3h) e(v(x+3h)) (x 3h) e(v(x3h)) = h(a0(e(v(x3h)) (x 3h)ve(v(x3h))) + a1(e(v(x2h)) + (x 2h)ve(v(x2h))) +a2(e(v(xh)) + (x h)ve(v(xh))) + a3(e(vx) + xve(vx)) +a4(e(v(x+h)) + (x + h)ve(v(x+h))) + a5(e(v(x+2h)) + (x + 2h)ve(v(x2h))) + a6(e(v(x+3h)) + (x + 3h)ve(v(x+3h)))) Chem. Modell., 2008, 5, 350–487 | 415 This journal is

c


4 final2: = combine(final2/exp(v*x)); final2 : =((x+ 3h) e(v(x+3h)) + ( x 3h) e(v(x3h)))e(vx) = e(vx) h(a0(e(v(x3h)) (x 3h)ve(v(x3h))) + a1(e(v(x2h)) + (x 2h)ve(v(x2h))) +a2(e(v(xh)) + (x h)ve(v(xh))) + a3(e(vx) + xve(vx)) +a4(e(v(x+h)) + (x + h)ve(v(x+h))) + a5(e(v(x+2h)) + (x + 2h)ve(v(x+2h))) + a6(e(v(x+3h)) + (x + 3h)ve(v(x+3h)))) 4 final2: = expand(final2); final2 :¼ðeðvhÞ Þ3 x þ 3ðeðvhÞ Þ3 h

x ðeðvhÞ Þ3

þ

3h

¼

ha0 ðeðvhÞ Þ3

þ

ha0 vx

ðeðvhÞ Þ3 3h a0 v ha1 ha1 vx 2h a1 v ha2 ha2 vx h2 a2 v þ þ þ þ ðvhÞ ðvhÞ 3 2 2 e e ðeðvhÞ Þ ðeðvhÞ Þ ðeðvhÞ Þ2 eðvhÞ ðeðvhÞ Þ 2

ðeðvhÞ Þ3 2

þ ha3 þ ha3 xv þ ha4 eðvhÞ þ ha4 veðvhÞ x þ h2 a4 veðvhÞ ha5 ðeðvhÞ Þ2 þ ha5 vðeðvhÞ Þ2 þ x þ 2h2 a5 vðeðvhÞ Þ2 þ ha6 ðeðvhÞ Þ3 þ ha6 vðeðvhÞ Þ3 x þ 3h2 a6 vðeðvhÞ Þ3 4 final2: = simplify(convert(final2,trig)); final2 : =2(x + 27h cosh(vh)2 72h cosh(vh)4 + 16x cosh(vh)6 + 16x cosh(vh)5 sinh(vh) + 48h cosh (vh)5 sinh (vh) + 48h cosh(vh)6 16x cosh(vh)3 sinh(vh) + 3x cosh(vh) sinh(vh) 48h cosh(vh)3 sinh(vh) + 9h cosh(vh) sinh(vh) 24x cosh(vh)4 + 9x cosh(vh)2/(cosh(vh) + sinh(vh))3 = h (6a6 cosh(vh) sinh(vh) 32a6 cosh(vh)3 sinh(vh) + a0 a2 + a4 a6 + 32a6 cosh(vh)5 sinh(vh) 4a4 cosh(vh) sinh(vh) + 16a5 cosh(vh)4 sinh(vh) + ha2v a2 vx 3ha6v a6vx + ha4v + a4vx + 4a3 cosh(vh)2 sinh(vh) 2ha2v cosh(vh) sinh(vh) + 2a2 cosh(vh) sinh(vh) + 8ha4v cosh(vh)3 sinh(vh) + a1 cosh(vh) + a1 sinh(vh) + 2a2 cosh(vh)2 + 4a3 cosh(vh)3 + 8a4 cosh(vh)4 + 16a5 cosh(vh)5 + 32a6 cosh(vh)6 + 8a4 cosh(vh)3 sinh(vh) + 8a4 vx cosh(vh)3 sinh(vh) 8a4 cosh(vh)2 3a3 cosh(vh) a3 sinh(vh) 12a5 sinh(vh) cosh(vh)2 + a5 sinh(vh) + 5a5 cosh(vh) + 18a6 cosh(vh)2 20a5 cosh(vh)3 48a6 cosh(vh)4 40h a5v cosh(vh)3 8a4vx cosh(vh)2 24ha5v cosh(vh)2 sinh(vh) + 18a6vx cosh(vh)2 12a5vx cosh(vh)2 sinh(vh) 32a6vx cosh(vh)3 sinh(vh) 96ha6v cosh(vh)3 sinh(vh) 48a6vx cosh(vh)4 + 6a4vx cosh(vh) sinh(vh) 4a4vx cosh(vh) sinh(vh) + 54ha6v cosh(vh)2 + a5vx sinh(vh) 144ha6v cosh(vh)4 4ha4v cosh(vh) sinh(vh) + 2ha5v sinh(vh) + 18ha6v cosh(vh) sinh(vh) 8ha4v cosh(vh)2 + 10ha5v cosh(vh) 3a3xv cosh(vh) 20a5vx cosh(vh)3 + 5a5vx cosh(vh) a3xv sinh(vh) 3ha0v + a0vx + a1vx cosh(vh) + a1vx sinh(vh) 2ha1v cosh(vh) 2ha1v sinh(vh) + 2a2vx cosh(vh)2 2ha2v cosh(vh)2 + 4a3xv cosh(vh)3 + 8a4vx cosh(vh)4 + 8ha4v cosh(vh)4 + 16a5vx cosh(vh)5 + 32ha5v cosh(vh)5 + 32a6vx cosh(vh)6 + 96ha6v cosh(vh)6 + 4a3xvcosh(vh)2 sinh(vh) +2a2vx cosh(vh) sinh(vh) + 16a5vx cosh(vh)4 sinh(vh) + 32ha5v cosh(vh)4 sinh(vh) + 32a6vx cosh(vh)5 sinh(vh) + 96ha6v cosh(vh)5 sinh(vh))/(cosh(vh) + sinh(vh))3 4 4 4 y[n]: = x*exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = (x+h)*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = (xh)*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = (x+2*h)*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = (x2*h)*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$1); 4 y[n+3]: = (x+3*h)*exp(v*(x+3*h)); 4 f[n+3]: = diff(y[n+3],x$1); 416 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 y[n3]: = (x3*h)*exp(v*(x3*h)); 4 f[n3]: = diff(y[n3],x$1); 4 4 final3: = y[n+3]y[n3] = h*(a[0]*f[n3]+a[1]*f[n2]+a[2]*f[n1]+ a[3]*f[n]+a[4]*f[n+1]+a[5]*f[n+2]+a[6]*f[n+3]); final3 : =(x + 3h) e(v(x+3h)) (x 3h) e(v(x3h)) = h (a0(e(v(x3h)) (x 3h)ve(v(x3h))) + a1(e(v(x2h)) (x 2h)ve(v(x2h))) +a2(e(v(xh)) + (x h)ve(v(xh))) a3(e(vx) xve(vx)) + a4(e(v(x+h)) (x+h)ve(v(x+h))) + a5(e(v(x+2h)) (x+2h)ve(v(x+2h))) + a6(e(v(x+3h)) (x+3h)ve(v(x+3h)))) 4 final3: = combine(final3/exp(v*x)); final3 : =((x+3h) e((x3h)v) + (x + 3h) e(v(x3h))) e(vx) = h(a1(e(v(x2h)) (x2h)ve(v(x2h))) + a2(e(v(xh)) (xh)ve(v(xh))) + a4(e(v(x+h)) (x+h)ve(v(xh))) + a3(e(vx) xve(vx)) + a4(e(v(x+h)) + (x+h)ve(v(x+h))) + a5(e(v(x+2h)) (x+2h)ve(v(x+2h))) + a0(e(v(x3h)) (x3h)ve(v(x3h))) + a6((x3h)ve((x3h)v) + e((x3h)v)) + a3(e(vx)xve(vx)))e(vx) 4 final3: = expand(final3); final3 :¼

x ðeðvhÞ Þ3

þ

3h ðeðvhÞ Þ3

ðeðvhÞ Þ3 x þ 3ðeðvhÞ Þ3 h ¼ ha1 ðeðvhÞ Þ2

ha1 v þ ðeðvhÞ Þ2 x þ 2h2 a1 vðeðvhÞ Þ2 þ ha2 eðvhÞ ha2 veðvhÞ x þ h2 a2 veðvhÞ þ

ha4 ha4 vx h2 a4 v ha5 ha5 vx 2h2 a5 v þ 2 2 eðvhÞ eðvhÞ eðvhÞ ðeðvhÞ Þ ðeðvhÞ Þ2 ðeðvhÞ Þ

þ ha0 ðeðvhÞ Þ3 ha0 ðeðvhÞ Þ3 x þ 3h2 a0 vðeðvhÞ Þ3 þ

ha6 ðeðvhÞ Þ3

ha6 vx ðeðvhÞ Þ3

3h2 a6 v ðeðvhÞ Þ3

þ ha3 ha3 xv

4 final3: = simplify(convert(final3,trig)); final3 : = 2(x 24x cosh(vh)4 + 9x cosh(vh)2 27h cosh(vh)2 16x cosh(vh)3 sinh(vh) + 3x cosh(vh) sinh(vh) + 48h cosh(vh)3 sinh(vh) + 72h cosh(vh)4 + 16x cosh(vh)6 + 16x cosh(vh)5 sinh(vh) 48h cosh(vh)5 sinh(vh) 48h cosh(vh)6 9h cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))3 = h(a0 a2 + a4 a6 2a4 cosh(vh) sinh(vh) ha2v + a2vx + 3ha6v + a6vx + 12a1 sinh(vh) cosh(vh)2 + 32a0 cosh(vh)3 sinh(vh) 6a0 cosh(vh) sinh(vh) ha4v a4vx 4a3 cosh(vh)2 sinh(vh) + 4ha2v cosh(vh) sinh(vh) + 4a2 cosh(vh) sinh(vh) + 16a1vx cosh(vh)4 sinh(vh) 32ha1v cosh(vh)4 sinh(vh) + 32a0vx cosh(vh)5 sinh(vh) + 8a2vx cosh(vh)3 sinh(vh) 8ha2v cosh(vh)3 sinh(vh) 12a1vx sinh(vh) cosh(vh)2 5a1 cosh(vh) a1 sinh(vh) + 8a2 cosh(vh)2 4a3 cosh(vh)3 + 32a0vx cosh(vh)6 96ha0v cosh(vh)6 32a0 cosh(vh)5 sinh(vh) 16a1 cosh(vh)4 sinh(vh) 2a4 cosh(vh)2 + 3a3 cosh(vh) + a3 sinh(vh) 8a2 cosh(vh)3 sinh(vh) 32a0 cosh(vh)6 16a1 cosh(vh)5 8a2 cosh(vh)4 a5 sinh(vh) a5 cosh(vh) + 48a0 cosh(vh)4 18a0 cosh(vh)2 + 20a1 cosh(vh)3 32a0vx cosh(vh)3 sinh(vh) + 96ha0v cosh(vh)3 sinh(vh) + 2a4vx cosh(vh)2 + 2a4vx cosh(vh) sinh(vh) + a5vx sinh(vh) + 2ha4v cosh(vh) sinh(vh) + 2ha5v sinh(vh) + 2ha4v cosh(vh)2 + 2ha5v cosh(vh) 3a3xv cosh(vh) + a5vx cosh(vh) 96ha0vcosh(vh)5 sinh(vh) + 16a1vx cosh(vh)5 32ha1v cosh(vh)5 + 8a2vx cosh(vh)4 8ha2v cosh(vh)4 a3xv sinh(vh) + 24ha1v cosh(vh)2 sinh(vh) 18ha0v cosh(vh) sinh(vh) + 6a0vx cosh(vh) sinh(vh) + 3ha0va0vx + 5a1vx cosh(vh) + a1vx sinh(vh) 10ha1v cosh(vh) 2ha1v sinh(vh) 8a2vx cosh(vh)2 + 8ha2v cosh(vh)2 + 4a3xv cosh(vh)3 + 4a3xv cosh(vh)2 sinh(vh) 4a2vx cosh(vh) sinh(vh) + 144ha0v cosh(vh)4 20a1vx cosh(vh)3 + 40ha1v cosh(vh)3 54ha0v cosh(vh)2 48a0vx cosh(vh)4 + 18a0vx cosh(vh)2)/ (cosh(vh) + sinh(vh))3 4 Chem. Modell., 2008, 5, 350–487 | 417 This journal is

c


4 eq3: = final2; eq3 : =2(x+27h cosh(vh)2 72h cosh(vh)4 + 16x cosh(vh)6 + 16x cosh(vh)5 sinh(vh) + 48h cosh(vh)5 sinh(vh) + 48h cosh(vh)6 16x cosh(vh)3 sinh(vh) + 3x cosh(vh) sinh(vh) 48h cosh(vh)3 sinh(vh) + 9h cosh(vh) sinh(vh) 24x cosh(vh)4 + 9x cosh(vh)2)/(cosh(vh) + sinh(vh))3 = h(6a6 cosh(vh) sinh(vh) 32a6 cosh(vh)3 sinh(vh) + a0a2+a4a6+32a6 cosh(vh)5 sinh(vh) 4a4 cosh(vh) sinh(vh) + 16a5 cosh(vh)4 sinh(vh) + ha2v a2vx 3ha6v a6vx + ha4v + a4vx + 4a3 cosh(vh)2 sinh(vh) 2ha2v cosh(vh) sinh(vh) + 2a2 cosh(vh) sinh(vh) + 8ha4v cosh(vh)3 sinh(vh) + a1 cosh(vh) + a1 sinh(vh) + 2a2 cosh(vh)2 + 4a3 cosh(vh)3 + 8a4 cosh(vh)4 + 16a5 cosh(vh)5 + 32a6 cosh(vh)6 + 8a4 cosh(vh)3 sinh(vh) + 8a4 vx cosh(vh)3 sinh(vh) 8a4 cosh(vh)2 3a3 cosh(vh) a3 sinh(vh) 12a5 sinh(vh) cosh(vh)2 + a5 sinh(vh) + 5a5 cosh(vh) + 18a6 cosh(vh)2 20a5 cosh(vh)3 48a6 cosh(vh)4 40ha5v cosh(vh)3 8a4vx cosh(vh)2 24ha5v cosh(vh)2 sinh(vh) + 18a6vx cosh(vh)2 12a5vx cosh(vh)2 sinh(vh) 32a6vx cosh(vh)3 sinh(vh) 96ha6v cosh(vh)3 sinh(vh) 48a6vx cosh(vh)4 + 6a6vx cosh(vh) sinh(vh) 4a4vx cosh(vh) sinh(vh) + 54ha6v cosh(vh)2 + a5vx sinh(vh) 144ha6v cosh(vh)4 4ha4v cosh(vh) sinh(vh) + 2ha5v sinh(vh) + 18ha6v cosh(vh)sinh(vh) 8ha4v cosh(vh)2 + 10ha5vcosh(vh) 3a3xv cosh(vh) 20a5vx cosh(vh)3 + 5a5vx cosh(vh) a3xv sinh(vh) 3ha0v + a0vx + a1vx cosh(vh) + a1vx sinh(vh) 2ha1v cosh(vh) 2ha1v sinh(vh) + 2a2vx cosh(vh)2 2ha2v cosh(vh)2 + 4a3xv cosh(vh)3 + 8a4vx cosh(vh)4 + 8ha4v cosh(vh)4 + 16a5vx cosh(vh)5 + 32ha5v cosh(vh)5 + 32a6vx cosh(vh)6 + 96ha6v cosh(vh)6 + 4a3xv cosh(vh)2 sinh(vh) + 2a2vx cosh(vh) sinh(vh) + 16a5vx cosh(vh)4 sinh(vh) + 32ha5v cosh(vh)4 sinh(vh) + 32a6vx cosh(vh)5 sinh(vh) + 96ha6vcosh(vh)5 sinh(vh))/(cosh(vh) + sinh(vh))3 4 eq4: = final3; eq4 : = 2 ( x 24x cosh(vh)4 + 9x cosh(vh)2 27h cosh(vh)2 16x cosh(vh)3 sinh(vh) + 3x cosh(vh) sinh(vh) + 48h cosh(vh)3 sinh(vh) + 72h cosh(vh)4 + 16x cosh(vh)6 + 16x cosh(vh)5 sinh(vh) 48h cosh(vh)5 sinh(vh) 48h cosh(vh)6 9h cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))3 = h(a0 a2 + a4 a6 2a4 cosh(vh) sinh(vh) ha2v + a2vx + 3ha6v+a6vx + 12a1 sinh(vh) cosh(vh)2 + 32a0 cosh(vh)3 sinh(vh) 6a0 cosh(vh) sinh(vh) ha4v a4vx 4a3 cosh(vh)2 sinh(vh) + 4ha2v cosh(vh) sinh(vh) + 4a2 cosh(vh) sinh(vh) + 16a1vx cosh(vh)4 sinh(vh) 32ha1v cosh(vh)4 sinh(vh) + 32a0vx cosh(vh)5 sinh(vh) + 8a2vx cosh(vh)3 sinh(vh) 8ha2v cosh(vh)3 sinh(vh) 12a1vx sinh(vh) cosh(vh)2 5a1 cosh(vh) a1 sinh(vh) + 8a2 cosh(vh)2 4a3 cosh(vh)3 + 32a0vx cosh(vh)6 96ha0v cosh(vh)6 32a0 cosh(vh)5 sinh(vh) 16a1 cosh(vh)4 sinh(vh) 2a4 cosh(vh)2 + 3a3 cosh(vh) + a3 sinh(vh) 8a2 cosh(vh)3 sinh(vh) 32a0 cosh(vh)6 16a1 cosh(vh)5 8a2 cosh(vh)4 a5 sinh(vh) a5 cosh(vh) + 48a0 cosh(vh)4 18a0 cosh(vh)2 + 20a1 cosh(vh)3 32a0vx cosh(vh)3 sinh(vh) + 96ha0v cosh(vh)3 sinh(vh) + 2a4vx cosh(vh)2 + 2a4vx cosh(vh) sinh(vh) + a5vx sinh(vh) + 2ha4v cosh(vh) sinh(vh) + 2ha5v sinh(vh) + 2ha4v cosh(vh)2 + 2ha5v cosh(vh) 3a3xv cosh(vh) + a5vx cosh(vh) 96ha0v cosh(vh)5 sinh(vh) + 16a1vx cosh(vh)5 32ha1v cosh(vh)5 + 8a2vx cosh(vh)4 8ha2v cosh(vh)4 a3xv sinh(vh) + 24ha1v cosh(vh)2 sinh(vh) 18ha0v cosh(vh) sinh(vh) + 6a0vx cosh(vh) sinh(vh) + 3ha0v a0vx + 5a1vx cosh(vh) + a1vx sinh(vh) 10ha1v cosh(vh) 2ha1v sinh(vh) 8a2vx cosh(vh)2 + 8ha2v cosh(vh)2 + 4a3xv cosh(vh)3 + 4a3xvcosh(vh)2 sinh(vh) 4a2vx cosh(vh) sinh(vh) + 144ha0v cosh(vh)4 20a1vx cosh(vh)3 + 40ha1v cosh(vh)3 54ha0v cosh(vh)2 48a0vx cosh(vh)4 + 18a0vx cosh(vh)2)/(cosh(vh) + sinh(vh))3 4 eq3i: = subs(v = I*v,eq3); eq3i : =2 (x + 27h cosh(vhI)2 72h cosh(vhI)4 + 16x cosh(vhI)6 + 16x cosh(vhI)5 sinh(vhI) + 48h cosh(vhI)5 sinh(vhI) + 48h cosh(vhI)6 16x cosh(vhI)3 sinh(vhI) + 3x cosh(vhI) sinh(vhI) 48h cosh(vhI)3 sinh(vhI) + 9h cosh(vhI) sinh(vhI) 24x cosh(vhI)4 + 9x cosh(vhI)2)/(cosh(vhI) + sinh(vhI))3 = h(a0 a2 + a4 a6 32a6 cosh(vhI)3 sinh(vhI) + 8a4 cosh(vhI)3 sinh(vhI) + 16a5 cosh(vhI)4 sinh(vhI) + 8Iha4v 418 | Chem. Modell., 2008, 5, 350–487 This journal is

c


cosh(vhI)4 12a5 sinh(vhI) cosh(vhI)2 + 6a6 cosh(vhI) sinh(vhI) + 32a6 cosh(vhI)5 sinh(vhI) 4a4 cosh(vhI) sinh(vhI) + 4a3 cosh(vhI)2 sinh(vhI) + 2a2 cosh(vhI) sinh(vhI) + a1 cosh(vhI) + a1 sinh(vhI) + 2a2 cosh(vhI)2 + 4a3 cosh(vhI)3 + 8a4 cosh(vhI)4 + 16a5 cosh(vhI)5 + 32a6 cosh(vhI)6 8a4 cosh(vhI)2 3a3 cosh(vhI) a3 sinh(vhI) + a5 sinh(vhI) + 5a5 cosh(vhI) + 18a6 cosh(vhI)2 20a5 cosh(vhI)3 48a6 cosh(vhI)4 + ha4vI + ha2vI+a4vxI 4Iha4v cosh(vhI) sinh(vhI) 24Iha5v cosh(vhI)2 sinh(vhI) a3xv sinh(vhI)I 2Iha1v cosh(vhI) 2Iha1v sinh(vhI) + 2Ia2vx cosh(vhI)2 2Iha2v cosh(vhI)2 + 4Ia3xv cosh(vhI)3 + 8Ia4vx cosh(vhI)4 + 16Ia5 vx cosh(vhI)5 + 32Iha5v cosh(vhI)5 + 32Ia6vx cosh(vhI)6 + 96Iha6v cosh(vhI)6 + 4Ia3xv cosh(vhI)2 sinh(vhI) + 2Ia2vx cosh(vhI) sinh(vhI) + 16Ia5vx cosh(vhI)4 sinh(vhI) + 32Iha5v cosh(vhI)4 sinh(vhI) + 32Ia6vx cosh(vhI)5 sinh(vhI) + 96Iha6v cosh(vhI)5 sinh(vhI) + a5vx sinh(vhI)I + a1vx cosh(vhI)I + a1vx sinh(vhI) I 2Iha2v cosh(vhI) sinh(vhI) + 8Iha4v cosh(vhI)3 sinh(vhI) + 8Ia4vx cosh(vhI)3 sinh(vhI) 40Iha5 v cosh(vhI)3 + 8Ia4vx cosh(vhI)2 + 18Ia6vx cosh(vhI)2 12Ia5vx cosh(vhI)2 sinh(vhI) 32Ia6vx cosh(vhI)3 sinh(vhI) 96Iha6v cosh(vhI)3 sinh(vhI) 48Ia6vx cosh(vhI)4 + 6Ia6vx cosh(vhI) sinh(vhI) 4Ia4vx cosh(vhI) sinh(vhI) + 54Iha6v cosh(vhI)2 144Iha6v cosh(vhI)4 + 2Iha5v sinh(vhI) + 18Iha6v cosh(vhI) sinh(vhI) 8Ih a4v cosh(vhI)2 + 10Iha5v cosh(vhI) 3Ia3xv cosh(vhI) 20Ia5vx cosh(vhI)3 +5Ia5vx cosh(vhI)+a0vxIa2vxI 3Iha6v a6vxI 3Iha0v)/(cosh(vhI) + sinh(vhI))3 4 eq4i: = subs(v = I*v,eq4); eq4i : = 2(x 24x cosh(vhI)4 + 9x cosh(vhI)2 27h cosh(vhI)2 16x cosh(vhI)3 sinh(vhI) + 3x cosh(vhI) sinh(vhI) + 48h cosh(vhI)3 sinh(vhI) + 72h cosh(vhI)4 + 16x cosh(vhI)6 + 16x cosh(vhI)5 sinh(vhI) 48h cosh(vhI)5 sinh(vhI) 48h cosh(vhI)6 9h cosh(vhI) sinh(vhI))/(cosh(vhI) + sinh(vhI))3 = h(12a1 sinh(vhI) cosh(vhI)2 + 32a0 cosh(vhI)3 sinh(vhI) + a0 a2 + a4 a6 16a1 cosh(vhI)4 sinh(vhI) + 32Ia0vx cosh(vhI)5 sinh(vhI) 2a4 cosh(vhI) sinh(vhI) 4a3 cosh(vhI)2 sinh(vhI) + 4a2 cosh(vhI) sinh(vhI) 6a0 cosh(vhI) sinh(vhI) 32a0 cosh(vhI)5 sinh(vhI) 8a2 cosh(vhI)3 sinh(vhI) 5a1 cosh(vhI) a1 sinh(vhI) + 8a2 cosh(vhI)2 4a3 cosh(vhI)3 2a4 cosh(vhI)2 + 3a3 cosh(vhI) + a3 sinh(vhI) a5 sinh(vhI) a5 cosh(vhI) 32a0 cosh(vhI)6 16a1 cosh(vhI)5 8a2 cosh(vhI)4 + 48a0 cosh(vhI)4 18a0 cosh(vhI)2 + 20a1 cosh(vhI)3 + a6vxI + a2vxI + 4Iha2v cosh(vhI) sinh(vhI) + 16Ia1vx cosh(vhI)4 sinh(vhI) 32Iha1v cosh(vhI)4 sinh(vhI) 4Ia2vx cosh(vhI) sinh(vhI) + 144Iha0v cosh(vhI)4 20Ia1vx cosh(vhI)3 + 40Iha1v cosh(vhI)3 54Iha0v cosh(vhI)2 48Ia0vx cosh(vhI)4 + 18Ia0vx cosh(vhI)2 a3xv sinh(vhI)I 2Iha1v sinh(vhI) + 4Ia3xv cosh(vhI)3 + 4Ia3xv cosh(vhI)2 sinh(vhI) ha2vI + 3Iha6v ha4vI a4vxI + 3Iha0v a0vxI + 8Ia2vx cosh(vhI)3 sinh(vhI) 8Iha2v cosh(vhI)3 sinh(vhI) 12Ia1 vxsinh(vhI) cosh(vhI)2 + 32Ia0vx cosh(vhI)6 96Iha0v cosh(vhI)6 32Ia0vx cosh(vhI)3 sinh(vhI) + 96Iha0v cosh(vhI)3sinh(vhI) + 2Ia4vx cosh(vhI)2 + 2Ia4vx cosh(vhI) sinh(vhI) + 2Iha4v cosh(vhI) sinh(vhI) + 2Iha4v cosh(vhI)2 + 2Iha5v cosh(vhI) 96Iha0v cosh(vhI)5 sinh(vhI) + 16Ia1vx cosh(vhI)5 32Iha1v cosh(vhI)5 + 8Ia2vx cosh(vhI)4 8Iha2v cosh(vhI)4 + 24Iha1v cosh(vhI)2 sinh(vhI) 18Iha0v cosh(vhI) sinh(vhI) + 6Ia0vx cosh(vhI) sinh(vhI) + 5Ia1vx cosh(vhI) 10Iha1v cosh(vhI) 8Ia2vx cosh(vhI)2 + 8Iha2v cosh(vhI)2 + a5vx sinh(vhI)I + a5vx cosh(vhI)I + a1vx sinh(vhI)I + 2Iha5v sinh(vhI) 3Ia3xv cosh(vhI))/(cosh(vhI) + sinh(vhI))3 4 4 eq3f: = simplify(evalc(Re(eq3i))); eq3f : =6 cos(vh)h (3 + 4 cos(vh)2) = h(4a6 cos(vh)3 a1 + a3 a5 + 2a1 cos(vh)2 3 cos(vh) a0 + 2a5 cos(vh)2 3a6 cos(vh) 4 cos(vh) ha1v sin(vh) 4 cos(vh)ha5v sin(vh) 2 cos(vh) a5vx sin(vh) + 2 cos(vh)a1vx sin(vh) + a2 cos(vh) 12 sin(vh)ha0v cos(vh)2 + a4 cos(vh) + 4a0 cos(vh)3 + 3 sin(vh)ha0v sin(vh) a0vx + sin(vh) a2vx sin(vh) ha2v sin(vh) a4vx + sin(vh)a6vx + 3 sin(vh) ha6v sin(vh) ha4v + 4 sin(vh) a0vx cos(vh)2 4 sin(vh) a6vx cos(vh)2 12 sin(vh)ha6v cos(vh)2) 4 eq4f: = simplify(evalc(Im(eq3i))); Chem. Modell., 2008, 5, 350–487 | 419 This journal is

c


eq4f : =2 sin(vh)x (1 + 4 cos(vh)2) = h (sin(vh) a0 sin(vh) a2 9ha6v cos(vh) + a4vx cos(vh) + cos(vh)ha4v + 2a1vx cos(vh)2 + 9 cos(vh)ha0v + cos(vh)a2vx + 2a5vx cos(vh)2 3 cos(vh)a6vx + 4ha5v cos(vh)2 4ha1v cos(vh)2 ha2v cos(vh) 3 cos(vh)a0vx 4 sin(vh)a0 cos(vh)2 + 4 sin(vh)a6 cos(vh)2 + 12ha6v cos(vh)3 sin(vh)a6 + sin(vh) a4 2 cos(vh)a1 sin(vh) + 4a6vx cos(vh)3 + 2 cos(vh)a5 sin(vh) + a3xv a1vx a5vx 2ha5v + 2ha1v 12ha0v cos(vh)3 + 4a0 vx cos(vh)3) 4 eq5f: = simplify(evalc(Re(eq4i))); eq5f : =6 cos(vh)h(3 + 4 cos(vh)2) = h(4a6 cos(vh)3 a1+a3a5 + 2a1 cos(vh)2 3 cos(vh) a0 + 2a5 cos(vh)2 3a6 cos(vh) 4 cos(vh) ha1v sin(vh) 4 cos(vh)ha5v sin(vh) 2 cos(vh) a5vx sin(vh) + 2 cos(vh) a1vx sin(vh) + a2 cos(vh) 12 sin(vh)ha0v cos(vh)2 + a4 cos(vh) + 4a0 cos(vh)3 + 3 sin(vh)ha0v sin(vh)a0vx + sin(vh)a2vx sin(vh)ha2v sin(vh)a4vx + sin(vh) a6vx + 3 sin(vh)ha6v sin(vh)ha4v+4 sin(vh)a0vx cos(vh)2 4 sin(vh)a6vx cos(vh)2 12 sin(vh)ha6v cos(vh)2) 4 eq6f: = simplify(evalc(Im(eq4i))); eq6f : =2 sin(vh)x(1 + 4 cos(vh)2) = h (sin(vh) a0sin(vh)a2 9ha6v cos(vh) + a4vx cos(vh) + cos(vh)ha4v + 2a1vx cos(vh)2 + 9 cos(vh) ha0v + cos(vh) a2vx + 2a5vx cos(vh)2 3 cos(vh)a6 vx + 4ha5v cos(vh)2 4ha1v cos(vh)2 ha2v cos(vh) 3 cos(vh)a0v x 4 sin(vh)a0 cos(vh)2 + 4 sin(vh)a6 cos(vh)2 + 12ha6v cos(vh)3 sin(vh)a6 + sin(vh)a4 2 cos(vh) a1 sin(vh) + 4a6vx cos(vh)3 + 2 cos(vh) a5 sin(vh) + a3xv a1vx a5vx 2ha5v + 2ha1v 12ha0v cos(vh)3 +4a0vx cos(vh)3) 4 simplify(eq3feq5f); 0=0 4 simplify(eq4f+eq6f); 0=0 4 4 4 y[n]: = x^n; 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = (x+h)^n; 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = (xh)^n; 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = (x+2*h)^n; 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = (x2*h)^n; 4 f[n2]: = diff(y[n2],x$1); 4 y[n+3]: = (x+3*h)^n; 4 f[n+3]: = diff(y[n+3],x$1); 4 y[n3]: = (x3*h)^n; 4 f[n3]: = diff(y[n3],x$1); 4 4 final5: = y[n+3]y[n3] = h*(a[0]*f[n3]+a[1]*f[n2]+a[2]*f[n1]+ a[3]*f[n]+ a[4]*f[n+1]+a[5]*f[n+2]+a[6]*f[n+3]); a0 ðx 3hÞn n a1 ðx 2hÞn n final5 :¼ðx þ 3hÞn ðx 3hÞn ¼ h þ x 3h x 2h n n n a2 ðx hÞ n a3 x n a4 ðx þ hÞ n a5 ðx þ 2hÞn n a6 ðx þ 3hÞn n þ þ þ þ þ xh x xþh x þ 2h x þ 3h 4 n: = 0; n : =0 4 eq2: = simplify(final5); 420 | Chem. Modell., 2008, 5, 350–487 This journal is

c


eq2 : =0 = 0 4 n: = 1; n : =1 4 eq2: = simplify(final5); eq2 : =6h = h(a0 + a1 + a2 + a3 + a4 + a5 + a6) 4 eq2: = simplify(eq2/h); eq2 : =6 = a0 + a1 + a2 + a3 + a4 + a5 + a6 4 eq7f: = eq2; eq7f : =6 = a0+a1+a2+a3+a4+a5+a6 4 n: = 2; n : =2 4 eq3: = simplify(final5); eq3 : =12xh = 2h(a0x3ha0 + a1x2ha1 + a2xha2 + a3x + a4x + ha4 + a5x + 2ha5 + a6x + 3ha6) 4 eq3: = simplify(eq32*eq2*x*h); eq3 : =0 = 6h2a04h2a1 2h2a2 + 2h2a4 + 4h2a5 + 6h2a6 4 eq3: = simplify(eq3/(h^2)); eq3 : =0 = 6a04a1 2a2 + 2a4 + 4a5 + 6a6 4 eq8f: = eq3; eq8f : =0 = 6a04a1 2a2 + 2a4 + 4a5 + 6a6 4 4 n: = 3; n : =3 4 eq4: = simplify(final5); eq4 = 18x2h + 54h3 = 3h(a0x2 6ha0x + 9h2a0 + a1x2 4ha1x + 4h2a1 + a2x2 2ha2x + h2a2 + a3x2 + a4x2 + 2ha4x + h2a4 + a5x2 + 4ha5x + 4h2a5 + a6x2 + 6ha6x + 9h2a6) 4 eq4: = simplify(eq43*eq2*x^2*h3*x*h^2*eq3); eq4 : =54h3 = 27h3a0 + 12h3a1 + 3h3a2 + 3h3a4 + 12h3a5 + 27h3a6 4 eq4: = simplify(eq4/(h^3)); eq4 : =54 = 27a0 + 12a1 + 3a2 + 3a4 + 12a5 + 27a6 4 eq9f: = eq4; eq9f : =54 = 27a0 + 12a1 + 3a2 + 3a4 + 12a5 + 27a6 4 4 4 eq1f; 0= vh sin(vh)(a0 + a2 a4 + a6 + 2a1 cos(vh) 2a5 cos(vh)2 4a6 cos(vh)2 + 4a0 cos(vh)2) 4 eq2f; 2 sin(vh)(1 + 4 cos(vh)2) = vh(a1 + a3 a5 + 4a0 cos(vh)3 + 2a1 cos(vh)2 3 cos(vh)a0 3a6 cos(vh) + a4 cos(vh) + a2 cos(vh) + 4a6 cos(vh)3 + 2a5 cos(vh)2) 4 eq3f; 6 cos(vh)h(3 + 4 cos(vh)2) = h(4a6 cos(vh)3 a1 + a3 a5 + 2a1 cos(vh)2 3 cos(vh)a0 + 2a5 cos(vh)2 3a6 cos(vh) 4 cos(vh)ha1v sin(vh) 4 cos(vh) ha5v sin(vh) 2 cos(vh) a5vx sin(vh) + 2 cos(vh) a1vx sin(vh) + a2 cos(vh) 12 sin(vh)ha0v cos(vh)2 + a4 cos(vh) + 4a0 cos(vh)3 + 3 sin(vh)ha0v sin(vh)a0vx + sin(vh)a2vx Chem. Modell., 2008, 5, 350–487 | 421 This journal is

c


sin(vh)ha2v sin(vh)a4vx + sin(vh)a6vx+3 sin(vh)ha6v sin(vh)ha4v + 4 sin(vh)a0vx cos(vh)2 4 sin(vh)a6vx cos(vh)2 12 sin(vh) ha6v cos(vh)2) 4 eq4f; 2 sin(vh)x (1 + 4 cos(vh)2) = h(sin(vh)a0 sin(vh)a2 9ha6v cos(vh) + a4vx cos(vh) + cos(vh)ha4v + 2a1vx cos(vh)2 + 9 cos(vh)ha0v + cos(vh)a2vx + 2a5vx cos(vh)2 3 cos(vh)a6vx + 4ha5v cos(vh)2 4ha1v cos(vh)2 ha2v cos(vh) 3 cos(vh)a0vx 4 sin(vh)a0 cos(vh)2 + 4 sin(vh) a6 cos(vh)2 + 12ha6v cos(vh)3 sin(vh)a6 + sin(vh) a4 2 cos(vh) a1 sin(vh) + 4a6vx cos(vh)3 + 2 cos(vh) a5 sin(vh) + a3xv a1vxa5vx 2ha5v + 2ha1v 12ha0v cos(vh)3 + 4a0 v x cos(vh)3) 4 eq7f; 6 = a0 + a1 + a2 + a3 + a4 + a5 + a6 4 eq8f; 0 = 6a0 4a1 2a2 + 2a4 + 4a5 + 6a6 4 eq9f; 54 = 27a0 + 12a1 + 3a2 + 3a4 + 12a5 + 27a6 4 4 4 solut: a[4],a[5],a[6]});

=

solve({eq1f,eq2f,eq3f,eq4f,eq7f,eq8f,eq9f},{a[0],a[1],a[2],a[3],

solut : ={a6 = 14 ( 4hv cos(vh)4 + 4 cos(vh)3 sin(vh) + 12hv cos(vh)3 vh cos(vh)2 4 cos(vh)2 sin(vh) cos(vh) sin(vh) + 9h2 v2 cos(vh) sin(vh) 9hv cos(vh) + sin(vh) 3v2 h2 sin(vh) + 2vh)/(v2 h2 sin(vh)(cos(vh)3 3 cos(vh)2 + 3 cos(vh) 1)), a5 = 12 (4 cos(vh)4 sin(vh) + 18hv cos(vh)3 + 4 cos(vh)3 sin(vh) 9 cos(vh)2 sin(vh) 6vh cos(vh)2 + 18h2 v2 cos(vh)2 sin(vh) 15hv cos(vh) cos(vh) sin(vh) + 2 sin(vh) + 3vh)/(v2 h2 sin(vh) (cos(vh)3 3 cos(vh)2 + 3 cos(vh) 1)), a4 = 14 (32 cos(vh)4 sin(vh) + 36hv cos(vh)4 + 36hv cos(vh)3 4 cos(vh)3 sin(vh) + 36h2 v2 cos(vh)3 sin(vh) + 36h2 v2 cos(vh)2 sin(vh) 39vh cos(vh)2 36 cos(vh)2 sin(vh) 39hv cos(vh) + 27h2 v2 cos(vh) sin(vh) + cos(vh) sin(vh) + 7 sin(vh) + 6vh 9v2 h2 sin(vh))/(v2 h2 sin(vh) (cos(vh)3 3 cos(vh)2 + 3 cos(vh) 1)), a3 = (12 cos(vh)4 sin(vh) + 16hv cos(vh)4 4 cos(vh)3 sin(vh) + 12h2 v2 cos(vh)3 sin(vh) + 6hv cos(vh)3 + 18h2 v2 cos(vh)2 sin(vh) 14vh cos(vh)2 11 cos(vh)2 sin(vh) 9hv cos(vh) + cos(vh) sin(vh) + 2 sin(vh) + vh)/(v2 h2 sin(vh) (cos(vh)3 3 cos(vh)2 + 3 cos(vh) 1)), a2 = 14(32 cos(vh)4 sin(vh) + 36hv cos(vh)4 + 36hv cos(vh)3 4 cos(vh)3 sin(vh) + 36h2 v2 cos(vh)3 sin(vh) + 36h2 v2 cos(vh)2 sin(vh) 39vh cos(vh)2 36 cos(vh)2 sin(vh) 39hv cos(vh) + 27h2v2 cos(vh) sin(vh) + 7 sin(vh) + 6vh 9v2 h2 sin(vh))/(v2 h2 sin(vh) (cos(vh)3 3 cos(vh)2 + 3 cos(vh) 1)), a1 = 12 (4 cos(vh)4 sin(vh) + 18hv cos(vh)3 + 4 cos(vh)3 sin(vh) 9 cos(vh)2 sin(vh) 6vh cos(vh)2 + 18h2 v2 cos(vh)2 sin(vh) 15hv cos(vh) cos(vh) sin(vh) + 2 sin(vh) + 3vh)/(v2 h2 sin(vh) (cos(vh)3 3 cos(vh)2 + 3 cos(vh) 1)), a0 = 14( 4hv cos(vh)4 + 4 cos(vh)3 sin(vh) + 12hv cos(vh)3 vh cos(vh)2 4 cos(vh)2 sin(vh) cos(vh) sin(vh) + 9h2 v2 cos(vh) sin(vh) 9hv cos(vh) + sin(vh) 3v2 h2 sin(vh) + 2vh)/(v2 h2 sin(vh) (cos(vh)3 3 cos(vh)2 + 3 cos(vh) 1))} 4 assign(solut); 4 h : =1; h : =1 4 a[0]: = combine(a[0]); a0 : =(v cos(4v) + 5v cos(2v) sin(4v) sin(2v) 6v cos(3v) + 2 sin(3v) 9v2 sin(2v) + 6v2 sin(v))/(v2 sin(4v) 14v2 sin(2v) + 6v2 sin(3v) + 14v2 sin(v)) 422 | Chem. Modell., 2008, 5, 350–487 This journal is

c



41 9 2 3 4 577 191 431 þ v þ v þ v6 þ v8 þ v10 140 700 3850 10510500 42042000 1021020000 852437 2479369 v12 þ v14 þ 20532303792000 594356162400000

4 a[1]: = combine(a[1]); a1 : =(sin(5v) 6 sin(3v) +sin(v) + 18v cos(3v) 6v cos(v) + 2 sin(4v) +2 sin(2v)12v cos(2v) + 18v2 sin(3v) + 18v2 sin(v))/(v2 sin(4v)14v2 sin(2v) + 6v2 sin(3v) + 14v2 sin(v)) 4 a1t: = convert(taylor(a[1],v = 0,24),polynom); a1t :¼

54 27 2 27 4 2341 83 17247 v þ v v6 v8 v10 35 350 15400 14014000 8624000 19059040000 4663753 19482877 v12 v14 54752810112000 2316465043200000

4 a[2]: = combine(a[2]); a2 : =(4 sin(5v) + 6 sin(3v) 4 sin(v) 9v cos(4v) + 3v cos(2v) 18v cos(3v) + 24v cos(v) +sin(4v) + sin(2v) 9v2 sin(4v) 45v2 sin(2v) 18v2 sin(3v))/(v2 sin(4v) 14v2 sin(2v) + 6v2 sin(3v) + 14v2 sin(v)) 4 a2t: = convert(taylor(a[2],v = 0,24),polynom); a2t :¼

27 27 2 27 4 61 67 171 þ v v þ v6 þ v8 þ v10 140 140 1925 350350 28028000 952952000 450869 5874133 v12 v14 13688202528000 1505702278080000

4 a[3]: = combine(a[3]); a3 : =(6 sin(5v) 4 sin(3v) + 6 sin(v) + 16v cos(4v) + 8v cos(2v) 4 sin(4v) 4 sin(2v) + 12v2 sin(4v) +24v2 sin(2v) + 12v cos(3v) 36v cos(v) + 36v2 sin(3v) + 36v2 sin(v))/(v2 sin(4v) 14v2 sin(2v) + 6v2 sin(3v) + 14v2 sin(v)) 4 a3t: = convert(taylor(a[3],v = 0,24),polynom); a3t :¼

68 9 2 177 4 521 359 7573 þ v v v6 þ v8 þ v10 35 35 7700 4204200 24024000 5717712000 12582191 147083219 þ v12 þ v14 82129215168000 9034213668480000

4 a[4]: = combine(a[4]); a4 : =(4 sin(5v) + 6 sin(3v) 4 sin(v) 9v cos(4v) + 3v cos(2v) 18v cos(3v) +24v cos(v) + sin(4v) + sin(2v) 9v2 sin(4v) 45v2 sin(2v) 18v2 sin(3v))/(v2 sin(4v) 14v2 sin(2v) + 6v2 sin(3v) + 14v2 sin(v)) 4 a4t: = convert(taylor(a[4],v = 0,24),polynom); a4t :¼

27 27 2 27 4 61 67 171 þ v v þ v6 v8 v10 140 140 1925 350350 28028000 952952000 450869 5874133 v12 v14 13688202528000 1505702278080000

4 a[5]: = combine(a[5]); a5 : =(sin(5v) 6 sin(3v) + sin(v) + 18v cos(3v) 6v cos(v) + 2 sin(4v) + 2 sin(2v) 12v cos(2v) + 18v2 sin(3v) + 18v2 sin(v))/(v2 sin(4v) 14v2 sin(2v) + 6v2 sin(3v) + 14v2 sin(v)) Chem. Modell., 2008, 5, 350–487 | 423 This journal is

c



54 27 2 27 4 2341 83 17247 v þ v v6 v8 v10 35 350 15400 14014000 8624000 19059040000 4663753 19482877 v12 v14 54752810112000 2316465043200000

4 a[6]: = combine(a[6]); a6 : =(v cos(4v) + 5v cos(2v) sin(4v) sin(2v) 6v cos(3v) + 2 sin(3v) 9v2 sin(2v) +6v2 sin(v))/(v2 sin(4v) 14v2 sin(2v) + 6v2 sin(3v) + 14v2 sin(v)) 4 a6t: = convert (taylor (a[6],v = 0,24),polynom); a6t :¼

41 9 2 3 4 577 191 431 þ v þ v þ v6 þ v8 þ v10 140 700 3850 10510500 42042000 1021020000 852437 2479369 v12 þ v14 þ 20532303792000 594356162400000

4 4 simplify(a[4]a[2]); 0 4 simplify(a[5]a[1]); 0 4 simplify(a[6]a[0]); 0 4 4 restart; 4 qnp3: = convert(taylor(q(x+3*h),h = 0,13),polynom); 4 qnp2: = convert(taylor(q(x+2*h),h = 0,13),polynom); 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(xh),h = 0,13),polynom); 4 qnm2: = convert(taylor(q(x2*h),h = 0,13),polynom); 4 qnm3: = convert(taylor(q(x3*h),h = 0,13),polynom); 4 snp3: = convert(taylor(diff(q(x+3*h),x$1),h = 0,13),polynom); 4 snp2: = convert(taylor(diff(q(x+2*h),x$1),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$1),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$1),h = 0,13),polynom); 4 snm2: = convert(taylor(diff(q(x2*h),x$1),h = 0,13),polynom); 4 snm3: = convert(taylor(diff(q(x3*h),x$1),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$1); 4 4 4 c[0]: = subs(v = v*h,41/140+9/700*v^2+3/3850*v^4+577/10510500*v^6+ 191/42042000*v^8+431/1021020000*v^10+852437/ 20532303792000*v^12+2479369/594356162400000*v^14); 4 c[1]: = subs(v = v*h,54/3527/350*v^2+27/15400*v^42341/ 14014000*v^683/8624000*v^817247/19059040000*v^104663753/ 54752810112000*v^1219482877/2316465043200000*v^14); 4 c[2]: = subs(v = v*h,27/140+27/140*v^227/1925*v^4+61/350350*v^667/ 28028000*v^8171/952952000*v^10450869/13688202528000*v^125874133/ 1505702278080000*v^14); 4 4 c[3]: = subs(v = v*h,68/359/35*v^2+177/7700*v^4521/4204200*v^6+ 359/24024000*v^8+7573/5717712000*v^10+12582191/82129215168000*v^12+ 147083219/9034213668480000*v^14); 424 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 4 lte: = simplify(qnp3qnm3h*(c[0]*(snm3+snp3) + c[1]*(snm2+snp2) + c[2]*(snm1+snp1) + c[3]*sn));

139055909 9 2 11 ð9Þ v14 h23 ðDð9Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 111190322073600000 2800 163315637 v12 h21 ðDð9Þ ÞðqÞðxÞ 13140674426880000 5461 9 2 9 ð7Þ v4 h17 ðDð13Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 3104640000 700 1 7538077 v2 h15 ðDð13Þ ÞðqÞðxÞ v12 h19 ðDð7Þ ÞðqÞðxÞ 36750 109505620224000 5191 339 v12 h17 ðDð5Þ ÞðqÞðxÞ v4 h13 ðDð9Þ ÞðqÞðxÞ 31667328000 1232000 9 47 ðDð9Þ ÞðqÞðxÞh9 ðDð11Þ ÞðqÞðxÞh11 1400 30800 1363 2297 v4 h15 ðDð11Þ ÞðqÞðxÞ v6 h13 ðDð7Þ ÞðqÞðxÞ 51744000 28028000 3 2 13 ð11Þ 9 4 9 ð5Þ v h ðD ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 8000 1400 1977 57 4 11 ð7Þ v8 h13 ðDð5Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 112112000 30800 1 6 11 ð5Þ 10597 v h ðD ÞðqÞðxÞ v6 h15 ðDð9Þ ÞðqÞðxÞ 6160 672672000 102461 67217 v6 h17 ðDð11Þ ÞðqÞðxÞ v6 h19 ðDð13Þ ÞðqÞðxÞ 60540480000 565044480000 719 401 v8 h15 ðDð7Þ ÞðqÞðxÞ v8 h17 ðDð9Þ ÞðqÞðxÞ 96096000 295680000 17883 v8 h19 ðDð11Þ ÞðqÞðxÞ 125565440000 288161 73 v8 h21 ðDð13Þ ÞðqÞðxÞ v10 h15 ðDð5Þ ÞðqÞðxÞ 29059430400000 44844800 79297 10469 v10 h17 ðDð7Þ ÞðqÞðxÞ v10 h19 ðDð9Þ ÞðqÞðxÞ 114354240000 83166720000 7623409 v10 h21 ðDð11Þ ÞðqÞðxÞ 576345369600000 6371171 v10 h23 ðDð13Þ ÞðqÞðxÞ 6916144435200000 308215841 v12 h23 ðDð11Þ ÞðqÞðxÞ 236532139683840000 3002395763 v12 h25 ðDð13Þ ÞðqÞðxÞ 33114499555737600000 606607 v14 h19 ðDð5Þ ÞðqÞðxÞ 36501873408000 1254178951 v14 h21 ðDð7Þ ÞðqÞðxÞ 180684273369600000 39768568349 v14 h25 ðDð11Þ ÞðqÞðxÞ 303549579260928000000 99580001333 v14 h27 ðDð13Þ ÞðqÞðxÞ 10927784853393408000000 23327 ðDð13Þ ÞðqÞðxÞh13 34496000

lte :¼


c


4 coeff(lte,h,9);

9 2 ð7Þ 9 9 4 ð5Þ v ðD ÞðqÞðxÞ ðDð9Þ ÞðqÞðxÞ v ðD ÞðqÞðxÞ 700 1400 1400

Appendix C 4 restart; 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$1); 4 4 4 final: = y[n+2]y[n2] = h*(a[0]*f[n2]+a[1]*f[n1]+a[2]*f[n]+ a[3]*f[n+1]+a[4]*f[n+2]); final : =e(v(x+2h)) e(v(x2h)) = h(a0ve(v(x2h)) + a1ve(v(xh)) + a2ve(vx) + a3ve(v(x+h)) + a4ve(v(x+2h))) 4 final: = combine(final/exp(v*x)); final : =e(vx)(e(v(x+2h)) e(v(x2h))) = e(vx)h(a0ve(v(x2h)) + a1ve(v(xh)) + a2ve(vx) + a3ve(v(x+h)) + a4ve(v(x+2h))) 4 final: = expand(final); final :¼ ðeðvhÞ Þ2

1 ðeðvhÞ Þ2

¼

ha0 v ðeðvhÞ Þ2

þ

ha1 v þ ha2 v þ ha3 veðvhÞ þ ha4 vðeðvhÞ Þ2 eðvhÞ

4 final: = simplify(convert(final,trig)); final :¼

4 coshðvhÞð2 coshðvhÞ3 þ2 coshðvhÞ2 sinhðvhÞ 2 coshðvhÞ sinhðvhÞÞ ðcoshðvhÞ þ sinhðvhÞÞ2

¼vhða0 þ a1 coshðvhÞ þ a1 sinhðvhÞ þ 2a2 coshðvhÞ2 þ 2a2 coshðvhÞ sinhðvhÞ a2 þ 4a3 coshðvhÞ3 þ 4a3 coshðvhÞ2 sinhðvhÞ 3a3 coshðvhÞ a3 sinhðvhÞ þ 8a4 coshðvhÞ4 þ 8a4 coshðvhÞ3 sinhðvhÞ 8a4 coshðvhÞ2 4a4 coshðvhÞ sinhðvhÞ þ a4 Þ=ðcoshðvhÞ þ sinhðvhÞÞ2 4 4 4 4 4 4 4 4 4 4 4 4

y[n]: = exp(v*x); f[n]: = diff(y[n],x$1); y[n+1]: = exp(v*(x+h)); f[n+1]: = diff(y[n+1],x$1); y[n1]: = exp(v*(xh)); f[n1]: = diff(y[n1],x$1); y[n+2]: = exp(v*(x+2*h)); f[n+2]: = diff(y[n+2],x$1); y[n2]: = exp(v*(x2*h)); f[n2]: = diff(y[n2],x$1);


c


4 4 4 final1: = y[n+2]y[n2] a[3]*f[n+1]+a[4]*f[n+2]);

=

h*(a[0]*f[n2]+a[1]*f[n1]+a[2]*f[n]+

final1 : =e(v(x+2h)) e(v(x2h)) = h(a0ve(v(x2h)) a1ve(v(xh)) a2ve(vx) a3ve(v(x+h)) a4ve(v(x+2h))) 4 final1: = combine(final1/exp(v*x)); final1 : =(e((x2h)v) e((x+2h)v))e(vx) = h(a0ve((x+2h)v) a1ve((x+h)v) a2ve(vx) a3ve(v(x+h)) a4ve((x2h)v))e(vx) 4 final1: = expand(final1); final1 :¼

1 ðeðvhÞ Þ2

ðeðvhÞ Þ2 ¼ ha0 vðeðvhÞ Þ2 ha1 veðvhÞ ha2 v

ha3 v ha4 v eðvhÞ ðeðvhÞ Þ2

4 final1: = simplify(convert(final1,trig)); final1 :¼

4 coshðvhÞð2 coshðvhÞ3 þ 2 coshðvhÞ2 sinhðvhÞ 2 coshðvhÞ sinhðvhÞÞ ðcoshðvhÞ þ sinhðvhÞÞ2

¼ vhð8a0 coshðvhÞ4 þ 8a0 coshðvhÞ3 sinhðvhÞ 8a0 coshðvhÞ2 4a0 coshðvhÞ sinhðvhÞ þ a0 þ 4a1 coshðvhÞ3 þ 4a1 coshðvhÞ2 sinhðvhÞ 3a1 coshðvhÞ a1 sinhðvhÞ þ 2a2 coshðvhÞ2 þ 2a2 coshðvhÞ sinhðvhÞ a2 þ a3 coshðvhÞ þ a3 sinhðvhÞ þ a4 Þ=ðcoshðvhÞ þ sinhðvhÞÞ2 4 4 eq1: = final; 4 eq2: = final1; 4 eq1i: = subs(v = I*v,eq1); 4 eq2i: = subs(v = I*v,eq2); 4 4 eq1f: = simplify(evalc(Re(eq1i))); 4 eq2f: = simplify(evalc(Im(eq1i))); 4 simplify(evalc(Re(eq2i))); 4 simplify(evalc(Im(eq2i))); 4 4 4 y[n]: = x^n; 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = (x+h)^n; 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = (xh)^n; 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = (x+2*h)^n; 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = (x2*h)^n; 4 f[n2]: = diff(y[n2],x$1); 4 4 4 final2: = y[n+2]y[n2] = +a[3]*f[n+1]+a[4]*f[n+2]);

h*(a[0]*f[n2]+a[1]*f[n1]+a[2]*f[n]

final2 :¼ðx þ 2hÞn ðx 2hÞn a0 ðx 2hÞn n a1 ðx hÞn n a2 xn n a3 ðx þ hÞn n a4 ðx þ 2hÞn n ¼h þ þ þ þ x 2h xh x xþh x þ 2h Chem. Modell., 2008, 5, 350–487 | 427 This journal is

c


4 n: = 0; n : =0 4 eq2: = simplify(final2); eq2 : =0 = 0 4 n: = 1; n : =1 4 eq2: = simplify(final2); eq2 : =4h = h(a0 + a1 + a2 + a3 + a4) 4 eq2: = simplify(eq2/h); eq2 : =4 = a0 + a1 + a2 + a3 + a4 4 eq3f: = eq2; eq3f : =4 = a0 + a1 + a2 + a3 + a4) 4 n: = 2; n : =2 4 eq3: = simplify(final2); eq3 : =8xh = 2h(a0x 2a0h + a1x a1h + a2x + a3x + a3h + a4x + 2a4h) 4 eq3: = simplify(eq32*eq2*x*h); eq3 : =0 = 4a0h2 2a1h2 + 2a3h2 + 4a4h2 4 eq3: = simplify(eq3/(h^2)); eq3 : =0 = 4a0 2a1 + 2a3 + 4a4 4 eq4f: = eq3; eq4f : =0 = 4a0 2a1 + 2a3 + 4a4 4 4 n: = 3; n : =3 4 eq4: = simplify(final2); eq4 : =12x2h + 16h3 = 3h(a0x2 4ha0x + 4a0h2 + a1x2 2ha1x + a1h2 + a2x2 + a3x2 + 2ha3x + a3h2+a4x2+4ha4x+4a4h2) 4 eq4: = simplify(eq43*eq2*x^2*h3*x*h^2*eq3); eq4 : =16h3 = 12a0h3 + 3a1h3 + 3a3h3 + 12a4h3 4 eq4: = simplify(eq4/(h^3)); eq4 : =16 = 12a0 + 3a1 + 3a3 + 12a4 4 eq5f: = eq4; eq5f : =16 = 12a0 + 3a1 + 3a3 + 12a4 4 4 4 428 | Chem. Modell., 2008, 5, 350–487 This journal is

c


solut: = solve({eq1f,eq2f,eq3f,eq4f,eq5f},{a[0],a[1],a[2],a[3],a[4]}); ( 1 3 cosðvhÞ sinðvhÞ vh þ 4vh cosðvhÞ solut : ¼ a4 ¼ ; 3 vhð1 þ cosðvhÞ2 2 cosðvhÞÞ a2 ¼

2 vh þ 2vh cosðvhÞ2 þ 8vh cosðvhÞ 9 cosðvhÞ sinðvhÞ ; 3 vhð1 þ cosðvhÞ2 2 cosðvhÞÞ

a3 ¼

4 3 cosðvhÞ sinðvhÞ þ vh þ 2vh cosðvhÞ2 ; 3 vhð1 þ cosðvhÞ2 2 cosðvhÞÞ

a0 ¼

1 3 cosðvhÞ sinðvhÞ vh þ 4vh cosðvhÞ ; 3 vhð1 þ cosðvhÞ2 2 cosðvhÞÞ

) 4 3 cosðvhÞ sinðvhÞ þ vh þ 2vh cosðvhÞ2 ; a1 ¼ 3 vhð1 þ cosðvhÞ2 2 cosðvhÞÞ 4 assign(solut); 4 h: = 1; h : =1 4 a[0]: = combine(a[0]); a0 :¼

3 sinð2vÞ þ 2v 8v cosðvÞ 9v þ 3v cosð2vÞ 12v cosðvÞ


14 8 2 1 4 1 97 139 þ v þ v þ v6 v8 v10 45 945 4725 311850 2043241200 20432412000 229 285689 v12 v14 595458864000 16631166071520000

4 a[1]: = combine(a[1]); a1 :¼

12 sinð2vÞ þ 16v þ 8v cosð2vÞ 9v þ 3v cosð2vÞ 12v cosðvÞ


64 32 2 4 4 2 97 139 v v v6 þ v8 þ v10 45 945 4725 155925 510810300 5108103000 229 285689 þ v12 þ v14 148864716000 4157791517880000

4 a[2]: = combine(a[2]); a2 :¼

4 cosð2vÞ 32v cosðvÞ þ 18 sinð2vÞ 9v þ 3v cosð2vÞ 12v cosðvÞ


8 16 2 2 4 1 97 139 þ v þ v þ v6 v8 v10 15 315 1575 51975 340540200 3405402000 229 285689 v12 v14 99243144000 2771861011920000

4 a[3]: = combine(a[3]); a3 :¼

12 sinð2vÞ þ 16v þ 8v cosð2vÞ 9v þ 3v cosð2vÞ 12v cosðvÞ Chem. Modell., 2008, 5, 350–487 | 429

This journal is

c



64 32 2 4 4 2 97 139 v v v6 þ v8 þ v10 45 945 4725 155925 510810300 5108103000 229 285689 v12 þ v14 þ 148864716000 4157791517880000

4 a[4]: = combine(a[4]); a4 :¼

3 sinð2vÞ þ 2v 8v cosðvÞ 9v þ 3v cosð2vÞ 12v cosðvÞ


14 8 2 1 4 1 97 139 þ v þ v þ v6 v8 v10 45 945 4725 311850 2043241200 20432412000 229 285689 v12 v14 595458864000 16631166071520000

4 simplify(a[4]a[0]); 0 4 simplify(a[3]a[1]); 0 4 restart; 4 qnp2: = convert(taylor(q(x+2*h),h = 0,13),polynom); 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(xh),h = 0,13),polynom); 4 qnm2: = convert(taylor(q(x2*h),h = 0,13),polynom); 4 snp2: = convert(taylor(diff(q(x+2*h),x$1),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$1),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$1),h = 0,13),polynom); 4 snm2: = convert(taylor(diff(q(x2*h),x$1),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$1); 4 a[0]: = subs(v = v*h,14/45+8/945*v^2+1/4725*v^4+1/311850*v^697/ 2043241200*v^8139/20432412000*v^10229/595458864000*v^12285689/ 16631166071520000*v^14); 4 a[1]: = subs(v = v*h,64/4532/945*v^24/4725*v^42/155925*v^6+97/ 510810300*v^8+139/5108103000*v^10+229/148864716000*v^12+285689/ 4157791517880000*v^14); 4 a[2]: = subs(v = v*h,8/15+16/315*v^2+2/1575*v^4+1/51975*v^697/ 340540200*v^8139/3405402000*v^10229/99243144000*v^12285689/ 2771861011920000*v^14); 4 a[3]: = subs(v = v*h,64/4532/945*v^24/4725*v^42/155925*v^6+97/ 510810300*v^8+139/5108103000*v^10+229/148864716000*v^12+285689/ 4157791517880000*v^14); 4 a[4]: = subs(v = v*h,14/45+8/945*v^2+1/4725*v^4+1/311850*v^697/ 2043241200*v^8139/20432412000*v^10-229/595458864000*v^12285689/ 16631166071520000*v^14); 4 4 lte: = simplify(qnp2qnm2h*(a[0]*snm2+a[1]*snm1+a[2]*sn+a[3]*snp1+ a[4]*snp2)); 17 1 v2 h13 ðDð11Þ ÞðqÞðxÞ v4 h13 ðDð9Þ ÞðqÞðxÞ 3572100 378000 31 229 v4 h17 ðDð13Þ ÞðqÞðxÞ þ v12 h19 ðDð7Þ ÞðqÞðxÞ 8573040000 3572753184000

lte :¼


c


31 1 v2 h15 ðDð13Þ ÞðqÞðxÞ v4 h11 ðDð7Þ ÞðqÞðxÞ 214326000 28350 1 v6 h11 ðDð5Þ ÞðqÞðxÞ 311850 8856359 þ v14 h27 ðDð13Þ ÞðqÞðxÞ 30175587720165888000000 139 1 4 9 ð5Þ v10 h15 ðDð5Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ þ 20432412000 4725 17 v16 h17 ðDð11Þ ÞðqÞðxÞ þ 9430344000 229 4 2 9 ð7Þ v12 h23 ðDð11Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ þ 1059216238080000 2835 8 17 23 ðDð7Þ ÞðqÞðxÞh7 ðDð9Þ ÞðqÞðxÞh9 ðDð11Þ ÞðqÞðxÞh11 945 14175 31850 1 2 11 ð9Þ 2363 v h ðD ÞðqÞðxÞ þ v10 h21 ðDð11Þ ÞðqÞðxÞ 9450 617876138880000 1649 v8 h19 ðDð11Þ ÞðqÞðxÞ þ 61787613888000 31 17 v6 h19 ðDð13Þ ÞðqÞðxÞ v4 h15 ðDð11Þ ÞðqÞðxÞ 565820640000 142884000 1 1 v6 h15 ðDð9Þ ÞðqÞðxÞ v6 h13 ðDð7Þ ÞðqÞðxÞ 24948000 1871100 8 2 7 ð5Þ 299 v h ðD ÞðqÞðxÞ ðDð13Þ ÞðqÞðxÞh13 945 56133000 97 97 v8 h15 ðDð5Þ ÞðqÞðxÞ þ v8 h15 ðDð7Þ ÞðqÞðxÞ þ 2043241200 12259447200 97 v8 h17 ðDð9Þ ÞðqÞðxÞ þ 163459296000 3007 þ v8 h21 ðDð13Þ ÞðqÞðxÞ 3707256833280000 139 139 v10 h17 ðDð7Þ ÞðqÞðxÞ v10 h19 ðDð9Þ ÞðqÞðxÞ þ 122594472000 1634592960000 4309 v10 h23 ðDð13Þ ÞðqÞðxÞ þ 37072568332800000 229 þ v12 h17 ðDð5Þ ÞðqÞðxÞ 595458864000 229 þ v12 h21 ðDð9Þ ÞðqÞðxÞ 47636709120000 7099 v12 h25 ðDð13Þ ÞðqÞðxÞ þ 1080400562841600000 285689 þ v14 h19 ðDð5Þ ÞðqÞðxÞ 1663116071520000 285689 þ v14 h21 ðDð7Þ ÞðqÞðxÞ 99786996429120000 285689 þ v14 h23 ðDð9Þ ÞðqÞðxÞ 1330493285721600000 285689 v14 h25 ðDð11Þ ÞðqÞðxÞ þ 29583909529574400000


c


4 coeff(lte,h,7);

8 8 2 ð5Þ ðDð7Þ ÞðqÞðxÞ v ðD ÞðqÞðxÞ 945 945

Appendix D 4 restart; 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$1); 4 4 final: = y[n+2]y[n2] +a[3]*f[n+1]+a[4]*f[n+2]);

=

h*(a[0]*f[n2]+a[1]*f[n1]+a[2]*f[n]

final : =e(v(x+ 2h)) e(v(x2h)) = h(a0ve(v(x2h)) + a1ve(v(xh)) + a2ve(vx) + a3ve(v(x+h)) + a4ve(v(x+2h))) 4 final: = combine(final/exp(v*x)); final : =e(vx)(e(v(x+2h)) e(v(x2h))) = e(vx)h(a0ve(v(x2h)) + a1ve(v(xh)) + a2ve(vx) + a3ve(v(x+h)) + a4ve(v(x+2h))) 4 final: = expand(final); final :¼ ðeðvhÞ Þ2

1 ðeðvhÞ Þ2

¼

ha0 v ðeðvhÞ Þ2

þ

ha1 v þ ha2 v þ ha3 veðvhÞ þ ha4 vðeðvhÞ Þ2 eðvhÞ


4 coshðvhÞð2 coshðvhÞ3 þ 2 coshðvhÞ2 sinhðvhÞ 2 coshðvhÞ sinhðvhÞÞ ðcoshðvhÞ þ sinhðvhÞÞ2 ¼ vhða0 þ a1 coshðvhÞ þ a1 sinhðvhÞ þ 2a2 coshðvhÞ2 þ 2a2 coshðvhÞ sinhðvhÞ a2 þ 4a3 coshðvhÞ3 þ 4a3 coshðvhÞ2 sinhðvhÞ 3a3 coshðvhÞ a3 sinhðvhÞ þ 8a4 coshðvhÞ4 þ 8a4 coshðvhÞ3 sinhðvhÞ 8a4 coshðvhÞ2 4a4 coshðvhÞ sinhðvhÞ þ a4 Þ=ðcoshðvhÞ þ sinhðvhÞÞ2

4 4 4 4 4 4 4 4 4 4 4 4 4

y[n]: = exp(v*x); f[n]: = diff(y[n],x$1); y[n+1]: = exp(v*(x+h)); f[n+1]: = diff(y[n+1],x$1); y[n1]: = exp(v*(xh)); f[n1]: = diff(y[n1],x$1); y[n+2]: = exp(v*(x+2*h)); f[n+2]: = diff(y[n+2],x$1); y[n2]: = exp(v*(x2*h)); f[n2]: = diff(y[n2],x$1);


c


4 final1: = y[n+2]y[n2] = h*(a[0]*f[n2]+a[1]*f[n1]+ a[2]*f[n]+ a[3]*f[n+1]+a[4]*f[n+2]); final1 : =e(v(x+2h)) e(v(x2h)) = h(a0ve(v(x2h)) a1ve(v(xh)) a2ve(vx) a3ve(v(x+h)) a4ve(v(x+2h))) 4 final1: = combine(final1/exp(v*x)); final1 : =(e((x2h)v) e((x+2h)v))e(vx) = h(a0ve((x+2h)v) a1ve((x+h)v) a2ve(vx) a3ve(v(x+h)) a4ve((x2h)v))e(vx) 4 final1: = expand(final1); final1 :¼

1 ðeðvhÞ Þ2

ðeðvhÞ Þ2 ¼ ha0 vðeðvhÞ Þ2 ha1 veðvhÞ ha2 v

ha3 v ha4 v þ eðvhÞ ðeðvhÞ Þ2

4 final1: = simplify(convert(final1,trig)); final1 :¼

4 coshðvhÞð2 coshðvhÞ3 þ 2 coshðvhÞ2 sinhðvhÞ 2 coshðvhÞ sinhðvhÞÞ ðcoshðvhÞ þ sinhðvhÞÞ2 4

¼ vhð8a0 coshðvhÞ þ 8a0 coshðvhÞ3 sinhðvhÞ 8a0 coshðvhÞ2 4a0 coshðvhÞ sinhðvhÞ þ a0 þ 4a1 coshðvhÞ3 þ 4a1 coshðvhÞ2 sinhðvhÞ 3a1 coshðvhÞ a1 sinhðvhÞ þ 2a2 coshðvhÞ2 þ 8a2 coshðvhÞ3 sinhðvhÞ a2 þ a3 coshðvhÞ2 a3 sinhðvhÞ þ a4 Þ=ðcoshðvhÞ þ sinhðvhÞÞ2 4 4 eq1: = final; 4 eq2: = final1; 4 eq1i: = subs(v = I*v,eq1); 4 eq2i: = subs(v = I*v,eq2); 4 4 eq1f: = simplify(evalc(Re(eq1i))); 4 eq2f: = simplify(evalc(Im(eq1i))); 4 simplify(evalc(Re(eq2i))); 4 simplify(evalc(Im(eq2i))); 4 simplify(eq1fsimplify(evalc(Re(eq2i)))); 4 simplify(eq2f+simplify(evalc(Im(eq2i)))); 4 4 4 4 y[n]: = x*exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = (x+h)*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = (xh)*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = (x+2*h)*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = (x2*h)*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$1); 4 4 4 final2: = y[n+2]y[n2] = h*(a[0]*f[n2]+a[1]*f[n1]+ a[2]*f[n] +a[3]*f[n+1]+a[4]*f[n+2]); final2 : =(x + 2h)e(v(x+2h)) (x 2h)e(v(x2h)) = h(a0(e(v(x2h)) + (x 2h)ve(v(x2h))) + a1(e(v(xh)) + (x h)ve(v(xh))) + a2(e(vx) + xve(vx)) + a3(e(v(x+h)) + (x + h)ve(v(x+h))) + a4(e(v(x+2h)) + (x + 2h)ve(v(x+2h)))) Chem. Modell., 2008, 5, 350–487 | 433 This journal is

c


4 final2: = combine(final2/exp(v*x)); final2 : =((x + 2h)e(v(x+2h)) + (x + 2h)e(v(x2h)))e(vx) = e(vx)h(a0(e(v(x2h)) + (x 2h)ve(v(x2h))) + a1(e(v(xh)) + (x h)ve(v(xh))) + a2(e(vx) + xve(vx)) + a3(e(v(x+h)) + (x + h)ve(v(x+h))) + a4(e(v(x+2h)) + (x + 2h)ve(v(x+2h)))) 4 final2: = expand(final2); final2 :¼ðeðvhÞ Þ2 x þ 2ðeðvhÞ Þ2 h 2

2h a0 v ðeðvhÞ Þ2

x ðeðvhÞ Þ2

þ

2h ðeðvhÞ Þ2

¼

ha0 ðeðvhÞ Þ2

þ

ha0 vx ðeðvhÞ Þ2

2

þ

ha1 ha1 vx h a1 v þ ðvhÞ ðvhÞ þ ha2 þ ha2 xv þ ha3 eðvhÞ e e eðvhÞ

þ ha3 veðvhÞ x þ h2 a3 veðvhÞ þ ha4 ðeðvhÞ Þ2 þ ha4 vðeðvhÞ Þ2 x þ 2h2 a4 vðeðvhÞ Þ2 4 final2: = simplify(convert(final2,trig)); final2 : =4(2x cosh(vh)4 + 2x cosh(vh)3 sinh(vh) 2x cosh(vh)2 x cosh(vh) sinh(vh) + 4h cosh(vh)4 + 4h cosh(vh)3 sinh(vh) 4h cosh(vh)2 2h cosh(vh) sinh(vh) + h)/(cosh(vh) + sinh(vh))2 = h(8a4 cosh(vh)3 sinh(vh) + a0 a2 + a4 4a4 cosh(vh) sinh(vh) +2a2 cosh(vh) sinh(vh) 4a4vx cosh(vh) sinh(vh) + 2a2xv cosh(vh) sinh(vh) + 8a4vx cosh(vh)3 sinh(vh) +4a3 cosh(vh)2 sinh(vh) + a1 cosh(vh) + a1 sinh(vh) + 2a2 cosh(vh)2+ 4a3 cosh(vh)3 + 8a4 cosh(vh)4 3a3 cosh(vh) a3 sinh(vh) 8a4 cosh(vh)2 + a0vx a2xv + a4vx + 2ha4v 2ha0v + a1vx cosh(vh) + a1vx sinh(vh) ha1v cosh(vh) ha1v sinh(vh) + 2a2xv cosh(vh)2 + 4a3vx cosh(vh)3 + 4ha3v cosh(vh)3 + 8a4vx cosh(vh)4 + 16ha4v cosh(vh)4 + 4a3vx cosh(vh)2 sinh(vh) + 4ha3v cosh(vh)2 sinh(vh) + 16ha4v cosh(vh)3 sinh(vh) 3a3vx cosh(vh) a3vx sinh(vh) ha3v sinh(vh) 16ha4v cosh(vh)2 8a4vx cosh(vh)2 3ha3v cosh(vh) 8ha4v cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 4 4 y[n]: = x*exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = (x+h)*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = (xh)*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = (x+2*h)*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = (x2*h)*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$1); 4 4 final3: = y[n+2]y[n2] = h*(a[0]*f[n2]+a[1]*f[n1]+ a[2]*f[n]+ a[3]*f[n+1]+a[4]*f[n+2]); final3 : =(x + 2h)e(v(x+2h)) (x 2h)e(v(x2h)) = h(a0(e(v(x2h)) (x 2h)ve(v(x2h))) + a1(e(v(xh)) (x h)ve(v(xh))) + a2(e(vx) xve(vx)) + a3(e(v(x+h)) (x + h)ve(v(x+h))) + a4(e(v(x+2h)) (x + 2h)ve(v(x+2h)))) 4 final3: = combine(final3/exp(v*x)); final3 : =((x + 2h)e((x2h)v) + (x + 2h)e((x+2h)v))e(vx) = h(a3(e(v(x+h)) (x + h)ve(v(x+h))) + a1((x + h)ve((x+h)v) + e((x+h)v)) + a4((x 2h)ve((x2h)v) + e((x2h)v)) a0((x + 2h)ve((x+2h)v) + e((x+2h)v)) + a2(e(vx) xve(vx))e(vx)) 434 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 final3: = expand(final3); final3 :¼

x ðeðvhÞ Þ2 þ

þ

2h ðeðvhÞ Þ2

ðeðvhÞ Þ2 x þ 2ðeðvhÞ Þ2 h ¼

ha3 ha3 vx ðvhÞ e eðvhÞ

h2 a3 v ha4 vx 2h2 a4 v ha1 veðvhÞ x þ h2 a1 veðvhÞ þ ha1 eðvhÞ 2 ðvhÞ e ðeðvhÞ Þ ðeðvhÞ Þ2 ha4 ðeðvhÞ Þ2

ha0 vðeðvhÞ Þ2 x þ 2h2 a0 vðeðvhÞ Þ2 þ ha0 ðeðvhÞ Þ2 þ ha2 ha2 xv

4 final3: = simplify(convert(final3,trig)); final3 : =4(h + 2x cosh(vh)4 + 2x cosh(vh)3 sinh(vh) 2x cosh(vh)2 x cosh(vh) sinh(vh) 4h cosh(vh)4 4h cosh(vh)3 sinh(vh) +4h cosh(vh)2 + 2h cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 = h(8a0 cosh(vh)3 sinh(vh) a0 + a2 a4 2a2 cosh(vh) sinh(vh) 4a1 cosh(vh)2 sinh(vh) + 2a2xv cosh(vh) sinh(vh) + 3a1 cosh(vh) + a1 sinh(vh) 2a2 cosh(vh)2 a3 cosh(vh) a3 sinh(vh) + 16ha0v cosh(vh)2 8a0vx cosh(vh)2 + 4a1vx cosh(vh)3 4ha1v cosh(vh)3 + 8a0vx cosh(vh)4 8a0 cosh(vh)4 4a1 cosh(vh)3 + 8ha0v cosh(vh) sinh(vh) 4a0vx cosh(vh) sinh(vh) + 4a0 cosh(vh) sinh(vh) + 8a0 cosh(vh)2 16ha0v cosh(vh)3 sinh(vh) 4ha1v cosh(vh)2 sinh(vh) + 8a0vx cosh(vh)3 sinh(vh) 16ha0v cosh(vh)4 + 4a1vx cosh(vh)2 sinh(vh) + a0vx a2xv + a4vx + 2ha4v 2ha0v 3a1vx cosh(vh) a1vx sinh(vh) + 3ha1v cosh(vh) + ha1v sinh(vh) + 2a2xv cosh(vh)2 + a3vx cosh(vh) + a3vx sinh(vh) + ha3v sinh(vh) + ha3v cosh(vh))/(cosh(vh) + sinh(vh))2 4 4 eq3: = final2; 4 eq4: = final3; 4 eq3i: = subs(v = I*v,eq3); 4 eq4i: = subs(v = I*v,eq4); 4 4 eq3f: = simplify(evalc(Re(eq3i))); 4 eq4f: = simplify(evalc(Im(eq3i))); 4 simplify(evalc(Re(eq4i))); 4 simplify(evalc(Im(eq4i))); 4 simplify(eq3fsimplify(evalc(Re(eq4i)))); 4 simplify(eq4f+simplify(evalc(Im(eq4i)))); 4 4 4 y[n]: = x^n; 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = (x+h)^n; 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = (xh)^n; 4 f[n1]: = diff(y[n1],x$1); 4 y[n+2]: = (x+2*h)^n; 4 f[n+2]: = diff(y[n+2],x$1); 4 y[n2]: = (x2*h)^n; 4 f[n2]: = diff(y[n2],x$1); 4 4 final5: = y[n+2]y[n2] = h*(a[0]*f[n2]+a[1]*f[n1]+a[2]*f[n]+ a[3]*f[n+1]+a[4]*f[n+2]); final5 :¼ðx þ 2hÞn ðx 2hÞn a0 ðx 2hÞn n a1 ðx hÞn n a2 xn n a3 ðx þ hÞn n a4 ðx þ 2hÞn n ¼h þ þ þ þ x 2h xh x xþh x þ 2h 4 n: = 0; Chem. Modell., 2008, 5, 350–487 | 435 This journal is

c


n : =0 4 eq5: = simplify(final5); eq5 : =0 = 0 4 n: = 1; n : =1 4 eq5: = simplify(final5); eq5 : =4h = h(a0 + a1 + a2 + a3 + a4) 4 eq5: = simplify(eq5/h); eq5 : =4 = a0 + a1 + a2 + a3 + a4 4 eq5f: = eq5; eq5f : =4 = a0 + a1 + a2 + a3 + a4 4 4 4 solut: = solve({eq1f,eq2f,eq3f,eq4f,eq5f},{a[0],a[1],a[2],a[3],a[4]}); solut : ={a4 = (hv cos(vh)3 + cos(vh)2 sin(vh) + 2hv cos(vh)2 cos(vh) sin(vh) vh + h2v2 sin(vh))/(h2v2 sin(vh)(cos(vh)2 2 cos(vh) + 1)), a3 = 2(cos(vh)3 sin(vh) + hv cos(vh)2+ 2cos(vh)h2v2 sin(vh) cos(vh) sin(vh) vh)/(h2v2 sin(vh)(cos(vh)2 2 cos(vh) + 1)), a2 = 2(vh + hv cos(vh)3 cos(vh)2 sin(vh) cos(vh) sin(vh) + h2v2 sin(vh) + 2h2v2 cos(vh)2 sin(vh) + 2 cos(vh)3 sin(vh))/(h2v2 sin(vh) (cos(vh)2 2 cos(vh) + 1)), a1 = 2(cos(vh)3 sin(vh) + hv cos(vh)2 + 2 cos(vh) h2v2 sin(vh) cos(vh) sin(vh) vh)/(h2v2 sin(vh)(cos(vh)2 2 cos(vh) + 1, a0 = (hv cos(vh)3 + cos(vh)2 sin(vh) + 2hv cos(vh)2 cos(vh) sin(vh) sin(vh) vh + h2v2 sin(vh))/ (h2v2 sin(vh) (cos(vh)2 2 cos(vh) + 1))} 4 assign(solut); 4 h: = 1; h : =1 4 a[0]: = combine(a[0]); a0 :¼

v cosð3vÞ þ 3v cosðvÞ sinð3vÞ sinðvÞ 4v cosð2vÞ þ 2 sinð2vÞ 4v2 sinðvÞ v2 sinð3vÞ 5v2 sinðvÞ þ 4v2 sinð2vÞ


14 16 2 19 4 37 114731 þ v þ v þ v6 þ v8 45 945 14175 311850 10216206000 67651 18413 þ v10 þ v12 61297236000 166728481920

4 a[1]: = combine(a[1]); a1 :¼

sinð4vÞ 2 sinð2vÞ þ 4v cosð2vÞ 4v þ 8v2 sinð2vÞ v2 sinð3vÞ 5v2 sinðvÞ þ 4v2 sinð2vÞ


64 64 2 44 4 8 71 317 v þ v v6 v8 v10 45 945 14175 155925 2554051500 7662154500 367 v12 208410602400

4 a[2]: = combine(a[2]); a2 : =(8v 2v cos(3v) 6v cos(v) + 2 sin(3v) + 2 sin(v) 12v2 sin(v) 4v2 sin(3v) 2 sin(4v))/(v2 sin(3v) 5v2 sin(v) + 4v2 sin(2v)) 436 | Chem. Modell., 2008, 5, 350–487 This journal is

c



8 32 2 2 4 1 6 38149 1447 þ v v v v8 v10 15 315 225 7425 1702701000 681080400 2323 v12 10687723200

4 a[3]: = combine(a[3]); a3 :¼

sinð4vÞ 2 sinð2vÞ þ 4v cosð2vÞ 4v þ 8v2 sinð2vÞ v2 sinð3vÞ 5v2 sinðvÞ þ 4v2 sinð2vÞ


64 64 2 44 4 8 71 317 v þ v v6 v8 v10 45 945 14175 155925 2554051500 7662154500 367 v12 208410602400

4 a[4]: = combine(a[4]); a4 :¼

v cosð3vÞ þ 3v cosðvÞ sinð3vÞ sinðvÞ 4v cosð2vÞ þ 2 sinð2vÞ 4v2 sinðvÞ v2 sinð3vÞ 5v2 sinðvÞ þ 4v2 sinð2vÞ


14 16 2 19 4 37 114731 67651 þ v þ v þ v6 þ v8 þ v10 45 945 14175 311850 10216206000 61297236000 18413 v12 þ 166728481920

4 simplify(a[4]a[0]); 0 4 simplify(a[3]a[1]); 0 4 4 4 4 restart; 4 qnp2: = convert(taylor(q(x+2*h),h = 0,13),polynom); 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(xh),h = 0,13),polynom); 4 qnm2: = convert(taylor(q(x2*h),h = 0,13),polynom); 4 snp2: = convert(taylor(diff(q(x+2*h),x$1),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$1),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$1),h = 0,13),polynom); 4 snm2: = convert(taylor(diff(q(x2*h),x$1),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$1); 4 4 4 a[0]: = subs(v = v*h,14/45+16/945*v^2+19/14175*v^4+37/311850*v^6+ 114731/10216206000*v^8+67651/61297236000*v^10+18413/166728481920*v^12); 4 a[1]: = subs(v = v*h,64/4564/945*v^2+44/14175*v^48/155925*v^671/ 2554051500*v^8317/7662154500*v^10367/208410602400*v^12); 4 a[2]: = subs(v = v*h,8/15+32/315*v^22/225*v^41/7425*v^638149/ 1702701000*v^81447/681080400*v^102323/10687723200*v^12); 4 Chem. Modell., 2008, 5, 350–487 | 437 This journal is

c


4 lte: = simplify(qnp2qnm2h*(a[0]*(snp2+snm2)+a[1]*(snp1+snm1) + a[2]*sn));

13 8 4 7 ð3Þ v4 h15 ðDð11Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 17146080 945 31 17 v2 h15 ðDð13Þ ÞðqÞðxÞ v2 h15 ðDð11Þ ÞðqÞðxÞ 107163000 1786050 1 2 11 ð9Þ 22339 v h ðD ÞðqÞðxÞ v10 h13 ðDð3Þ ÞðqÞðxÞ 4725 5108103000 103 40877 v4 h17 ðDð13Þ ÞðqÞðxÞ v12 h17 ðDð5Þ ÞðqÞðxÞ 449064000 277880803200 1541 29 4 9 ð5Þ v10 h17 ðDð7Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 7858620000 14175 16 2 7 ð5Þ 490891 v h ðD ÞðqÞðxÞ v12 h19 ðDð7Þ ÞðqÞðxÞ 945 25009272288000 8 17 23 ðDð7Þ ÞðqÞðxÞh7 ðDð9Þ ÞðqÞðxÞh9 ðDð11Þ ÞðqÞðxÞh11 945 14175 311850 8 1 4 11 ð7Þ v6 h11 ðDð5Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 51975 4050 2 6 9 ð3Þ 8 2 9 ð7Þ v h ðD ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 4725 2835 263 409 v6 h17 ðDð11Þ ÞðqÞðxÞ v4 h13 ðDð9Þ ÞðqÞðxÞ 3929310000 23814000 7 197 v6 h13 ðDð7Þ ÞðqÞðxÞ v6 h15 ðDð9Þ ÞðqÞðxÞ 334125 130977000 41 1 v6 h19 ðDð13Þ ÞðqÞðxÞ v8 h11 ðDð3Þ ÞðqÞðxÞ 20207880000 22275 152951 89 v8 h13 ðDð5Þ ÞðqÞðxÞ v8 h15 ðDð7Þ ÞðqÞðxÞ 10216206000 44579808 116551 v8 h17 ðDð9Þ ÞðqÞðxÞ 817296480000 1958071 v8 h19 ðDð11Þ ÞðqÞðxÞ 308938069440000 39161491 v8 h21 ðDð13Þ ÞðqÞðxÞ 203899125830400000 8999 144301 v10 h15 ðDð5Þ ÞðqÞðxÞ v10 h19 ðDð9Þ ÞðqÞðxÞ 6129723600 10297935648000 2886337 v10 h21 ðDð11Þ ÞðqÞðxÞ 4634071041600000 769711 v10 h23 ðDð13Þ ÞðqÞðxÞ 40779825166080000 899 v12 h15 ðDð3Þ ÞðqÞðxÞ 2043241200 1963931 v12 h21 ðDð9Þ ÞðqÞðxÞ 1400519248128000 51347 v12 h25 ðDð13Þ ÞðqÞðxÞ 823834851840000 2417287 v12 h25 ðDð13Þ ÞðqÞðxÞ 1279859128289280000 299 ðDð13Þ ÞðqÞðxÞh13 56133000

Ite :¼


c


4 coeff(lte,h,7);

8 4 ð3Þ 16 2 ð5Þ 8 v ðD ÞðqÞðxÞ v ðD ÞðqÞðxÞ ðDð7Þ ÞðqÞðxÞ 945 945 945

Appendix E 4 restart; 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 4 final: = y[n+1]y[n1] = h*(a[0]*f[n1]+a[1]*f[n]+a[2]*f[n+1]); final : =e(v(x+h)) e(v(xh)) = h(a0ve(v(xh))+a1ve(vx) + a2ve(v(x+h))) 4 final: = combine(final/exp(v*x)); final : =e(vx)(e(v(x+h)) e(v(xh))) =e(vx) h(a0ve(v(xh))+a1ve(vx) + a2ve(v(x+h))) 4 final: = expand(final); final :¼ eðvhÞ

1 eðvhÞ

¼

ha0 v þ ha1 v þ ha2 veðvhÞ eðvhÞ


2ðcoshðvhÞ2 þ coshðvhÞ sinhðvhÞ 1Þ coshðvhÞ þ sinhðvhÞ ¼

4 4 4 4 4 4 4 4 4 4

vhða0 þ a1 coshðvhÞ þ a1 sinhðvhÞ þ 2a2 coshðvhÞ2 þ 2a2 coshðvhÞ sinhðvhÞ a2 Þ coshðvhÞ þ sinhðvhÞ

y[n]: = exp(v*x); f[n]: = diff(y[n],x$1); y[n+1]: = exp(v*(x+h)); f[n+1]: = diff(y[n+1],x$1); y[n1]: = exp(v*(xh)); f[n1]: = diff(y[n1],x$1); final1: = y[n+1]y[n1] = h*(a[0]*f[n1]+a[1]*f[n]+a[2]*f[n+1]);

final1 : =e(v(x+h)) e(v(xh)) = h(a0ve(v(xh))+a1ve(vx) a2ve(v(x+h))) 4 final1: = combine(final1/exp(v*x)); final1 : =e(vx)(e(v(x+h)) e(v(xh))) = e(vx)h(a0ve(v(xh))a1ve(vx) a2ve(v(x+h))) 4 final1: = expand(final1); final1 :¼

1 ha2 v eðvhÞ ¼ ha0 veðvhÞ ha1 v þ ðvhÞ eðvhÞ e

4 final1: = simplify(convert(final1,trig)); final1 : ¼ ¼

2ðcoshðvhÞ2 þ coshðvhÞ sinhðvhÞ 1Þ coshðvhÞ þ sinhðvhÞ vhð2a0 coshðvhÞ2 þ 2a0 coshðvhÞ sinðvhÞ a0 þ a1 coshðvhÞ þ a1 coshðvhÞ þ a1 sinhðvhÞ þ a2 Þ coshðvhÞ þ sinhðvhÞ


c


4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

eq1: = final; eq2: = final1; eq1i: = subs(v = I*v,eq1); eq2i: = subs(v = I*v,eq2); eq1f: = simplify(evalc(Re(eq1i))); eq2f: = simplify(evalc(Im(eq1i))); simplify(evalc(Re(eq2i))); simplify(evalc(Im(eq2i))); simplify(eq1fsimplify(evalc(Re(eq2i)))); simplify(eq2f+simplify(evalc(Im(eq2i))));

y[n]: = x^n; f[n]: = diff(y[n],x$1); y[n+1]: = (x+h)^n; f[n+1]: = diff(y[n+1],x$1); y[n1]: = (xh)^n; f[n1]: = diff(y[n1],x$1); final5: = y[n+1]y[n1] = h*(a[0]*f[n1]+a[1]*f[n]+a[2]*f[n+1]); n: = 0; eq5: = simplify(final5); n: = 1; eq5: = simplify(final5); eq5: = simplify(eq5/h); eq5f: = eq5; solut: = solve({eq1f,eq2f,eq5f},{a[0],a[1],a[2]}); sinðvhÞ þ vh 2ðvh cosðvhÞ sinðvhÞÞ solut :¼ a2 ¼ ; a1 ¼ ; vhðcosðvhÞ 1Þ vhðcosðvhÞ 1Þ sinðvhÞ þ vh a0 ¼ vhðcosðvhÞ 1Þ

4 assign(solut); 4 h: = 1; h : =1 4 a[0]: = combine(a[0]); a0 :¼

sinðvÞ v v cosðvÞ v

4 a0t: = convert(taylor(a[0],v = 0,20),polynom); 1 1 1 4 1 1 691 1 a0t :¼ þ v2 þ v þ v6 þ v8 þ v10 þ v12 3 90 2520 75600 2395008 54486432000 2668723200 3617 43867 þ v14 þ v16 333456963840000 141919283810304000 4 a[1]: = combine(a[1]); a1 :¼

2v cosðvÞ 2 sinðvÞ v cosðvÞ v


c


4 a1t: = convert(taylor(a[1],v = 0,20),polynom); 4 1 1 4 1 1 691 1 a1t :¼ v2 v v6 v8 v10 v12 3 45 1260 37800 1197504 27243216000 1334361600 3617 43867 v14 v16 166728481920000 70959641905152000

4 a[2]: = combine(a[2]); a2 :¼

sinðvÞ v v cosðvÞ v

4 a2t: = convert(taylor(a[2],v = 0,20),polynom); 1 1 1 4 1 1 691 1 a2t :¼ þ v2 þ v þ v6 þ v8 þ v10 þ v12 3 90 2520 75600 2395008 54486432000 2668723200 3617 43867 þ v14 þ v16 333456963840000 141919283810304000

4 simplify(a[2]a[0]); 0 4 4 restart; 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(xh),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$1),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$1),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$1); 4 4 a[0]: = subs(v = v*h,1/3+1/90*v^2+1/2520*v^4+1/75600*v^6+1/ 2395008*v^8+691/54486432000*v^10+1/2668723200*v^12+3617/ 333456963840000*v^14+43867/141919283810304000*v^16); 4 a[1]: = subs(v = v*h,4/31/45*v^21/1260*v^41/37800*v^61/ 1197504*v^8691/27243216000*v^101/1334361600*v^123617/ 166728481920000*v^1443867/70959641905152000*v^16); 4 4 lte: = simplify(qnp1qnm1h*(a[0]*(snp1+snm1) + a[1]*sn)); Ite :¼

43867 v16 h29 ðDð13Þ ÞðqÞðxÞ 33989782007994856243200000 691 1 v10 h21 ðDð11Þ ÞðqÞðxÞ v6 h13 ðDð7Þ ÞðqÞðxÞ 98860182220800000 27216000 691 1 2 7 ð5Þ v10 h15 ðDð5Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 653837184000 1080 1 4 7 ð3Þ 3617 v h ðD ÞðqÞðxÞ v14 h27 ðDð13Þ ÞðqÞðxÞ 2520 79863209605251072000000 1 1 v12 h15 ðDð3Þ ÞðqÞðxÞ v6 h17 ðDð11Þ ÞðqÞðxÞ 2668723200 137168640000 43867 v16 h23 ðDð7Þ ÞðqÞðxÞ 51090942171709440000 1 1 v8 h19 ðDð11Þ ÞðqÞðxÞ ðDð5Þ ÞðqÞðxÞh5 4345502515200 90 1 1 1 ðDð7Þ ÞðqÞðxÞh7 ðDð9Þ ÞðqÞðxÞh9 ðDð11Þ ÞðqÞðxÞh11 1890 90720 7484400 Chem. Modell., 2008, 5, 350–487 | 441 This journal is

c


1 1 v8 h13 ðDð5Þ ÞðqÞðxÞ ðDð13Þ ÞðqÞðxÞh13 28740096 718502400 1 1 v6 h19 ðDð13Þ ÞðqÞðxÞ v8 h11 ðDð3Þ ÞðqÞðxÞ 18106260480000 2395008 1 1 v8 h15 ðDð7Þ ÞðqÞðxÞ v8 h17 ðDð9Þ ÞðqÞðxÞ 862202880 48283361280 1 691 v8 h21 ðDð13Þ ÞðqÞðxÞ v10 h13 ðDð3Þ ÞðqÞðxÞ 573606332006400 5448643200 691 691 v10 h17 ðDð7Þ ÞðqÞðxÞ v10 h19 ðDð9Þ ÞðqÞðxÞ 19615115520000 1098446469120000 691 v10 h23 ðDð13Þ ÞðqÞðxÞ 13049544053145600000 1 1 v12 h17 ðDð5Þ ÞðqÞðxÞ v12 h19 ðDð7Þ ÞðqÞðxÞ 32024678400 960740352000 1 1 v12 h21 ðDð9Þ ÞðqÞðxÞ v12 h23 ðDð11Þ ÞðqÞðxÞ 53801459712000 4842131374080000 1 v12 h25 ðDð13Þ ÞðqÞðxÞ 639161341378560000 3617 v14 h19 ðDð5 ÞðqÞðxÞ 4001483566080000 3617 v14 h21 ðDð7Þ ÞðqÞðxÞ 120044506982400000 3617 v14 h23 ðDð9Þ ÞðqÞðxÞ 6722492391014400000 3617 v14 h25 ðDð11Þ ÞðqÞðxÞ 605024315191296000000 43867 v16 h19 ðDð3Þ ÞðqÞðxÞ 141919283810304000 43867 v16 h21 ðDð5Þ ÞðqÞðxÞ 1703031405723648000 43867 v16 h25 ðDð9Þ ÞðqÞðxÞ 2861092761615728640000 43867 1 v16 h27 ðDð11Þ ÞðqÞðxÞ v2 h5 ðDð3Þ ÞðqÞðxÞ 2574983485454155776000000 90 1 1 2 9 ð7Þ 2 13 ð11Þ v h ðD ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 32400 163296000 1 1 v2 h15 ðDð13Þ ÞðqÞðxÞ v4 h9 ðDð5Þ ÞðqÞðxÞ 21555072000 30240 1 1 v4 h11 ðDð7Þ ÞðqÞðxÞ v4 h13 ðDð9Þ ÞðqÞðxÞ 907200 50803200 1 1 v4 h15 ðDð11Þ ÞðqÞðxÞ v4 h17 ðDð13Þ ÞðqÞðxÞ 4572288000 603542016000 1 1 v2 h11 ðDð9Þ ÞðqÞðxÞ v6 h11 ðDð5Þ ÞðqÞðxÞ 1814400 907200 1 1 v6 h9 ðDð3Þ ÞðqÞðxÞ v6 h15 ðDð9Þ ÞðqÞðxÞ 75600 1524096000 3617 v14 h17 ðDð3Þ ÞðqÞðxÞ 333456963840000


c


4 coeff(lte,h,5);

1 1 ðDð5Þ ÞðqÞðxÞ v2 ðDð3Þ ÞðqÞðxÞ 90 90

Appendix F 4 restart; 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$1); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$1); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$1); 4 4 final: = y[n+1]y[n1] = h*(a[0]*f[n1]+a[1]*f[n]+a[2]*f[n+1]); final : =e(v(x+h)) e(v(xh)) = h(a0ve(v(xh)) + a1ve(vx) + a2ve(v(x+h))) 4 final: = combine(final/exp(v*x)); final : =e(vx)(e(v(x+h)) e(v(xh))) = e(vx)h(a0ve(v(xh)) + a1ve(vx) + a2ve(v(x+h))) 4 final: = expand(final); final :¼ ðeðvhÞ Þ

1 ha0 v ¼ þ ha1 v þ ha2 veðvhÞ eðvhÞ eðvhÞ


2ðcoshðvhÞ2 þ coshðvhÞ sinhðvhÞ 1Þ ¼ vhða0 þ a1 coshðvhÞ coshðvhÞ þ sinhðvhÞ þ a1 sinhðvhÞ þ 2a2 coshðvhÞ2 þ 2a2 coshðvhÞ sinhðvhÞ a2 Þ=ðcoshðvhÞ þ sinhðvhÞÞ

4 4 4 4 4 4 4 4 4

y[n]: = exp(v*x); f[n]: = diff(y[n],x$1); y[n+1]: = exp(v*(x+h)); f[n+1]: = diff(y[n+1],x$1); y[n1]: = exp(v*(xh)); f[n1]: = diff(y[n1],x$1); final1: = y[n+1]y[n1] = h*(a[0]*f[n1]+a[1]*f[n]+a[2]*f[n+1]);

final1 : =e(v(x+h)) e(v(xh)) = h(a0ve(v(xh)) a1ve(vx) a2ve(v(x+h))) 4 final1: = combine(final1/exp(v*x)); final1 : =(e((xh)v)e((x+h)v))e(vx)=h(a0ve((x+h)v) a1ve(vx) a2ve((xh)v))e(vx) 4 final1: = expand(final1); final1 :¼

1 eðvhÞ

eðvhÞ ¼ ha0 veðvhÞ ha1 v

ha2 v eðvhÞ

4 final1: = simplify(convert(final1,trig)); 2ðcoshðvhÞ2 þ coshðvhÞ sinhðvhÞ 1Þ ¼ vhð2a0 coshðvhÞ2 coshðvhÞ þ sinhðvhÞ þ 2a0 coshðvhÞ sinhðvhÞ a0 þ a1 coshðvhÞ þ a1 sinhðvhÞ þ a2 Þ=ðcoshðvhÞ þ sinhðvhÞÞ

final1 :¼

4 4 eq1: = final; 4 eq2: = final1; Chem. Modell., 2008, 5, 350–487 | 443 This journal is

c


4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

eq1i: = subs(v = I*v,eq1); eq2i: = subs(v = I*v,eq2); eq1f: = simplify(evalc(Re(eq1i))); eq2f: = simplify(evalc(Im(eq1i))); eq3f: = simplify(evalc(Re(eq2i))); eq4f: = simplify(evalc(Im(eq2i))); simplify(eq1feq3f); simplify(eq2f+eq4f); y[n]: = x* exp (v*x); f[n]: = diff(y[n],x$1); y[n+1]: = (x+h)* exp (v*(x+h)); f[n+1]: = diff(y[n+1],x$1); y[n1]: = (xh)* exp (v*(xh)); f[n1]: = diff(y[n1],x$1); final2: = y[n+1]y[n1] = h*(a[0]*f[n1]+a[1]*f[n]+a[2]*f[n+1]);

final2 : =(x + h)e(v(x+h)) (x h)e(v(xh)) = h(a0e(v(xh)) + (x h) ve(v(xh))) + a1(e(vx) + xve(vx) + a2(e(v(x+h)) + (x + h)ve(v(x+h)))) 4 final2: = combine(final2/exp(v*x)); final2 : =((x + h)e(v(x+h)) + (x + h)e(v(xh)))e(vx) = e(vx)h(a0(e(v(xh)) + (x h) ve(v(xh)) + a1(e(vx) + xve(vx)) + a2(e(v(x+h)) + (x + h)ve(v(x+h)))) 4 final2: = expand(final2); h ha0 ha0 vx h2 a0 v þ ¼ þ ðvhÞ ðvhÞ þ ha1 eðvhÞ eðvhÞ eðvhÞ e e ðvhÞ ðvhÞ þ ha1 xv þ ha2 e þ ha2 ve x þ h2 a2 vðvhÞ

final2 :¼eðvhÞ x þ eðvhÞ h

x

4 final2: = simplify(convert(final2,trig)); 2ðx coshðvhÞ2 þ x coshðvhÞ sinhðvhÞ x þ h coshðvhÞ2 þ h coshðvhÞ sin coshðvhÞ þ sinhðvhÞ ¼hða0 þ a0 vx ha0 v þ a1 coshðvhÞ þ a1 sinhðvhÞ þ a1 xv coshðvhÞ

final2 :¼

þ a1 xv sinhðvhÞ þ 2a2 coshðvhÞ2 þ 2a2 coshðvhÞ sinhðvhÞ a2 þ 2a2 vx coshðvhÞ2 þ 2a2 vx coshðvhÞ sinhðvhÞ a2 vx þ 2ha2 v coshðvhÞ2 þ 2ha2 coshðvhÞ sinhðvhÞ ha2 vÞ=ðcoshðvhÞ þ sinhðvhÞÞ 4 4 4 4 4 4 4 4 4 4

y[n]: = x*exp(v*x); f[n]: = diff(y[n],x$1); y[n+1]: = (x+h)*exp(v*(x+h)); f[n+1]: = diff(y[n+1],x$1); y[n1]: = (xh)*exp(v*(xh)); f[n1]: = diff(y[n1],x$1); final3: = y[n+1]y[n1] = h*(a[0]*f[n1]+a[1]*f[n]+a[2]*f[n+1]);

final3 : =(x + h) e(v(x+h)) (x h) e(v(xh)) = h(a0e(v(xh)) (x h) ve(v(xh))) + a1(e(vx) xve(vx)) + a2(e(v(x+h))(x + h) ve(v(x+h)))) 4 final3: = combine(final3/exp(v*x)); 444 | Chem. Modell., 2008, 5, 350–487 This journal is

c


final3 : =((x + h) e((xh)v) + (x + h) e((x+h)v))e(vx) = h(a2((x h) ve((xh)v) + e((xh)v)) + a0((x + h) ve((x+h)v) + e((x+h)v)) + a1(e(vx) xve(vx))) e(vx) 4 final3: = expand(final3); final3 :¼

ha2 vx h2 a2 v ha2 ðvhÞ þ ðvhÞ e eðvhÞ e ha0 veðvhÞ x þ h2 a0 veðvhÞ þ ha0 eðvhÞ þ ha1 ha1 xv x

eðvhÞ

þ

h

eðvhÞ

eðvhÞ x þ eðvhÞ h ¼

4 final3: = simplify(convert(final1,trig)); final3 : = 2( x + x cosh (vh)2 + x cosh (vh) sinh (vh) h cosh (vh)2 h cosh (vh) sinh (vh))/(cosh (vh) + sinh (vh)) = h(a2 vx + ha2 v a2 + 2a0 vx cosh (vh)2 + 2a0 vx cosh (vh) sinh (vh) a0 vx 2ha0 v cosh (vh)2 2ha0 v cosh (vh) sinh (vh) + ha0 v 2a0 cosh (vh)2 2a0 cosh (vh) sinh (vh) + a0 a1 cosh (vh) a1 sinh (vh) + a1 xv cosh (vh) + a1 xv sinh (vh))/(cosh (vh) + sinh (vh)) 4 4 eq5: = final2; 4 eq6: = final3; 4 eq5i: = subs(v = I*v,eq5); 4 eq6i: = subs(v = I*v,eq6); 4 4 eq5f: = simplify(evalc(Re(eq5i))); 4 eq6f: = simplify(evalc(Im(eq5i))); 4 eq7f: = simplify(evalc(Re(eq6i))); 4 eq8f: = simplify(evalc(Im(eq6i))); 4 simplify(eq5feq7f); 4 simplify(eq6f+eq8f); 4 4 solut: = solve({eq1f,eq2f,eq5f,eq6f},{a[0],a[1],a[2]}); 2ðvh cosðvhÞ sinðvhÞÞ hv cosðvhÞ sinðvhÞ solut :¼ a1 ¼ ; a0 ¼ ; v2 h2 sinðvhÞ v2 h2 sinðvhÞ hv cosðvhÞ sinðvhÞ a2 ¼ v2 h2 sinðvhÞ 4 assign(solut); 4 h: = 1; h : =1 4 a[0]: = combine(a[0]); a0 :¼

v cosðvÞ þ sinðvÞ v2 sinðvÞ

4 a0t: = convert(taylor(a[0],v = 0,20),polynom); 1 1 2 4 1 6 2 1382 4 a0t :¼ þ v2 þ v þ v þ v8 þ v10 þ v12 3 45 945 4725 93555 638512875 18243225 3617 87734 v14 þ v16 þ 162820783125 38979295480125 4 a[1]: = combine(a[1]); a1 :¼

2v sinð2vÞ v2 sinðvÞ Chem. Modell., 2008, 5, 350–487 | 445

This journal is

c


4 a1t: = convert(taylor(a[1],v = 0,20),polynom); 4 2 13 4 1 6 647 176639 v þ v þ v8 þ v10 a1t :¼ v2 þ 3 45 1890 2700 14968800 40864824000 2867 284899 1437422989 v12 þ v14 þ v16 þ 6538371840 6412633920000 319318388573184000 4 a[2]: = combine(a[2]); a2 :¼

v cosðvÞ þ sinðvÞ v2 sinðvÞ

4 a2t: = convert(taylor(a[2],v = 0,20),polynom); 1 1 2 4 1 6 2 1382 4 v þ v þ v8 þ v10 þ v12 a2t :¼ þ v2 þ 3 45 945 4725 93555 638512875 18243225 3617 87734 þ v14 þ v16 162820783125 38979295480125 4 simplify(a[2]a[0]); 0 4 4 restart; 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$1),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$1),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$1); 4 4 a[0]: = subs(v = v*h,1/3+1/45*v^2+2/945*v^4+1/4725*v^6+2/93555*v^8+ 1382/638512875*v^10+4/18243225*v^12+3617/162820783125*v^14+87734/ 38979295480125*v^16); 4 a[1]: = subs(v = v*h,4/32/45*v^2+13/1890*v^4+1/2700*v^6+647/ 14968800*v^8+176639/40864824*v^10+2867/6538371840*v^12+284899/ 6412633920000*v^14+1437422989/319318388573184000*v^16); 4 4 lte: = simplify(qnp1qnm1h*(a[0]*(snp1+snm1) + a[1]*sn)); 4 1 v12 h15 ðDð3Þ ÞðqÞðxÞ v4 h11 ðDð7Þ ÞðqÞðxÞ 18243225 170100 1 3617 v14 h23 ðDð9Þ ÞðqÞðxÞ v4 h5 ðDÞðqÞðxÞ 90 3282466987800000 1 2 7 ð5Þ 1 6 9 ð3Þ v h ðD ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 540 4725 1 2 4 7 ð3Þ v2 h15 ðDð13Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 10777536000 945 9683 1 v14 h15 DðqÞðxÞ v2 h11 ðDð9Þ ÞðqÞðxÞ 108972864000 907200 1 1 v8 h17 ðDð9Þ ÞðqÞðxÞ v4 h15 ðDð11Þ ÞðqÞðxÞ 943034400 857304000 1 1 v6 h15 ðDð9Þ ÞðqÞðxÞ v4 h13 ðDð9Þ ÞðqÞðxÞ 95256000 9525600 1 3617 v2 h9 ðDð7Þ ÞðqÞðxÞ v14 h17 ðDð3Þ ÞðqÞðxÞ 16200 162820783125

lte :¼


c


1 1 v12 h17 ðDð5Þ ÞðqÞðxÞ v2 h13 ðDð11Þ ÞðqÞðxÞ 54729675 81648000 1 4 9 ð5Þ 691 v h ðD ÞðqÞðxÞ v10 h23 ðDð13Þ ÞðqÞðxÞ 5670 76462172186400000 1 1 1 ðDð5Þ ÞðqÞðxÞh5 ðDð7Þ ÞðqÞðxÞh7 ðDð9Þ ÞðqÞðxÞh9 90 1890 90720 1 691 ðDð11Þ ÞðqÞðxÞh11 v10 h15 ðDð5Þ ÞðqÞðxÞ 7484400 3831077250 1441031 37 v16 h17 DðqÞðxÞ v10 h11 DðqÞðxÞ 160059342643200 4276800 3617 1 v14 h25 ðDð11Þ ÞðqÞðxÞ þ v6 h11 ðDð5Þ ÞðqÞðxÞ 29542202890200000 56700 87734 1 v16 h19 ðDð3Þ ÞðqÞðxÞ v2 h5 ðDð3Þ ÞðqÞðxÞ 38979295480125 45 9557 1 v12 h13 DðqÞðxÞ v4 h17 ðDð13Þ ÞðqÞðxÞ 10897286400 11316418000 1 1 v6 h13 ðDð7Þ ÞðqÞðxÞ v6 h17 ðDð11Þ ÞðqÞðxÞ 1701000 8573040000 1 2 v6 h19 ðDð13Þ ÞðqÞðxÞ v8 h11 ðDð3Þ ÞðqÞðxÞ 1131641280000 93555 1 1 v8 h13 ðDð5Þ ÞðqÞðxÞ v8 h15 ðDð7Þ ÞðqÞðxÞ 561330 16839900 1 1 v8 h19 ðDð11Þ ÞðqÞðxÞ v8 h21 ðDð13Þ ÞðqÞðxÞ 84873096000 11203248672000 1382 691 v10 h13 ðDð3Þ ÞðqÞðxÞ v10 h17 ðDð7Þ ÞðqÞðxÞ 638512875 114932317500 691 1 v10 h19 ðDð9Þ ÞðqÞðxÞ ðDð13Þ ÞðqÞðxÞh13 6436209780000 718502400 691 1 v10 h21 ðDð11Þ ÞðqÞðxÞ v12 h19 ðDð7Þ ÞðqÞðxÞ 579258880200000 1641890250 1 1 v12 h21 ðDð9Þ ÞðqÞðxÞ v12 h23 ðDð11Þ ÞðqÞðxÞ 91945854000 8275126860000 1 v12 h25 ðDð13Þ ÞðqÞðxÞ 1092316745520000 3617 v14 h19 ðDð5Þ ÞðqÞðxÞ 1953849397500 3617 v14 h21 ðDð7Þ ÞðqÞðxÞ 58615481925000 3617 v14 h27 ðDð13Þ ÞðqÞðxÞ 38995707815064000000 43867 v16 h21 ðDð5Þ ÞðqÞðxÞ 233875772880750 43867 v16 h23 ðDð7Þ ÞðqÞðxÞ 7016273186422500 43867 v16 h25 ðDð9Þ ÞðqÞðxÞ 392911298439660000 43867 v16 h27 ðDð11Þ ÞðqÞðxÞ 35362016859569400000 43867 1 6 7 v16 h29 ðDð13Þ ÞðqÞðxÞ v h DðqÞðxÞ 4667786225463160800000 1260 13 v8 h9 DðqÞðxÞ 151200


c


4 coeff(lte,h,5);

1 4 1 1 v DðqÞðxÞ ðDð5Þ ÞðqÞðxÞ v2 ðDð3Þ ÞðqÞðxÞ 90 90 45

4 4

Appendix G 4 restart; 4 ‘‘Development of the method’’ 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final:=y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2]=h^2*(b[0]*(f[n+2]+ f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final : =e(v(x+2h)) + ae(v(x+h)) (2 + 2a)e(vx) + ae(v(xh)) + e(v(x2h)) = h2(b0(v2e(v(x+2h)) + v2e(v(x2h))) + b1(v2e(v(x+h)) + v2e(v(xh))) + b2v2e(vx)) 4 final: = combine(final/exp(v*x)); final : =(ae(v(x+h)) + (2 2a)e(vx) + ae(v(xh)) + e(v(x+2h)) + e(v(x2h)))e(vx) = e(vx)h2(b0(v2e(v(x+2h)) + v2e(v(x2h))) + b1(v2e(v(x+h)) + v2e(v(xh))) + b2v2e(vx)) 4 final: = expand(final); final :¼aeðvhÞ 2 2a þ ¼h2 b0 v2 ðeðvhÞ Þ2 þ þ

a 1 þ ðeðvhÞ Þ2 þ eðvhÞ ðeðvhÞ Þ2

h2 b0 v2 ðeðvhÞ Þ2

þ h2 b1 v2 eðvhÞ

h2 b1 v2 þ h2 b2 v2 eðvhÞ

4 final: = simplify(convert(final,trig)); final : =2(2a cosh(vh)3) + 2a cosh(vh)2 sinh(vh) a cosh(vh) 6 cosh(vh)2 4 cosh(vh) sinh(vh) + 2 2a cosh(vh)2 2a cosh(vh) sinh(vh) + a + 4 cosh(vh)4 + 4 cosh(vh)3 sinh(vh))/(cosh(vh) + sinh(vh))2 = h2v2(8b0 cosh(vh)4 + 8b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh)2 4b0 cosh(vh) sinh(vh) + 2b0 + 4b1 cosh(vh)3 + 4b1 cosh(vh)2 sinh(vh) 2b1 cosh(vh) + 2b2 cosh(vh)2 + 2b2 cosh(vh) sinh(vh) b2)/ (cosh(vh) + sinh(vh))2 4 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 448 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final1: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]);

=

final1 : =e(v(x+2h)) + ae(v(x+h)) (2 + 2a)e(vx) + ae(v(xh)) + e(v(x2h)) = h2(b0(v2e(v(x+2h)) + v2e(v(x2h))) + b1(v2e(v(x+h)) + v2e(v(xh))) + b2v2e(vx)) 4 final1: = combine(final1/exp(v*x)); final1 : =(ae((xh)v) + (2 2a)e(vx) + ae((x+h)v) + e((x2h)v) + e((x+2h)v))e(vx) = h2(b1(v2e((xh)v)+v2e((x+h)v))+b0(v2e((x2h)v) + v2e((x+2h)v)) + b2v2e(vx))e(vx) 4 final1: = expand(final1); final1 :¼aeðvhÞ 2 2a þ ¼h2 b0 v2 ðeðvhÞ Þ2 þ þ

a 1 þ ðeðvhÞ Þ2 þ ðvhÞ eðvhÞ ðe Þ2


þ h2 b1 v2 eðvhÞ


4 final1: = simplify(convert(final1,trig)); final1 : =2(2a cosh(vh)3) + 2a cosh(vh)2 sinh(vh) a cosh(vh) 6 cosh(vh)2 4 cosh(vh) sinh(vh) + 2 2a cosh(vh)2 2a cosh(vh) sinh(vh) + a + 4 cosh(vh)4 + 4 cosh(vh)3 sinh(vh))/(cosh(vh) + sinh(vh))2 = h2v2(8b0 cosh(vh)4 + 8b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh)2 4b0 cosh(vh) sinh(vh) + 2b0 + 4b1 cosh(vh)3 + 4b1 cosh(vh)2 sinh(vh) 2b1 cosh(vh) + 2b2 cosh(vh)2 + 2b2 cosh(vh) sinh(vh) b2)/ (cosh(vh) + sinh(vh))2 4 simplify(finalfinal1); 4 eq1: = final; 4 eq1i: = subs(v = I*v,eq1); 4 4 y[n]: = x^n; 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = (x+h)^n; 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = (xh)^n; 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = (x+2*h)^n; 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = (x2*h)^n; 4 f[n2]: = diff(y[n2],x$2); 4 4 final2: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] = h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final2 :¼ðx þ 2hÞn þ aðx þ hÞn ð2 þ 2aÞxn þ aðx hÞn þ ðx 2hÞn 2

¼h

b0

þ b1

ðx þ 2hÞn n2 ðx þ 2hÞ2 ðx þ hÞn n2 ðx þ hÞ2

ðx þ 2hÞn n ðx þ 2hÞ2

ðx þ hÞn n

ðx þ hÞ2 n 2 x n xn n 2 þ b2 x x2

þ

þ

ðx 2hÞn n2 ðx 2hÞ2

ðx hÞn n2 ðx hÞ2

! ðx 2hÞn n

ðx 2hÞ2 ! ðx hÞn n ðx hÞ2


c


4 n: = 0; n : =0 4 eq2: = simplify(final2); eq2 : =0 = 0 4 n: = 2; n : =2 4 eq2: = simplify(final2); eq2 : =8h2 + 2ah2 = 2h2(2b0 + 2b1 + b2) 4 eq2: = simplify(eq2/h^2); eq2 : =8 + 2a = 4b0 + 4b1 + 2b2 4 n: = 4; n : =4 4 eq3: = simplify(final2); eq3 : =48x2h2+32h4+12ax2h2+2ah4 = 12h2(2b0x2+8b0h2+2b1x2+2b1h2 + b2x2) 4 eq3: = simplify(eq36*eq2*x^2*h^2); eq3 : =32h4 + 2ah4 = 96h4b0 + 24h4b1 4 eq3: = simplify(eq3/(h^4)); eq3 : =32 + 2a = 96b0 + 24b1 4 4 ‘‘Stability of the method’’; 4 stab: = k[n+2]+a*k[n+1] (2+2*a)*k[n]+a*k[n1]+k[n2]+ H^2*(b[0]*(k[n+2]+k[n2]) + b[1]*(k[n+1]+k[n1]) + b[2]*k[n]); stab : =k6 + ak5 (2 + 2a)k4 + ak3 + k2 + H2(b0(k6 + k2) + b1(k5 + k3) + b2k4) 4 AH: = coeff(stab,k[n+2]); AH : =1 + H2b0 4 BH: = coeff(stab,k[n+1]); BH : =a + H2b1 4 CH: = coeff(stab,k[n]); CH : =2 2a + H2b2 charact: = AAH*(l^2+l^(2)) + BBH*(l^1+l^(1)) + CCH; charact1: = subs(l=exp(I*H),charact)=0; charact1: = factor(simplify(convert(charact1,trig))); charact2: = subs(l=exp(I*H),charact)=0; charact2: = factor(simplify(convert(charact2,trig))); charact3: = subs(l=exp(I*H),charact)=0; charact3: = factor(simplify(convert(charact3,trig))); charact4: = subs(l=exp(I*H),charact)=0; charact4: = factor(simplify(convert(charact4,trig))); simplify(charact1charact4); simplify(charact2charact3); charact1: = subs(AAH=AH,BBH=BH,CCH=CH,charact1); charact2: = subs(AAH=AH,BBH=BH,CCH=CH,charact2);

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 eq4: = subs(H=v*h,charact1);

eq4 : =4(1 + h2b0v2) cos(vh)2 2(a + h2b1v2) cos(vh) 4 2h2b0v2 2a + h2b2v2 =0 4 eq5: = subs(H=v*h,charact2); 450 | Chem. Modell., 2008, 5, 350–487 This journal is

c


eq5 : =4(1 + h2b0v2) cos(vh)2 2(a + h2b1v2) cos(vh) 4 2h2b0v2 2a + h2b2v2 =0 4 eq1ir: = combine(evalc(Re(eq1i))); eqlir : =2a cos(vh) 2 + 2 cos(2vh) 2a = 2h2v2 cos(2vh)b0 2h2b1v2 cos(vh) h2b2v2 4 eq1ii: = combine(evalc(Im(eq1i))); 4 4 4 4 4 4 4 4

eqlii : =0 = 0 solution: = solve({eq1ir,eq2,eq3},{b[0],b[1],b[2]}); assign(solution); eq4: = simplify(eq4); eq5: = simplify(eq5); solution2: = solve({eq4,eq5},{a}); assign(solution2); ‘‘Coefficients of the method’’; h: = 1; h : =1

4 b[0]: = simplify(b[0]); b0 :¼

12 cosðvÞ2 cosðvÞ2 v2 þ 12 11v2 þ 3v4 v2 ðcosðvÞ2 v2 þ 12 cosðvÞ2 12 þ 11v2 Þ

4 b0t: = convert(taylor(b[0],v = 0,26),polynom); b0t :¼

1 31 2 2 4061 6926 1163978 þ v þ v4 þ v6 v8 þ v10 15 4725 23625 245581875 6841209375 15084866671875 31868924 166642196489 v12 þ v14 3846641001328125 253243610322437109375 2807016106 111075102732218 v16 þ v18 46896964874525390625 21045661269016534541015625

4 b[1]: = simplify(b[1]); b1 :¼

16ðcosðvÞ2 v2 þ 3 cosðvÞ2 3 þ 2v2 Þ v2 ðcosðvÞ2 v2 þ 12 cosðvÞ2 12 þ 11v2 Þ


16 544 2 656 4 205664 6 1178656 v þ v v þ v8 15 4725 70875 245581875 15962821875 97475072 199892816 12729850365536 v10 þ v12 v14 15084866671875 349694636484375 253243610322437109375 2403112310752 139583101228894144 v16 v18 þ 542664879262365234375 357776241573281087197265625

4 b[2]: = simplify(b[2]); b2 :¼

6ð5 cosðvÞ2 v2 þ 2v4 cosðvÞ2 þ 12 cosðvÞ2 v4 12 þ 7v2 Þ v2 ðcosðvÞ2 v2 þ 12 cosðvÞ2 12 þ 11v2 Þ


26 446 2 6836 4 623278 6 33129508 8 307069244 v þ v v þ v v10 15 525 70875 81860625 47888465625 5028288890625 20522239256 13292551401874 þ v12 v14 3846641001328125 28138178924715234375 8303101039036 436323427317448988 v16 v18 þ 199929166044029296875 119258747191093695732421875 Chem. Modell., 2008, 5, 350–487 | 451 This journal is

c


4 a: = simplify(a); a :¼

16ðcosðvÞ2 v2 þ 3 cosðvÞ2 3 þ 2v2 Þ cosðvÞ2 v2 þ 12 cosðvÞ2 12 þ 11v2

4 at: = convert(taylor(a,v = 0,26),polynom); at :¼

16 2 544 4 656 6 205664 8 1178656 v þ v v þ v v10 15 4725 70875 245581875 15962821875 97475072 199892816 12729850365536 þ v12 v14 þ v16 15084866671875 349694636484375 253243610322437109375 2403112310752 139583101228894144 v18 þ v20 542664879262365234375 357776241573281087197265625

4 plot([b[0]],v = 100..100,labels=[v,b_0],thickness=3,title=‘‘bahavior of the coefficient b_0’’); 4 plot([b[1]],v = 50..50,labels=[v,b_1],thickness=3,title=‘‘bahavior of the coefficient b_1’’); 4 plot([b[2]],v = 50..50,labels=[v,b_2],thickness=3,title=‘‘bahavior of the coefficient b_2’’); 4 plot([a],v = 50..50,labels=[v,a_],thickness=3,title=‘‘bahavior of the coefficient a_’’); 4 4 4 restart; 4 ‘‘Local Truncation Error of the method’’; 4 qnp2: = convert(taylor(q(x+2*h),h = 0,13),polynom); 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(xh),h = 0,13),polynom); 4 qnm2: = convert(taylor(q(x2*h),h = 0,13),polynom); 4 snp2: = convert(taylor(diff(q(x+2*h),x$2),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$2),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$2),h = 0,13),polynom); 4 snm2: = convert(taylor(diff(q(x2*h),x$2),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$2); 4 b[0]: = subs(v = v*h,1/15+31/4725*v^2+2/23625*v^4+4061/ 245581875*v^66926/6841209375*v^8+1163978/ 15084866671875*v^1031868924/3846641001328125*v^12+166642196489/ 253243610322437109375*v^14); 4 b[1]: = subs(v = v*h,16/15544/4725*v^2+656/70875*v^4205664/ 245581875*v^6+1178656/15962821875*v^897475072/ 15084866671875*v^10+199892816/349694636484375*v^1212729850365536/ 253243610322437109375*v^14); 4 b[2]: = subs(v = v*h,26/15446/525*v^2+6836/70875*v^4623278/ 81860625*v^6+33129508/47888465625*v^8307069244/ 5028288890625*v^10+20522239256/3846641001328125*v^1213292551401874/ 28138178924715234375*v^14); 4 a: = subs(v = v*h, 16/15*v^2+544/4725*v^4656/70875*v^6+205664/ 245581875*v^81178656/15962821875*v^10+97475072/ 15084866671875*v^12199892816/349694636484375*v^14+12729850365536/ 253243610322437109375*v^162403112310752/ 542664879262365234375*v^18+139583101228894144/ 357776241573281087197265625*v^20); 4 lte: = simplify(qnp2+a*qnp1 (2+2*a)*qn+ a*qnm1+ qnm2h^2*(b[0]*(snp2+snm2) + b[1]*(snp1+snm1) + b[2]*sn)); 452 | Chem. Modell., 2008, 5, 350–487 This journal is

c


300389038844 v18 h24 ðDð6Þ ÞðqÞðxÞ 24419919566806435546875 99061 2 2 8 ð6Þ v8 h16 ðDð8Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ þ 6033946668750 945 2 19 79 ðDð8Þ ÞðqÞðxÞh8 ðDð10Þ ÞðqÞðxÞh10 ðDð12Þ ÞðqÞðxÞh12 945 56700 3742200 439 1577 h16 v2 ðDð14Þ ÞðqÞðxÞ h18 v4 ðDð14Þ ÞðqÞðxÞ 3929310000 1060913700000 15557 h20 v6 ðDð14Þ ÞðqÞðxÞ 55697969250000 6095197 h22 v8 ðDð14Þ ÞðqÞðxÞ þ 358416432123750000 3040481 h24 v10 ðDð14Þ ÞðqÞðxÞ 2352107835812109375 8021018233 h26 v12 ðDð14Þ ÞðqÞðxÞ þ 57579599820680437500000 1909 19 v2 h14 ðDð12Þ ÞðqÞðxÞ v2 h12 ðDð10Þ ÞðqÞðxÞ 523908000 243000 17 2 10 ð8Þ 197 v h ðD ÞðqÞðxÞ v4 h14 ðDð10Þ ÞðqÞðxÞ 18900 133953750 313 8 v4 h12 ðDð8Þ ÞðqÞðxÞ v4 h10 ðDð6Þ ÞðqÞðxÞ 8930250 14175 15343 3161 v6 h16 ðDð10Þ ÞðqÞðxÞ v6 h14 ðDð8Þ ÞðqÞðxÞ 88409475000 2946982500 1082 1853 v6 h12 ðDð6Þ ÞðqÞðxÞ v4 h16 ðDð12Þ ÞðqÞðxÞ þ 49116375 35363790000 2636 66167 v8 h14 ðDð6Þ ÞðqÞðxÞ v6 h18 ðDð12Þ ÞðqÞðxÞ 1064188125 7426395900000 51511 53492 þ v10 h18 ðDð8Þ ÞðqÞðxÞ þ v10 h16 ðDð6Þ ÞðqÞðxÞ 90509200031250 232074871875 12763691 v8 h20 ðDð12Þ ÞðqÞðxÞ þ 23894428808250000 1747693 v8 h20 ðDð10Þ ÞðqÞðxÞ þ 181018400062500 101097013 v10 h22 ðDð12Þ ÞðqÞðxÞ 2508915024866250000 633569 v10 h20 ðDð10Þ ÞðqÞðxÞ 905092000312500 2048524 v12 h18 ðDð10Þ ÞðqÞðxÞ 109904028609375 563196240707 v14 h24 ðDð10Þ ÞðqÞðxÞ 91167699716077359375000 5780434663 v14 h22 ðDð8Þ ÞðqÞðxÞ 1012974441289748437500 87304376498 v14 h20 ðDð6Þ ÞðqÞðxÞ þ 50648722064487421875 247688143 þ v12 h24 ðDð12Þ ÞðqÞðxÞ 56450588059490625000 397807823923 þ v16 h24 ðDð8Þ ÞðqÞðxÞ 159543474503135378906250 1591231295692 v16 h22 ðDð6Þ ÞðqÞðxÞ þ 11395962464509669921875

lte :¼


c


3182462591384 v16 h20 ðDð4Þ ÞðqÞðxÞ 759730830967311328125 12729850365536 v16 h18 ðDð2Þ ÞðqÞðxÞ þ 253243610322437109375 883380115561 v14 h26 ðDð12Þ ÞðqÞðxÞ 2552695592050166062500000 17447887653611768 v20 h26 ðDð6Þ ÞðqÞðxÞ þ 16099930870797648923876953125 34895775307223536 v20 h24 ðDð4Þ ÞðqÞðxÞ þ 1073328724719843261591796875 139583101228894144 v20 h22 ðDð2Þ ÞðqÞðxÞ þ 357776241573281087197265625 75097259711 v18 h30 ðDð12Þ ÞðqÞðxÞ 4061521022351246360156250000 75097259711 v18 h28 ðDð10Þ ÞðqÞðxÞ 30769098654176108789062500 75097259711 v18 h26 ðDð8Þ ÞðqÞðxÞ 341878873935290097656250 211438316431 h28 v14 ðDð4Þ ÞðqÞðxÞ 19145216940376245468750000 2180985956701471 v20 h32 ðDð12Þ ÞðqÞðxÞ þ 1338870251215532484509607421875000 2180985956701471 þ v20 h28 ðDð8Þ ÞðqÞðxÞ 112699516095583542467138671875 2180985956701471 v20 h30 ðDð10Þ ÞðqÞðxÞ þ 10142956448602518822042480468750 49813726 v12 h20 ðDð8Þ ÞðqÞðxÞ þ 242338383083671875 600778077688 v18 h22 ðDð4Þ ÞðqÞðxÞ 1627994637787095703125 1169213051 v12 h22 ðDð10Þ ÞðqÞðxÞ þ 14540302985020312500 2403112310752 v18 h20 ðDð2Þ ÞðqÞðxÞ 542664879262365234375 397807823923 v16 h28 ðDð12Þ ÞðqÞðxÞ þ 1895376477097248301406250000 397807823923 v16 h26 ðDð10Þ ÞðqÞðxÞ þ 14358912705282184101562500 257 h14 ðDð14Þ ÞðqÞðxÞ 224532000 þ

4 coeff(lte,h,8); 4 4 4 4 4

2 2 ð6Þ 2 v ðD ÞðqÞðxÞ ðDð8Þ ÞðqÞðxÞ 945 945

restart; ‘‘Stability analysis of the method’’; AH: = 1+v^2*b[0]; BH: = a+v^2*b[1]; CH: = 22*a+v^2*b[2];


c


4 b[0]: = (v^2*cos(v)^212*cos(v)^2+1211*v^2+3*v^4)/ (v^2*(12*cos(v)^212+v^2*cos(v)^2+11*v^2)); 4 b[1]: = 16*(v^2*cos(v)^2+2*v^2+3*cos(v)^23)/(v^2*(12*cos(v)^2 12+v^2*cos(v)^2+11*v^2)); 4 b[2]: = 6*(2*v^4*cos(v)^2+5*v^2*cos(v)^2+12*cos(v)^212 v^4+7*v^2)/ (v^2*(12*cos(v)^212+v^2*cos(v)^2+11*v^2)); 4 a: = (16*v^2*cos(2*v)80*v^248*cos(2*v) + 48)/(12*cos(2*v)12+v^2* cos(2*v) + 23*v^2); 4 AH; 4 BH; 4 CH; 4 P1H: = simplify(2*AH2*BH+CH); P1H :¼

12v4 sinðvÞ2 2

12 cosðvÞ 12 þ v2 cosðvÞ2 þ 11v2

4 P2H: = simplify(12*AH2*CH); P2H :¼

24ðcosðvÞ2 þ 1Þv4 12 cosðvÞ2 12 þ v2 cosðvÞ2 þ 11v2

4 P3H: = simplify(2*AH+2*BH+CH); P3H :¼

12v4 sinðvÞ2 12 cosðvÞ2 12 þ v2 cosðvÞ2 þ 11v2

4 plot([P1H,P2H,P3H], v = 0..100, color=[black,red,blue], style=[line,line,point], linestyle=[SOLID, DOT, DASHDOT], title=‘‘Stability Polynomias for the New Method’’); 4 plot([NH], v = 0..100, color=[black], style=[line], linestyle=[SOLID], title=‘‘Stability Polynomias for the New Method-Polynomial N(H,w)’’); 4 restart; 4 final: = k[n+2]+a*k[n+1](2+2*a)*k[n]+a*k[n1]+k[n2]+ H^2*(b[0]*(k[n+2]+k[n2]) + b[1]*(k[n+1]+k[n1]) + b[2]*k[n]); 4 AH: = combine(coeff(final,k[n+2])); 4 BH: = combine(coeff(final,k[n+1])); 4 CH: = combine(coeff(final,k[n])); 4 P1H: = simplify(2*AH2*BH+CH); 4 P2H: = simplify(12*AH2*CH); 4 P3H: = simplify(2*AH+2*BH+CH); 4 b[0]: = (cos(v)^2*v^212*cos(v)^211*v^2+12+3*v^4)/v^2/ (12*cos(v)^2+cos(v)^2*v^2+11*v^212); 4 b[1]: = 16*(cos(v)^2*v^2+3*cos(v)^23+2*v^2)/v^2/(12*cos(v)^2+ cos(v)^2*v^2+11*v^212); 4 b[2]: = 6*(12*cos(v)^2+5*cos(v)^2*v^2+2*v^4*cos(v)^212+ 7*v^2v^4)/ v^2/(12*cos(v)^2+cos(v)^2*v^2+11*v^212); 4 a: = 16*(cos(v)^2*v^2+3*cos(v)^23+2*v^2)/(12*cos(v)^2+ cos(v)^2*v^2+11*v^212); 4 P1H: = combine(P1H); P1H : =(64H2v2 cos(2v) 320H2v2 192H2 cos(2v) + 192H2 + 12H2v4 64v4 cos(2v) + 320v4 + 192v2 cos(2v) 192v2 12H2v4 cos(2v))/(12v2 cos(2v) 12v2 + v4 cos(2v) + 23v4) 4 P2H: = combine(P2H); P2H :¼

48v2 cosð2vÞ þ 48v2 þ 48H 2 cosð2vÞ 48H 2 þ 72H 2 v2 þ 24H 2 v2 cosð2vÞ 12 cosð2vÞ 12 þ v2 cosð2vÞ þ 23v2 Chem. Modell., 2008, 5, 350–487 | 455 This journal is

c


4 P3H: = combine(P3H); P3H :¼

12H 2 v2 cosð2vÞ þ 12H 2 v2 12 cosð2vÞ 12 þ v2 cosð2vÞ þ 23v2

4 NH: = combine(P2H^24*P1H*P3H); NH : =(9216v4 cos(2v) + 768H2v4 cos(4v) + 6912v4 + 4608H2v2 cos(4v) + 13824H2v2 16128H2v4 768H4v2 cos(4v) 18432H2v2 cos(2v) + 15360H2v4 cos(2v) 15360H4 cos(2v)v2 + 9216H4v4 cos(2v) 20736H4 + 27648H4 cos(2v) + 576 cos(2v) + 24v2 cos(4v) 1080v2 + 1056v2 cos(2v) + v4 cos(4v) + 1059v4 + 92v4 cos(2v))

Appendix H 4 restart; 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 4 final: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]);

=

final : =e(v(x+2h)) + ae(v(x+h)) (2+2a)e(vx)+ae(v(xh)) + e(v(x2h)) =h2(b0(v2e(v(x+2h)) + v2 e(v(x2h))) + b1 (v2e(v(x+h)) + v2e(v(xh))) + b2v2e(vx)) 4 final: = combine(final/exp(v*x)); final : =(ae(v(x+h)) + (22a)e(vx)+ae(v(xh))+e(v(x+2h)) +e(v(x2h)) + e(vx) \= e(vx)h2 (b0(v2e(v(x+2h)) + v2e(v(x2h))) + b1(v2e(v(x+h)) + v2e(v(xh))) + b2v2e(vx)) 4 final: = expand(final); final :¼aeðvhÞ 2 2a þ ¼h2 b0 v2 ðeðvhÞ Þ2 þ


h2 b0 v2 ðeðvhÞ Þ

þ h2 b1 v2 eðvhÞ þ 2


4 final: = simplify(convert(final,trig)); final : =2 (2a cosh(vh)3 + 2a cosh(vh)2 sinh(vh) a cosh(vh) 6 cosh(vh)2 4 cosh(vh) sinh(vh) + 2 2a cosh(vh)2 2a cosh(vh) sinh(vh) + a + 4 cosh(vh)4 + 4 cosh(vh)3 sinh(vh))/(cosh(vh) + sinh(vh))2 = h2v2(8b0 cosh(vh)4 + 8b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh)2 4b0 cosh(vh) sinh(vh) + 2b0 + 4b1 cosh(vh)3+4b1 cosh(vh)2 sinh(vh) 2b1 cosh(vh) + 2b2 cosh(vh)2 + 2b2 cosh(vh) sinh(vh) b2)/ (cosh(vh) + sinh(vh))2 4 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = exp(v*(xh)); 456 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]);

=

final : =e(v(x+2h)) + ae(v(x+h)) (2+2a)e(vx) + ae(v(xh)) + e(v(x2h)) = h2 (b0(v2e(v(x+2h))+v2e(v(x2h))) + b1 (v2 e(v(x+h))+v2e(v(xh))) + b2v2e(vx)) 4 final: = combine(final/exp(v*x)); final : =(ae((xh)v) + (22a)e(vx) + ae((x+h)v)+e((x2h)v) + e((x+2h)v))e(vx) = h2(b1(v2e((xh)v) + v2e((x+h)v)) + b0(v2e((x2h)v) + v2e((x+2h)v)) + b2v2e(vx))e(vx) 4 final: = expand(final); final :¼aeðvhÞ 2 2a þ ¼h2 b0 v2 ðeðvhÞ Þ2 þ

a 1 þ ðeðvhÞ Þ2 þ eðvhÞ ðeðvhÞ Þ2


þ h2 b1 v2 eðvhÞ þ


4 final: = simplify(convert(final,trig)); final : =2(2a cosh(vh)3 + 2a cosh(vh)2 sinh(vh) a cosh(vh) 6 cosh(vh)2 4 cosh(vh) sinh(vh) + 22a cosh(vh)2 2a cosh(vh) sinh(vh) + a + 4 cosh(vh)4 + 4 cosh(vh)3 sinh(vh))/(cosh(vh) + sinh(vh))2 = h2v2(8b0 cosh(vh)4+8b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh)2 4b0 cosh(vh)sinh(vh) + 2b0 + 4b1 cosh(vh)3 + 4b1 cosh(vh)2 sinh(vh) 2b1 cosh(vh) + 2b2 cosh(vh)2 + 2b2 cosh(vh) sinh(vh) b2)/ (cosh(vh) + sinh(vh))2 4 4 eq1: = final; 4 eq1i: = subs(v = I*v,eq1); 4 eq1i: = simplify(eq1i); 4 4 y[n]: = x*exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = (x+h)*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = (xh)*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = (x+2*h)*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = (x2*h)*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 4 final: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] = h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final : =(x + 2h)e(v(x+2h)) + a(x + h)e(v(x+h)) (2 + 2a)xe(vx) + a(x h)e(v(xh)) + (x 2h)e(v(x2h))=h2(b0(2ve(v(x+2h)) + (x + 2h)v2e(v(x+2h)) + 2ve(v(x2h)) + (x2h)v2e(v(x2h))) + b1(2ve(v(x+h)) + (x+h)v2e(v(x+h)) + 2ve(v(xh)) + (xh)v2e(v(xh))) + b2(2ve(vx) + xv2e(vx))) 4 final: = combine(final/exp(v*x)); final : =(a(x + h)e(v(x+h)) + a(x h)e(v(xh)) + (x+2h)e(v(x+2h)) + (x 2h)e(v(x2h)) + ( 2 2a)xe(vx))e(vx) = e(vx)h2(b0(2ve(v(x+2h)) + (x+2h)v2e(v(x+2h)) +


c


2ve(v(x2h)) + (x 2h)v2e(v(x2h))) + b1(2ve(v(x+h)) + (x + h)v2e(v(x+h)) + 2ve(v(xh)) + (x h)v2e(v(xh))) + b2(2ve(vx) + xv2e(vx))) 4 final: = expand(final); final :¼aeðvhÞ x þ aeðvhÞ h þ

2h ðeðvhÞ Þ2

ax ah x þ ðeðvhÞ Þ2 x þ 2ðeðvhÞ Þ2 h þ eðvhÞ eðvhÞ ðeðvhÞ Þ2

2x 2xa ¼ 2h2 b0 vðeðvhÞ Þ2 þ h2 b0 v2 ðeðvhÞ Þ2 x

þ 2h3 b0 v2 ðeðvhÞ Þ2 þ

2h2 b0 v

h2 b0 v2 x

2h3 b0 v2

þ 2h2 b1 veðvhÞ ðeðvhÞ Þ2 2h2 b1 v h2 b1 v2 x h3 b1 v2 þ h2 b1 v2 eðvhÞ x þ h3 b1 v2 eðvhÞ þ ðvhÞ þ ðvhÞ ðvhÞ þ 2h2 b2 v e e e þ h2 b2 xv2 ðeðvhÞ Þ2

þ

ðeðvhÞ Þ2

4 final: = simplify(convert(final,trig)); final : =2(2x ah sinh(vh) + 8h cosh(vh)3 sinh(vh) 4h cosh(vh) sinh(vh) + 2xa cosh(vh)2 sinh(vh) + 4x cosh(vh)4 + 8h cosh(vh)4 6x cosh(vh)2 8h cosh(vh)2 + 2xa cosh(vh)3 + 2ah cosh(vh)3 xa cosh(vh) 2ah cosh(vh) 4x cosh(vh) sinh(vh) 2xa cosh(vh)2 + 4x cosh(vh)3 sinh(vh) + xa +2ah cosh(vh)2 sinh(vh) 2xa cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 = h2v(4b02b2 + 4b2 cosh(vh) sinh(vh) + 2b0vx b2xv + 16hb0v cosh(vh)3 sinh(vh) + 4b1vx cosh(vh)2 sinh(vh) + 8b0vx cosh(vh)3 sinh(vh) + 4hb1v cosh(vh)2 sinh(vh) + 2b2xv cosh(vh) sinh(vh) 8b0vx cosh(vh)2 16hb0v cosh(vh)2 + 8b0vx cosh(vh)4 + 16hb0v cosh(vh)4 + 4b1vx cosh(vh)3 + 4hb1v cosh(vh)3 2b1vx cosh(vh) 4hb1v cosh(vh) 2hb1v sinh(vh) +2b2xv cosh(vh)2 + 16b0 cosh(vh)3 sinh(vh) + 8b1 cosh(vh)2 sinh(vh) + 16b0 cosh(vh)4 + 8b1 cosh(vh)3 4b1 cosh(vh) + 4b2 cosh(vh)2 16b0 cosh(vh)2 8b0 cosh(vh) sinh(vh) 4b0vx cosh(vh) sinh(vh) 8hb0v cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 4 4 eq2: = final; 4 eq2i: = subs(v = I*v,eq2); 4 eq2i: = simplify(eq2i); 4 eq2rl: = evalc(Re(4*x2*x*a+2*x*a*cos(v*h) + 4*x*cos(v*h)^2+8*I*h*cos(v*h)*sin(v*h) + 2*I*a*h*sin(v*h))); 4 eq2rr: = evalc(Re (h^2*v*(2*b[0]*v*x b[2]*x*v2*I*h*v*b[1]*sin (v*h)8*I*h*v*b[0]*cos(v*h)*sin(v*h) + 8*I*b[0]*cos(v*h)^2+4*I*b[1]*cos(v*h) + 2*I*b[2]4*b[0]*v*x*cos(v*h)^22*b[1]*v*x*cos(v*h)4*I*b[0]))); 4 eq2r: = simplify(eq2rl=eq2rr); 4 eq2il: = evalc(Im(4*x2*x*a+2*x*a*cos(v*h) + 4*x*cos(v*h)^2+8*I*h*cos(v*h)*sin(v*h) +2*I*a*h*sin(v*h))); 4 eq2ir: = evalc(Im(h^2*v* (2*b[0]*v*xb[2]*x*v 2*I*h*v*b[1]*sin(v*h)8*I*h*v*b[0]*cos(v*h)* sin(v*h) + 8*I*b[0]*cos(v*h)^2+4*I*b[1] *cos(v*h) + 2*I*b[2]4*b[0]*v*x*cos(v*h)^22*b[1]*v*x*cos(v*h)4*I*b[0]))); 4 eq2i: = simplify(eq2il=eq2ir); 4 4 y[n]: = x*exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = (x+h)*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = (xh)*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = (x+2*h)*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = (x2*h)*exp(v*(x2*h)); 458 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 f[n2]: = diff(y[n2],x$2); 4 4 final: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]);

=

final : =(x + 2h)e(v(x+2h)) + a(x + h)e(v(x+ h)) (2 + 2a)xe(vx) + a(x h)e(v(xh)) + (x 2h)e(v(x2h)) = h2 (b0(2ve(v(x+2h)) + (x + 2h)v2e(v(x + 2h)) 2ve(v(x 2h)) + (x 2h)v2e(v(x 2h)) + b1 (2v e(v(x+h)) + (x + h)v2e(v(x+h))2v e(v(xh)) + (x h)v2e(v(xh))) + b2(2ve(vx) + xv2e(vx))) 4 final: = combine(final/exp(v*x)); final : =(a(x + h)e((xh)v) + a(x h)e((x+h)v) + (x + 2h)e((x2h)v) + (x 2h)e((x+2h)v) + (2 2a)xe(vx)))e(vx) = h2 (b1((x + h)v2e((xh)v) + (x h)v2e((x+h)v) 2ve((xh)v) 2ve((x+h)v)) + b0 ((x + 2h)v2e((x2h)v) + (x 2h)v2e((x+2h)v) 2ve((x2h)v)2ve((x+2h)v))+b2(2ve(vx) + xv2e(vx)))e(vx) 4 final: = expand(final); final :¼

ax ah x 2h þ þ aeðvhÞ x aeðvhÞ h þ þ þ ðeðvhÞ Þ2 x eðvhÞ eðvhÞ ðeðvhÞ Þ2 ðeðvhÞ Þ2 h2 b1 v2 x h3 b1 v2 þ ðvhÞ þ h2 b1 v2 eðvhÞ x h3 b1 v2 eðvhÞ e eðvhÞ 2h2 b1 v h2 b0 v2 x 2h3 b0 v2 2 ðvhÞ ðvhÞ 2h b1 ve þ þ þ h2 b0 v2 ðeðvhÞ Þ2 x e ðeðvhÞ Þ2 ðeðvhÞ Þ2 2h2 b0 v 2h2 b0 vðeðvhÞ Þ2 2h2 vb2 þ h2v2 xb2 2h3 b0 v2 ðeðvhÞ Þ2 ðeðvhÞ Þ2

2ðeðvhÞ Þ2 h 2x 2xa ¼

4 final: = simplify(convert(final,trig)); final : =2 (2x 8h cosh(vh)4 6x cosh(vh)2 + xa + 2xa cosh(vh)2 sinh(vh) 2ah cosh(vh)2 sinh(vh) 2xa cosh(vh) sinh(vh) + 4x cosh(vh)4 + 2ah cosh(vh) + ah sinh(vh) + 2xa cosh(vh)3 2ah cosh(vh)3 + 4x cosh(vh)3 sinh(vh) 8h cosh(vh)3 sinh(vh) 4x cosh(vh) sinh(vh) 2xa cosh(vh)2 + 8h cosh(vh)2 + 4h cosh(vh) sinh(vh) xa cosh(vh))/(cosh(vh) + sinh(vh))2 = h2v(4b0 + 2b2 4b2 cosh(vh) sinh(vh) + 2b0 vx b2 xv 16hb0v cosh(vh)3 sinh(vh) + 4b1vx cosh(vh)2 sinh(vh) + 8b0 vx cosh(vh)3 sinh(vh) 4h b1 v cosh(vh)2 sinh(vh) + 2b2 xv cosh(vh) sinh(vh) 8b0vx cosh(vh)2 + 16h b0v cosh(vh)2 + 8b0vx cosh(vh)4 16h b0v cosh(vh)4 + 4b1vx cosh(vh)3 4h b1v cosh(vh)3 2b1vx cosh(vh) + 4h b1v cosh(vh) + 2h b1v sinh(vh) + 2b2 xv cosh(vh)2 16b0 cosh(vh)3 sinh(vh) 8b1 cosh(vh)2 sinh(vh) 16b0 cosh(vh)4 8b1 cosh(vh)3 + 4b1 cosh(vh) 4b2 cosh(vh)2 + 16b0 cosh(vh)2 + 8b0 cosh(vh) sinh(vh) 4b0 vx cosh(vh) sinh(vh) + 8hb0v cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 4 eq3: = final; eq3i: = subs(v = I*v,eq3); eq3i: = simplify(eq3i); 4 eq3rl: = evalc(Re(4*x2*x*a+4*x*cos(v*h)^2+2*x*a*cos(v*h) 2*I*a*h*sin(v*h)8*I*h*cos(v*h)*sin(v*h))); eq3rr: = evalc(Re(h^2*v*(2*b[0]*v*x +b[2]*x*v8*I*h*v* b[0]* cos(v*h)*sin(v*h)2*I*h*v*b[1]*sin(v*h) + 8*I*b[0]*cos(v*h)^2 + 4*I*b[1]* cos(v*h) + 4*b[0]*v*x*cos(v*h)^2 + 2*b[1]*v*x*cos(v*h) 4*I*b[0]+2*I*b[2]))); eq3r: = simplify(eq3rl=eq3rr); eq3il: = evalc(Im(4*x2*x*a + 4*x*cos(v*h)^2 + 2*x*a*cos(v*h) 2*I*a* h*sin(v*h)8*I*h*cos(v*h)*sin(v*h))); eq3ir: = evalc(Im(h^2*v*(2*b[0]*v*x+b[2]*x*v8*I*h*v*b[0]* cos(v*h)*sin(v*h)2*I*h*v*b[1]*sin(v*h) + 8*I*b[0]*cos(v*h)^2 + 4*I*b[1]*cos(v*h) + 4*b[0]*v*x*cos(v*h)^2 + 2*b[1]*v*x*cos(v*h)4*I*b[0] + 2*I*b[2]))); Chem. Modell., 2008, 5, 350–487 | 459 This journal is

c


4 eq3i: = simplify(eq3il=eq3ir); simplify(eq2req3r); simplify(eq2i+eq3i); 4 y[n]: = x^n; f[n]: = diff(y[n],x$2); y[n+1]: = (x+h)^n; f[n+1]: = diff(y[n+1],x$2); y[n1]: = (xh)^n; f[n1]: = diff(y[n1],x$2); y[n+2]: = (x+2*h)^n; f[n+2]: = diff(y[n+2],x$2); y[n2]: = (x2*h)^n; f[n2]: = diff(y[n2],x$2); 4 4 4 final: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] = h^2*(b[0]* (f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final :¼ðx þ 2hÞn þ aðx þ hÞn ð2 þ 2aÞxn þ aðx hÞn þ ðx 2hÞn 2

¼h

b0

ðx þ 2hÞn n2 ðx þ 2hÞ2


þ


! ðx 2hÞn n

ðx 2hÞ2 ! n 2 ! ðx þ hÞn n2 ðx þ hÞn n ðx hÞn n2 ðx hÞn n x n xn n þ 2 þ b2 þb x x2 ðx þ hÞ2 ðx hÞ2 ðx þ hÞ2 ðx hÞ2

4 n: = 0; n : =0 4 eq4: = simplify(final); eq4 : =0=0 4 n : =2; n : =2 4 eq4: = simplify(final); eq4 : =8h2 + 2ah2 = 2h2 (2b0 + 2b1 + b2) 4 eq4: = simplify(eq4/h^2); eq4 : =8 + 2a = 4b0 + 4b1 + 2b2 4 4 ‘‘Characteristic Equation of the new method’’; 4 final: =k[n+2]+a*k[n+1] (2+2*a)*k[n]+a*k[n1] +k[n2] +H^2*(b[0]*(k[n+2]+k[n2]) + b[1]*(k[n+1]+k[n1]) + b[2]*k[n]); final : =k6 + ak5 (2 + 2a) k4+ak3+k2+H2 (b0 (k6 + k2) + b1 (k5 + k3) + b2 k4) 4 AH: = coeff(final,k[n+2]); AH : =1 + H2 b0 4 BH: = coeff(final,k[n+1]); BH : =a + H2 b1 4 CH: = coeff(final,k[n]); CH : =22a + H2 b2 4 charact: = AAH*(l^2+l^(2)) + BBH*(l^1+l^(1)) + CCH; 4 charact1: = subs(l=exp(I*H),charact)=0; 460 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

charact1: = factor(simplify(convert(charact1,trig))); charact2: = subs(l=exp(I*H),charact)=0; charact2: = factor(simplify(convert(charact2,trig))); charact3: charact3: charact4: charact4: charact1: charact2:

= = = = = =

subs(l=exp(I*H),charact)=0; factor(simplify(convert(charact3,trig))); subs(l=exp(I*H),charact)=0; factor(simplify(convert(charact4,trig))); subs(AAH=AH,BBH=BH,CCH=CH,charact1); subs(AAH=AH,BBH=BH,CCH=CH,charact2);

eq6: = subs(H=v*h,charact1); eq7: = subs(H=v*h,charact2); solution: = solve({eq1i,eq2i,eq4},{b[0],b[1],b[2]}); assign(solution); h: = 1; eq6: = simplify(eq6); eq7: = simplify(eq7); solution2: = solve({eq6,eq7},{a}); assign(solution2); h: = 1; b[0]: = simplify(b[0]);

b0 :¼

2 cosðvÞ2 sinðvÞ þ v3 cosðvÞ2 cosðvÞ sinðvÞv2 þ cosðvÞv3 þ sinðvÞv2 2 sinðvÞ ð2 cosðvÞ2 sinðvÞ þ v3 cosðvÞ2 þ cosðvÞv3 2 sinðvÞÞv2


1 41 2 229 4 169453 2921123 þ v þ v þ v6 þ v8 15 4725 283500 1964655000 340540200000 1697050559 174666574507 v10 þ v12 þ 1930862934000000 1969480192680000000 2335855834422281 56749712127519331 v14 þ v16 þ 259321456970175600000000 62237149672842144000000000 1083684122457639105553 v18 þ 1172361188387274665280000000000

4 b[1]: = simplify(b[1]); b1 :¼

4ðcosðvÞv3 þ cosðvÞ2 sinðvÞ sinðvÞÞ ðcosðvÞv3 þ 2 sinðvÞ cosðvÞ 2 sinðvÞÞv2


16 584 2 31 4 259543 6 20497933 v þ v v þ v8 15 4725 3375 491163750 766215450000 621522329 30105814033 v10 þ v12 482715733500000 4923700481700000000 187673953738211 710910702715663 v14 þ v16 64830364242543900000000 5186429139403512000000000 1728544227315682013 v18 266445724633483515120000000000

4 b[2]: = simplify(b[2]); Chem. Modell., 2008, 5, 350–487 | 461 This journal is

c


b2 : =2 (4 sin(v) cos(v)3 + 2 cos(v)3 sin(v) v2 + 2 cos(v)2 sin(v) 2 cos(v)2 sin(v)v2 + 3v3 cos(v)2 4 sin(v) cos(v) cos(v) sin(v)v2 + 3 cos(v)v3 2 sin(v) + sin(v) v2)/ ((2 cos(v)2 sin(v) + v3 cos(v)2 + cos(v)v3 2 sin(v))v2) 4 b2t: = convert(taylor(b[2],v = 0,26),polynom); b2t :¼

26 1318 2 14687 4 2718047 6 701492321 v þ v v þ v8 15 1575 141750 327442500 1532430900000 927346549 972815720521 v10 þ v12 35756721000000 984740096340000000 3171075028973219 24802857234233993 v14 þ v16 43220242828362600000000 31118574836421072000000000 603704556959681910947 v18 1953935313978879110880000000000

4 a: = combine(a); a :¼

4 cosðvÞv3 sinð3vÞ þ 3 sinðvÞ cosðvÞv3 þ sinð2vÞ 2 sinðvÞ

4 at: = convert(taylor(a,v = 0,26),polynom); 16 2 584 4 31 6 259543 8 20497933 v þ v v þ v v10 15 4725 3375 491163750 766215450000 621522329 30105814033 v12 v14 þ 482715733500000 492370048170000000 187673953738211 710910702715663 v16 v18 þ 64830364242543900000000 5186429139403512000000000 1728544227315682013 v20 þ 266445724633483515120000000000

at :¼

4 4 plot([b[0]],v = 100..100,labels=[v,b_0],thickness=3,title=‘‘bahavior of the coefficient b_0’’); 4 plot([b[1]],v = 50..50,labels=[v,b_1],thickness=3,title=‘‘bahavior of the coefficient b_1’’); 4 plot([b[2]],v = 50..50,labels=[v,b_2],thickness=3,title=‘‘bahavior of the coefficient b_2’’); 4 plot([a],v = 50..50,labels=[v,a_],thickness=3,title=‘‘bahavior of the coefficient a_’’); 4 4 restart; 4 qnp2: = convert(taylor(q(x+2*h),h = 0,13),polynom); 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(xh),h = 0,13),polynom); 4 qnm2: = convert(taylor(q(x2*h),h = 0,13),polynom); 4 snp2: = convert(taylor(diff(q(x+2*h),x$2),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$2),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$2),h = 0,13),polynom); 4 snm2: = convert(taylor(diff(q(x2*h),x$2),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$2); 4 4 b[0]: = subs(v = v*h,1/15+41/4725*v^2+229/283500*v^4+169453/196 4655000*v^6+2921123/340540200000*v^8+1697050559/19308629340 00000*v^10+174666574507/1969480192680000000*v^12+2335855834422281/ 259321456970175600000000*v^14); 462 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 b[1]: = subs(v = v*h,16/15584/4725*v^2+31/3375*v^4259543/ 491163750*v^6+20497933/766215450000*v^8621522329/ 482715733500000*v^10+30105814033/4923700481700000 00*v^12187673953738211/64830364242543900000000*v^14); 4b[2]: = subs(v = v*h,26/151318/1575*v^2+14687/141750*v^42718047/ 327442500*v^6+701492321/1532430900000*v^8927346549/ 35756721000000*v^10+972815720521/9847400963400000 00*v^123171075028973219/43220242828362600000000*v^14); 4 a: = subs(v = v*h, 16/15*v^2+584/4725*v^431/3375*v^6+259543/ 491163750*v^820497933/766215450000*v^10+621522329/ 482715733500000*v^1230105814033/ 492370048170000000*v^14+187673953738211/6483036 4242543900000000*v^16710910702715663/ 5186429139403512000000000*v^18+1728544227315682013/ 266445724633483515120000000000*v^20); 4 4 lte: = simplify(qnp2+a*qnp1(2+2*a)*qn+a*qnm1+qnm2 h^2*(b[0]*(snp2+snm2) + b[1]*(snp1+snm1) + b[2]*sn));

Ite :¼

1728544227315682013 v20 h22 ðDð2Þ ÞðqÞðxÞ 266445724633483515120000000000 63307 v6 h16 ðDð10Þ ÞðqÞðxÞ 58939650000 187673953738211 v16 h20 ðDð4Þ ÞðqÞðxÞ þ 777964370910526800000000 1728544227315682013 v20 h24 ðDð4Þ ÞðqÞðxÞ þ 3197348695601802181440000000000 2 4 8 ð4Þ 26941567501 v h ðD ÞðqÞðxÞ h22 v8 ðDð14Þ ÞðqÞðxÞ 945 183509213247360000000 6691 v8 h12 ðDð4Þ ÞðqÞðxÞ 392931000 187673953738211 þ v16 h28 ðDð12Þ ÞðqÞðxÞ 15526924100380658085120000000000 45629 2 v4 h14 ðDð10Þ ÞðqÞðxÞ ðDð8Þ ÞðqÞðxÞh8 4286520000 945 19 79 ðDð10Þ ÞðqÞðxÞh10 ðDð12Þ ÞðqÞðxÞh12 þ 56700 3742200 629013647 v12 h18 ðDð6Þ ÞðqÞðxÞ 5251947180480000 187673953738211 v16 h26 ðDð10Þ ÞðqÞðxÞ þ 117628212881671652160000000000 89 v2 h12 ðDð10Þ ÞðqÞðxÞ 850500 5333969437686359 v14 h24 ðDð10Þ ÞðqÞðxÞ 46677862254631608000000000 12383845068989 257 v14 h20 ðDð6Þ ÞðqÞðxÞ h14 ðDð14Þ ÞðqÞðxÞ 1037285827880702400000 224532000 2911 130259 v4 h12 ðDð8Þ ÞðqÞðxÞ v4 h16 ðDð12Þ ÞðqÞðxÞ 17860500 282910320000


c


6973 21377 h16 v2 ðDð14Þ ÞðqÞðxÞ h18 v4 ðDð14Þ ÞðqÞðxÞ 47151720000 1543147200000 5250313 h20 v6 ðDð14Þ ÞðqÞðxÞ 3564670032000000 14847506411 h24 v10 ðDð14Þ ÞðqÞðxÞ 988126532870400000000 178888678109201 h26 v12 ðDð14Þ ÞðqÞðxÞ 117923020432753536000000000 72476627287717501 h28 v14 ðDð14Þ ÞðqÞðxÞ 470512851526686608640000000000 11 6 10 ð4Þ 3791 v h ðD ÞðqÞðxÞ v6 h12 ðDð6Þ ÞðqÞðxÞ 18900 39293100 675271 5753947 v6 h14 ðDð8Þ ÞðqÞðxÞ v6 h18 ðDð12Þ ÞðqÞðxÞ 47151720000 118822334400000 299099 303736663 v8 h14 ðDð6Þ ÞðqÞðxÞ v8 h16 ðDð8Þ ÞðqÞðxÞ 24518894400 193086293400000 5095698107 v8 h18 ðDð10Þ ÞðqÞðxÞ 46340710416000000 14844935597 v8 h20 ðDð12Þ ÞðqÞðxÞ 3058486887456000000 248389 4398091 v10 h14 ðDð4Þ ÞðqÞðxÞ v10 h16 ðDð6Þ ÞðqÞðxÞ 55724760000 3861725868000 339833503 v10 h18 ðDð8Þ ÞðqÞðxÞ 2206700496000000 7723799927 v10 h20 ðDð10Þ ÞðqÞðxÞ 695110656240000000 7857010321 v10 h22 ðDð12Þ ÞðqÞðxÞ 15858820897920000000 7597 71 2 10 ð8Þ v2 h14 ðDð12Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 1571724000 56700 4 2 8 ð6Þ 119170973 v h ðD ÞðqÞðxÞ v12 h16 ðDð4Þ ÞðqÞðxÞ 945 386172586800000 7877661178427 v12 h20 ðDð8Þ ÞðqÞðxÞ 496309008555360000000 33605167988257 v12 h22 ðDð10Þ ÞðqÞðxÞ 29778540513321600000000 2894937696283 v12 h24 ðDð12Þ ÞðqÞðxÞ 57805402172918400000000 15058968179 v14 h18 ðDð4Þ ÞðqÞðxÞ 393896038536000000 9935148184603567 v14 h22 ðDð8Þ ÞðqÞðxÞ 6223714967284214400000000 79709526694381819 v14 h26 ðDð12Þ ÞðqÞðxÞ 15683761717556220288000000000 187673953738211 þ v16 h18 ðDð2Þ ÞðqÞðxÞ 64830364242543900000000 187673953738211 þ v16 h22 ðDð6Þ ÞðqÞðxÞ 23338931127315804000000000 187673953738211 þ v16 h24 ðDð8Þ ÞðqÞðxÞ 1306980143129685024000000000


c


710910702715663 v18 h20 ðDð2Þ ÞðqÞðxÞ 5186429139403512000000000 710910702715663 v18 h22 ðDð4Þ ÞðqÞðxÞ 62237149672842144000000000 710910702715663 v18 h24 ðDð6Þ ÞðqÞðxÞ 1867114490185264320000000000 710910702715663 v18 h26 ðDð8Þ ÞðqÞðxÞ 104558411450374801920000000000 710910702715663 v18 h28 ðDð10Þ ÞðqÞðxÞ 9410257030533732172800000000000 710910702715663 v18 h30 ðDð12Þ ÞðqÞðxÞ 1242153928030452646809600000000000 1728544227315682013 v20 h26 ðDð6Þ ÞðqÞðxÞ þ 95920460868054065443200000000000 1728544227315682013 v20 h28 ðDð8Þ ÞðqÞðxÞ þ 5371545808611027664819200000000000 1728544227315682013 v20 h30 ðDð10Þ ÞðqÞðxÞ þ 483439122774992489833728000000000000 1728544227315682013 v20 h32 ðDð12Þ ÞðqÞðxÞ þ 63813964206299008658052096000000000000 17 4 10 ð6Þ v h ðD ÞðqÞðxÞ 11340

4 coeff(lte,h,8);

2 4 ð4Þ 2 4 2 ð6Þ v ðD ÞðqÞðxÞ ðDð8Þ ÞðqÞðxÞ v ðD ÞðqÞðxÞ 942 945 945

4 restart; 4 AH: = 1+v^2*b[0]; 4 BH: = a+v^2*b[1]; 4 CH: = 22*a+v^2*b[2]; 4 b[0]: = (2*cos(v)^2*sin(v) + cos(v)^2*v^3+v^3*cos(v)v^2*cos(v)*sin(v) 2*sin(v)+v^2*sin(v))/(2*cos(v)^2*sin(v)+cos(v)^2*v^3+v^3*cos(v)2*sin(v))/v^2; 4 4 b[1]: = 4*(cos(v)^2*sin(v) + v^3*cos(v)sin(v))/(2*cos(v)*sin(v) + v^3*cos(v)2*sin(v))/v^2; 4 b[2]: = 2*(2*v^2*cos(v)^3*sin(v) + 4*cos(v)^3*sin(v) + 2*cos(v)^2*sin(v) + 3*cos(v)^2*v^32*v^2*cos(v)^2*sin(v) + 3*v^3*cos(v)4*cos(v)*sin(v)v^2*cos(v)*sin(v)2*sin(v) + v^2*sin(v))/(2*cos(v)^2*sin(v) + cos(v)^2*v^3+v^3*cos(v)2*sin(v))/v^2; 4 a: = (sin(3*v) + 3*sin(v)4*v^3*cos(v))/(sin(2*v) + v^3*cos(v)2*sin(v)); 4 AH; 4 BH; 4 CH; 4 P1H: = combine(2*AH2*BH+CH); P1H : =(14v2 8v2 cos(v) + 12v2 cos(3v) 4v5 sin(2v) 17v2 cos(2v) + 2v2 cos(4v) + v5 sin(3v) + 2v5 sin(v) v5 sin(5v) + 2v5 sin(4v) + v2 cos(6v) 4v2 cos(5v))/ (2 cos(v) cos(5v) + 2 sin(4v)v3 4v3 sin(2v) 8 cos(2v) + 2 cos(4v) + 3 cos(3v) + 6 + 3v6 cos(v) + v6 cos(3v) + 2v6 cos(2v) + 2v6) 4 P2H: = combine(12*AH2*CH); Chem. Modell., 2008, 5, 350–487 | 465 This journal is

c


P2H : =(52v2 48v2 cos(v) + 40v2 cos(3v) 24v5 sin(2v) 30v2 cos(2v) 20v2 cos(4v) + 14v5 sin(3v) + 12v5 sin(v) + 2v5 sin(5v) 4v5 sin(4v) 2v2 cos(6v) + 8v2 cos(5v))/(2 cos(v) cos(5v) + 2 sin(4v)v3 4v3 sin(2v) 8 cos(2v) + 2 cos(4v) + 3 cos(3v) + 6 + 3v6 cos(v) + v6 cos(3v) + 2v6 cos(2v) + 2v6) 4P3H: = combine(2*AH+2*BH+CH); P3H : =(14v2 8v2 cos(v) + 12v2 cos(3v) 4v5 sin(2v) 17v2 cos(2v) + 2v2 cos(4v) + v5 sin(3v) + 2v5 sin(v) v5 sin(5v) + 2v5 sin(4v) + v2 cos(6v) 4v2 cos(5v))/(2 cos(v) cos(5v) + 2 sin(4v)v3 4v3 sin(2v) 8 cos(2v) + 2 cos(4v) + 3 cos(3v) + 6 + 3v6 cos(v) + v6 cos(3v) + 2v6 cos(2v) + 2v6) 4 NH: = combine(P2H^24*P1H*P3H); NH : =(7040v4 9216v4 cos(v) 896v4 cos(2v) + 8192v4 cos(3v) 7680v7 sin(2v) + 5376v7 sin(v) + 5632v7 sin(3v) 1536v7 sin(4v) 8704v4 cos(4v) 1536v7 sin(5v) + 832v4 cos(6v) + 4096v4 cos(5v) 1920v7 sin(7v) 1280v10 cos(v) + 256v10 cos(7v) + 256v10 cos(3v) + 832v10 128v7 sin(9v) + 512v10 cos(2v) 768v10 cos(4v) + 768v7 sin(8v) 512v10 cos(6v) + 768v10 cos(5v) + 64v4 cos(10v) + 1664v4 cos(8v) 64v10 cos(8v) + 2560v7 sin(6v) 2560v4 cos(7v) 512v4 cos(9v))/(154 56 cos(v) 238 cos(2v) + 112 cos(3v) + 8v9 sin(6v) 112v3 sin(2v) + 56v3 sin(v) + 112 sin(4v)v3 + 104 cos(4v) 32v3 sin(5v) 19 cos(6v) 80 cos(5v) 24v6 cos(2v) + 30v6 + 28 cos(7v) + cos(10v) 2 cos(8v) 4 cos(9v) + 22v12 24 cos(4v) v6 24v9 sin(2v) + 40v12 cos(v) + 20v12 cos(3v) + 31v12 cos(2v) + 8v3 sin(8v) + 4v9 sin(5v) 12v9 sin(3v) + 24v6 cos(6v) + 10v12 cos(4v) 12v9 sin(v) + 4v9 sin(7v) + 4v12 cos(5v) 4v3 sin(9v) 6v6 cos(8v) + 20v3 sin(7v) 48v3 sin(6v) + v12 cos(6v)) 4 plot([P1H], v = 0..10, color=[red], style=[line], thickness=3, title=‘‘Stability Polynomias for the New Method - P1H’’); 4 plot([P2H], v = 0..10, color=[blue], style=[line], thickness=3, title= ‘‘Stability Polynomias for the New Method - P2H’’); 4 plot([P3H], v = 0..10, color=[black], style=[line], thickness=3, title= ‘‘Stability Polynomias for the New Method - P3H’’); 4 plot([NH], v = 0..10, color=[green], style=[line], thickness=3, title= ‘‘Stability Polynomias for the New Method - NH’’); 4 restart; 4 final: = k[n+2]+a*k[n+1](2+2*a)*k[n]+a*k[n1] +k[n2]+ H^2*(b[0]*(k[n+2]+k[n2]) + b[1]*(k[n+1]+k[n1]) + b[2]*k[n]); 4 AH: = coeff(final,k[n+2]); 4 BH: = coeff(final,k[n+1]); 4 CH: = coeff(final,k[n]); 4 b[0]: = (2*cos(v)^2*sin(v) + cos(v)^2*v^3+v^3*cos(v)v^2*cos(v)*sin(v) 2*sin(v)+v^2*sin(v))/(2*cos(v)^2*sin(v)+cos(v)^2*v^3+v^3*cos(v)-2*sin(v))/v^2; 4 b[1]: = 4*(cos(v)^2*sin(v) + v^3*cos(v)sin(v))/(2*cos(v)*sin(v) + v^3*cos(v)-2*sin(v))/v^2; 4 b[2]: = 2*(2*v^2*cos(v)^3*sin(v) + 4*cos(v)^3*sin(v) + 2*cos(v)^2*sin(v) + 3*cos(v)^2*v^32*v^2*cos(v)^2*sin(v) + 3*v^3*cos(v) 4*cos(v)*sin(v)v^2*cos(v)*sin(v)2*sin(v) + v^2*sin(v))/(2*cos(v)^2*sin(v) + cos(v)^2*v^3+v^3*cos(v)2*sin(v))/v^2; 4 a: = (-sin(3*v) + 3*sin(v)4*v^3*cos(v))/(sin(2*v) + v^3*cos(v)2*sin(v)); 4 P1H: = simplify(2*AH2*BH+CH); 4 P2H: = simplify(12*AH2*CH); 4 P3H: = simplify(2*AH+2*BH+CH); 4 NH: = combine(P2H^24*P1H*P3H);

Appendix I 4 restart; 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$2); 466 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]);

=

final : =e(v(x+2h)) + ae(v(x+h)) (2+2a)e(vx) + ae(v(xh)) + e(v(x2h)) = h2 (b0(v2e(v(x+2h)) + v2e(v(x2h))) + b1v2e(v(x+h)) + v2e(v(xh))) + b2v2e(vx)) 4 final: = combine(final/exp(v*x)); final : =(ae(v(x+h)) + (2 2a)e(vx) + ae(v(xh)) + e(v(x+2h)) + e(v(x2h))) e(vx) = e(vx)h2 (b0(v2e(v(x+2h)) + v2e(v(x2h))) + b1(v2e(v(x+h)) + v2e(v(xh))) + b2v2e(vx)) 4 final: = expand(final); final :¼aeðvhÞ 2 2a þ ¼h2 b0 v2 ðeðvhÞ Þ2 þ


h2 b0 v2 ðeðvhÞ Þ

þ h2 b1 v2 eðvhÞ þ 2


4 final: = simplify(convert(final,trig)); final : =2 (2a cosh(vh)3 + 2a cosh(vh)2 sinh(vh) a cosh(vh) 6 cosh(vh)2 4 cosh(vh) sinh(vh) + 2 2a cosh(vh)2 2a cosh(vh) sinh(vh) + a + 4 cosh(vh)4 + 4 cosh(vh)3 sinh(vh))/(cosh(vh) + sinh(vh))2 = h2v2(8b0 cosh(vh)4 + 8b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh)2 4b0 cosh(vh) sinh(vh) + 2b0 + 4b1 cosh(vh)3 + 4b1 cosh(vh)2 sinh(vh) 2b1 cosh(vh) + 2b2 cosh(vh)2 + 2b2 cosh(vh) sinh(vh) b2)/ (cosh(vh) + sinh(vh))2 4 4 y[n]: = exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final1: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] = h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final1 : =e(v(x+2h)) + ae(v(x+h)) (2+2a)e(vx) + ae(v(xh)) + e(v(x2h)) = h2 (b0 (v2e(v(x+2h)) + v2e(v(x2h))) + b1v2e(v(x+h)) + v2e(v(xh))) + b2v2e(vx)) 4 final1: = combine(final1/exp(v*x)); final1 : =(ae((xh)v) + (2 2a)e(vx) + ae((x+h)v) + e((x2h)v) + e((x+2h)v)) e(vx) = h2 (b1(v2e((xh)v) + v2e((x+h)v)) + b0(v2e((x2h)v) + v2e((x+2h)v)) + b2v2e(vx)) e(vx))e(vx) Chem. Modell., 2008, 5, 350–487 | 467 This journal is

c


4 final1: = expand(final1); final1 :¼aeðvhÞ 2 2a þ ¼h2 b0 v2 ðeðvhÞ Þ2 þ



þ h2 b1 v2 eðvhÞ þ


4 final1: = simplify(convert(final1,trig)); final1 : =2 (2a cosh(vh)3 + 2a cosh(vh)2 sinh(vh) a cosh(vh) 6 cosh(vh)2 4 cosh(vh) sinh(vh) + 2 2a cosh(vh)2 2a cosh(vh) sinh(vh) + a + 4 cosh(vh)4 + 4 cosh(vh)3 sinh(vh))/(cosh(vh) + sinh(vh))2 = h2v2(8b0 cosh(vh)4 + 8b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh)2 4b0 cosh(vh) sinh(vh) + 2b0 + 4b1 cosh(vh)3 + 4b1 cosh(vh)2 sinh(vh) 2b1 cosh(vh) + 2b2 cosh(vh)2 + 2b2 cosh(vh) sinh(vh) b2)/ (cosh(vh) + sinh(vh))2 4 4 eq1: = final; 4 eq1a: = final1; 4 4 simplify(eq1eq1a); 4 eq1i: = subs(v = I*v,eq1); 4 4 y[n]: = x*exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = (x+h)*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = (xh)*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = (x+2*h)*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = (x2*h)*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final2: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] = h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final2 : =(x+2h) e(v(x+2h))+a(x+h)e(v(x+h)) (2 + 2a)x e(vx) + a(x h)e(v(xh)) + (x 2h)e(v(x2h)) = h2 (b0 (2ve(v(x+2h)) + (x+2h)v2e(v(x+2h)) + 2ve(v(x2h))+(x2h)v2e(v(x2h))) +b1 (2ve(v(x+h)) + (x+h) v2e(v(x+h)) +2ve(v(xh))+(xh)v2e(v(xh))) + b2 (2ve(vx) + xv2 e(vx))) 4 final2: = combine(final2/exp(v*x)); final2 : =e(vx) ((x + 2h) e(v(x + 2h)) + a(x + h)e(v(x+h)) (2 + 2a)x e(vx) + a(x h)e(v(xh))+ (x 2h)e(v(x2h))) = e(vx) h2 (b0 (2ve(v(x + 2h)) + (x + 2h)v2e(v(x +2h)) + 2ve(v(x2h))+(x2h)v2e(v(x2h))) + b1 (2ve(v(x+h)) + (x+h) v2e(v(x+h)) +2ve(v(xh))+(xh)v2e(v(xh))) + b2 (2ve(vx) + xv2 e(vx))) 4 final2: = expand(final2); final2 :¼ðeðvhÞ Þ2 x þ 2ðeðvhÞ Þ2 h þ aeðvhÞ x þ aeðvhÞ h 2x 2xa þ þ

x ðeðvhÞ Þ2 2h2 b0 v

2h ðeðvhÞ Þ2

ax ah eðvhÞ eðvhÞ

¼ 2h2 b0 vðeðvhÞ Þ2 þ h2 b0 v2 ðeðvhÞ Þ2 x þ 2h3 b0 v2 ðeðvhÞ Þ2

h2 b0 v2 x

2h3 b0 v2

þ 2h2 b1 veðvhÞ þ h2 b1 v2 eðvhÞ x ðeðvhÞ Þ2 2h2 b1 v h2 b1 v2 x h3 b1 v2 þ h3 b1 v2 eðvhÞ þ ðvhÞ þ ðvhÞ ðvhÞ þ 2h2 b2 v þ h2 b2 xv2 e e e þ

ðeðvhÞ Þ2

þ

ðeðvhÞ Þ2


c


4 final2: = simplify(convert(final2,trig)); final2 : =2 (2x + 2ah cosh(vh)3 8h cosh(vh)2 xa cosh(vh) 4x cosh(vh) sinh(vh) 2ah cosh(vh) + 4x cosh(vh)3 sinh(vh) + 8h cosh(vh)3 sinh(vh) + xa + 2xa cosh(vh)2 sinh(vh) + 2ah cosh(vh)2 sinh(vh) + 2xa cosh(vh)3 2xa cosh(vh)2 ah sinh(vh) + 4x cosh(vh)4 + 8h cosh(vh)4 6x cosh(vh)2 2xa cosh(vh) sinh(vh) 4h cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 = h2 v(b2 xv + 4b0 2b2 + 16b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh) sinh(vh) 4b0 vx cosh(vh) sinh(vh) + 8b1 cosh(vh)2 sinh(vh) 16h b0 v cosh(vh)2 8b0 vx cosh(vh)2 + 4b2 cosh(vh) sinh(vh) + 4h b1 v cosh(vh)2 sinh(vh) + 16b0 cosh(vh)4 +8b1 cosh(vh)3 4b1 cosh(vh) + 4b2 cosh(vh)2 16b0 cosh(vh)2 8h b0 v cosh(vh) sinh(vh) + 2b0 vx + 8b0 vx cosh(vh)4 + 16h b0 v cosh(vh)4 + 4b1 vx cosh(vh)3 + 4h b1 v cosh(vh)3 2b1 vx cosh(vh) 4h b1 v cosh(vh) 2h b1 v sinh(vh) + 2b2 xv cosh(vh)2 + 8b0 vx cosh(vh)3 sinh(vh) + 16h b0 v cosh(vh)3 sinh(vh) + 4b1 vx cosh(vh)2 sinh(vh) + 2b2 xv cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 4 4 y[n]: = x*exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = (x+h)*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = (xh)*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = (x+2*h)*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = (x2*h)*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final3: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] = h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final3 : =(x + 2h) e(v(x+2h)) + a(x + h)e(v(x+h)) (2 + 2a)x e(vx) + a(x h) e(v(xh))+ (x 2h)e(v(x2h)) = h2 (b0 ( 2ve(v(x+2h)) + (x+2h)v2e(v(x+2h)) 2ve(v(x2h))+(x 2h)v2e(v(x2h))) +b1 ( 2v e(v(x+h)) + (x + h) v2e(v(x+h)) 2v e(v(xh))+(x h)v2e(v(xh))) + b2 ( 2v e(vx) + xv2 e(vx))) 4 final3: = combine(final3/exp(v*x)); final3 : =((x + 2h) e((x2h)v) + a(x + h) e((xh)v) (2 + 2a)x e(vx) + a(x h) e((x+h)v)+ (x 2h) e((x+2h)v)) e(vx) = h2 (b1 ((x+h)v2 e((xh)v/it>) + (xh) v2e((x+h)v) 2ve((xh)v) 2v e((x+h)v)) + b0((x+2h)v2e((x2h)v) + (x2h)v2e((x+2h)v) 2v e((x2h)v) 2v e((x+2h)v)) + b2 (2v e(vx) + xv2 e(vx))) e(vx) 4 final3: = expand(final3); final3 :¼

x ðeðvhÞ Þ2

þ

2h ðeðvhÞ Þ2

þ

ax ah þ 2x 2xa þ aeðvhÞ x aeðvhÞ h eðvhÞ eðvhÞ

h2 b1 v2 x h3 b1 v2 þ ðvhÞ þ h2 b1 v2 eðvhÞ x h3 b1 v2 eðvhÞ eðvhÞ e 2h2 b1 v h2 b0 v2 x 2h3 b0 v2 2 ðvhÞ ðvhÞ 2h b1 ve þ þ þ h2 b0 v2 ðeðvhÞ Þ2 x e ðeðvhÞ Þ2 ðeðvhÞ Þ2 2h2 b0 v 2h2 b0 vðeðvhÞ Þ2 2h2 b2 v þ h2 b2 xv2 2h3 b0 v2 ðeðvhÞ Þ2 ðeðvhÞ Þ2 þ ðeðvhÞ Þ2 x 2ðeðvhÞ Þ2 h ¼

4 final3: = simplify(convert(final3,trig)); final3 : =2 (2x 2ah cosh(vh)3 + 8h cosh(vh)2 xa cosh(vh) 4x cosh(vh) sinh(vh) + 2ah cosh(vh) + 4x cosh(vh)3 sinh(vh) 8h cosh(vh)3 sinh(vh) + xa + 2xa


c


cosh(vh)2 sinh(vh) 2ah cosh(vh)2 sinh(vh) + 2xa cosh(vh)3 2xa cosh(vh)2 + ah sinh(vh) + 4x cosh(vh)4 8h cosh(vh)4 6x cosh(vh)2 2xa cosh(vh) sinh(vh) + 4h cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 = h2 v(b2 xv 4b0 2b2 16b0 cosh(vh)3 sinh(vh) + 8b0 cosh(vh) sinh(vh) 4b0 vx cosh(vh) sinh(vh) 8b1 cosh(vh)2 sinh(vh) + 16h b0 v cosh(vh)2 8b0 vx cosh(vh)2 + 4b2 cosh(vh) sinh(vh) 4h b1 v cosh(vh)2 sinh(vh) 16b0 cosh(vh)4 8b1 cosh(vh)3 + 4b1 cosh(vh) 4b2 cosh(vh)2 + 16b0 cosh(vh)2 + 8h b0 v cosh(vh) sinh(vh) + 2b0 vx + 8b0 vx cosh(vh)4 16h b0 v cosh(vh)4 + 4b1 vx cosh(vh)3 4h b1 v cosh(vh)3 2b1 vx cosh(vh) + 4h b1 v cosh(vh) + 2h b1 v sinh(vh) + 2b2 xv cosh(vh)2 + 8b0 vx cosh(vh)3 sinh(vh) 16h b0 v cosh(vh)3 sinh(vh) + 4b1 vx cosh(vh)2 sinh(vh) + 2b2 xv cosh(vh) sinh(vh))/(cosh(vh) + sinh(vh))2 4 4 eq2: = final2; 4 eq2a: = final3; 4 eq2i: = subs(v = I*v,eq2); 4 eq3i: = subs(v = I*v,eq2a); 4 4 4 y[n]: = x^2*exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = (x+h)^2*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = (xh)^2*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = (x+2*h)^2*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = (x2*h)^2*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final4: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] = h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final4 : =(x + 2h)2 e(v(x+2h)) + a(x + h)2 e(v(x+h)) (2 + 2a) x2 e(vx) + a(x h)2 e(v(xh)) + (x 2h)2 e(v(x2h)) = h2 (b0(2e(v(x+2h)) + 4(x + 2h)v e(v(x+2h)) + (x + 2h)2 v2e(v(x+2h)) + 2e(v(x2h)) + 4(x 2h)v e(v(x2h)) + (x 2h)2v2e(v(x2h))) + b1 (2e(v(x+h)) + 4 (x + h) v e(v(x+h)) + (x + h)2 v2e(v(x+h)) + 2e(v(xh)) + 4 (x h) v e(v(xh)) + (x h)2 v2e(v(xh))) + b2 (2e(vx) + 4xv e(vx) + x2 v2e(vx))) 4 final4: = combine(final4/exp(v*x)); final4 : =e(vx) ((x + 2h)2 e(v(x+2h)) + a (x + h)2 e(v(x+h)) (2 + 2a)x2 e(vx) + a(x h)2 e(v(xh)) + (x 2h)2 e(v(x2h))) = h2 (b1 ((4x + 4h)v e(v(x+h)) + (x+h)2 v2e(v(x+h)) + (4x 4h)v e(v(xh)) + (x h)2 v2e(v(xh)) + 2e(v(x+h))+2e(v(xh))) + b0((4x+8h)v e(v(x+2h)) + (x + 2h)2 v2e(v(x+2h)) + ( 8h + 4x)v e(v(x2h))+(x2h)2 v2e(v(x2h))+2e(v(x+2h))+2e(v(x2h))) + b2(2e(vx) + 4xv e(vx) + x2 v2e(vx))) e(vx) 4 final4: = expand(final4); final4 :¼ðeðvhÞ Þ2 x2 þ 4ðeðvhÞ Þ2 xh þ 4ðeðvhÞ Þ2 h2 þ aeðvhÞ x2 þ 2aeðvhÞ xh þ aeðvhÞ h2 2x2 2x2 a þ þ

4h2 ðeðvhÞ Þ2

ax2 2axh ah2 x2 4xh ðvhÞ þ ðvhÞ þ ðvhÞ ðvhÞ e e e ðe Þ2 ðeðvhÞ Þ2

¼ 4h2 b1 veðvhÞ x þ

þ 2h2 b0 ðeðvhÞ Þ2 þ

h4 b1 v2 2h3 b1 v2 x 2 þ 2h b þ 2h2 b1 eðvhÞ 2 eðvhÞ eðvhÞ

2h2 b1 2h2 b0 þ þ h2 b1 v2 eðvhÞ x2 þ 4h3 b1 veðvhÞ eðvhÞ ðeðvhÞ Þ2


c


4h2 b1 vx 4h3 b1 v h2 b1 v2 x2 ðvhÞ þ eðvhÞ eðvhÞ e 2 2 2 2 2 ðvhÞ 2 3 ðvhÞ 2 þ 4h b2 xv þ h b2 x v þ 4h b0 vðe Þ x þ 8h b0 vðe Þ þ 2h3 b1 v2 eðvhÞ x þ h4 b1 v2 eðvhÞ þ

þ h2 b0 v2 ðeðvhÞ Þ2 x2 þ 4h3 b0 v2 ðeðvhÞ Þ2 x þ 4h4 b0 v2 ðeðvhÞ Þ2 2

þ

4h b0 vx ðeðvhÞ Þ2

2

þ

2 2

h b0 v x ðeðvhÞ Þ2

3

2

4h b0 v x ðeðvhÞ Þ2

4

þ

8h3 b0 v ðeðvhÞ Þ2

2

4h b0 v

ðeðvhÞ Þ2

4 final4: = simplify(convert(final4,trig)); final4 : =2 (4h2 + 2x2 + 4axh cosh(vh)2 sinh(vh) + x2a + 2ah2 cosh(vh)3 + 4x2 cosh(vh)3 sinh(vh) + 16xh cosh(vh)4 + 16h2 cosh(vh)3 sinh(vh) + 2x2a cosh(vh)3 4x2 cosh(vh) sinh(vh) 2x2a cosh(vh)2 x2a cosh(vh) ah2 cosh(vh) + 4x2 cosh(vh)4 + 16h2 cosh(vh)4 6x2 cosh(vh)2 2axh sinh(vh) + 16xh cosh(vh)3 sinh(vh) + 2x2a cosh(vh)2 sinh(vh) + 4axh cosh(vh)3 + 2ah2 cosh(vh)2 sinh(vh) 2x2a cosh(vh) sinh(vh) 4axh cosh(vh) 16xh cosh(vh)2 8h2 cosh(vh) sinh(vh) 8xh cosh(vh) sinh(vh) 16h2 cosh(vh)2)/(cosh(vh) + sinh(vh))2=h2(4b2xv + 4b0 2b2 + 16b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh) sinh(vh) 16b0vx cosh(vh) sinh(vh) + 8b1 cosh(vh)2 sinh(vh) 64hb0v cosh(vh)2 32b0vx cosh(vh)2 + 4b2 cosh(vh) sinh(vh) 16h2b0v2 cosh(vh) sinh(vh) 16hb0v2x cosh(vh) sinh(vh) 32hb0v2x cosh(vh)2 32h2b0v2 cosh(vh)2 + 16hb1v cosh(vh)2 sinh(vh) b2x2v2 4b0v2x2 cosh(vh) sinh(vh) 8b0v2x2 cosh(vh)2 +16b0 cosh(vh)4 + 8b1 cosh(vh)3 4b1 cosh(vh) + 4b2 cosh(vh)2 b0 cosh(vh)2 8hb1v2x cosh(vh) + 32h2b0v2 cosh(vh)3 sinh(vh) 32hb0v cosh(vh) sinh(vh) + 8b0vx + 32b0vx cosh(vh)4 + 64hb0v cosh(vh)4 + 16b1vx cosh(vh)3 + 16hb1v cosh(vh)3 8b1vx cosh(vh) 16hb1v cosh(vh) 8hb1v sinh(vh) + 8b2xv cosh(vh)2 + 32b0vx cosh(vh)3 sinh(vh) + 64hb0v cosh(vh)3 sinh(vh) + 16b1vx cosh(vh)2 sinh(vh) + 8b2xv cosh(vh) sinh(vh) + 8h2b0v2 + 2b0v2x2 + 4h2b1v2 cosh(vh)2 sinh(vh) 2h2b1v2 cosh(vh) +4h2b1v2 cosh(vh)3 + 32h2b0v2 cosh(vh)4 2b1v2x2 cosh(vh) + 2b2x2v2 cosh(vh)2 + 4b1v2x2 cosh(vh)3 + 8b0v2x2 cosh(vh)4 4hb1v2x sinh(vh) + 8hb1v2x cosh(vh)3 + 8hb1v2x cosh(vh)2 sinh(vh) + 2b2x2v2 cosh(vh) sinh(vh) + 4b1v2x2 cosh(vh)2 sinh(vh) + 8b0v2x2 cosh(vh)3 sinh(vh) + 32hb0v2x cosh(vh)4 + 32hb0v2x cosh(vh)3 sinh(vh))/(cosh(vh) + sinh(vh))2 4 4 y[n]: = x^2*exp(v*x); 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = (x+h)^2*exp(v*(x+h)); 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = (xh)^2*exp(v*(xh)); 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = (x+2*h)^2*exp(v*(x+2*h)); 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = (x2*h)^2*exp(v*(x2*h)); 4 f[n2]: = diff(y[n2],x$2); 4 4 final5: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] = h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]); final5 : =(x+2h)2 e(v(x+ 2h)) + a(x + h)2 e(v(x+h)) (2 + 2a)x2 e(vx) + a(x h)2 e(v(xh)) + (x 2h)2 e(v(x 2h)) = h2(b0(2e(v(x + 2h)) 4(x + 2h)ve(v(x+ 2h)) + (x + 2h)2v2e(v(x+ 2h))+2e(v(x 2h))4(x 2h)ve(v(x 2h)) + (x 2h)2v2e(v(x 2h))) + b1(2e(v(x+h)) 4(x + h)ve(v(x+h)) + (x + h)2v2e(v(x+h)) + 2e(v(xh)) 4(x h)ve(v(xh)) + (x h)2v2e(v(xh))) + b2(2e(vx) 4xve(vx) + x2v2e(vx))) Chem. Modell., 2008, 5, 350–487 | 471 This journal is

c


4 final5: = combine(final5/exp(v*x)); final5 : =(a(x + h)2e((xh)v) (2 + 2a)x2e(vx) + a(x h)2e((x + h)v) + (x + 2h)2e((x 2h)v) + (x 2h)2e((x+ 2h)v))e(vx) = h2(b1((4x 4h)ve((xh)v) + (4x + 4h)ve((x+h)v) + (x + h)2v2e((xh)v) + (x h)2v2e((x+h)v) + 2e((xh)v) + 2e((x+h)v)) + b0((4x 8h)ve((x 2h)v) + (4x + 8h)ve((x+ 2h)v) + (x + 2h)2v2e((x 2h)v) + (x 2h)2v2e((x+ 2h)v) + 2e((x 2h)v) + 2e((x+ 2h)v)) + b2(2e(vx) 4xve(vx) + x2v2e(vx)))e(vx) 4 final5: = expand(final5); ax2 2axh ah2 þ þ 2x2 2x2 a þ aeðvhÞ x2 2aeðvhÞ xh þ aeðvhÞ h2 eðvhÞ eðvhÞ eðvhÞ x2 4xh 4h2 þ þ þ þ ðeðvhÞ Þ2 x2 4ðeðvhÞ Þ2 xh þ 4ðeðvhÞ Þ2 h2 ¼ 2 2 ðvhÞ ðvhÞ ðe Þ ðe Þ ðeðvhÞ Þ2 h4 b1 v2 2h3 b1 v2 x þ 2h2 b1 eðvhÞ þ 2h2 b0 ðeðvhÞ Þ2 4h2 b1 veðvhÞ x þ ðvhÞ þ 2h2 b2 þ eðvhÞ e 2h2 b1 2h2 b0 þ ðvhÞ þ þ h2 b1 v2 eðvhÞ x2 þ 4h3 b1 veðvhÞ 2h3 b1 v2 eðvhÞ x e ðeðvhÞ Þ2

final5 :¼

4h2 b1 vx 4h3 b1 vx h2 b1 v2 x2 ðvhÞ þ 4h2 b2 xv þ h2 b2 x2 v2 eðvhÞ e eðvhÞ 4h2 b0 vðeðvhÞ Þ2 x þ 8h3 b0 vðeðvhÞ Þ2 þ h2 b0 v2 ðeðvhÞ Þ2 x2

þ h4 b1 v2 eðvhÞ

4h3 b0 v2 ðeðvhÞ Þ2 x þ 4h4 b0 v2 ðeðvhÞ Þ2 þ

4h3 b0 v2 x ðeðvhÞ Þ2

þ

8h3 b0 v ðeðvhÞ Þ2

4h2 b0 vx ðeðvhÞ Þ2

þ

h2 b0 v2 x2 ðeðvhÞ Þ2

4h4 b0 v2 ðeðvhÞ Þ2

4 final5: = simplify(convert(final5,trig)); final5 : =2(4h2 + 2x2 4axh cosh(vh)2 sinh(vh) + x2a + 2ah2 cosh(vh)3 + 4x2 cosh(vh)3 sinh(vh) 16xh cosh(vh)4 + 16h2 cosh(vh)3 sinh(vh) + 2x2a cosh(vh)3 4x2 cosh(vh) sinh(vh) 2x2a cosh(vh)2 x2a cosh(vh) ah2 cosh(vh) + 4x2 cosh(vh)4 + 16h2 cosh(vh)4 6x2 cosh(vh)2 + 2axh sinh(vh) 16xh cosh(vh)3 sinh(vh) + 2x2a cosh(vh)2 sinh(vh) 4axh cosh(vh)3 + 2ah2 cosh(vh)2 sinh(vh) 2x2a cosh(vh) sinh(vh) + 4axh cosh(vh) + 16xh cosh(vh)2 8h2 cosh(vh) sinh(vh) + 8xh cosh(vh) sinh(vh) 16h2 cosh(vh)2)/(cosh(vh) + sinh(vh))2 = h2(4b2xv + 4b0 2b2 + 16b0 cosh(vh)3 sinh(vh) 8b0 cosh(vh) sinh(vh) + 16b0vx cosh(vh) sinh(vh) + 8b1 cosh(vh)2 sinh(vh) 64hb0v cosh(vh)2 + 32b0vx cosh(vh)2 + 4b2 cosh(vh) sinh(vh) 16h2b0v2 cosh(vh) sinh(vh) + 16hb0v2x cosh(vh) sinh(vh) + 32hb0v2x cosh(vh)2 32h2b0v2 cosh(vh)2 + 16hb1v cosh(vh)2 sinh(vh) b2x2v2 4b0v2x2 cosh(vh) sinh(vh) 8b0v2x2 cosh(vh)2 + 16b0 cosh(vh)4 +8b1 cosh(vh)3 4b1 cosh(vh) + 4b2 cosh(vh)2 16b0 cosh(vh)2 + 8hb1v2x cosh(vh) + 32h2b0v2 cosh(vh)3 sinh(vh) 32hb0v cosh(vh) sinh(vh) 8b0vx 32b0vx cosh(vh)4 + 64hb0v cosh(vh)4 16b1vx cosh(vh)3 + 16hb1v cosh(vh)3 + 8b1vx cosh(vh) 16hb1v cosh(vh) 8hb1v sinh(vh) 8b2xv cosh(vh)2 32b0vx cosh(vh)3 sinh(vh) + 64hb0v cosh(vh)3 sinh(vh) 16b1vx cosh(vh)2 sinh(vh) 8b2xv cosh(vh) sinh(vh) + 8h2b0v2 + 2b0v2x2 + 4h2b1v2 cosh(vh)2 sinh(vh) 2h2b1v2 cosh(vh) +4h2b1v2 cosh(vh)3 + 32h2b0v2 cosh(vh)4 2b1v2x2 cosh(vh) + 2b2x2v2 cosh(vh)2 + 4b1v2x2 cosh(vh)3 + 8b0v2x2 cosh(vh)4 + 4hb1v2x sinh(vh) 8hb1v2x cosh(vh)3 8hb1v2x cosh(vh) sinh(vh) + 2b2x2v2 cosh(vh) sinh(vh) + 4b1v2x2 cosh(vh)2 sinh(vh) + 8b0v2x2 cosh(vh)3 sinh(vh) 32hb0v2x cosh(vh)4 32hb0v2x cosh(vh)3 sinh(vh))/(cosh(vh) + sinh(vh))2 4 4 eq4: = final4; 4 eq4a: = final5; 4 eq4i: = subs(v = I*v,eq4); 472 | Chem. Modell., 2008, 5, 350–487 This journal is

c


4 eq5i: = subs(v = I*v,eq4a); 4 4 4 eq1ir: = combine(evalc(Re(eq1i))); 4 eq1ii: = combine(evalc(Im(eqli))); 4 eq2ir: = combine(evalc(Re(eq2i))); 4 eq2ii: = combine(evalc(Im(eq2i))); 4 eq3ir: = combine(evalc(Re(eq3i))); 4 eq3ii: = combine(evalc(Im(eq3i))); 4 eq4ir: = combine(evalc(Re(eq4i))); 4 eq4ii: = combine(evalc(Im(eq4i))); 4 eq5ir: = combine(evalc(Re(eq5i))); 4 eq5ii: = combine(evalc(Im(eq4i))); 4 simplify(eq2ireq3ir); 4 simplify(eq2ii+eq3ii); 4 simplify(eq4iieq5ii); 4 simplify(eq4ireq5ir); 4 4 y[n]: = x^n; 4 f[n]: = diff(y[n],x$2); 4 y[n+1]: = (x+h)^n; 4 f[n+1]: = diff(y[n+1],x$2); 4 y[n1]: = (xh)^n; 4 f[n1]: = diff(y[n1],x$2); 4 y[n+2]: = (x+2*h)^n; 4 f[n+2]: = diff(y[n+2],x$2); 4 y[n2]: = (x2*h)^n; 4 f[n2]: = diff(y[n2],x$2); 4 final6: = y[n+2]+a*y[n+1](2+2*a)*y[n]+a*y[n1]+y[n2] h^2*(b[0]*(f[n+2]+f[n2]) + b[1]*(f[n+1]+f[n1]) + b[2]*f[n]);

=

final6 :¼ðx þ 2hÞn þ aðx þ hÞn ð2 þ 2aÞxn þ aðx hÞn þ ðx 2hÞn 2

¼h þb1

b0

ðx þ 2hÞn n2 ðx þ 2hÞ2


þ


! ðx 2hÞn n

ðx 2hÞ2 ! n 2 ! ðx þ hÞn n2 ðx þ hÞn n ðx hÞn n2 ðx hÞn n x n xn n þ þ b 2 x2 x2 ðx þ hÞ2 ðx hÞ2 ðx þ hÞ2 ðx hÞ2

4 n: = 0; n : =0 4 eq7: = simplify(final6); eq7 : =0=0 4 n: = 2; n : =2 4 eq7: = simplify(final6); eq7 : =8h2 + 2ah2=2h2(2b0 + 2b1 + b2) 4 eq7: = simplify(eq7/h^2); 4 4 4 ‘‘Characteristic Equation of the new method’’; 4 final: = k[n+2]+a*k[n+1](2+2*a)*k[n]+ a*k[n1]+k[n2]+ H^2*(b[0]*(k[n+2]+k[n2]) + b[1]*(k[n+1]+k[n1]) + b[2]*k[n]); final : =k6+ak5(2 + 2a)k4 + ak3 + k2 + H2 (b0(k6 + k2) + b1 (k5 + k3) + b2k4) Chem. Modell., 2008, 5, 350–487 | 473 This journal is

c


4 AH: = coeff(final,k[n+2]); AH : =1 + H2b0 4 BH: = coeff(final,k[n+1]); BH : =a + H2b1 4 CH: = coeff(final,k[n]); 4 4 4 4 4 4 4 4 4 4 4 4

CH : = 2 2a + H2b2 charact: = AAH*(l^2+l^(2)) + BBH*(l^1+l^(1)) + CCH; charact1: = subs(l=exp(I*H),charact)=0; charact1: = factor(simplify(convert(charact1,trig))); charact2: = subs(l=exp(I*H),charact)=0; charact2: = factor(simplify(convert(charact2,trig))); charact3: = subs(l=exp(I*H),charact)=0; charact3: = factor(simplify(convert(charact3,trig))); charact4: = subs(l=exp(I*H),charact)=0; charact4: = factor(simplify(convert(charact4,trig))); charact1: = subs(AAH=AH,BBH=BH,CCH=CH,charact1); charact2: = subs(AAH=AH,BBH=BH,CCH=CH,charact2); eq9: = subs(H=v*h,charact1);

eq9 : =4(1+h2b0v2) cos(vh)22(a+v2h2b1) cos(vh) 4 2a+h2v2b22h2b0v2=0 4 eq10: = subs(H=v*h,charact2); eq10 : =4 (1 + h2b0v2) cos(vh)2 + 2(a + v2h2b1) cos(vh) 4 2a + h2v2b2 2h2b0v2=0 4 solution: = solve({eq1ir,eq2ir,eq2ii,eq4ir,eq4ii},{b[0],b[1], b[2]}); 4 assign(solution); 4 solution2: = solve({eq9,eq10},{a}); 4 assign(solution2); 4 h: = 1; h : =1 4 b[0]: = simplify(b[0]); b0 :¼

sinðvÞ cosðvÞ þ 2v cosðvÞ 2 sinðvÞ þ v v2 ð3 sinðvÞ cosðvÞ þ 2v cosðvÞ þ vÞ

4 b[0]: = combine(b[0]); b0 :¼

sinð2vÞ 4v cosðvÞ þ 4 sinðvÞ 2v 3v2 sinð2vÞ þ 4v3 cosðvÞ þ 2v3


1 17 2 163 4 60607 1697747 þ v þ v þ v6 þ v8 15 1575 94500 218295000 37837800000 519335027 12254045443 609739626367891 þ v10 þ v12 þ v14 71513442000000 10420530120000000 3201499468767600000000 23701757945389571 241048102702471793021 þ v16 þ v18 768359872504224000000000 48245316394540224960000000000 2342557239622309932869 þ v20 2894718983672413497600000000000

4 b[1]: = simplify(b[1]); b1 :¼

2ð2 cosðvÞ3 v 3 cosðvÞ2 sinðvÞ þ 2v cosðvÞ2 þ v cosðvÞ 3 sinðvÞ cosðvÞ þ vÞ ð3 sinðvÞ cosðvÞ þ 2v cosðvÞ þ vÞv2


c


4 b[1]: = combine(b[1]); b1 :¼

2v cosð3vÞ þ 10v cosðvÞ 3 sinð3vÞ 3 sinðvÞ þ 4 cosð2vÞv þ 8v 6 sinð2vÞ 3v2 sinð2vÞ þ 4v3 cosðvÞ þ 2v3


16 208 2 247 4 4154 1790779 606383 v þ v þ v6 þ v8 þ v10 15 1575 23625 27286875 28378350000 62511750000 28752907369 6387572067797 þ v12 þ v14 18235927710000000 25011714599746875000 7946098717130549 40407052991328497737 v16 þ v18 þ 192089968126056000000000 6030664549317528120000000000 785374879934365420061 v20 þ 723679745918103374400000000000

4 b[2]: = simplify(b[2]); b2 : =2(4 cos(v)3 v 4 cos(v)3 sin(v) 2 cos(v)2 sin(v) + 4v cos(v)2 sin(v) cos(v) + v 2 sin(v))/(v2(3 sin(v) cos(v) + 2v cos(v) + v)) 4 b[2]: = combine(b[2]); b2 : =(4v cos(3v) 12v cos(v) + 2 sin(4v) + 6 sin(2v) + 2 sin(3v) + 10 sin(v) 8 cos(2v)v 12v)/(3v2 sin(2v) + 4v cos(v) + 2v3) 4 b2t: = convert(taylor(b[2],v = 0,26),polynom); b2t :¼

26 1298 2 727 4 1003979 6 13137323 8 94972363 v þ v v þ v v10 15 1575 6750 109147500 56756700000 2750517000000 127664236097 979315457827727 v12 v14 36471855420000000 1600749734383800000000 37789929189709687 384562270272719754337 v16 v18 384179936252112000000000 24122658197270112480000000000 3737147592950136298993 v20 1447359491836206748800000000000

4 a: = simplify(a); a :¼

2ð2 cosðvÞ3 v þ 2 cosðvÞ2 3 cosðvÞ2 sinðvÞ þ v cosðvÞ 3 sinðvÞ cosðvÞ þ vÞ 3 sinðvÞ cosðvÞ þ 2v cosðvÞ þ v

4 a: = combine(a); a :¼

2v cosð3vÞ 10v cosðvÞ 4 cosð2vÞv 8v þ 3 sinð3vÞ þ 3 sinð3vÞ þ 6 sinð2vÞ 3 sinð2vÞ þ 4v cosðvÞ þ 2v

4 at: = convert(taylor(a,v = 0,26),polynom); 16 2 208 4 247 6 4154 1790779 606383 v þ v v v8 v10 v12 15 1575 23625 27286875 28378350000 62511750000 28752907369 6387572067797 v14 v16 18235927710000000 25011714599746875000 7946098717130549 40407052991328497737 v18 v20 192089968126056000000000 6030664549317528120000000000 785374879934365420061 v22 723679745918103374400000000000

at :¼

4 plot([b[0]],v = 100..100,labels=[v,b_0],thickness=3,title=‘‘bahavior of the coefficient b_0’’); 4 plot([b[1]],v = 50..50,labels=[v,b_1],thickness=3,title=‘‘bahavior of the coefficient b_1’’); Chem. Modell., 2008, 5, 350–487 | 475 This journal is

c


4 plot([b[2]],v = 50..50,labels=[v,b_2],thickness=3,title=‘‘bahavior of the coefficient b_2’’); 4 plot([a],v = 50..50,labels=[v,a_],thickness=3,title=‘‘bahavior of the coefficient a_’’); 4 4 restart; 4 qnp2: = convert(taylor(q(x+2*h),h = 0,13),polynom); 4 qnp1: = convert(taylor(q(x+h),h = 0,13),polynom); 4 qnm1: = convert(taylor(q(xh),h = 0,13),polynom); 4 qnm2: = convert(taylor(q(x2*h),h = 0,13),polynom); 4 snp2: = convert(taylor(diff(q(x+2*h),x$2),h = 0,13),polynom); 4 snp1: = convert(taylor(diff(q(x+h),x$2),h = 0,13),polynom); 4 snm1: = convert(taylor(diff(q(xh),x$2),h = 0,13),polynom); 4 snm2: = convert(taylor(diff(q(x2*h),x$2),h = 0,13),polynom); 4 qn: = q(x); 4 sn: = diff(q(x),x$2); 4 b[0]: = subs(v = v*h,1/15+17/1575*v^2+163/94500*v^4+60607/218295000 *v^6 + 1697747/37837800000*v^8 + 519335027/71513442000000 *v^10+12254045443/10420530120000000*v^12+609739626367891/3201 499468767600000000*v^14+23701757945389571/76835987250422400 0000000*v^16+241048102702471793021/482453163945402249600000 00000*v^18+2342557239622309932869/2894718983672413497600000 000000*v^20); 4 b[1]: = subs(v = v*h,16/15208/1575*v^2+247/23625*v^4+4154/ 27286875*v^6+1790779/28378350000*v^8+606383/62511750000* v^10+28752907369/18235927710000000*v^12+6387572067797/ 25011714599746875000*v^14+7946098717130549/ 192089968126056000000000*v^16+40407052991328497737/ 6030664549317528120000000000*v^18+785374879934365420061/ 723679745918103374400000000000*v^20); 4 b[2]: = subs(v = v*h,26/151298/1575*v^2+727/6750*v^41003979/ 109147500*v^6+13137323/56756700000*v^894972363/ 2750517000000*v^10127664236097/36471855420000000*v^12979315457827727/ 1600749734383800000000*v^1437789929189709687/ 384179936252112000000000*v^16384562270272719754337/ 24122658197270112480000000000*v^183737147592950136298993/ 1447359491836206748800000000000*v^20); 4 a: = subs(v = v*h, 16/15*v^2+208/1575*v^4247/23625*v^64154/ 27286875*v^81790779/28378350000*v^10606383/ 62511750000*v^1228752907369/18235927710000000*v^146387572067797/ 25011714599746875000*v^167946098717130549/ 192089968126056000000000*v^1840407052991328497737/ 6030664549317528120000000000*v^20785374879934365420061/ 723679745918103374400000000000*v^22); 4 4 lte: = simplify(qnp2+a*qnp1 (2+2*a)*qn+ a*qnm1+ qnm2 h^2*(b[0]*(snp2+snm2) + b[1]*(snp1+snm1) + b[2]*sn)); 1284533 457674401 v10 h14 ðDð4Þ ÞðqÞðxÞ v10 h16 ðDð6Þ ÞðqÞðxÞ 29189160000 42908065200000 26605220168191289256587 v20 h30 ðDð10Þ ÞðqÞðxÞ 2574597119595699534336000000000000 440378958732374405037143 v20 h32 ðDð12Þ ÞðqÞðxÞ 962899322728791625841664000000000000

lte :¼


c


3919 6802345787 v4 h12 ðDð8Þ ÞðqÞðxÞ v10 h18 ðDð8Þ ÞðqÞðxÞ 11907000 5148967824000000 2070403 v6 h18 ðDð12Þ ÞðqÞðxÞ 13202481600000 785374879934365420061 v22 h32 ðDð10Þ ÞðqÞðxÞ 1313044530993806762511360000000000000 2 19 79 ðDð8Þ ÞðqÞðxÞh8 ðDð10Þ ÞðqÞðxÞh10 ðDð12Þ ÞðqÞðxÞh12 945 56700 3742200 38021293 92293 v12 h16 ðDð4Þ ÞðqÞðxÞ v4 h16 ðDð12Þ ÞðqÞðxÞ 5363508150000 94303440000 2 2 8 ð6Þ 257 v h ðD ÞðqÞðxÞ h14 ðDð14Þ ÞðqÞðxÞ 315 224532000 223 9467 v2 h12 ðDð10Þ ÞðqÞðxÞ v2 h14 ðDð12Þ ÞðqÞðxÞ 1701000 1571724000 13 2 10 ð8Þ 2 4 8 ð4Þ 53 4 10 ð6Þ v h ðD ÞðqÞðxÞ v h ðD ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 8100 315 18900 31933 2 6 8 ð2Þ v4 h14 ðDð10Þ ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 1428840000 945 121 6 10 ð4Þ 53951 v h ðD ÞðqÞðxÞ v6 h12 ðDð6Þ ÞðqÞðxÞ 56700 130977000 790567 1668641 v6 h14 ðDð8Þ ÞðqÞðxÞ v6 h16 ðDð10Þ ÞðqÞðxÞ 15717240000 471517200000 17 8 10 ð2Þ 16717 2 12 ð4Þ v h ðD ÞðqÞðxÞ v h ðD ÞðqÞðxÞ 28350 65488500 13384649 4322359 v8 h14 ðDð6Þ ÞðqÞðxÞ v8 h16 ðDð8Þ ÞðqÞðxÞ 204324120000 529729200000 2950255843 6868411783 v8 h18 ðDð10Þ ÞðqÞðxÞ v12 h18 ðDð6Þ ÞðqÞðxÞ 5148967824000000 3978747864000000 983055373499 v12 h20 ðDð8Þ ÞðqÞðxÞ 4595453782920000000 16561592265791 v12 h22 ðDð10Þ ÞðqÞðxÞ 1102908907900800000000 7839337771 v12 h24 ðDð12Þ ÞðqÞðxÞ 11795817196800000000 39831713 v14 h16 ðDð2Þ ÞðqÞðxÞ 21454032600000 50269620847 v14 h18 ðDð4Þ ÞðqÞðxÞ 43766226504000000 537085263024383 v14 h20 ðDð6Þ ÞðqÞðxÞ 1920899681260560000000 7986217135364971 v14 h22 ðDð8Þ ÞðqÞðxÞ 230507961751267200000000 1293684403127741 v14 h24 ðDð10Þ ÞðqÞðxÞ 531941450195232000000000 20840974266406639 v14 h26 ðDð12Þ ÞðqÞðxÞ 193626687871064448000000000 11634157 v16 h18 ðDð2Þ ÞðqÞðxÞ 38594556000000 5955987976517 v16 h20 ðDð4Þ ÞðqÞðxÞ 320149946876760000000


c


632656660291181 v16 h20 ðDð6Þ ÞðqÞðxÞ 13970179500079800000000 543270330017726183 v16 h24 ðDð8Þ ÞðqÞðxÞ 96813343935532224000000000 352016233855041947 v16 h26 ðDð10Þ ÞðqÞðxÞ 893661636327989760000000000 13367129101245771611 v16 h28 ðDð12Þ ÞðqÞðxÞ 766761683969415214080000000000 46886977753529 v18 h20 ðDð2Þ ÞðqÞðxÞ 960449840630280000000 13891645302341687 v18 h22 ðDð4Þ ÞðqÞðxÞ 461015923502534400000000 11175133699026299017 v18 h24 ðDð6Þ ÞðqÞðxÞ 1523536037196007104000000000 3157194649927030649501 v18 h26 ðDð8Þ ÞðqÞðxÞ 3473662780406896197120000000000 6648615603062501355073 v18 h28 ðDð10Þ ÞðqÞðxÞ 104209883412206885913600000000000 90629598711667624225999 v18 h30 ðDð12Þ ÞðqÞðxÞ 32096644090959720861388800000000000 1822679070604949 v20 h22 ðDð2Þ ÞðqÞðxÞ 230507961751267200000000 35320026494699603303 v20 h24 ðDð4Þ ÞðqÞðxÞ 7236797459181033744000000000 121379200592810813441 v20 h26 ðDð6Þ ÞðqÞðxÞ 102166552364908711680000000000 26847014079210919788781 v20 h28 ðDð8Þ ÞðqÞðxÞ 182367295971362050348800000000000 785374879934365420061 v22 h24 ðDð2Þ ÞðqÞðxÞ 723679745918103374400000000000 785374879934365420061 v22 h26 ðDð4Þ ÞðqÞðxÞ 8684156951017240492800000000000 785374879934365420061 v22 h28 ðDð6Þ ÞðqÞðxÞ 260524708530517214784000000000000 785374879934365420061 v22 h30 ðDð8Þ ÞðqÞðxÞ 14589383677708964027904000000000000 4774771817 4093 v10 h20 ðDð10Þ ÞðqÞðxÞ v10 h12 ðDð2Þ ÞðqÞðxÞ 51489678240000000 65488500 8617575143 1195837 v8 h20 ðDð12Þ ÞðqÞðxÞ v12 h14 ðDð2Þ ÞðqÞðxÞ 339831876384000000 102162060000 585779653139 v10 h22 ðDð12Þ ÞðqÞðxÞ 142729388081280000000 785374879934365420061 v22 h34 ðDð12Þ ÞðqÞðxÞ 173321878091182492651499520000000000000 4339 167159 h16 v2 ðDð4Þ ÞðqÞðxÞ h18 v14 ðDð14Þ ÞðqÞðxÞ 23575860000 5658206400000 1410679 h20 v6 ðDð14Þ ÞðqÞðxÞ 297055836000000


c


474297233 h22 v8 ðDð14Þ ÞðqÞðxÞ 617876138880000000 265986246593 h24 v10 ðDð14Þ ÞðqÞðxÞ 2140940821219200000000 87865750642793 h26 v12 ðDð14Þ ÞðqÞðxÞ 4367519275287168000000000 1774368692349119 h28 v14 ðDð14Þ ÞðqÞðxÞ 544575059637368760000000000 2207140566799641023 h30 v16 ðDð14Þ ÞðqÞðxÞ 4182336458014992076800000000000 123457035636656886524489 h32 v18 ðDð14Þ ÞðqÞðxÞ 1444348984093187438762496000000000000 2399563988253179736677917 h34 v20 ðDð14Þ ÞðqÞðxÞ 173321878091182492651499520000000000000

4 coeff(lte,h,8);

2 2 2 ð6Þ 2 4 ð4Þ 2 2 ð6Þ ðDð8Þ ÞðqÞðxÞ v ðD ÞðqÞðxÞ v ðD ÞðqÞðxÞ v ðD ÞðqÞðxÞ 945 315 315 945

4 4 4 4 4 restart; 4 ‘‘Stability Analysis’’ 4 AH: = 1 + v^2*b[0]; 4 BH: = a + v^2*b[1]; 4 CH: = 22*a + v^2*b[2]; 4 b[0]: = (sin(2*v)4*v* cos(v) + 4* sin(v)2*v)/(3*v^2* sin(2*v) + 4*v^3* cos(v) + 2*v^3); 4 b[1]: = (2*v* cos(3*v) + 10*v* cos(v)3* sin(3*v)3* sin(v) + 4* cos(2*v)*v + 8*v6* sin(2*v))/(3*v^2* sin(2*v) + 4*v^3* cos(v) + 2*v^3); 4 b[2]: = (4*v* cos(3*v)12*v* cos(v) + 2* sin(4*v) + 6* sin(2*v) + 2* sin(3*v) + 10* sin(v)8* cos(2*v)*v12*v)/(3*v^2* sin(2*v) + 4*v^3* cos(v) + 2*v^3); 4 a: = (2*v* cos(3*v)10*v* cos(v)4* cos(2*v)*v8*v + 3* sin(3*v) + 3* sin(v) + 6* sin(2*v))/(3* sin(2*v) + 4*v* cos(v) + 2*v); 4 AH: = combine(AH); 4 BH: = combine(BH); 4 CH: = combine(CH); 4 P1H: = simplify(2*AH2*BH + CH); P1H :¼

8 sinðvÞ3 ðcosðvÞ 1Þ 3 sinðvÞ cosðvÞ þ 2v cosðvÞ þ v

4 P2H: = simplify(12*AH2*CH); P2H :¼

16ðcosðvÞ 1 cosðvÞ2 þ cosðvÞ3 Þ sinðvÞ 3 sinðvÞ cosðvÞ þ 2v cosðvÞ þ v Chem. Modell., 2008, 5, 350–487 | 479

This journal is

c


4 P3H: = simplify(2*AH + 2*BH + CH); P3H :¼

8 sinðvÞ3 ðcosðvÞ 1Þ 3 sinðvÞ cosðvÞ þ 2v cosðvÞ þ v

4 SNH: = simplify(P2H^24*P1H*P3H); SNH : = 1024 sin(v)2 cos(v)2 (cos(v)2 2 cos(v) + 1)/(6 sin(v) v cos(v) 12 sin(v) v cos(v)2 + v2 + 9 cos(v)2 + 4v2 cos(v)2 + 4v2 cos(v) 9 cos(v)4) 4 plot([P1H,P2H], v = 0..100, color=[black,red], style=[line,line,line,line], thickness=3,linestyle=[SOLID, SOLID, SOLID, SOLID], title=‘‘Stability Polynomias for the New Method-P1H and P2H’’); 4 plot([P3H], v = 0..100, color=[blue], style=[line], thickness=3,linestyle= [SOLID], title=‘‘Stability Polynomias for the New Method-P3H’’); 4 plot([SNH], v = 0..100, color=[green], style=[line], thickness=3,linestyle= [SOLID], title=‘‘Stability Polynomias for the New Method-NH’’); 4 4 restart; 4 final: = k[n + 2] + a*k[n + 1](2 + 2*a)*k[n] + a*k[n1] + k[n2] + H^2*(b[0]*(k[n + 2] + k[n2]) + b[1]*(k[n + 1] + k[n1]) + b[2]*k[n]); 4 AH: = combine(coeff(final,k[n + 2])); 4 BH: = combine(coeff(final,k[n + 1])); 4 CH: = combine(coeff(final,k[n])); 4 P1H: = simplify(2*AH2*BH + CH); 4 P2H: = simplify(12*AH2*CH); 4 P3H: = simplify(2*AH + 2*BH + CH); 4 b[0]: = (sin(2*v)4*v* cos(v) + 4* sin(v)2*v)/(3*v^2* sin(2*v) + 4*v^3* cos(v) + 2*v^3); 4 b[1]: = (2*v* cos(3*v) + 10*v* cos(v) 3* sin(3*v) 3* sin(v) + 4* cos(2*v)*v + 8*v 6* sin(2*v))/(3*v^2* sin(2*v) + 4*v^3* cos(v) + 2*v^3); 4 b[2]: = (4*v* cos(3*v)12*v* cos(v) + 2* sin(4*v) + 6* sin(2*v) + 2* sin(3*v) + 10* sin(v)8* cos(2*v)*v12*v)/(3*v^2* sin(2*v) + 4*v^3* cos(v) + 2*v^3); 4 a: = (2*v* cos(3*v)10*v* cos(v)4* cos(2*v)*v8*v + 3* sin(3*v) + 3* sin(v) + 6* sin(2*v))/(3* sin(2*v) + 4*v* cos(v) + 2*v); 4 4 P1H: = combine(P1H); P1H =(32v3 24v2 sin(2v) + 40v3 cos(v) 40H2 v cos(v) 8H2 v cos(3v) 16H2 cos(2v) v + 20H2 sin(2v) + 24H2 sin(v) 32H2 v + 8v3 cos(3v) 12v2 sin(3v) 12v2 sin(v) + 16v3 cos(2v) + 8H2 sin(3v) + 2H 2 sin(4v))/(3v2 sin(2v) + 4v3 cos(v) + 2v3) 4 P2H: = combine(P2H); P2H : =(24v2 sin(2v) + 24v3 cos(v) 24H2 v cos(v) + 28H2 sin(v) 8v3 cos(3v) + 12v2 sin(3v) + 12v2 sin(v) 16v3 cos(2v) + 8H2v cos(3v) 4H2 sin(4v) 4H2 sin(3v) + 16H2 cos(2v) v)/(3v2 sin(2v) + 4v3 cos(v) + 2v3) 4 P3H: = combine(P3H); P3H :¼

4H 2 sinð2vÞ þ 12H 2 sinðvÞ 4H 2 sinð3vÞ þ 2H 2 sinð4vÞ 3v2 sinð2vÞ þ 4v3 cosðvÞ þ 2v3

4 NH: = combine(P2H^24*P1H*P3H); NH : =(384v4 H2 cos(2v) 768H2 v2 cos(v) 192H4 v sin(6v) + 256H4 v2 cos(5v) + 1024v4H2 cos(v) + 256v4H2 cos(4v) 128H4 v2 cos(4v) + 96H2 v2 cos(2v) 1792H2v3 sin(4v) 1728H4 v sin(2v) + 512H4v sin(v) + 512H4v sin(3v) + 640H4v sin(4v) + 384H2 v3 sin(2v) + 1024H2 v3 sin(v) 128v4H2 cos(6v) + 256v6 cos(5v)


c


576H 2 v 2 cos(4v) + 384v3H2 sin(6v) + 864v4 1792v4H2 + 192H4 v2 cos(2v) 128H4 cos(v) 96H4 cos(4v) + 192v6 cos(2v) + 16H4 cos(2v) + 64v6 cos(6v) + 384H2 v2 cos(3v) + 144v4 cos(2v) 864v4 cos(4v) + 896H4v2 + 1024H2v3 sin(3v) 512v4H2 cos(5v) + 576H 2 v2 + 384H2 v2 cos(5v) 192v5 sin(6v) + 1344v5 sin(2v) 512v6 cos(v) 768v6 cos(3v) + 64H4 cos(3v) + 64H4 cos(5v) + 896v6 + 64H4v2 cos(6v) 96H2 v2 cos(6v) 768H4v2 cos(3v) + 96H4 1536v5 sin(3v) 1536v5 sin(v) 1152v4 cos(v) + 576v4 cos(5v) + 576v4 cos(3v) + 1152v5 sin(4v) 128v6 cos(4v) + 1536v4H2 cos(3v) 512H4v2 cos(v) 144v4 cos(6v) 16H4 cos(6v))/(9v4 9v4 cos(4v)24v5 sin(3v)24v5 sin(v)24v5 sin(2v)+16v6 cos(2v)+24v6+32v6 cos(v))

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