Chemical Modelling: Vol. 2: Applications and Theory (Specialist Periodical Reports)

Chemical Modelling Applications and Theory Volume 2 A Specialist Periodical Report Chernical Modelling Applications...

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Chemical Modelling Applications and Theory

Volume 2

A Specialist Periodical Report

Chernical Modelling Applications and Theory Volume 2

A Review of the Literature Published between June 1999 and May 2001 Senior Reporter A. Hinchliffe, Department of Chemistry, UMIST, Manchester, UK Rep0rters D. Babic, Rudjer BoSkovit University, Zagreb, Croatia D.M. Heyes, University of Surrey, Guildford, UK D.J. Klein, Texas A & M University at Galveston, Galveston, Texas, USA R.A. Lewis, Lilly Research Centre, Windlesham, Surrey, UK P.L.A. Popelier, UMIST, Manchester, UK D. Pugh, Universrty of Strathclyde, Glasgow, UK T. Simos, Democritus University of Thrace, Konthi, Greece

P.J. Smith, UMIST, Manchester, UK M. Springborg, University of Saarland, Soarbrucken, Germany N. Trinajstic, Rudjer 80ikovii University, Zagreb, Croatia S. Wilson, Rutherford Appleton Laboratory, Chilton, Oxfordshire, UK

ROYAL SOCIETY OF CHEMISTRY

ISBN 0-85404-259-8 ISSN 1472-0965

0The Royal Society of Chemistry 2002 All rights reserved Apart from any fair dealing for the purposes of research or private study, or criticism or review as permitted under the terms of the UK Copyright, Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted, in any form or by any means, without the prior permission in wirting 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. Enquires 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 OWF, UK Registered Charity Number 207890 For further information see our web site at www.rsc.org Typeset by Keytec Typesetting Ltd Printed by Athenaeum Press Ltd, Gateshead, Tyne and Wear, UK

Preface

I laid the foundations for this Series in Volume 1. The ground rules are very simple: colleagues reporting on new topics are asked to give the rest of us an easily understandable historical perspective together with their own critical comments on the literature for the period under review. Colleagues reporting on continuing topics are simply expected to give a critical review of the literature for the period. The period under consideration for this volume is June 1999 to May 2001, and subsequent volumes will give biennial coverage of the literature to May 2003 n, where n = 0, 2, 4, 6, . . . Note my repeated use of the word ‘critical’. When the RSC and I market researched this new SPR title, it quickly became apparent that colleagues were not interested in a dull and uncritical compilation of literature references. Several of them remarked (rather unkindly, I thought) that they could ask their PhD students to sit at a networked PC, dial up Web of Science, and produce such a comprehensive list by the end of a single afternoon. What they wanted was critical insight into the recent literature. That is what we are trying to give. There are still many gaps in coverage, and I’m sure you will have your own ideas as to what is good and bad with this very new title. Rather than grumbling to your colleagues and writing acidic book reviews, why not volunteer your own expertise? I am always willing to listen to constructive suggestions, and can be reached at

+

[email protected] Volume 2 consists of eight contributions. Several are continuations from the topics treated in Volume 1, some are new. A couple of existing Reporters in Volume 1 asked to be excused for Volume 2, but will reappear in Volume 3. The contributions are not in any particular order, other than the ‘new’ topics are towwds the start of the volume. The molecular simulation of liquids is now a vast field of human endeavour, and we open with a contribution on ‘Simulation of the Liquid State’ by David Heyes. David captures the spirit of the SPR exactly when he writes ‘. . .The ready availability of fast computers has meant that there are many more researchers working in this ever expanding field . . .[and] . . .I have restricted my discussion to . . . areas that have interested me’. Several people pointed out a gap in the coverage of Volume 1, namely the field of enumeration. Nenad TrinajstiC and his co-workers have written our first chapter

vi

Chemical Modelling: Applications and Theory, Voltrme 2

in this field, which strikes a nice balance between historical perspective and upto-the-minute literature. Michael Springborg continues to report on the growth of density functional theory. Theodore Simos reported on the current status of atomic structure calculations in Volume 1. He has broadened the scope a little for Volume 2, and reports on progress in the solution of lD, 2D and 3D differential equations in chemistry. It was always my intention to include industrial applications. Much of this kind of work never reaches the primary journals because of confidentiality restrictions and commercial forces. I am very pleased to tell you that Richard Lewis has been able to give us a fascinating glimpse into the world of commercial computer-aided drug design, without apparently breaching a single one of his employer’s trade secrets. David Pugh continues his coverage of electric and magnetic properties. David also gives us a historical insight into those rare beasts magnetizability and hypermagne t izabi1ity. Steven Wilson continues his coverage of many body perturbation theory, whilst Paul Popelier and Paul Smith continue the story of recent advances in the theory of quantum topological atoms. Alan Hinchliffe Manchester, 200 1

Contents

Chapter 1 Simulation of the Liquid State By D.M. Heyes Introduction Simple Liquids 2.1 Dynamics 2.2 Thermodynamics 2.3 Mixtures Water and its Solutions 3.1 Pure Water 3.2 Aqueous Solutions Organic Liquids 4.1 Alkanes 4.2 Oxygen Containing Molecules Non-equilibrium Molecular Dynamics (NEMD) Glasses 6.1 Phenomenology 6.2 Structural Models for Supercooled Liquids 6.3 Ageing 6.4 Rheology 6.5 Glasses in Confined Geometries Liquid Surfaces 7.1 Liquid-Vapour Interfaces 7.2 Liquid-Liquid Interfaces 7.3 Liquid-Solid Interfaces 7.4 Tribology 7.5 Two-dimensional Liquids 7.6 Droplets Dissipative Particle Dynamics Computational Techniques 9.1 Introduction 9.2 Periodic Boundary Conditions 9.3 Long-range Coulomb Forces 9.4 Integrators and Thermostatting Chemical Modelling: Applications and Theory, Volume 2 (Q The Royal Society of Chemistry, 2002

1 1 2 2 8 10 11 11 15 16 16 19 19 24 24 26 32 34 35 36 36 39 40 41 42 42 43 45 45 45 46 48

...

Vlll

Chemical Modelling: Applications and Theory, Volume 2

9.5 Ergodicity and Sampling of Rare Events References

49 50

Chapter 2 Enumeration in Chemistry By D.J Klein, D,BabiC and N. TrinajstiC 1 Introduction and Historical Review 1.1 Early History: Isomer Enumeration 1.2 Further Enumerations 1.3 Why Enumerate? 2 Enumeration Methods 2.1 Enumeration under Group Equivalences 2.2 Linear Recursive Methods - Kekule Structure Counting 2.3 Transfer Matrix Methods 2.4 Exhaustive Generation (Brute Force) Methods 2.5 Other Methods 3 Current Results 3.1 Isomers: Enumeration and Generation 3.2 Fullerenes and Related Objects 3.3 Counts of Resonance Structures and Related Items 3.4 Walks, Connected Subgraphs and Vertices at a Given Distance 3.5 Other Enumerations 4 Conclusion References

56

Chapter 3 Density Functional Theory By Michael Springborg 1 Introduction 2 Basic Principles 3 Functionals 4 Semi-empirical Methods 5 Order-N Methods 6 Heterogeneous Catalysis 7 Descriptions of Chemical Reactions 8 Quantum Treatment of Other Particles 9 Problems with l l r Potentials 10 Exact-exchange Methods 11 Time-dependent Density-functional Theory 12 Polarizability and Hyperpolarizability 13 Conclusions Acknowledgements References

96

Chapter 4 Numerical Methods for the Solution of lD, 2D, and 3D Differential Equations Arising in Chemical Problems By TE. Simos 1 Introduction

56 56 57 59 60

61 64 67 70 71 73 73 81 84 85 86 88 88

96 98 101 103 111 119 127 135 140 142 148 158 164 166 166 170 170

ix

Contents

2 Adapted, Exponentially Fitted and Trigonometrically Fitted Symplectic Integrators 2.1 Case m = 0 2.2 Case m = 1 2.3 Runge-Kutta-Nystrom Method with FSAL Property 2.4 Trigonometrically Fitted Symplectic Intergrators 2.5 Exponentially Fitted Symplectic Intergrators 2.6 Exponentially Fitted and Trigonometrically Fitted Symplectic Linear Symmetric Multistep Methods 2.6.1 First Family of Methods - Case to = -2 2.6.2 Second Family of Methods - Case to # -2 2.6.3 Stability Analysis 2.7 Numerical Examples 2.7.1 Inhomogeneous Equation 2.7.2 Duffin’s Equation 2.7.3 An Orbit Problem Studied by Stiefel and Bettis 3 Dissipative Methods 3.1 Phase-lag of Non-symmetric (Dissipative) Two-step Methods 3.2 Dissipative Methods Developed in the Literature 3.3 Generator of Dissipative Numerov-type Methods 3.4 Exponentially Fitted Dissipative Numerov-type Methods 3.4.1 New Exponentially Fitted Dissipative Two-step Method. Case I 3.4.2 New Trigonometrically Fitted Dissipative Two-step Method. Case I 3.4.3 New Exponentially Fitted Dissipative Two-step Method. Case I1 3.4.4 New Trigonometrically Fitted Dissipative Two-step Method. Case I1 4 Numerical Illustrations for Linear Multistep Methods and Dissipative Methods 4.1 Resonance Problem 4.1.1 The Woods-Saxon Potential 4.1.2 Modified Woods-Saxon Potential 4.2 The Bound-states Problem 4.3 Remarks and Conclusions 5 New Developments on Numerical Methods with Constant Coefficients and on the Methods with Coefficients Dependent on the Frequency of the Problem 5.1 Methods with Constant Coefficients (Generators of Numerical Methods) 5.2 Methods with Coefficients Dependent on the Frequency of the Problem 5.2.1 Exponentially Fitted Hybrid Methods 5.2.2 Bessel-fitted and Neumann-fitted Methods

171 173 173 174 175 177 178 178 188 194 196 196 199 20 1 204 206 207 215 2 16 219 220 22 1 222 224 224 225 228 230 23 1

23 1 23 1 23 8 238 240

X


5.3 Runge-Kutta Exponentially Fitted Methods 5.4 Modified Runge-Kutta Phase-fitted Methods 5.5 Modified Runge-Kutta-Nystrom Phase-fitted Methods 6 Numerical Illustration on Variable-step Methods 6.1 Coupled Differential Equations 7 General Comments Appendix A Appendix B Appendix C References Chapter 5 Computer-aided Drug Design 2000-2001 By Richard A . Lewis 1 Introduction 2 3D-QSAR 3 Pharmacophores 4 Library Design 5 ADME/Tox 6 Docking and Scoring 7 Cheminformatics 8 Structure-based Drug Design 9 Reviews 10 Conclusions References Chapter 6 Electric Multipoles, Polarizabilities, Hyperpolarizabilities and Analogous Magnetic Properties By David Pugh 1 Introduction 2 Response of Closed Shell Molecules to Magnetic Fields 2.1 Magnetic Susceptibility 2.2 Nuclear Shielding 2.3 Interaction of Molecules with Electromagnetic Fields: Higher Order Terms 2.4 Gauge Invariance 2.4.1 Change of Origin 2.4.2 Gauge Invariant Atomic Orbitals (London Atomic Orbitals) (GIAOs or LAOS) 2.4.3 Other Approaches to Gauge Invariance 2.5 Ab initio Calculations of Magnetic Response to 1999 2.6 Current Density Functional Theory (CDFT) 3 Review of Literature on Response of Molecules to Magnetic Fields: June 1999-May 200 1 3.1 Ab initio Calculations 3.2 Density Functional Calculations

243 245 246 247 247 249 25 1 262 266 268

271 27 1 271 274 275 278 280 284 285 287 288 288

294

294 294 295 296 297 298 299 299 301 301 302 303 303 304

xi

Contents

4

Review of Literature on Response of Molecules to Electric Fields: June 1999-May 200 1 4.1 New Schemes for Calculation and Analysis of Properties 4.2 A6 initio Calculations on Atoms 4.3 A 6 initio and DFT Calculations on Diatomic Molecules 4.4 A6 initio and DFT Calculations on Small and Mediumsized Molecules 4.4.1 Water 4.4.2 03,SO2, Se02 and Te02 4.4.3 Other Molecules 4.5 Semi-empirical Calculations on Molecules 4.6 Vibrational Effects 4.7 Calculations on Complexes, Dimers, Clusters and Excited States 4.8 Fullerenes 4.9 Polymers 4.10 Crystals References

Chapter 7 Many-body Perturbation Theory and Its Application to the Molecular Electronic Structure Problem By S. Wilson 1 Introduction 2 Many-body Perturbation Theory through Second Order 2.1 Rayleigh- Schrodinger Perturbation Theory through Second Order 2.2 Marller-Plesset Perturbation Theory 2.3 Partitioning and the Remainder Term 2.4 The Choice of Zero-order Hamiltonian 2.5 Scaling of the Zero-order Hamiltonian 2.6 Multireference Second-order Many-body Perturbation Theory, Intruder States and Brillouin- Wigner Perturbation Theory through Second Order 3 Some Applications of Second-order Many-body Perturbation Theory with a Marller-Plesset Reference Hamiltonian 3.1 Publications with the String ‘MP2’ in Their Title 3.2 Publications with the String ‘MP2’ in Their Title and/or Keywords 3.2.1 Journal of Chemical Physics 3.2.2 Chemical Physics Letters 3.2.3 Journal of Physical Chemistry A 3.2.4 Journal of Physical Chemistry B 4 Summary and Prospects

305 305 307 307 308 308 309 309 312 315 317 318 3 19 320 32 1

329 329 331 332 337 344 349 350

354

360 361 364 365 367 370 376 377

xii


Acknowledgements References

Chapter 8 Quantum Topological Atoms By PL.A. Popelier and PJ Smith 1 Introduction 2 Theoretical 2.1 Alternative Partitioning 2.2 Electron Correlation 2.3 Algorithms and Software 2.4 Transferability 2.5 Pseudopotential 2.6 Intermolecular Interaction 2.7 Transfer Probability 2.8 Entropy 2.9 General Extensions 2.10 Quantum Monte Carlo 2.1 1 Magnetic Coupling 3 Chemical Bonding 3.1 Theory 3.2 Heavy Atom Group Elements 3.3 Surface Science 3.4 Fluorides 3.5 Transition Metals 3.6 van de Waals 3.7 Agostic 3.8 Radicals 3.9 Alkali and Alkaline Earth Oxides and Halides 3.10 Organic 3.1 1 Aromaticity 3.12 Minerals 3.13 Populations 3.14 Bond and Valence Indices 3.15 Solid State 3.16 Organometallics 3.17 Chemical Shift 3.18 Biological 3.19 Noble Gases 3.20 Zeolites 3.2 1 Hypervalency 3.22 Polymers 4 X-Ray Diffraction 4.1 Organic Compounds 4.2 Minerals 4.3 Metals 4.4 Hydrogen Bonding

378 378 391

39 1 398 398 399 400 400 40 1 40 1 402 402 402 403 403 403 403 404 406 407 407 410 41 1 41 1 412 412 414 415 415 416 416 417 417 417 418 418 418 419 419 419 419 420 420

xiii

Contents

4.5 Comparison between Theory and Experiment 4.6 Transition Metals 4.7 Biological 5 Laplacian of the Electron Density 5.1 Surface Science 5.2 Theory: Electron Pair Localization 5.3 Transition Metals 5.4 Heavy Metal Group 5.5 Non-linear Optics 6 Hydrogen Bonding 6.1 Reviews 6.2 Dihydrogen Bond 6.3 Groups 13/15 6.4 C-H.. .X 6.5 Organic 6.6 Cooperative Effect 6.7 Blue-shifted 6.8 Biochemical 6.9 With Ions 6.10 Isotope Effects 6.11 Low Barrier 6.12 Intramolecular 6.13 n-Systems 6.14 Kinetic Energy Density 6.15 Organometallic 7 Topology of Other Functions 7.1 ELF 7.2 Electrostatic Potential 7.3 Intracule -Extracule 8 Reactions 8.1 Organic 8.2 Inorganic 8.3 Transition Metals 8.4 Mass Spectrometry 8.5 Rotation Barrier 8.6 Biological 9 Ionic Materials 9.1 Thermodynamics 9.2 Phase Change 9.3 ImpurityAIoping 10 Spectroscopy 11 Opinions and Plans 12 Conclusion 13 Disclaimer References

420 422 422 423 423 423 424 424 424 425 425 425 425 426 426 427 427 427 427 428 428 428 429 429 429 430 430 432 432 433 433 436 436 437 437 437 437 437 437 438 438 438 439 439 440

I Simulation of the Liquid State BY D.M. HEYES

1 Introduction

The molecular simulation of liquids is a now vast field of research, and as with many others in recent years it is becoming increasingly difficult to keep abreast of all of the significant developments that are taking place. The ready availability of fast computers has meant that there are many more researchers working in this ever expanding field of applications, producing ever larger amounts of work to assimilate! This poses something of a problem when it comes to writing a review, especially one with the rather ambitious title of ‘Computer Simulation of Liquids’. I am not going to attempt to cover all the branches of this field. Rather, in my review of the developments between 1999 and 2001 I have restricted my discussion to a few areas that have interested me. The choice is inevitably somewhat subjective, but hopefully by adopting this approach I will have a better chance of producing a useful document, rather than a gallop through many topics with only the briefest of discussion about each, which I am sure would be of little use to the scientific community. There are a number of molecular simulation books that describe the standard techniques, and these are recommended as background material for the present article, e.g. refs. 1-7. I am therefore not going to discuss the ‘nuts and bolts’ of molecular simulation, except to mention an often overlooked fact, which is the reason for much of the success of these approaches. Most simulations are carried out still typically for less than a thousand molecules, and if it was not for the use of periodic boundary conditions (PBC) it would not be possible to simulate bulk systems with this number of molecules. These systems would have such a high surface to volume ratio that the results would be dominated by surface effects. The PBC procedure is illustrated in Figure 1, which shows a two-dimensional square cell in which the molecules are surrounded by image cells. A molecule near the cell boundary interacts with the ‘real’ molecules in the central cell and with image cell molecules. Molecules leaving the cell re-enter through the opposite face with the same velocity.

~~

~

~~

~

Chemical Modelling: Applications and Theory, Volume 2 0 The Royal Society of Chemistry, 2002

2


Figure 1 Schematic representation of the Periodic Boundaries construction in two dimensions. A particle from the central cell moves across a boundary (dejined by the arrow) and is replaced by an image of itseg The image then becomes a ‘real particle and the simulation continues uninterrupted (Courtesy of Dr. C.A. Bearchell, Department of Chemistry, University of Surrey)

2 Simple Liquids

The study of simple liquids can be said to be the beginning of Molecular Dynamics and Monte Carlo in the 1950s and 60s. Although the scope of molecular simulation, as a field or discipline, has widened dramatically since then, there is nevertheless a continual interest in simple liquids. In fact, this is partly due to the fact that the so-called ‘simple’ liquids are far from simple! One of the motivations for the continual interest in the simple liquids is that, because of the basic nature of the interparticle interactions, an improved understanding of these systems should lead to better theoretical models, which can be extended to more complex molecular liquids. Also, the rapid growth of interest in colloids and polymers (so-called ‘complex’ liquids) in recent years has provided new areas where the theories of simple liquids can be applied, especially those associated with local structure and thermodynamics. In the latter case, phase equilibria and the location of phase boundaries feature prominently. In this section, some of the recent advances in our understanding of simple liquids are covered.

2.1 Dynamics. - Of course, within the category of ‘simple liquids’ studied by statistical mechanics and molecular simulation, there are model liquids that are, strictly speaking, not found in nature. For example, the ubiquitous hard sphere fluid, where the pair potential has the form

1: Simulation of the Liquid State

3

(2.1) is a case in point. The energy is infinite on contact of the spheres (at a) and zero for larger separations. This is the energy of interaction which would approximate that of two macroscopically sized elastic spheres with high elastic modulus, say two snooker or billiard balls. In these cases the length-scale of the particle interactions is many orders of magnitude smaller than the particle diameter. One of the main features of hard spheres is the co-ordination number. It has been shown recently that for spheres at random close packing, the mean number of particle contacts is 4.8, which is somewhat lower than has often been assumed before (6, and even 12, have been used).8 Interestingly these authors also performed a computer ‘simulation’ in which they took a random test sphere, and placed immobile point contacts on its surface. They determined the mean number of points required on the surface of the sphere to eradicate the possibility of translation of the test particle, which was found to be 2D + 1, where D is the space dimension. Therefore, in 3D this ‘co-ordination number’ is 7, which is lower than the value of 4.8, indicating that in states where the contacts are ‘correlated’ (i.e. in a dense liquid or glassy state) translational diffusion can be removed by fewer contacts. The procedure for carrying out Metropolis Monte Carlo of hard spheres is particularly simple, as the Boltnnann factor does not require specific evaluation - just overlap detection. Jater proposed an improved Metropolis Monte Carlo algorithm to simulate hard core systems, in which they replaced the usual sequence of single particle trial displacements by a collective trial ‘move’ of a chain of particles.’ The hard-sphere system is widely used in statistical mechanics as a reference state in theories of liquids and solids. Its uses have traditionally been quite broad, extending from equations of state, the structure of molecular liquids and dynamical properties. As mentioned already, it has also found a new lease of life as a model reference system for some colloids and granular materials. Simple molecules (e.g. water) interact with an appreciable attractive tail extending beyond the hard core, and which usually has more or less the same range as the core. It is not possible to find a simple molecule that does not have an appreciable attractive or van der Waals region as well as a hard repulsive core. In contrast, on the micron and larger scale, for these systems, the hard-sphere can be an even more realistic representation of the effective pair potential, which can be steeply repulsive and have a negligible attractive component. It must be borne in mind, however, that the hard-sphere particle potential is hdamentally unrealistic in that its pair potential is discontinuous and non-differentiable unlike those for all real systems. Considerable care is therefore required in extrapolating from any steeply repulsive potential, made progessively steeper, to the hard-sphere potential. This is because many quantities diverge either to zero or infinity according to the order in which the limit is made, of potential steepness and number of particles etc., any of which factors may be significant. Non-physical results such as purely exponential and delta function time correlation functions may be generated. These

4


issues have been considered recently by Powles, Rickayzen and Heyes in several publication^.'^'^ These authors have considered, amongst other forms, the generalised soft-sphere potential

where cr and E represent the particle diameter and energy scale, respectively. With n = 12, this potential is no other than the repulsive part of the Lennard-Jones potential. If n is taken much larger than 12, above 100 say, then the fluid behaves in many respects more like the hard-sphere fluid. In the large n limit they referred to this repulsive potential as the ‘steeply repulsive potential’, or SRP for short. As n --+ 00 the properties approach those of the hard sphere. In the publications, the dynamical relaxation and transport coefficients as a function of n were explored, and in particular the behaviour of these systems as they become more like hard spheres. In addition to being of intrinsic interest, these fluids furnish new perspectives on the behaviour of the hard-sphere system itself. A useful link between the SRP results and those of a hard-sphere system can be established by ascribing an effective hard-sphere diameter, o H Swhich , can then be used to specify equivalent hard-sphere packing fractions in the hard-sphere analytic formulae (e.g. Carnahan-Starling equation of state3). The Barker and Henderson formula for oHs,which is based on a free energy optimisation, was the prescription used in these studies,

where p = l / k B T as usual. They showed that at short times the time correlation functions relevant to the collective-property transport coefficients (i. e. shear and bulk viscosities and thermal conductivity) all obey the same simple scaling behaviour. This is in terms of a renormalised ‘time’ constructed by multiplying time by the parameter characterising the steepness of the potential, n. The time t is replaced by nt. An example of this scaling is shown in Figure 2. The reason for this can be shown to be rigorous in the hard-sphere limit by formal expansion of the time correlation function about t = 0. For a time correlation function, C(t) (normalised so that C(0) = 1) we find C ( t ) = 1 - x2+ . . . 0(x4)

where x = n ( k B T / ~ ) ” 2 ( ~ / m c r 2 )and ’ / 2 tm is the mass of the SRP particle, which is a non-dimensional time that includes the temperature and the potential stiffness parameter, n, as already mentioned. Note that the correlation function is independent of density, which is not surprising as in the large n limit the short time decay of C(t) is dominated by binary collisions. It is tempting to extrapolate this series in the following closure,

5


1

0.9 0.8 0.7 0.6 0.5

Thermal

\

'\

\

\

\

0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

nt

The normalised shear stress, bulk pressure and heat j7ux (thermal conductivity) function for a SRP fruid with n = 1152 at the effective hard sphere packing fraction of 0.45 at a reduced temperature of T * = I .0 (D.M. Heyes, Department of Chemistry, University of Surrey, unpublished work)

C ( t )=

1 ~

1 +x2

=1

-

x2

+ x4 - 0(x6)

which fits the C ( t ) data for the heat flux, shear stress and pressure correlation functions down to at least C ( t ) = 0.5 (especially for the pressure case) quite well. Using the Green-Kubo formulae (relating these time correlation functions to the transport coefficients) one can write the transport coefficient as the product of a modulus, M,, and a relaxation time, z, which is the integral of C ( t ) from t = 0 to oc. M , can be calculated as a static average, and in the hard-sphere limit has a simple analytic form involving the equation of state of the equivalent hard-sphere fluid, and n . M , diverges as n and z goes as n-' as n -+ 00. Although the hardsphere fluid has been very successful as a reference fluid, for example, in developing analytical equations of state it has serious deficiencies in accounting for the dynamical relaxation processes in real systems. The hard-sphere fluid, in fact, is not a good reference fluid for the short time ('p') viscoelastic relaxation aspects of rheology.'* There are potential technical problems in simulating steeply repulsive potential fluids by MD, and in response a new molecular dynamics algorithm capable of integrating the equations of motion of impulsive-continuous potential systems (e.g. such as a hard sphere core combined with an Y - " tail) has been derived using operator splitting techniques by Houndonougbo et al. l 4 Anento et al. used MD to investigate the dependence of the dynamical properties of simple liquids on the softness of the potential core.15 They found that the longitudinal modes associated with density fluctuations propagate to

6

Chemical Modelling: Applications and Theory, klume 2

higher wave numbers in liquids with softer potential cores. In contrast, the propagating transverse modes are weakly influenced by the softness of the potential core. Verdaguer and Padro also examined velocity cross-correlations and momentum transfer between concentric shells around a typical particle.16 They concluded that velocity cross-correlations are not only the consequence of direct interactions, but also indirect interactions caused by the neighbours countering local density fluctuations. Wu et al. investigated by MD the instantaneous normal modes of simple Lennard-Jones-like and repulsive LJ liquids, focussing on the effects of the potential on localised resonant modes.17 Itagaki et al. used MD on a tapered LJ potential systems to calculate the self-part of van Hove's spatiotemporal distribution function,

where Ari(t) is the displacement of particle i over a time t.18 In the intermediate time regime between 1 and 500 ps they found a 'pre-Gaussian' distribution of the general form,

The function y ( t ) decreases to zero as time elapses and passes over into the socalled macroscopic time regime. Ohta et al. performed MD simulations of the wave dispersion relationships of Yukawa systems." The Yukawa potential has a screened Coulomb form,

where kD is the inverse screening length, q is the charge and E~ is the dielectric constant of free space. The Yukawa potential could represent, for example, a dust particle particle in a plasma or a colloid particle in an electrolyte, so this potential form is quite ubiquitous. The Yukawa system is characterised by two dimensionless parameters, K = kDa, where a is the mean interparticle distance, a = ( 3 / 4 ~ p ) l where /~ p is the average number density. There is also the inverse temperature, r = q2/4moaT, where T is the temperature. The system is said to be strongly coupled if I'* = r exp(-K) is greater than unity. MD simulations of the wave dispersion relationships showed that there is a truncation of the transverse wave dispersion at long wavelength. l 9 The search for an analytic expression for the velocity autocorrelation function, Z ( t ) = (1/3)(v(O).v(t)), where v(t) is the velocity of an arbitrary molecule at time t, has been of interest since the start of molecular simulation. The selfdiffusion coefficient can be obtained from Z ( t ) using the Green-Kubo formula,

I: Simulation of the Liquid State

7

D

=

1;

Z( t)dt

A quasi-solid model of the liquid has proved popular. This idea is illustrated well by the treatment of Zwanzig, who derived the following expression for Z ( t ) in a simple liquid based on the ‘hopping’ of molecules between potential energy minima in the energy landscape,20

kT Z( t) = - exp( - t/zm) m

(2.10)

where p ( o ) is the density of normal modes and the exponential pre-factor represents the hopping of the molecules between neighbouring sites with an associated characteristic relaxation timescale, t,. The normal mode distribution is most appropriately determined by quenching the system to the nearest local energy minimum, and expanding the potential energy to second order to determine the frequency distribution for that configuration. Chisolm et al. used this approach more directly, from the set of normal modes and frequencies, Their final formula for Z(t) was (2.1 1)

where the mI are arbitrarily chosen vectors. This formula gave excellent agreement with the Z ( t ) determined by MD for liquid sodium. Molecular Dynamics has been used to calculate the self-difision coefficient and shear viscosity of liquid transition and noble metals using effective pair potentials obtained from the embedded atom model, with reasonable success.22 A longstanding ambition has been to obtain dynamical information from Monte Carlo simulations. Of course, formally MC has no inherent timescale and is a, albeit sophisticated, phase space sampling procedure. Huitma and van Eerden have looked at this issue again.23 They ascribed the following definition of the physical time per MC step,

where (Dr)MC is the mean square displacement of the particles per step and & is the self-diffision coefficient measured by MD. They conclude that MC gives realistic dynamics for processes whose defining timescales are in excess of one ps (using argon) such as crystal growth from the melt, but for processes where the rate determining step is less than about 1 ps Monte Carlo may show deviations from molecular dynamics results. Another deficiency of Monte Carlo is that temperature is an input parameter, and unlike MD there has not been an independent check on the temperature prevailing within the simulated system. Butler et al. showed that this (‘configurational’) temperature can be obtained from fluctuations in the molecular potential energy, y24

8

Chemical Modelling: Applications and Theory, Volume 2 (2.13)

More recently, Jepps et al.25 and Rickayzen and Powles26 have generalised this treatment to (2.14) where B is a vector in phase space. For example, B = (0, . . . 0, p1 . . . p 3 N )gives the kinetic energy definition of temperature. The choice, B = (ql . . . q 3 N0, . . . 0) gives the Clausius virial theorem, k g T = -(CqlFi)/3N , where Fj is the force. A derivation of the configurational temperature for orientational motion was made by Chialvo et al.27 Delhommelle and Evans computed the configurational temperature in atomic and molecular liquids in confined slit pores at equilibrium and undergoing Poiseuille They explored two forms of the configurational temperature, one based on the forces and mean-square forces, written as eq. (2.13) with the averaging performed separately for the numerator and denominator terms. The second definition had the averaging, as in eq. (2.14), performed on the ratio. The former approach was shown to give better agreement with the kinetic temperature in equilibrium and non-equilibrium situations. 2.2 Thermodynamics. - There has been renewed interest in the incorporation of three-body interaction energies in molecular simulation. Although three-body interactions can make a significant contribution to intermolecular interactions, their effects have traditionally been included using an effective pair potential. Liquid-vapour phase equilibria calculations were carried out by Marcelli and Sadus using the NVT Gibbs ensemble technique using explicit three body terms in the potential energy surface.30 When compared with the liquid-vapour coexistence envelope of argon, it was shown that the bare (‘true’) two-body term gives rise to a narrower liquid regime, which was largely corrected for by the inclusion of the three-body terms. It was noted that the ratio of the three-body energy, E3, to the two-body energy, E2, for the various noble gas fluids followed a linear relationship of the form

(2.15) where p is the density and v is the energy parameter in the (Axilrod-Teller) three-body interaction, see Figure 3, which shows a plot of the ratio of the threebody energy to the two-body energy as a function of density. This prompted the authors to carry out simulations with only the two-body potential, then adding in the three-body energy simply using the above formula. This gave liquid coexistence values which were almost as good as those generated with the threebody potential included explicitly in the simulation force field (indicating that the three-body terms were not seriously affecting the configurational distributions).

1 : Simulation of the Liquid State

0.0

-

-0.2

-

9

EU’E, El v

-0.4

-

-0.6

0.0

0.2

0.4

0.6

0.8

P* Figure 3

The ratio of three-body and two-body energies obtained from molecular simulation at different reduced densities. Results are shown for argon (A), kzypton (+) and xenon (0).The line through the points was obtained from eg. (6) (Reprinted with permission from G. Marcelli and R.J.Sadus, J Chem. Phys. 2000, 112, pp. 6382-6385. Q 2000, American Institute of Physics)

Another study including three-body terms in the potential was carried out by Bomont et al. who performed MD simulations of supercritical argon including three body forces.31 Yoon and Ohr derived an expression for the compressibility of hard-sphere fluids in terms of the radial free space distribution function, [(r ), which is the probability of acceptance of a (radial) displacement, r, in a Monte Carlo ~irnulation.~~ Alemany et al. performed MD simulations of liquid C60 and calculated its difision coefficient and shear viscosity.33They simulated a system of 13 72 C60molecules interacting through Girifalco’s potential,


10

where r is the centre-to-centre separation of two Cb0 molecules (in units of their The parameters a and p were obtained from enthalpies of diameter, 7.1 sublimation and the lattice constant of fcc fullerite ( a = 0.04677 and 8.485 X eV). The intermediate scattering function, F(q, t ) was computed as a function of time and wavenumber. Interestingly, unlike simple liquids, there was no evidence of collective behaviour, as illustrated in Figure 4.

A).

2.3 Mixtures. - There has been an interest in simple liquid mixtures almost from the start of molecular simulation. This has continued in the last few years, although the principal interest appears to have shifted away from simple molecular mixtures of the argonkrypton variety to semi-continuous distributions of particle sizes which have relevance to real systems such as polydisperse colloidal mixtures. Sear has performed MD simulations of a dense fluid of He generated a so-called 'hat' distribution of sizes in polydisperse hard a range of 5- 10% about a mean value. With increasing volume fraction into the glassy regime he found that the dynamics became 'heterogeneous', by which is meant that the particles with similar mobilities are clustered together, the slow dynamics showing strong finite size effects. Zhang et af. also looked at

1

0.8

0.6

0.4

0.2 n -

Figure 4

0

2

4 t (PSI

6

8

Normalised intermediate scattering function F "(4, t) = F(q, t)/F(q, 0) for liquid C6,, in the state (T = 1900 K, p = 0.74 nm--l) for several wave numbers q. The inset shows the normalised transverse current correlation funFtion Cy(q, t) = C,(q, t)/C,(q, 0) of liquid C,, in the same state for q = 0.145 A-' (Reprinted with permission from M.M.G.Alemany, C . Rey, 0. Dieguez and L.J. Gallego, J Chem. Phys., 2000, 112, pp. 1071 1-10713. 0 2000, American Institute of Physics)

1: Simulution of the Liquid State

11

polydisperse hard-sphere fluids, generating a distribution function of particles at fixed total volume using NPT Monte Carlo. In addition to the trial particle and total ‘volume’ moves, they also implemented a third type of ‘move’ to sample the polydispersity of the system. They selected two particles at random, and attempted to exchange an increment of volume (in the range [-AV,,,, AVmu]) between them. Above a certain volume fiaction (or P* > 2.0) they found the distribution developed essentially a bimodal distribution of particle sizes, in which there were a few large particles embedded within a continuous phase of much smaller ones. Borowko et al. performed scaled particle Monte Carlo on binary LJ mixtures.36This is in the tradition of the Widom method for calculating chemical potential . The computational procedure involved attempting ‘two-step’ insertions of particles, which are initially smaller than the typical, and then ‘grown’ in size. For particles larger than the average, a different procedure was invoked. In this case, an existing particle was randomly chosen, and an attempted enlargement move attempted.

3 Water and its Solutions There are many simulations of water and aqueous solutions being published each year, undoubtedly more than for any other liquid. There have been many interesting developments in this area in the last two years, particularly in the the construction of pair potentials and also in the fascinating phase behaviour of water.

3.1 Pure Water. - Water is undoubtedly by far the most studied liquid, and yet its physical properties are still not well understood, and it is still providing many surprises. A detailed microscopic picture still eludes us. The density maximum at 4 “C and the rapidly increasing heat capacity in the supercooled regime37>38 are two aspects of this behaviour. There are anomalous increases of thermodynamic quantities and divergences in dynamic properties on approaching a temperature of ca. 227 K. There is also a liquid-liquid transition in the supercooled water between a low (LDA) and high density (HDA) amorphous form. Although these features must be associated with the preferred tetrahedral co-ordination of the water molecule, establishing a more detailed explanation has proved problematical. The behaviour of supercooled liquid water has been widely studied in recent years, by experiment and molecular simulation. The LDA supercooled water is thought to be the ‘connected’ water molecules in normal water separating out as part of a liquid-liquid phase transition. This is consistent with the recent observation that LDA transforms to Ice Ih (and not HDA) in the pressure range 3-7 kbar.39MD simulations of supercooled and glassy water were carried out by ’ found that there was not a dramatic structural difference Starr et ~ 1 . ~They between the liquid and glassy states. In agreement with experiment, the simulated liquid structure was nearly independent of temperature at high pressures (ca. 500 Mpa). A detailed analysis of the ftee energy surface of SPCIE model water from simulations by Scala et al. gave a prediction of the liquid-liquid critical

12


point in supercooled water to be T, = 130 f 5 K, P, = 290 f30 MPa and p c = 1.10 f 0.03 gcmP3, although the authors suggest that the glass transition would intervene to pre-empt this.4’ In this study they confirmed other workers’ conclusions, in this case for supercooled water, that the isochoric potential energy scales with temperature to the power 3/5 over a wide density range.41 The existence of low density (higher proportion of four co-ordinated molecules) and high density (higher proportion of five co-ordinated molecules) seems to be a common feature of tetrahedrally co-ordinated molecular liquids. This ‘polyamorphism’ as it has been termed, would appear to be present in liquid silica as well, as shown by MD simulation^.^^ In this study, the critical point associated with this liquid-liquid phase transition has been estimated to be ca. 730 K, which is well below the glass transition temperature of silica (1450 K) and one would not expect to observe it in practice therefore. In the last two years, molecular simulations have carried out to investigate the structural, thermodynamic and dynamical properties of bulk liquid water. For example, Tanaka investigated the ‘connectivity’ of supercooled water in terms of the network distribution of four-coordinated water The minimum energy structures obtained by quenching instantaneous configurations into their local energy minima were obtained. Going down in temperature from 223 to 193 K it was observed that there was a dramatic increase in the connectedness of the four-coordinated water molecules. The dynamics of the tetrahedrally coordinated water molecules were examined by MD using the pair-density van Hove function resolved in the molecular frame.& There was evidence of a translational mode at 200 cm-’ and a collective relaxation mode at ca. 10 cm-I. There was also a 0...O.. .O ‘flexing’ mode at ca. 60 cm-’. The time dependent statistics of the breaking of hydrogen bonds was explored by Stan et aZ.45Their definition of the hydrogen bond was that the interaction energy was less than -10 kJmol-’ and the 0 - H . . .O angle less than 30”. They calculated hydrogen bond life-time autocorrelation functions (which allowed for for breaking and reformation at intermediate times) over a range of temperatures which scaled with a stretched exponential consistent with mode coupling theory (MCT). Marti perfomed a similar study of supercritical water, SW,46They used MD to investigate the extent to which SW can be said to have a hydrogen-bonded network. The ‘intermittent’ hydrogen bond lifetime is typically less than 0.5 ps in the supercritical water (as opposed to ca. 2 ps in bulk water under ambient conditions). The supercritical quantities of water from the SPC/E model are T, = 640 (647) K, p c = 0.29 (0.322) g ~ m and - ~ Pc = 160 (221) bar, where the experimental values are given in bracket^.^' There have been many water-water potentials proposed (e.g. rigid and flexible SPC and TIP4P) since the first simulations of water carried out by Rahman and Stillinger in 1971.48The popular SPC/E potential treats water as a rigid unit with three point charges located at the centres of the oxygen and hydrogen atoms, which have an OH distance of 1.0 and HOH bond angle of 109.47”, the tetrahedral angle. Each hydrogen has a charge, qH = 0.4238e (where e is the absolute of the charge on the electron) and the charge on the oxygen atom is -24H.49 Polarisable water models for water have also been developed.’’ Refine-

A

13

1: Simulation qJ'the Liquid State

ment of the rigid molecule potential model has been carried out by Mahoney and Jorgensen." They obtained an improved non-polarisable model called TIP5P. It is a simple point charge model, but with five interaction centres, although there is no charge on the oxygen atom itself (only a Lennard-Jones interaction is present between the oxygen atoms). The molecule is tetrahedral, with the two hydrogen atoms having +0.241e charges and the two lone pairs with -0.241e charges. The two (OH and lone-pairloxygen) 'bond lengths' are 0.9572 and 0.70 respectively. In generic form the potential between two water molecules, a and h, is

A,

This potential leads to an improved prediction of the thermodynamics over a wide density and temperature range, and, as seen in Figure 5, the density maximum at 4 "C. (The popular SPC/E water model exhibits a density maximum at -26 O C 4 7 . The model also produces a good value for the dielectric constant at 81.5 f 1.5 under ambient conditions. Starr et al. performed MD simulations on 216 water molecules interacting via the SPC/E potential. The dramatic change in the water molecule's dynamics and bulk liquid structure were found to be in agreement with Mode Coupling Theory p(T) for simple water models 1.05

1.04

1.03 1.02

1.01

0.99 0.98 0.97

0.96 0.95

-.50

Figure 5

-25

0

25

50

75

100

Dens@ of TiPnP water models vs. experiment as a function of temperature at

I atm (Reprinted with permission from M.W. Mahoney and W.L. Jorgensen, J Chem. Ph-vs.. 2000. 112, pp. 891 0-8922. Q 2000, American Institute of Physics)


14

(MCT). For example, the temperature dependence of the diffusion coefficient, D, obeyed very well the MCT form,

where above temperatures of T, M 1.2 Tg, where Tg is the glass transition temperature. T, can be thought of as a 'cross-over' temperature, at which the dynamics goes from being dominated by density fluctuations (above) to being controlled by 'activated' processes (below T,). This has been demonstrated in the simulation results, as shown in Figure 6 . A hrther analysis of TIP5P by Mahoney and Jorgenson was carried using MD.52The calculated diffusion coefficient at 25 "C and 1 atm pressure was found to be 2.62 f 0.04 X lop5cm2s-I which compares quite well with the experimental value of 2.30 X lop5cm2s-'. The pressure dependence was also in quite good agreement with experiment. The density maximum of water at 4 "C was also reproduced quite well by the TIP5P model, as may be seen in Figure 5. The thermodynamic and structural properties of water were determined in terms of the inherent structures or 'basins of attraction' by Starr et aZ.53

10'

10"

1o'

P

1o-¶

1o-3

Tn,- 1

TR,- 1

T/T,- 1

Figure 6 Fit of each isochore to the power law D M ( T / T c - 1)' predicted by MCT We include the results of ref 43 along the p = 1 .O g cm isochore (Reprinted with permission from F.W. Starr, F. Sciortino and H.E. Stanley, Phys. Rev. E , 1999, 60, pp. 6757-6768. @J(1999) by the American Physical Society)


15

MD simulations of supercooled water by Chen et al. focussed on the selfintermediate scattering function of the centre of mass of the water molecule^,^^

F,( k , t ) = (exp( -ik. r(O))exp(k . r( t ) ) )

(3.3)

where k is the scattering vector. The function showed two well-defined decay regimes, separated by a definite plateau [the so-called p (short-time) and a (longtime) decays]. The long-time decay fitted well to a stretched exponential, as suggested by MCT. The k-dependence of the shear viscosity, q(k), was determined through the wave-vector dependent transverse-current correlation function, C,( k , t), and the Mori-Zwanzig projection formalism,55

where

+ u2k 2 ) where a is a state-dependent constant enabled the zero-wave vector Newtonian viscosity to be calculated. The fit gave 9.71 mP which compares well with the experimental value (8.9 mP). Guo and Zhang used equilibrium molecular dynamics to calculate the shear and bulk viscosities of liquid water.56 Using the SPC/E model they found that these were 6.5 and 15.3 mP, respectively, as opposed to the experimental values 8.9 and 21.3 mP

An extrapolation of the form q ( k ) = q(O)/(1

3.2 Aqueous Solutions. - Over the years MD has been applied to investigate the solute structure and dynamics of alkali halides in water. It is well-known that in water at 25 "C the residence time of the water molecules around the ions decreases dramatically as the ion increases in size. MD simulations have been carried out to explore the hydration of ions in supercritical water solution^.^' They calculated a residence time correlation function,

where Ol(r, t ) is the Heaviside step function, which is 1 if a water molecule i is within a spherical region of radius Y within the first hydration shell of the ion, and 0 if it is not within this shell. N , is the average number of water molecules in this region at t = 0. The residence time, t,,is the relaxation time of this function, for example, when fitted to an exponential. Simulations were carried out at densities from 0.2 to 0.75 g ~ m - The ~ . t, values decreased with increasing density, and were largest for Lit (3.5 and 2.6 ps at 0.2 and 0.75 gcmP3 respectively). The

Chemical Modelling: Applications and Theoty, Volume 2

16

electrical conductances decreased with increasing density and were smallest for the strongly hydrated ions (e.g. Li+). MD simulations of sodium ions in water in cylindrical channels were carried out by Allen et The objective was to model some of the characteristics of biological channels. The channels were hydrophobic and hydrophilic to varying degrees, and were typically 40-60 long and 4- 12 in diameter. They found that for narrow channels the water radial density profiles were relatively independent of the surface type. They found that the local structure and diffwsion coefficients were sensitive to pore radius and the extent of affinity of the walls for the water molecules. The ions tended to congregate in the centres of the channels, along the channel axis. MD simulations of dilute O2 solutions were carried out by Fois et al.59 Oxygen, being an apolar gas, falls into the category of low solubility structure makers, because of an unfavourable entropic contribution to the Gibbs energy of solution. The simulations confirmed this picture. They found oscillations in the density and electrostatic charges around the solute. The hydrogen bond’s lifetime was found to be shorter in the first co-ordination shell, although the water translational and rotational diffusion co-efficient was lower in the first hydration shell than in the bulk.

A

A

4 Organic Liquids

4.1 Alkanes. - Molecular simulation continues to be applied and optimised to study the properties of organic liquids. Alkanes are perhaps the most used testbed for methodological developments. One aspect of this work has been to develop intra- and inter-molecular potentials, especially so-called united atom (UA) versions in which methyl and methylene groups are treated as a single interaction centre. This obviates the necessity to follow the dynamics of hydrogen atoms, which would require very small time steps. The Anisotropic United Atom, AUA scheme merges the hydrogen and carbon centres of force onto a single site at rcJi,where r, is the carbon position, and d is the displacement of the position of the interaction from the carbon atom position, in direction,

A total potential energy is the sum of various intra and inter-molecular which has the following typical form,6’ bonds

+

angles 1

i

cc

atoms atoms

i

j>i

POyrij)

17

I : Simulation of the Liquid State

The first two terms are the bond stretch and bond bending energy. The third term ( V , . . . V 3 )is the torsional energy for each four-atom group along the chain. The last term represents the non-bonding interactions, and usually has the LennardJones form. There is a continual process of refinement of the parameters in these polyolefin alkanes. Nath et al. performed Gibbs ensemble Monte Carlo simulations on 1-octene, 1-hexene, 1-pentene, 1-butene and propene.62 The liquidvapour co-existence envelope and vapour pressure were found to be in excellent agreement with experiment, as shown in Figure 7. Another theme often explored with model alkanes is the relative merits of equilibrium MD (EMD) and non-equilibrium MD (NEMD) as methods for obtaining transport coefficients. Dysthe et al. explored some aspects of the methodology used to obtain transport coefficients by MD.63 They applied the Green-Kubo formalism to flexible multicentre models from linear and branched

500 A

450

400

h

b

350

300

250

0.00

0.20

0.40

0.60

Density ( g k c )

Figure 7

Orthobaric densities for small alkenes. The .filled circles are simulation results from present work, thejlled triangles are from Spyriouni et al. (ref 12), open symbols are experimental results (ref 17), and the lines show an Ising Jit to the simulation points. The error bars near the critical point are less than three times the size of the symbols. Error bars at lower temperatures are less than twice the size of the symbols (Reprinted with permission from S.K. Nath, B.J. Babaszak and J.J. de Pablo, J Chem. Phvs., 2001, 114, p p . 3612-3616. 0 2001, American Institute of Physics)

18


alkanes in the gas and liquid states. They investigated details of the potential, the EMD vs. NEMD issue and system relaxation times as a basis for sub-averaging and assessment of statistical uncertainties. In another work, Kutteh derived the socalled GSHAKE algorithm for enhanced algorithmic stability when incorporating non-holonomic (velocity restrictions) and holonomic (bond and angle restrictions) constraints into MD sir nu la ti on^.^^ The relationship between molecular architecture and transport coefficients and rheology has been a major theme of EMD and NEMD simulations in recent years. Moore et al. used EMD and NEMD to predict the rheology of C30 isomers as model lubricant base stock^.^^ Calculations were caried out of n-triacontane, 9- noctyldocosane and squalene at temperatures in the 31 1-372 K range using a united atom model. Compared to experiment, the simulations underpredicted the kinematic viscosities of the first two compounds, but captured the temperature dependence much better. Dysthe et al. performed EMD simulations of the transport properties of n-butane, n-decane, n-hexadecane and 2-meth~lbutane.~~ This work used AUA parameters previously benchmarked against selected thermodynamic properties such as vapour pressures, enthalpies and They found that with increasing density the models progressively underestimated the viscosities and overestimated the diffusion coefficients by about a factor of 2, which is really quite good. The same trend was found with increasing chain length. The authors attributed this to deficiencies in the torsional potential. Bedrov et al. used EMD to compute the liquid shear viscosity and self-diffusion coefficient of the high explosive HMS (octahydro- 1,3,5,7-tetranitro-1,3,5,7-tetrazocine) over the temperature range 550-800 K.68 The viscosity decreased from 0.45 to 0.006 Pa s over this temperature range. The self-difision coefficient obeyed Arrhenius behaviour over this temperature range. McCabe et al. carried out equilibrium and non-equilibrium MD simulations of the viscosity of 9~ctylheptadecane.~~ The viscosity calculated was somewhat lower than experiment, although its pressure dependence was in quite good agreement. Simulations of alkane phase equilibria have been carried out. The solid-fluid phase diagrams of flexible hard-sphere united atom models of n-alkanes have been determined by Monte Carlo ~imulations.~~ These data were used as reference states in a van der Waals model. For certain torsional potentials, the theory correctly described the dependence of the ratio of triple point to critical point temperature as a function of chain length (has a minimum at about n = 3). Polson and Frenkel calculated the melting line of n-octane using an atomistic model for the molecule which included the usual internal degrees of f r e e d ~ m . ~Using ' ideal gas and Einstein crystal reference points, and thermodynamic integration coupled with Gibbs-Duhem integration, they found quite good agreement with the experimental co-existence data. Over the 1- 100 MPa pressure range the co-existence temperature was within two or three degrees of the experimental values. NPT MD simulations were carried out of the melting of n-CsHls to n-C16H34.72 With increasing chain length they found that the melting point increasingly underestimated the experimental value. For example, for n-CSHl8the melting range was 200-210 K whereas the experimental value is 216.4 K, and for n-C16H34 the figures were 250-260 and 291.3 K respectively. Picu et al. carried out non-equilibrium

1: Simulation of’ the Liquid Starr

19

molecular dynamics (NEMD) of linear chain molecules with N covalent bonds up to They were interested in the relative roles of bonded and non-bonded potentials in their contributions to the shear viscosity. It was found that the nonbonded potential had the principal contribution to the stress while the bonded contribution was negative, which is counter to the usual assumption that nonbonded interactions contribute little to the viscosity of a melt. Simulations of alkanes in the networks in porous solids have been carried out. n-Butane-methane mixtures in the zeolite silicalite were modelled by MD, and comparisons made with the results of quasi-elastic neutron scattering experiments. The two techniques gave self-diffision coefficients, D, that are statistically in agreement, and the value of D decreasing with the level of ‘loading’. Jump diffusion, for example, between adjacent channel segments was shown to be a key mechanism in pore tran~port.’~ An alternative approach to zeolite diffusion was invented by Vlugt et al.75Diffusion of isobutane in silicalate was followed by a so-called transition path sampling procedure which efficiently computed the hopping rates between stable states. Marchi and Procacci, investigated the use of constant pressure MD applied to long chain alkanes.76They showed that both atomic and molecular definitions of the pressure were statistically equivalent. Nymand et al. calculated by MD the temperature dependence of the dielectric function of liquid benzene.77

4.2 Oxygen Containing Molecules. - MD simulations of the dielectric properties of liquid ethanol have been carried out using a four interactions site model of the molecule (two for the methyls and one each for the oxygen and hydrogen in the hydroxyl Self-consistency with classical theories was found. The short time decay of the total system dipole of the system was found to be dominated by the single molecule librational motion, whereas the long time decay was more collective in origin. The relaxation functions had relaxation times typically in the order of 5 ps. MD simulations of supercooled model glycerol have also been carried out recently.79The simulations overestimated the glass transition temperature by about 20 K. Saiz et al. performed MD simulations of liquid ethylene glycol (HOCH2CH20H) using several force fields.80The molecule has three torsional internal degrees of freedom and therefore 27 possible conformational states. They found many gauche conformations in their simulations and a 3D network structure in which each molecule had slightly fewer than four hydrogen bonds. Hanson camed out MD simulations of a single strand of polydimethylsiloxane (PDMS) or (-OSi(CH3)2-)n- interacting with a model silica surface, to study the bonding properties of thus polymer with silica filler^.^' It was found that the ‘pull-off’ force was more or less constant, irrespective of the extent of attachment.

5 Non-equilibrium Molecular Dynamics (NEMD)

Non-equilibrium Molecular Dynamics, NEMD, is the application of molecular dynamics simulation to investigate the response of liquids to the effects of

20


external perturbations. It has a relatively old history, in the area of liquids, starting in the early 1970s when ad hoc schemes were devised to impose shear flow on a liquids. In the 1980s and since then, these approaches have been placed on a sounder statistical mechanics footing, pincipally by Evans and Morriss, and Hoover and co-workers. The basic strategy is as follows. Consider a N-particle system with co-ordinates and peculiar momenta {ql, q2 . . . q N , pl, . . . p N }= { q , p } = r. The internal energy of the system, Ho, is,

where @(q)is the potential energy, which depends on the molecular co-ordinates, q . K is the kinetic energy which is defined in terms of the peculiar momenta (i.e. for each particle, its momentum with the average drift momentum at its location subtracted off). In the presence of an external field, F,, the generic thermostatted equations of motion are

where F,(q) = -&D(q)/aq, and a is the thermostat multiplier. C, and 0, incorporate the coupling of the system to the field. The dissipative flux is J = -(dH,/dt)/ VF, and the transport coefficient is y = J/F,. There are synthetic schemes for all of the transport coefficients. A novel recent application of this construction was devised by Maillet et a/. who applied it to model uniaxial shock waves passing through a They proposed an equizibrium MD method centred around a uniaxial ‘Hugoniostat’ which simulated the final state of a shocked crystal. This method homogeneously and uniaxially compressed the crystal at time zero to the final shocked volume, and then coupled the system to a thermostat that guarenteed that the final Hugoniot state was achieved. We have in this scheme,

where the heat flow rate is daldt, (which depends on the instantaneous values of the total energy, E), the pressure tensor component along the uniaxial compression direction is P,,. Po and Eo are the initial pressure and energy in the uncompressed state, and Y is a frequency associated with the heat flow rate. E~ is the initial compressive strain.

21

1: Simulation of' the Liquid Starr

In the last few years there have been a number of important developments in NEMD techniques and applications. A novel method for generating steady-state planar elongational flow was developed independently by Todd and Daiviss3 and by Baranyai and Cumming~,'~ The orthogonal contraction and elongation are performed at a 'magic' angle with respect to the usual rectangular periodic boundary conditions. The exponential growth in deformation rate associated with this process leads to instability problems in the algorithm, which have been circumvented using a numerical correction procedure.8s A number of NEMD-related themes continue to be popular, the effect of shear rate on the anisotropic distortion of the radial distribution function in a simple liquid being one of them. Kalyuzhnyi el al. compared a non-equilibrium distribution function theory of g ( r , Q), where Q is a generalised angle, with those produced directly by NEMD.8h They also compared the distribution functions of simple fluids under shear8' with the predictions of the theory of Gan and Eu,**who derived an angular dependent pair potential which can be used in an equilibrium simulation to give the non-equilibrium distribution functions. The advantage of the Gan and Eu approach is that the usual tools of equilibrium statistical mechanics (e.g. integral equations and perturbation theories) can be brought to bear on these non-equilibrium systems. The agreement of the theory at high shear rates with homogeneous shear flow NEMD is reasonable and does offer promise for making further improvements in the theory. Another topic of recurring interest associated with the simulation of shear flow in atomic liquids is the shear-rate dependence of the viscosity, pressure and other thermodynamic quantities. Marcelli et al. performed homogeneous shear flow NEMD on a fluid interacting with two-body, G 2 , and three-body, G3, terms in the interaction p ~ t e n t i a lThe . ~ ~ three-body potential term was @I(

ijk) = Y

( 1 + 3 cos 0, cos 0, cos 6 , ) ( Y I ,rrh r,k l3

(5.7)

which is the popular Axiltod-Teller form, where 8, is an internal angle at atom i , of the triangle formed by molecules i , j and k. They found that the strain rate dependence of the pressure goes as the square of the shear rate, rather than to the power 1.5, as has often previously been found. They discovered that the origin of this behaviour lay in the form of the two-body potential rather than in the inclusion of three-body effects per se. This result does inter alia illustrate that the shear rate dependence of thermodynamic and transport properties is probably not 'universal' but is sensitive to the molecular interactions. Lacks performed an instantaneous normal mode spectrum analysis of a Lennard-Jones fluid undergoing shear flow by NEMD.90Shear flow was shown to increase the fraction of imaginary frequencies, indicating a shear-enhanced eradication of the local energy minima."' Todd et al. discussed and compared various NEMD techniques for generating shear flow and hence shear vi~cosity.~' They compared (a) homogeneous shear, HS. within the 3D periodic boundary condition framework, (b) in a slit pore geometry, with the atomic fluids sandwiched between parallel atomic walls; thermostatting takes place only at the

22

Chemical Modelling: Applications and Theov, Volume 2

walls (sliding boundary, SB) and (c) as for (b) except that an homogeneous thermostat is applied to the walls and fluid atoms. They pointed out a hndamental difference of (b) compared with (a) and (c); that is the density and temperature vary across the film in (b). The thermodynamic state point is therefore varying across the pore, and any average velocity from such a calculation is only an effective one. Therefore care should be exercised when comparing (b) and (c). New ways of applying non-equilibrium states in molecular dynamics continue to be invented. Arya et al. devised a novel NEMD method for extracting the shear viscosity.92 The method follows the decay of a Gaussian velocity profile in a simulation cell, u,(y, 0) = a. exp(-boy2). An analytic solution of the NavierStokes equation leads to a decay of the central or peak velocity, up, according to

where to = 1/4vbo and v is the kinematic velocity. By focussing on the peak decay (which is at the furthest point from the boundaries), the effects of phonon traversal effects are minimised. Muller-Plathe devised a method to obtain the shear viscosity in which the periodic cell has a triangular velocity profile established in the x-dire~tion.~~ The maximum velocities are at the boundaries and in the centre (but in opposite directions though). The procedure was to interchange the velocities of the two particles with the largest velocities in the opposite direction to the average streaming velocity. This way a stress and strain rate are established in the simulation cell, and hence the viscosity can be computed using the usual formal definition. Heat flow and thermal conductivity continue to attract interest. Daivis and Coelho derived the leading terms of a generalised Fourier law for heat conduction in strong non-uniform shear Bedrov and Grant applied a novel NEMD method to obtain the thermal conductivity to n-butane and liquid The method sets up a heat flux and from the generated temperature gradient, the thermal conductivity is computed from the formal classical definition. The heat flux was created by exchanging velocities of particles between cold and hot layers located in the middle and adjacent to one of the periodic cell boundaries. The cold layer swaps the kinetic energy of the particle with the highest kinetic energy with the particle in the cold slab with the lowest kinetic energy. One advantage of this approach is that the heat flux does not need to be computed, which can be complicated for molecular liquids. Good agreement with experiment for n-butane and liquid water was obtained. New developments in the theory of non-equilibrium fluids continues to bring out new and fascinating formulae. Searles and Evans developed further a fluctuation theorem which gives the probability of observing second law violating dynamical fluctuations in finite sized non-equilibrium systems.96The key formula is

23


which in words means that the probability of getting the average flux (commensurate with the applied field, Fe), J, P r ( J ) to -J (the flux opposite to the expected value for an infinite system) P r ( - J ) is given by the above formula. The authors generalised this formula to many different NEMD ensembles. Another aspect of non-equilibrium relaxation was considered by Barrat and 2

1.5

1

0.5

0 0

0.4

0.2

0.6

0.8

1

c, ( t ) 2

+o I

= 0.00005 +*= 0.001 "y = 0.01

L. ..

1.5

)I(

a 0

A

n W 'u

$

1

0.5

0 0

0.2

0.6

0.4

0.8

1

c, ( t ) Figure 8 Parametric plots for (a) T = 0.3 and (b) T = 0.5, and various shear rates. In both ,figures, the dashed line is the FD7: and has a slope of -1IT. The full lines are linearfits to the data for y > 0 (Reprinted with permission from J.-L. Barrat and L. Berthier, Phys. Rev. E, 63, (2000) by the American Physical Society) 2000,012503.

Chemiccil Modelling: Applications and Theory, Volume 2

24

Berthier.97 They applied SLLOD NEMD to shear a Lennard-Jones particle mixture fluid to investigate the breakdown of the fluctuation dissipation theorem of glassy states out of equilibrium (in both senses). This work was carried out in the context of the wider and rapidly developing subject of ‘ageing’ which occurs widely in many materials. If one studies the correlation function, C ( t ) = ( A ( to t)B(to)) between observables A and B, the associated response function x ( t ) = d(A(to t ) ) / d h ( t o )where H is the field conjugated to B. At equilibrium these two functions are related through the Fluctuation Dissipation theorem, FDT, which states that aC( t ) / a t = - kTx( t). Out of equilibrium, this treatment requires some generalisation, by substituting the temperature, T, with an ‘effective’ temperature, Teff,which in the limit of zero shear rate converges to i? However, when T is below the glass transition temperature, when the viscosity goes to infinity, then the effective temperature is that of the system ageing at the same temperature. Their simulations of the integrated response function, M 4 (t ) , showed that sheared liquids obey the two-time scale, two-temperature phenomenology associated previously with glassy systems. As may be seen in Figure 8, in short time scales the FDT temperature is equal to the equilibrium or kinetic temperature, whereas at long times a different equivalent temperature is observed which is only weakly dependent on the bath (‘kinetic’) temperature. Barsky and Robbins performed NEMD simulations of the interfacial structure and rheology of binary blends of symmetric polymers which were made immiscible to various degrees by adjusting the cross-interaction parameters.” A liquid film was sheared by sliding boundaries. They found that there was a difference in velocity of the two species at the interface, suggesting a partial slip boundary condition. They attributed this to a difference in the positions of the centres of mass of the species on opposite sides of the interface. The viscosity in the interfacial region was lower than the bulk viscosity.

+

+

6 Glasses

There is much current simulation activity in studying supercooled liquids and the glass transition. There are a number of themes of investigation which are highlighted below.

6.1 Phenomenology. - When a liquid is cooled there is a dramatic increase in viscosity. If the freezing transition can be avoided, by rapid cooling into the socalled metastable supercooled regime, the viscosity will increase from a value typical of simple liquids, mPa s, to 10l2Pa s, at which point the glass transition intervene^.^^ The relaxation behaviour of glass-forming liquids shows a number of common features, such as a temperature dependence of the viscosity, q, represented by the Vogel-Fulcher-Tammann (VFT) empirical equation,

1: Simulation of the Liquid Stute

25

where To is the extrapolated temperature at which the data indicate that the viscosity would be infinite, qo is a reference viscosity and the parameter D is a measure of the structural ‘strength’ of the system. The largest values of D (==:20-100) are typical for those liquids whose microstructure is relatively insensitive to supercooling (they are said to be, structurally ‘simple’). These liquids display a near-Arrhenius viscosity- temperature dependence, and are termed ‘strong’ glass formers, after Angell. Covalently bonded liquids such as Si02 have D = 100, while Bz03 have D = 35. The so-called ‘fragile’ glasses have D zz 3-5, and these exhibit a rapid change in viscosity in the supercooled regime close to the glass transition, which indicates a much greater degree of structural reorganisation in the vicinity of the glass transition. The structural relaxation times are proportional to the viscosity, and eventually a glass is formed that has characteristic relaxation time far in excess of accessible laboratory time scales. Glasses are, for all intents and purposes, solids as far as their mechanical properties are concerned, but they are not crystalline, lacking the long-range atomistic or molecular order characteristic of a crystal. In some respects the ‘glass transition’ could be viewed as a (second order) thermodynamic phase transition, say in exhibiting a sharp decrease in the heat capacity at the glass transition temperature, Tg. However, the glass transition temperature itself depends on the cooling rate, and the glass is a solid only in the sense that the time scale for liquid-like flow phenomena is large compared with normal experimental timescales. Therefore one cannot consider it as a fixed point in the phase diagram, as would emerge from the application of Gibbs’ phase rule. These flow phenomena are the result of dynamical relaxation processes on the molecular scale that are occurring on a wide range of timescales, even below the glass transition. The discussion about whether the glass transition is a thermodynamic transition or a kinetic phenomenon is a perennial issue of debate, which shows no sign of abating. The difference in entropy between the supercooled liquid and the crystal, AS, decreases with temperature. Kauzmann plotted AS/A,S, where AmS is the value of A S at the melting point, as a function of temperature in the supercooled region.’”” He found that for some glass-formers this ratio tended to zero at a finite temperature, now called Kauzmann’s temperature, T K , which is below the experimentally measured glass transition temperature. This would suggest that for T < TK the entropy of the supercooled liquid would be lower than that of the crystal, which is not possible. This is known as Kauzmann’s paradox. In practice, the glass transition, where the system falls out of equilibrium, ‘intervenes’ to prevent the system evolving structurally into a state where there is this entropy ‘catastrophe’. Essentially the system falls out of quasi-equilibrium at Tg and the molecular microstructure is frozen in for all practical purposes. As mentioned already, some glass-formers show a large decrease in heat capacity and a nonArrhenius viscosity on approaching Tg. These are called ‘fiagile’ glass-formers (e.g. ZnCl,). They can be distinguished from ‘strong’ glass-formers (e.g. silica) which have small heat capacity change and Arrhenius temperature-viscosity dependence down to Tg. ‘Fragility’ can be ascertained from a thermodynamic perspective in terms of the rate of descent of AS/AmS close to Tg or from the

26


kinetic perspective, in terms of the relaxation time steepness index, rn = [dln(t,)/d(Tg/T)]T=Tg,where t, is the structural relaxation time of the arelaxation processes (e.g. v/Gm, where is the shear viscosity and G, is the infinite frequency shear modulus). The larger is rn, the more fragile is the liquid. Generally these two definitions are consistent, although not in all cases.'O' Despite the fact that experimental cooling rates are many orders of magnitude smaller than can be achieved in a computer, supercooled liquids have proved to be one of the most studied features of the liquid state by molecular simulators. Even within the period of this review, the number of papers published using MD to investigate supercooled liquids and the glassy state must be over a hundred. It clearly is not possible to do justice to them all in this review. MD simulation has been used many times in recent years to study the dynamical properties and structural relaxation of highly supercooled liquids near the glass transition temperature. A variety of model systems have been considered, and the complex network explored by spatial and temporal probes, such as the intermediate scattering function, F ( k , t ) . Some glassy systems have been simulated in two dimensions, 2D (e.g. hard disc mixtures"*) but most have been in 3D. The procedure used is to gradually or suddenly cool a MD liquid below its normal freezing point. Monte Carlo has also been used, and in this case a more sophisticated multimove MC procedure has been developed, involving 'steps' consisting of many particle displacements and particle swaps.'03 The usual structural probes, such as the radial distribution function, g ( r ) , which is so useful for liquids at temperatures above the melting point, are a relatively insensitive probe for supercooled liquids (in which relaxation times can increase Much of the contemporary interest is focussed on the mode by a factor of coupling temperature, T, > Tg which in broad terms separates the supercooled regime into two parts. For T < T, activated crossing of barriers dominates the dynamics, while for temperatures above T,, distinct barrier crossings can be ignored. In the latter case, the potential energy landscape (see below) is relatively flat.lo4It is thought that T, coincides with the crossover temperature below which discrete hopping processes become important part of the structural evolution, an idea postulated many years ago by Goldstein."' There are several recurring themes that appear in and have motivated the simulation studies. The first is that there is some distinguishing feature of the potential energy landscape which can lead to a quasi-equilibrium or solid model for the glassy state. This perspective on the glassy state is discussed in the next section. 6.2 Structural Models for Supercooled Liquids. - This approach focuses on the molecular order in the supercooled liquid. The potential energy surface or 'landscape' @ ( r N )where r N = r1, r2 . . . rN is a 3N-dimensional vector in phase space has a characteristic appearance in the various states of matter. In a liquid, the @-landscape has a distribution of many shallow energy minima. In a crystal there are a few steep and deep minima representing the collapse of the system into the crystalline states with long-range order. In a liquid simulation, each of these deep minima, or 'inherent' structures as they are called, will be surrounded

27


by a distribution of r." configurations, which on energy minimisation or 'quenching', say by steepest-descents, will alight on them. This is why they are also called 'basins of attraction'. The idea is to split configuration space into a series of inherent structures, IS, an idea originally proposed by Stillinger and Weber.'06 About each IS there is surrounded a basin of attraction of thermalised states or configurations, as sketched in Figure 9. Any point in configuration space can be described as the starting-point for a steepest-descent operation in the potential energy surface. The end-point of the steepest descent is the IS of the starting point. The motivation for performing this analysis of phase space is that one can assume that, to a large extent, the 'thermal' fluctuations about these ISs are of secondary importance, and could be incorporated in a model as mean-field approximation. The statistics of these inherent structures in a quenched binary Lennard-Jones type structure was analysed by Biichner and Heuer.'" During the MD simulation at constant temperature the system's energy and r"' were regularly energy-minimised. Then the simulation was continued starting from the configuration before the minimisation started. The dynamics were described in terms of the incoherent scattering function S(q, t ) , .

h

On increasing the depth of the quench, or on waiting longer after the quench, the relaxation in S ( q , t ) developed a plateau. The relaxation was distinguished by two widely separated characteristic time scales, fast (p) and slow ( a ) processes, corresponding to the initial rapid decay of density fluctuations and the much slower subsequent relaxation associated with collective dynamics, as revealed in Figure 10. This slowing down of the characteristic dynamics is associated with a more sharply structured appearance in the landscape of inherent structures. The system was found to spend increasingly longer periods of time oscillating between just a few ISs. The minima in the distribution of energies of the IS shifted to Potential energy landscape

Thermalised state

Inherent Structure or basin

-+

Figure 9 Schematic diagram illustrating the quenching of an MD conjguration into its nearest basin of attraction on the potential energy landscape

28


1 .o

0.8

0.6 W

=i 0.4

v)

0.2

0.0

-2

0

2

4

Figure 10 The temperature dependence of the incoherent scattering function SA,4(q,r) for N = 60 (Reprinted with permission from S. Buchner and A. Heuer, Phys. Rev. E, 1999, 60, 6507. (1999) by the American Physical Society)

more negative values, but only less than 0.5 k g T . The picture that emerged is that close to the glass transition, the system stays close to the ISs and the dynamics can be described by a superposition of local vibrations and isolated hopping processes. In a subsequent paper, Biichner and Heuer took this treatment further. They demonstrated that there was a link between this IS configurational space topography and the non-exponential relaxation processes and dynamic heterogeneities.Io8 In particular, they found that when the system was trapped in a IS valley the mobilities were especially small. The structure of the potential energy landscape was linked, at least qualitatively, to the slowing down in the structural relaxations in the supercooled liquid. Oligschleger and Schober also investigated the intermittent diffusion processes in supercooled liquids.*09They generated a glass using a modified soft-sphere potential,

where the interaction truncation was at r / a = 3.0, A / & = 2.54 X and B / E = -3.43 X lop3.The last two terms in the equation ensure the potential and force go to zero at the truncation radius. They observed localised jumps of ten or

I : Simulation qf'the Liquid State

29

more particles moving one after the other, collectively, in chainlike formations. The evidence now is rather strong that in supercooled the molecular dynamics is spatially 'heterogeneous'. Rearrangements of atoms/molecules in glasses occur co-operatively in concerted events of very short duration and involving many particles. These events, which one should consider the equivalent of a relaxation in the glassy state, are suppressed in small simulation samples. This was revealed recently in a simulation of N = 108-lo4 soft sphere particles, as seen in Figure 11 . ' I 0 The structural relaxation is slowed down because the number of directions connecting the local minima decreases dramatically with temperature. One of the interesting aspects of supercooled liquids is that although rearrangements of particle configurations in glassy materials are co-operative, involving many atoms, the difision coefficient, 0, of a tagged particle is much larger than predicted by the Stokes-Einstein relationship. Further MD simulations by Yamamoto and Onuki helped reconcile this paradox."' They found that its origin lies in the coexistence of inactive and active domains in which the diffusion coefficients are quite different. This is consistent with the emerging picture that the dynamics in a

Figure 11 Spatial distribution of particle displacements having the cluster size n 2 5 at T = 0.267 for N = lo". The arrows indicate individual particle displacements (Reprinted with permission from K. Kim and R. Yamamoto, Phys. Rev. E, 2000, 61, R41-R44. Q (2000) by the American Physical Society)

30


highly supercooled liquids system becomes increasingly 'heterogeneous'. Further light was cast on these dynamical heterogeneities by Donati et al."' They found that the highly 'mobile' regions formed string-like structures, whereas the immobile regions formed relatively compact clusters. Poole et al. invented a novel function to probe any growth of domains or lengthscales in the super~ooling."~ Another perspective on the time-dependent microstructural evolution in supercooled liquids was presented by a new resolution of the radial distribution function. The concept of 'neighbourship' was invented by K e y e ~ . "The ~ g ( r ) was resolved into a series of separate radial distributions for the first neighbour, second neighbour and so As the temperature decreased it was found that these shells became more radially separated and that the dynamics of the first shell departed from a simple difisive model, which is indicative of the onset of slow collective cluster dynamics.'14 Further support for the view that the dynamics of supercooled liquids are governed by the geometry of the configuration space comes from the work of Nave et a1.I" who performed an instantaneous normal mode, INM, analysis of supercooled SPC/E water, in the vicinity of the so-called Mode Coupling Temperature, MCT or T,. The basic idea is that liquids are solid-like at short times and that they can be considered to vibrate about a short-lived equilibrium configuration. This is broken up by periodic 'jumps' to new energy minima on the potential energy landscape. The INM method uses the eigenvectors (normal modes) and eigenvalues of the Hessian matrix (the second derivative of the potential energy) to provide a representation of the topology of the potential energy surface. For an N-particle system at a given temperature, one chooses a configuration comprising a 3N-dimensional vector of atomic co-ordinates, &. The total potential is expanded as a Taylor series about &.

where the 3N-dimensional force vector, F, and the 3N dynamical matrix, K, are given by

X

3N dimensional

where i is the atom index, rn, is the mass and F,, is the a-component of the force on the i-th atom,

Standard normal mode analysis of the matrix K is not necessarily positive definite, since the system is chosen from the trajectory of a system at non-zero temperature. Also F is also not necessarily zero. Diagonalisation of K yields the


31

instantaneous normal modes (eigenvectors) and the eigenmodes are the corresponding frequencies. Negative eigenmodes give imaginary frequencies (the square root of the eigenmodes). Simulations show that the INM of ‘normal’ and supercooled liquids always have positive and negative eigenvalues (ie. real and imaginary frequencies). The positive eigenmodes represent harmonic vibrations in well defined wells. Negative eigenmodes represent negative curvature in the phase space. The authors calculated the fraction of imaginary or unstable modes, and distinguished within this category those modes in which the negative curvature is the result of local anharmonicity (‘shoulder modes’), and the so-called double-well modes which only include those modes (‘direction of motion’) connecting two potential energy minima. They found a strong correlation between the increase in the number of double well modes and the MCT, indicating that the decrease in system mobility is reduced because the number of directions connecting local minima is decreasing. The number of modes that can lead to a new local minimum decreases with decreasing temperature and vanishes at T,. Above this temperature there is at least one unstable mode current, whereas at temperatures below T, the system mainly occupies a particular energy (metastable) minimum. At T, the system has sufficient thermal energy to traverse the lowest energy saddle point between energy minima in the IS surface. Bembebek and Laird performed an INM of supercooled silica, which was chosen because of its technological importance and it being an archetypal strong glass former.”6 The fraction of imaginary modes decreased monotonically as a function of temperature. At temperatures below the glass transition temperature, the imaginary frequency region was found to comprise of coupled rotations within chain-like groups of the Si04 units. Another MD simulation of silica focused on testing mode coupling theory in terms of the wave vector dependence of the Debye- Waller factor.’ The decoupling of the molecular reorientation in the supercooled regime has also been noted in MD of Lennard-Jones dumbbells. There is a reduced tendency to freeze the rotational degrees of freedom.”’ The are various ways of characterising this increasing entrapment of the atoms in a supercooled liquid. Allegrini et aE. calculated the mean-first-passage-time, MFPT, which is the average time it takes a particle to travel a prescribed distance.”’ This simple function helps identify characteristic length and time scales in a liquid. They defined a ‘dynamic entropy’ which is the inverse of this time. They found that the dynamic entropy for traversals of the order of one molecular diameter tended to zero as the Mode Coupling or MC temperature is approached, indicating increasing ‘entrapment’ of the molecules, and consistent with the various other approaches discussed in this section. A common feature of the slow (‘a’) structural relaxation in glass-forming liquids is an apparent universal analytic form for the long time decay, which is described by a stretched exponential. For example, for the intermediate scattering function, ISF,

’’

32


where the exponent, pq < 1, T, is the relaxation time and fq accounts for the initial decay of correlations due to phonons and fast relaxation processes. A more complete representation of the ISF, including short time kinetic effects, was proposed Liao et aZ.lzo

The short time relaxation part was computed using the so-called @dependent triple relaxation time (QTRT) kinetic model which is an approximation to the generalised Enskog model. This functional form was shown to reproduce well the MD generated simulation data in the supercooled regime. An interesting idea, to give another perspective on the structural disorder, was the calculation of an effective local ‘pressure’, P,, separately for each particle and its distribution,121

where r is the radius of the sphere ( i e . 0 for the LJ potential). The pi showed a wide distribution of values in a supercooled LJ system. 6.3 Ageing. - A glass is a dynamically and structurally evolving system, albeit very slowly, and its properties change with time. It shares this property with other disordered and frustrated systems such as foams, gels, concentrated colloids and granular materials. MD simulations have been carried out to follow these processes in model supercooled liquids. A typical simulation consisted of quenching a liquid state configuration to various temperatures in the supercooled region. Then a property, usually a time correlation function or response function, was calculated, starting the ‘time origin’ at different times after the quench. These ‘waiting’ times, t,, are sufficiently well-spaced to encompass the early stages of the ageing processes taking place in the system. The systems used were often mixtures of Lennard-Jones, LJ, particles, which do not crystallise. For example, Kob et aZ. used a binary (80:20) mixture of particles which interact with the LJ potential, 12’

(6.10) where a and j3 indicate either of the two species. They chose E,, = 1.0, oaa= 1.0, E , ~= 1.5, aa8= 0.8, E~~ = 0.5 and aSB= 0.88. The potential was shifted and truncated at the various distances, a,p. It is generally the case that when a system is driven out of thermodynamic equilibrium, its dynamical properties exhibit ageing effects. This means that quantities that at equilibrium were time independent now become functions of time. Time correlation fimctions, which at equilibrium only depend on a time

33


difference, now depend on two times, and so on. In the work of Kob et al. the non-equilibrium dynamics were characterised by the two-time correlation function, CA(t,, t t,) = (A*( t,)A( t, + t ) ) where A is a microscopically derived observable. The brackets, (. . .) refer to an average over many quench histories starting from a distribution of equilibrium states of the initial equilibrium system. The incoherent scattering function is a popular choice for CA.The simulations show that the systems are continually evolving and that the relaxation time increases with the age of the system. As Figure 12 reveals, the longer the waiting time, the longer it takes the system to 'forget' its initial configuration.'22 In contrast, the one-time properties, such as co-ordination numbers, are relatively insensitive to time. The collective-property time correlation functions show, in general, a strong waiting-time dependence. However, for long waiting times, this waiting-time dependence is not found at short times (i.e. t / t , t,. By an ingenious analysis of the underlying response functions underpinning CA(tw, t t,), Barrat and Kob124’125 showed that the short-time part of the decay can be associated with the thermal (‘heat-bath’) temperature, whereas the plateau region evolves with an effective temperature somewhat higher in the tradition of the ‘fictive’ temperature concept, applied to glasses for many years. The effective temperature of the a-relaxation processes is different from the thermostat. The picture that has emerged is that, unlike in normal liquids, the slow structural rearrangment of the atomic cages, which forms the basis of a-relaxation, is not constant with time but becomes slower with elapsed time from the quench. One can treat these out-of-equilibrium states as being in ‘quasi-equilibrium’. The two different temperatures, one controlled by the external bath and the other internally selected by the system, can be used to characterise the quasi-equilibrium state. Sciortino and Tartaglia showed that the latter, which characterises the response of the system to an external perturbation, can be calculated within the inherent structure thermodynamic formalism. 126

+

+

6.4 Rheology. - More recently Barrat and co-workers and others have turned their attention to the more general non-equilibrium situation of non-linear rheology of supercooled liquids. This is more complicated because not only is there ageing of the glassy material, but this now is affected by any imposed shear strain hist01-y.’~~ Although still largely qualitative at this stage they have established the general trends one expects in a sheared supercooled liquid, glass and indeed any ‘jammed’ system such as a foam or powder. The application of the shear strain is expected to suppress the ageing process and reduce the a-relaxation times, z, in a way which actually looks like the usual shear thinning curves. Below Tgthey expected z, to level off to a finite value in the limit of zero shear.


35

Below the glass transition temperature, this plateau disappears and the relaxation time diverges towards zero shear rate, as a power law (similar to the appearance of a yield stress effect in the appearance of the viscosity). Yamamoto and Onuki found that an externally applied shear strain rate caused strong shear thinning when the shear rate exceeded the inverse relaxation time associated with the slow (a-process) relaxation.”’ They investigated the disruption of the microstructure (measured by close neighbour separations or ‘bonds’) of the supercooled liquid by the imposition of a shear rate. The bond breakage time, zB,in ambient and sheared conditions can be fitted to z,’(j) = Z,’(O)

+A j

(6.11)

where A is a state point dependent constant. Frey and Lacks used MD to explore the effects of shear on alkane glasses.128 They showed that the shear strain caused the local energy minima and energy barriers on the potential energy landscape to become smaller in magnitude. The amorphous structure became unstable with respect to new local energy minima associated with plastic flow. The gradual changes in material (e.g. mechanical) properties occurring during the ageing of many amorphous materials is technologically important. It is known that, for example, the yield stress of an amorphous polymer increases with time on ageing, but after flow the yield stress drops to a value which is independent of time. The shear acts as a kind of ‘rejuvenation’ agent. Utz et al. performed Monte Carlo simulations of a binary Lennard-Jones mixture.’29 They found that after cooling a system at different rates, the application of a shear strain linear with time caused a rejuvenation of the system, reflected in the fact that the energy of the intrinsic structure became more positive. A feature of ‘frustrated’ systems in general, such as granular beds, glasses and foams which show ‘jamming’ transitions, is the observation that the distribution of normal interparticle forces, P(F), shows some common features. One of these is a near-exponential distribution of forces for the larger forces. O’Hern et al. carried out MD simulations of repulsive Lennard-Jones binary mixtures and calculated the P ( F ) as a function of temperature in the supercooled regime.’3o P ( F ) was found to develop a peak at the low force end of the distribution as temperature decreased. Granular systems, shown in the same study, had a plateau in P ( F ) in the limit of zero force. The presence of a peak or plateau, it was argued, is a signature of a large number of forces near the average value, which is consistent with the emergence of ‘force chains’ in which the forces between element along the chain are more or less equal (which must be the case to provide local force balance).

6.5 Glasses in Confined Geometries. - Some simulations have been carried out that have investigated glass formation in confined geometries. Nemeth and Lowen simulated hard spheres in smooth and atomistically rough cavities containing 134000 sphere~.’~’ They found that the glass transition decreased from a volume


36

fraction of ca. 0.58 in the thermodynamic limit to 0.35 for cavities containing only 13 spheres.

7 Liquid Surfaces

Almost at the beginning of molecular simulation it was realised that the basic approach used for bulk systems could be modified to investigate surfaces. In this section some of the recent developments and applications in this field are discussed. 7.1 Liquid-Vapour Interfaces. - The isotropy of local molecular structure in a liquid is broken in the vicinity of an interface. Perhaps the most basic, and studied, surface is the liquid-vapour interface, where there is a dramatic change of density (at least an order of magnitude) over a few molecular diameters. This can be modelled in a molecular simulation by introducing periodic boundary conditions that are restricted to two directions only (say, in the x and y directions). The molecules are allowed to relax and come to equilibrium in the third direction. In this way a thin nanometre-scale film is produced. Molecular Dynamics and Monte Carlo simulations of thin liquid films have been popular for thirty years, and this continues. The density and pressure profiles perpendicular to the surface are oRen computed. The density profile is

+

where Nz is the number of particles between z - A z / 2 and z Az/2, and the cross-sectional A = L,L,, the MD cell side-lengths in the x and y directions (parallel to the interface). The angular brackets refer to a time average. The surface tension, y , is calculated from the Kirkwood-Buff virial expression,

(c x) 34 7) (G

=

4A

-

d@g

i<J

The intermolecular distance between molecules i and j is rV and its components in the three directions are xo, y, and zg. The surface tension of the Lennard-Jones fluid has been calculated using grand canonical Monte Carlo simulations with finite size scaling corrections. 132 They extrapolate the temperature dependence of the surface tension and obtain an estimate of the LJ critical point temperature (i.e. where y + 0) of 1.3 1, which is within the accepted range of v a l ~ e s . ~ An important paper by Trokhymchik and Alejandre identified and removed previous observed discrepancies between MC and MD when applied to the simulation of truncated Lennard-Jones liquid films in equilibrium with the v a p 0 ~ r . IFigure ~~ 13 shows some examples in the density and pressure profiles across the film which illustrate the differences between the MC and MD standard

37


*Q

0.8

0.3 nz wa

0.1

3

c/)

2

-0.1

0.3 . I

n!!

g 3

v, c/) W LI L

*?-!

0.1 -0.1

0.3 z

e

w-

0.1

U

3 v,

3

-0.1

U

a,

0

10

20

30

40

z* Figure 13 Local density distribution, p* (z*), normal, PE(z*), tangential, PF (z*), and difference, PE (z*)-P: (z*), pressure pm$les for truncated W potential dejined b-y eq. (2). i%e cut-offradius, R = 2.50, reduced temperature, T" = 0.92. The solid and short-dashed lines correspond to MC and NMD data, respectively. In both simulations derivative, dU, (r)/dr, has been calculated according to eq. (3) [the equations referred to are those given in the original publication] (Reprinted with permission from A. Trokhymchuk and J. Alejandre, J Chem. Phys., 111, 1999, 8510-8523. 0 1999, American Institute of Physics)


38

truncation procedures. The standard approach in MC is to truncate the LennardJones potential at a certain distance, R. The simulations are performed using a spherically truncated LJ potential (ST),

where O(x) is the unit step function (O(x) = 0, x < 0, O(x) = 1, x 3 0). Instead MD simulations are typically carried out with a spherically truncated force,

, = 0,

r S R

(7.4)

r>R

By integration one obtains the potential associated with this force,

which gives

This is a spherically truncated and shifted potential, known as STS, which is the potential corresponding to the truncated force. The fact that MC and MD are, in fact, carried out with different potentials is the origin of the observed differences between the density profiles. The STS potential is shallower than the ST potential, and this gives rise to a lower density in the centre of the film. MD can be carried out with an ST potential, if the following modified force scheme is used,

= 0,

r > R (7.7)

where d(x) is the Kronecker delta function. The last term in the force, AF, is an impulse received by the particle when its interacting particle crosses the truncation radius, R. It can be approximated by

AF=-------@LJ(R)[#(r- R) - O(r - R - Ar)] Ar where Ar determines the thickness of two annuli of radius R and R

+ Ar

in


39

+

which a pair of particles are found. This pair force is included if R Y d R AT-.A value of A Y = 0.010 was found to be optimal. Simulations of thin Lennard-Jones films have shown that as the film gets thinner the two-liquid interfacial regions begin to overlap and any liquid-phase molecules in the central region suffer increasing tension in the planes parallel to the interface. This was considered to be one of the key features associated with the destabilisation of a liquid film and which eventually leads to the onset of film The pressure profiles are more sensitive to the film thickness than the surface tension values. One of the concerns in simulating liquid films in a vapour has been the suppression of capillary waves by the periodic boundaries. The mean square fluctuation in the z-coordinate of the interface, (2) is given by,

where y is again the surface tension, L is the periodicity of the simulation cell parallel to the surface and Bo is a constant. Capillary wave effects have recently been seen in a 1.24 million Lennard-Jones particle simulation of a liquid layer with two liquid-vapour surfaces.135They gave evidence that the interfacial width parameter does increase logarithmically with L in accord with eqn. (7.9). Yeh et al. performed MD simulations of the acetone/vapour interface.'36As for water, the strong molecular interactions, in this case C=O. . .C-0 and CH.. .O lead to a highly ordered interphase with the plane of the molecule cutting through the surface, and parallel to the surface normal, with one of the methyl groups pointing away from the bulk.

(4)

7.2 Liquid-Liquid Interfaces. - There has been increasing interest in simulating liquid-liquid surfaces in recent years. For example, a simulation of the planar interface between two immiscible liquids was carried out using adjusted LennardJones interaction^.'^' They found that unlike the monotonic decay of surface tension found for the liquid-vapour interface, that for the liquid-liquid interface there was a maximum at a certain temperature. MD simulations of the thermodynamics and dynamics of sodium chloride at a water/ 1,2-dichloroethane, a polarlnon-polar liquid boundary, were carried out.I3*They found that the ion pair was more stable at the interface than in bulk water. Bresme and Quirke used MD to simulate the wetting characteristics of a nanometre scale liquid lens at the boundary between two immiscible LJ-like interfaces. The general potential used is (7.10)

where the parameter aii controls the degree of miscibility of the liquids. For a l l = aZ2= 1 and aI2= 0 we have complete immiscibility. For the third, lens, component, 3, they adjusted a13 = a23 to vary the wetting ability of the lens. They

40


found that the wetting characteristics, in particular the contact angle as a function of the various surface tensions, followed well the classical predictions given by Neumann's equation. 139 Michael and Benjamin performed MD simulations of two solute molecules near a water-hydrophobic solvent boundary.'41 Four hydrophobic solvents were considered, ranging from n-nonane to 1-octanol to span a range of polarities. A pair of solute molecules acting as a charge-transfer system was adsorbed at the boundary between the two liquids. They were interested in the structural relaxation of the solute pair after a sudden change of their charges. The relaxation was found to be much slower in the organic phase and quite sensitive to the location of the pair with respect to the Gibbs Dividing Surface. 7.3 Liquid-Solid Interfaces. - Since the 1980s its has been known that the structure of thin liquid films confined between smooth walls is different from that of the bulk. For a liquid adjacent to a solid boundary, the density, p(r), oscillates as a function of distance from the surface with a periodicity roughly corresponding to a molecular diameter and a decay length of a few molecular diameters. This has been established by experiment for a range of liquid types, such as liquid metals and organic liquids. For liquids in a narrow gap between two smooth surfaces, the density oscillations originating from the two surfaces overlap, giving rise to so-called 'solvation' forces which oscillate in magnitude, reflecting a series of layering transitions as the two surfaces are progressively pressed closer together. Solid-liquid interfaces have been studied by molecular simulation. A recent example of this is a MD simulation of the boundary between crystalline copper faces, (100) and (1 1 l), and liquid aluminium, both modelled using n-body p0tentia1s.l~' The liquid is layered in the neighbourhood of the interface. Interestingly, a crystal-like order exists in these liquid layers was found, despite the large misfit between the atomic diameters of the Cu and Al. Diffusion in these layers appears to occur via a jump-diffusion mechanism between vacancies. Boda et al. carried out MC simulations of the double layer capacitance of a charged hard sphere representation of a molten They found that, close to the point of zero charge, the capacitance increased with temperature, in accordance with the experimental trend. MD calculations of the liquid water-NaC1( 100) interface have been carried Both water and NaCl were represented by variablecharge polarisable potentials. The structural properties of the water interface were similar to those using the fixed charge SPC/E model. The ordering effect of the crystal on submonolayer coverages was also established. Gallo et al. used MD to follow the single particle dynamics of model SPC/E water confined in a computer model of Vycor glass.14 Pronounced layering was observed, owing to the hydrophilicity of the substrate, which manifested dynamical signatures akin to supercooled water. Yeh and Berkowitz performed MD simulations of liquid water sandwiched between two Pt( 1 11) surfaces at high external electric fields.145They showed that, even with a relatively low field of 50 m V k l , the dielectric constant of water is reduced by about a half. There were also large electric field oscillations which mirror those in the density profile normal and close to the solid surface.


41

Hayward and Haymet performed MD simulations of ice-water interface^.'^^ They found that the higher index faces had thinner interfacial regions, and that the molecules were in states that were neither ice-like or water-like. Molecular Dynamics and Monte carlo simulations have been used to study the structure and dynamics of interlayer solutions in colloidal sodium laponite at 277 K.'47 The As would be expected, they observed density oscillations layer spacing was 34 adjacent to the clay planes, in order of proximity to the wall, water hydrogens, oxygens and then further out, the sodium counterions. The innermost layer was caused by the hydrogen bonds next to the surface oxygen atoms. Analysis of the radial distribution functions showed that the water structure inside the cavity was distinguishable from bulk (model) water. Shroll and Smith performed MD simulations in the Grand Canonical Ensemble of clay mineral swelling.14* Diffusion is anisotropic, being faster parallel to the clay sheets. Barlow et al. carried out Reverse Monte Carlo simulations of three different lipid and surfactant monolayers.149 Huitma et al. applied semigrand-isobaric-isothermal MC to investigate crystal growth on the face-centred-cubic ( 100) direction from a Lennard-Jones binary mixture.'50 By lowering the interaction strength between the solute and solvent paticles, or by lowering the temperature, they were able to change the growth mechanism from 'flat' to 'rough'. The transition zone on the liquid side in the former case was sharper and narrower both structurally and dynamically (as measured by the self-diffusion coefficient). Li and Ymamoto performed MD simulations of the melting of n-n~nadecane.'~' Melting took place preferentially in layers, with the molecular axes perpendicular to the surface plane.

A.

7.4 Tribology. - Lubricants, of course, are examples of liquids between two solid

surfaces. There has been much interest over the years in applying molecular simulation to issues of lubrication or 'tribological ' importance. In micromachinery the thickness of the lubricant film can be on the order of a nanometre and the shear rates can be as high as 10" s-'. The nanoscale resolution of MD makes it an ideal probe of these so-called 'boundary lubrication' issues - where the liquid film is only a few molecular diameters thick. One popular simulation procedure has been to set the two confining walls in relative sliding motion. The rheological and microstructural response of the intervening liquid is then examined. An alternative popular scheme is not to shear the liquid but to explore the static internal structure of tribologically-relevant liquids near the solid boundary. Interesting phenomena, such as wall slip (where the liquid velocity at the wall is not equal to the wall velocity, as would be expected for laminar flow) and stickslip dynamics (where the flow takes place in repeated pulses) have been the focus of much attention by molecular simulation (see, for example, ref. 152). Grand Canonical Monte Carlo has been used to model a monolayer of the molecule octamethylcyclotetrasiloxane (OMCTS) confined between mica-like surfaces, to understand the relationship between the intralayer structure and the atomic structure of the mica surfaces.'53 The molecule was found to be large enough that sliding of the surface layers was not able to break up the in-plane order of the fluid molecules. Wall slip in sheared thin films of hexadecane has

42

Chemical Modelling: Applications and Theory, Yolume 2

been explored by non-equilibrium MD.154Slip was examined at various shear rates and for films of different thicknesses. Liquid density had little effect on the degree of slip. The walls consisted of three layers taken out of a body-centred cubic lattice. The atoms were harmonically restrained to the lattice sites. The force constant, and therefore the wall ‘stiffness’, was adjusted. The extent of wall slip increased with wall stiffness, but decreased with increase in the LennardJones interaction strength ( E ) between wall and chain atoms. In a subsequent paper, these authors investigated the effect of wall roughness in wall ~1ip.l’~ Experimentally it has been found that rough walls reduce wall slip. The boundary conditions depend on the relative size of the particles, surface aspherities and film or tube thickness. The slip increased with periodicity length of the roughness, but only significantly for lengths about twice the length of the hexadecane molecule. The slip decreased and the apparent viscosity increased with the amplitude of the roughness. Travis and Gubbins performed NEMD simulations of Lennard- Jones and repulsive Lennard-Jones (WCA) fluids undergoing planar Poiseuille flow (which has a quadratic velocity profile) in a slit pore.156At a pore width of five molecular diameters, the presence of attractive forces has a relatively small affect on the density profile across the slit. An analysis of the ‘local’ shear viscosity and thermal conductivity across the slit revealed singularities in the former. They conclude that the Navier-Stokes equations are not applicable for pore widths lower than about five molecular diameters. Diestler and Schoen applied Grand Canonical Monte Carlo to the structure of a Lennard-Jones liquid film trapped between two parallel but nano-depth furrowed walls.157The length of the firrows were perpendicular to any sliding direction. They found that this indenting introduced layering not only normal to the surface planes but also normal to the direction of the furrows (which also produced an increase in the tension within the film). 7.5 Two-dimensional Liquids. - The order of the melting transition in two dimensions (2D) has been a perennial topic of study by molecular simulation. In fact, two-dimensional phases are in some respects quite different from their 3D analogues. The radial distribution function decays algebraically with distance, indicating that the long-range order inherent in 3D crystals is absent in 2D. Bates and Frenkel used hard-disc MD to investigate the nature of the melting transition in 2D.15*They calculated the elastic constants and equation of state in the vicinity of the supposed fluid-solid phase transition. Although the calculations were state of the art, it still proved not possible to rule out a weak first-order phase transition between the solid and isotropic fluid, rather than a two-step transition via a hexatic phase.

7.6 Droplets. - Liquid droplets and curved liquid surfaces in general have been the subject of some attention by MD and MC for about 30 years. A recent example of a simulation in this category was carried out by Ikoshoji et aZ.’59They gradually cooled binary Lennard-Jones mixtures until they ‘crystallised’ into icosahedrons or fcc structures, depending on the system size.


43

8 Dissipative Particle Dynamics In the last few years there has been an increasing interest in so-called mesoscale or ‘multi-scale’ simulation methods which can describe various types of soft condensed matter (e.g. polymer solutions, melts, colloidal liquids and liquid crystals) on their natural timescales rather than on the atomic timescales. These systems have many important time- and length-scales that are longer than those characterising simple molecular dynamics (typically picoseconds and nanometres) but often shorter than the macroscopic scales (e.g. centimetres and seconds). It is on this scale that the chemical variations need to impact on material properties. The simulation focuses on the mesoscale, with a coarse-grained representation of the chemistry being included in the model in terms of effective interactions. In this category, the Dissipative Particle Dynamics, DPD, method is probably the method most used. It was invented by Hoogerbrugge and Koelman at Shell Research Amsterdam. 160 DPD is a novel and economical technique for modelling at a coarse-grained level the dynamics of so-called complex liquids (e.g. polymer solutions, melts and colloidal liquids). It can simulate any dispersed particles and the fluid hydrodynamics on the mesoscale, without taking into account the enormous number of underlying atom-atom interactions and individual atom trajectories. Importantly, the technique is not so coarse-grained that statistical mechanics cannot be applied, and it includes thermal fluctuations fiom which a certain thermodynamic equilibrium can be said to exist. Being an off-lattice technique it does not suffer from the resolution restrictions and boundary limitations of lattice gas automata and lattice Boltzmann approaches, where the particles are confined to move on lattice sites. The interest in DPD shows no signs of abating and it is continually being developed. The original formulation of DPD did not satisfy the fluctuationdissipation theorem. This was corrected by Espaiiol and Warren.16’ It shares much in common with Molecular Dynamics, as far as the technical implementation is concerned, but the attributes of the particles are quite different. As for MD we have a manageable number of interacting particles which even follow Newton’s equations of motion (albeit with a velocity dependent force and a random term in the velocities),

where e, = ry/r,,, ry = r, - q, r, = Iryl and v , = v , - v,. The first term on the right is the conservative or thermodynamic force. The next term involving the constant, y , is the dissipative or drag term, and the final term includes the Brownian force. The noise amplitude o = (21tTyrn)’/~ where Tis the temperature links the dissipative and Brownian forces, thereby ensuring compliance with the fluctuation dissipation theorern.l6’ The random displacements generated from the d W,, (= d W,,) are independent elements of the Wiener process,

44

Chemical Modelling: Applications and Theoiy, Volume 2

dW, is an infinitesimal of order and can be written 8,Jdt where 8, = €Iji is a random variable with zero mean and unit variance. The dimensionless weight function W ( Y ) is normalised to the average number density, p = N / V ,

The function W ( Y ) is chosen to ensure incompressibility of the particles and there is no unique choice for its analytic form. The form

is popular.'62 The system consists of N particles which move in continuum space in discrete time steps. At each time step, h, the particles' momenta are updated so that the total momentum of the system is conserved. The numerical implementation of DPD can be cast into a form that is essentially traditional Molecular Dynamics, MD, using a modified version of the velocity-Verlet algorithm. Let us consider the position, ri, and velocity, vj,of DPD particle i, at times t and t h, where h is the simulation timestep; then, following Groot and Warren,'63we have

+

vj( t + h) = Yi( t ) + M i ( t)h

(8.7)

The force on the i-th particle, Fi, is velocity dependent, and consequently the algorithm involves a prediction step for the velocity at the next time step. The predicted velocity is v', given in eq. (8.7), which is evaluated just before the force, as it is needed for the force at time t h. The parameter d is equal to in the velocity-Verlet algorithm, but owing to the presence of the random force this value is not necessarily optimum (although it is usually adopted in practice'@). The DPD units of length, mass and time are specific to a particular application and the relationship between those of the model and real systems can be determined by considering the dimensionless groups relevant to the system. The foundations of DPD have been considered in a number of publicat i o n ~ . ~The ~ ~rules , ~ ~of ~dissipative - ~ ~ ~ particle dynamics were derived from the underlying molecular interactions by a systematic coarse graining procedure. '66~168 Evans derived expressions for the self-diffision coefficient and shear viscosity of the DPD particles in the form of the Green-Kubo time correlation function^.'^^ DPD can be used to model arbitrarily shaped objects made up of fused spheres by

+

I : Simulation ojthe Liquid State

45

incorporating the rigid body dynamics as for Molecular Dynamics. 170~171 The technique has recently been applied to polymers in solution and in the melt state by introducing bead-and-spring models for the polymer particles. The DPD polymer in a good solvent shows agreement with scaling and IOrkwood theory. The polymer melt has behaviour that agrees with the predictions of Rouse theory.I7* It has also been used to follow phase separation of immiscible liquids; the scaling exponents for the domain growth were broadly consistent with theoretical and lattice-based simulation predi~ti0ns.I~~ Groot and Warren used DPD to compute the surface tension between immiscible polymer melts, and related the DPD interactions to the Flory-Huggins x parameter^.'^^ DPD has been used to model a sheared liquid drop next to a solid The DPD method has also been extended to include a viscoelastic host I think it is fair to say that the merits and demerits of DPD are still debated. In my opinion, the DPD technique does have a ‘problem’ with the hydrodynamics, which ‘relaxes’ in the same time and distance scale as the dissolved particles. In reality, because of the near incompressibility of the solvent, the hydrodynamics relaxes essentially instantaneously on that particle’s timescale of structural evolution. One other ‘problem’ of the technique, as pointed out by Marsh and Yeomans, is that the temperature of the system depends on the value of the time step (as the dissipative force is inversely proportional to the square root of the time In an interesting article, Lowe looked at DPD from the perspective of another thermostatting procedure, but which conserves momentum and enhances the Besold et al. examined the various integration schemes used in DPD and found differences in the response hnctions and transport ~0efficients.I~~ These artefacts can be largely suppressed by using velocity-Verlet-based schemes in which the velocity dependence of the dissipative forces is taken into account.

9 Computational Techniques 9.1 introduction. - Technical improvements in molecular simulation have continued to be made over recent years. A number of old themes are still very popular, especially the important issue of how to deal with long-range forces. Interest on equation of motion integration algorithms (‘integrators’) has incidentally grown dramatically in recent years, with the establishment of objective measures. In the next sections we summarise some of the main advances and new approaches.

9.2 Periodic Boundary Conditions. - Computer simulations of interfacial systems have traditionally employed 2D periodicity and implemented a single layer embedded within a vacuum, or, with 3D periodicity, a lamellar structure. Wong and Pettitt devised a new boundary condition which contains only one interface, using an asymmetric unit of space group Pb. The lower half of the simulation cell is obtained by applying to the asymmetric unit a combination of reflection and translation operations across the intervening surface boundary plane.

46


9.3 Long-range Coulomb Forces. - Coulombic interactions are present in many molecular liquids and these play a key role in determining the molecular structure and the physical properties of these systems. It is therefore important to represent them as accurately as possible. The problem is that for charge-charge coulomb interactions, which decay as r - ’ , simple truncation is not possible and if carried out leads to unrealistic distortions in the structure and the dynamics. Traditionally this has been most often avoided by implementation of the Ewald summation method. The original summation is

where rii is the vector between charges i and j , and qi and q, are the corresponding charges. n is the cell vector and L is the cell sidelength (for simplicity here we will assume a cubic unit cell). The solution is to rewrite the r-l interaction as

where f(r) is a function which is much more rapidly decaying than r - ’ . The idea is that the rapidly varying part of r-’ is retained as a modified real-space sum (the first term in the above formula). The second term in the above formula ‘captures’ the slowly varying part of r - ’ . A slowly converging function in real space is rapidly converging in reciprocal space, and so the second term in the above recasting of r-’ can be implemented as a reciprocal lattice summation. This is the basic principle of the Ewald method. There are exact formulae for 3D lattices which are periodic in three and two dimensions. The details can be found in many textbooks (e.g. ref. 3). The one feature that has found to be important in the reduced dimensional situations is the lattice truncation boundary condition and the assumed dielectric constant of the surrounding medium, which gives a further contribution to the potential. This correction term depends on the total dipole moment of the unit or simulation cell. When a spherical geometry is used for the summation this correction energy has the form

where V is the volume of the cell. If the surrounding space has an infinite dielectric constant ( E , = oo),which is the case for an infinitely conducting metal (this is the so-called ‘tinfoil’ boundary condition), then this term vanishes. The other obvious case is if the sphere of cells is terminated by a vacuum ( E , = 1). This term was found to be important when calculating the coulomb forces in systems with slab geometry, and is necessary to get agreement with the 2D Ewald formula. In this case the component of the dipole moment perpendicular to the surface plane is required.179


47

One of the main factors in determining the accuracy of the Ewald method is the choice of the 'assignment parameter', a, which determines the rate of decay of the 'tuning' function f(r), and its optimal choice can improve the accuracy of, say the forces, by orders of magnitude. It is worth noting that the level of accuracy for a given value of a depends somewhat on what property is being calculated (e.g. potential or forces). Hummer et al. considered the calculation of the pressure by the Ewald method using the virial formula and the 'thermodynamic' definition based on the derivative of the potential energy and the They found that for finite N the two formulae give different values, with the thermodynamic route to the pressure showing a much smaller system size dependence. One problem with the Ewald method is that the computer time scales as O ( N 2 ) where N is the number of charges. For small systems, the real space part can be made O( N ) by various neigbourhood-list book-keeping schemes and although the reciprocal space series can be coded O(N), as the system size increases, the real space term or the number of reciprocal lattice vectors that have to be taken grows unfavourably (if one keeps the same real space cut-of€). The computational time can be hrther reduced by use of a Fast Fourier Transform on the reciprocal lattice series term. This modification of the usual 'off-lattice' Ewald method performs the Fourier transform of the potential by a Fast Fourier Transform, FFT, procedure. This is by no means a trivial procedure, as has been discussed recently by Deserno and Holm'*' and Toukmaji et al.'" There are delicate issues relating to the procedure of charge assignment to the grid points (necessary for FFT), numerical differentiation of the derived potential at the grid points. Then there is the assignment of the forces calculated on the mesh, mapped back onto the original off-grid charges in a way that maintains adherence to Newton's laws of motion to machine precision. If one of these steps dominates, there is little point in optimising unduly the others. The assignment of the charges to grid points and subsequent FFT treatment is a procedure invented by Hockney and Eastwood, and is known as the particle-particle-particle-mesh (P3M) method. 183 Some recent optimisation of this technique has been carried out by Hiinenberger.'84 Other analytic forms for the charge smearing function were tried. Polynomial formulae with finite spatial range were shown to offer improvements in computational efficiency and ease of implementation. Another way of making the Ewald method O(N(log(N)) was invented by Duan and Krasny, who accelerated the real space term using a tree code in which the interactions between clusters and distant particles are approximated by multipole expan~ions.''~Ewald sums for the Yukawa or screen-Coulomb potential (= exp(-icr)/r, where ic is a constant) have also been derived by Salin and Caillol.'86 The shear viscosity of strongly coupled Yukawa systems has been calculated by nonequilibrium molecular dynamics. They found a minimum in the viscosity at intermediate ic and r (inverse temperature) values.' 87 An alternative route to the long range forces is to return to the original sum and spherically truncate in a more sophisticated way. Wolf et al. investigated this further and showed that the main problem of the original sum is that inside the sphere around a particular charge the total charge is not generally zero.188If this

48


feature is eradicated, by summing up 'molecular' pairings of charges, then the accuracy of a truncation procedure can be considerably improved. Applications of the Ewald method and truncation method to systems in which the electrostatics of the interacting molecules are described by partial charges, point dipoles and anisotropic polarizabilities were made by Nymand and Linse.Is9

9.4 Integrators and Thermostatting. - Control of temperature in a MD simulation is a desirable feature in many applications, particularly away from equilibrium. Woodcock was the first to devise a thermostatting procedure, by velocity res~aling."~Since then, many thermostatting procedures have been devised. One of the most popular has been due to Berendsen, encapsulated in the following equations of motion,

4 = P/"

(9.4a)

P=F-Cp

(9.4b) (9.4c)

where q and p are the coordinates and momenta, rn is the mass, F is the interaction force, I; is the thermostatting parameter, To is the desired running temperature, T is the instantaneous kinetic temperature and z is a relaxation time. This thermostat is still used, although until recently the statistical mechanical ensemble it generated was not known."' The Nos&-Hoover thermostat is more useful (and therefore widely used) in generating a canonical ensemble and basically involves replacing 5 above by its first time derivative in the last equation above. The thermostatting multiplier then has its own parallel equation of motion, which most be integrated alongside those of the atomic positions, to obtain 5 as a function of time. This method has been generalised in the so-called Nos&-Hoover Chain (NHC) method, in which there is a hierarchy of thermostats. The NHC method has recently been extended to apply to non-equilibrium MD simulations (with shear) by Branka and W o j c i e ~ h o w s k i . ' ~Another ~ , ' ~ ~ method of thermostatting, originally due to Anderson, was to couple the dynamics to a stochastic heat bath. This idea has been developed further, in the fiamework of multiple time step symplectic (phase-space occupancy conserving) integrators, by Izaguirre et al. and Liu and T ~ c k e r m a n . 'Kutteh ~~ and Jones developed a thermostatting procedure based on non-holonomic constraints for rigid body molecular dynamics.'96 Sturgeon and Laird devised a new algorithm for isothermal-isobaric MD using an extended Hamiltonian with an Andersen piston combined with a No&-Poincare t h e r m o ~ t a t . In ' ~ ~passing, the development of symplectic algorithms, which allows them to be devised in a rational case-specific way, has been one of the most important steps forward in recent years, and advances in this area continue to be made.'98 Zhou et al. proposed a new method of integrating molecular systems, in which the constrained bond lengths or angles are adjusted each time step so that


49

the total energy is minimised with respect to the constrained distances (i.e. in response to external and centripetal forces). 199 Other new variants of MD that have been developed in recent years. They include a MD version of the Reverse Monte Carlo, RMC, technique.2w RMC does not have an interaction potential as an input parameter but uses a sequence of biased molecule displacements based on criteria to minimise the difference between an experimental and calculated structure factor for the various species in the system, T6th and Baranyai have developed the MD equivalent of this.2ooA fictitious potential field is introduced which is proportional to the mean square difference between the experimental and computed structure factors.

9.5 Ergodicity and Sampling of Rare Events. - Many important processes occur on a time scale that is not accessible by ‘standard’ implementations of molecular simulation techniques. There is a continual interest in expanding the time range or ability to explore phase space of simulation methods. Given enough computer time (measured as either Monte Carlo steps or Molecular Dynamics timesteps) the simulation will visit every allowed point in configuration space. For a finite walk, a limited region of phase space is explored, and the system is said to possess limited ergodicity. The approach to ergodicity in Monte Carlo simulations was studied by Neirotti et al., who used an energy metric, d k , as a measure of the ‘limited ergodicity’.20’ A series of A4 ‘walks’ was simulated. If Y i , is the average of the accumulated energy of the system up to step k for walk i, then we have the following definition,

If the walk is ergodic, d k for the k steps must decay to zero. Melchionna devised a method for the enhanced sampling (and energy barrier crossing) along a specified reaction co-ordinate, by enhancing the thermal noise along that direction.202 Another version of so-called ‘hyperdynamics’ was proposed by Sarrensen and Voter.203The objective of this approach is to follow activated rare events (between basins of attraction separated by high energy barriers), which can take place on timescales many orders of magnitude longer than is accessible by ordinary MD. By carrying out the simulation at a temperature much higher than the desired one, a time ordered sequence of possible barrier crossings, which it is assumed is still valid at the lower temperature, can be established. From the determined barrier crossing information so obtained the transition state statistics at the lower temperature can be derived. Phillips et al. developed Digitally Filtered Molecular Dynamics, DFMD, in which digital filter theory was applied to MD, enabling certain degrees of freedom to be enhanced or suppressed during the simulation based solely on frequency.2o4 There have been developments in the calculation of entropy and free energy. Schafer et al. derived a way of calculating the absolute entropies of molecular systems from the covariance matrix of the atomic position fluctuation^.^'^ Miller

50


and Reinhardt derived a generalised thermodynamic integration method, in which a metric scaling procedure was introduced to maintain the system close to the equilibrium state at each

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202. 203. 204. 205.

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2 Enumeration in Chemistry BY D. J. KLEIN, D. BABIC AND N. TRINAJSTIC

1 Introduction and Historical Overview Overall one might break down the applications of enumeration in chemistry into two broad areas, first an early historically important one concerning isomer enumeration, and second a variety of other enumeration problems arising in detailed descriptions of individual substances. Thence in this section we briefly recall these two broad categories of enumeration, and thereafter attempt a broad view of chemical enumeration, and its future. Thereafter in the ensuing sections we focus more tightly on methodology and recent results.

1.1 Early History: Isomer Enumeration. - The classic type of chemical enumeration concerns the enumeration of possible molecular structures. Indeed the subject predates the understanding of molecular structures, with Humboldt' (in 1799) enunciating the possibility of different chemical substances with the same elemental composition. WohleI-2 (in 1828) made history converting an inorganic compound (ammonium cyanate) to an isomeric organic compound (urea), thereby setting back a then popular vitalistic philosophy. Then later (in 1835) Berzelius3 coined the term isomer to describe the circumstance of multiple substances with the same elemental composition. With the proposal of (essentially modern) constitutional formulae (in 1864 by Crum Brown4) it was simultaneously noted that different structural formulae associate to different chemical substances. Indeed this was one key point in the acceptance of structural formulae (independent of geometrical embedding). For a decade various structural formulae were generated in an individual manner, without any announced mathematical systematics to the generation of the structures, such as perhaps is quite reasonable for the simpler molecules. Then in 1874 Cayley published his seminal paper5 concerning the enumeration of alkanes, with Flavitsky6 having made (earlier) independent listings. Discrepancies persisted in the enumeration, in part because of differences in understanding of just exactly what structural formulae might be. But following Cayley some mild activity continued, with finally Heme and Blair7 (around 1932) taking the subject of enumeration quite seriously, with tabulations for alkanes and related derivatives for up to around two dozen carbon atoms, with their results appearing in a half-dozen papers. Seemingly astoundingly large Chemical Modelling: Applications and Theory, Volume 2 0The Royal Society of Chemistry, 2002

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isomer counts of up to over lo2’ arose, though again errors persisted, in this case because of the tediousness of the involved computations, then done by hand. Then in 1937 Polya wrote his seminal pape? (of > 100 pages) concerning enumeration under group action. That is, Polya considered the enumeration of combinatorial mappings from one finite set to another, with there being equivalences amongst the mappings as mediated by group actions on the range set and (more especially) on the image set. In application to the molecular structure problem, for instance, the mappings could be that of C1 atoms and H atoms onto the six ligand positions of hexagonal benzene, with all C1 atoms being equivalent (under any permutation), all H atoms being equivalent (under any permutation), and the six ligand positions being associated to a permutation group corresponding to the rotation and reflections of the hexagonal-symmetry ring of benzene. Polya’s theory was described in a quite general mathematical manner, and made a tremendous impact, with his theory now being a standard staple of mathematical combinatorics texts, which often may not mention any chemical examples, though Polya’s motivation for the theory was for the purpose of isomer enumeration. Indeed Polya considered the enumeration of alkanes and also published a short illustrative paper’ for consumption by chemists. As a side note it is amusing to note that a substantial portion of Polya’s formal mathematical results were anticipated in 1927 by J. Redfield. However, Redfield’s work was long overlooked,” presumably because Redfield used an unusual notation (including astrological symbols) and did not have neat examples of application of the theory. Indeed in 1940 when Redfield submitted a second paper, the journal did not realize any connection with other work and judged the area to be so uninteresting that the paper was rejected. Only after Polya’s theory had become established and applied by a number of mathematicians was Redfield’s fist paper noted (around 1960), and even later was the rejected manuscript found, and finally published” in 1984, along with a brief survey of Redfield’s life and work. Finally in 2000 a yet m h e r more expository manuscript of Redfield’s was published.I2 In any event various elaborations and applications of Polya’s theory have continued over the years. A comprehensive mathematical monographI3 on Applied Finite Group Actions by Kerber has recently appeared, with its focus being not only on enumeration under group action, but also on the generation of representatives of the enumerated combinatoric structures. Moreover, this monograph notably unifies various approaches to the enumeration and generation problems, and indeed makes prominent mention of chemical applications, such as seem to have been a prime motivating factor in Kerber’s own research. Beyond the enumeration of isomers in isomer classes, Polya’s ideas have been applied to other problems of chemical interest - to enumerate: sub- and super-classes of isomers; chemical rearrangement processes; various types of quantum chemical classes of energy levels; and various interaction diagrams. 1.2 Further Enumerations. - Another, somewhat separate area of chemical enumeration concerns the counting of resonance structures for the purpose of gauging the extent of ‘resonance’. Often these structures may have much the flavor of different isomeric structures, though the enumeration is usually taken to

58


be relevant without regard to symmetry equivalences, thereby making some of the underlying mathematical aspects simpler than for the case of isomer enumeration. Again in the earlier work the enumeration was imagined to be done by hand. Pauling's ma~ter-work'~ on The Nature of the Chemical Bond was written with this view. Especially for the circumstance of conjugated n-electron networks, the numbers could become quite large, say for benzenoids of a dozen or more rings. Thence a more systematic approach became desirable, with a paper by Gordon and Davison'' (in 1952) providing an early attempt to enumerate Kekule structures for more extended n-networks. With a burdgeoning interest in chemical graph theory in the 1970s and thereafter, a few hundred papers enumerating Kekule structures were written, with partial reviews being found in the books by Cyvin and Gutman.'"'' The matter of making quantitative estimates of resonance energy for conjugated n-networks also came to be addressed in terms of a related enumeration problem of conjugated circuits, following the foundational papers of H e r n d ~ n ' and ~ ? ~of~Randid21.22 in the 1970s. Another very broad area of chemical enumeration may be found in the area of statistical mechanics. Thus in the 1930s M a ~ e 9developed ~ an elaborate theory of real gases, with graphs characterizing the interactions (and thence the deviations from ideality), and their enumeration being crucial to understanding condensation. Moreover there has now developed a whole area of lattice statistics which entails numerous types of graph enumeration problems. Often the primary focus is on computing associated enumerative generating functions, which themselves are identifiable as statistical-mechanical partition functions, and various (logarithmic) derivatives thereof giving different thermodynamic properties. Still occasionally there has been focus on explicit enumerations such as in Fowler and Rushb r o ~ k e ' s ~(1936) ~ enumeration of dimer coverings (as can incidentally be identified with Kekule structures); in M ~ n t r o l l ' s(~1950) ~ enumeration of partly self-avoiding random walks; in Wall et al.'s26Monte Carlo estimation of counts (and extents) of self-avoiding walks; or in Uhlenbeck and Ford's (1962) review2' of statistical mechanical graph theory and relevant enumeration procedures. Indeed, over the last few decades there have been a few hundred papers dealing with the self-avoiding embedding of linear chains on some regular lattice; a single chain represents a polymer in dilute solution, while more chains represent mixtures, and also other types of graphs to be embedded represent other conceivable (branched) polymer structures (e.g. dendrimers). The logarithm of such an enumeration represents a conformational entropy, and again different sorts of weightings and generating functions (and derivatives thereof) are of ultimate interest. Enumerations also arise in the area of quantum chemistry. Electron-pairing diagrams closely correspondent to chemical structure were early emphasized by Rumer2' (1932), and the computation in terms of these diagrams was soon taken up by P a ~ l i n g(1933) ~ ~ who devised a quantitative scheme entailing the enumeration of certain subgraphs (termed 'islands') obtained by the superposition of electronic pairing diagrams. The resulting valence-bond theory was immediately applied by Pauling and Wheland3" (1933) and thereafter by many others. Here the enumeration problems grow even more rapidly than in simply counting Kekule

2: Enumeration in Chemist?

59

structures, in as much as superpositions of pairs of Kekule structures are made, and for each superposition edge and cycle subgraphs are separately enumerated. The ensuant computational difficulty was presumably one of the reasons that valence-bond theory fell into disfavor for a period of time. Now though powerful computational procedures, essentially avoiding the graphical framework, have come to be used in the ab initio And in the semi-empirical realm powerful graph-theoretic techniques have been developed to deal even with quite large systems, such as of relevance in high-temperature superconduction where resonating valence-bond descriptions have been proposed32 ( 1986) to be of relevance. These graph-theoretic techniques typically focus not on the enumerations but the computation of associated enumerative generating functions, which turn out to give the relevant overall matrix elements. Within the molecular orbital approach, especially with correlation, there are Feynman diagrams33 and such, which could be enumerated, though typically at issue is their evaluation rather than their enumeration. There remain yet other chemical enumeration problems. There are various types of knots or links representing knotted or linked molecules, and such may be enumerated, though here too the interest typically goes beyond enumeration (to generation). One may enumerate different types of spectral lines (e.g. as expected in a molecular NMR spectrum), though typically more than the number of lines is of interest. That is, in molecular spectroscopies much of the focus typically is on the locations and intensities of the lines and, especially when there are enormous numbers of lines, they may not all be resolved. In the context of various quantitative structure-property relationships (QSPR) and structure-activity relationships (QSAR) there are enumerations of different types of subgraphs which may be used as graph invariants in these correlations. Indeed this area ends up bordering on the vast area of combinatorial chemistry and chemical informatics, again where the primary focuses typically are beyond enumeration. 1.3 Why Enumerate?. - Clearly enumeration has played an important role in the history of chemistry. But does it still? Are the noted enumerations just historical anachronisms? Is enumeration irrelevant for modern interests in quantitative descriptions of different substances? Indeed in all the areas we have noted, one may indeed argue that enumeration is but a first step towards a more comprehensive characterization and undertaking. Combinatorial formulae often merely identify two different enumerations to have equal values, with one of the enumerations being the easier to perform. We may note for instance that isomer enumeration in Polya theory identifies this enumeration to that of the enumeration of certain equivalence classes of functions. With the counts for two different sets of objects being equal, there often is a natural bijection (i.e. a one-to-one correspondence) between the two sets, so that the objects of one set may be used to represent (or even name) those of the other. Thence for the case of chemical isomers again, the mathematical set of objects offers a nomenclature for the isomers. Conversely too, granted a nomenclature, a possibility for enumeration is offered: one seeks to enumerate the names (which presumably exhibit some systematic structure). In some sense then a sensible nomenclature and enumera-

60


tions can be seen as intimately mutually inter-related. But with systematic labelling (i. e. nomenclature) one may anticipate that characterization of the enumerated elements might be systematically made also. In such a case a combinatorial chemistry of (even enormous) virtual libraries of chemical substances might be efficiently handled. In a number of contexts it seems that transformed enumerations are actually of more central interest. In particular rather than the numbers # N of some objects of a size N , interest may focus on an associated generating function

with x a real variable. In statistical mechanical enumerations such an F(x) may be a partition function with x being a simple function of the ration of temperature and an interaction strength. For instance, for the Ising model on a lattice network, in a standard combinatorial approach (e.g. as explained in ref. 34) the counts # N may be identified to the number of (not generally connected) subgraphs of N edges such that every vertex is of even degree, whence x = tanh(J/kT), with J the (nearest-neighbor) interaction strength, T the absolute temperature, and k Boltzmann’s constant. Indeed in such statistical mechanical frameworks, while F(x) might be the partition function of focus, the numbers # N might be interpretable as the partition functions in some sort of suitable microcanonicallike ensemble. More generally there can be more than one size characteristic and thence more than one variable. Derivatives of the partition function then lead to various thermodynamic quantities (energies, pressures, specific heats, etc.). In closer correspondence with isomer enumeration, the enumeration of conformations is the underlying foundation of polymer statistics, with F ( x ) an associated statistical-mechanical partition function. For suitable enumerations the generating function F ( x ) may be identified as an overlap for a many-body cluster-expanded wave-function, and derivatives of F(x) then giving different matrix elements, including that of the Hamiltonian; see, e.g., ref. 35. In any event enumeration or transformed enumerations underlie quite modern problems. Generally enumeration may be seen as a beginning wedge into a much wider context of questions and problems. Commonly the motive for enumeration might be efficient means by which to view whole ensembles, the properties of the ensemble, and perhaps properties of selected subcategories. Ultimately enumeration becomes hard to distinguish from combinatorics in general. Thence enumeration has and will remain fundamental to many aspects of chemistry.

2 Enumeration Methods

Methods of enumeration may be sought to be divided into different broad categories. The classical case of enumeration of isomers offers some formal challenge in order to account properly for the different sorts of classifications under various symmetries. Many other types of enumerations offer challenges in

61


extension to large systems, perhaps infinite (representing fluids or solids). In this section enumeration methodologies are broadly discussed, with some attention to specific examples. Section 3 goes on to describe recent work. 2.1 Enumeration under Group Equivalences. - The enumeration of isomer structures provides the classic example here. Consider a skeleton on which various ligands are to be substituted. Then different substitution patterns may be judged to be equivalent if one may be changed into the other under a symmetry transformation of the skeleton. These transformations may be viewed as permutations on the substitution sites of the skeleton, and include transformations corresponding to skeletal point-group symmetries, and possibly also permutations corresponding to non-rigidities in the molecular skeleton, Thus for an ethane skeleton with six substituent sites, the symmetry group (typically) includes the possibility of pseudo-rotation of the two ends relative to one another. Whether permutations corresponding to improper rotations (reflections, inversions etc. - if the skeleton exhibits such) are included or not depends on whether one wishes to distinguish chiral isomers or not. Thence the interest is in enumeration of equivalence classes (each generally containing different numbers of substitution patterns). Most generally there is a symmetry associated to the ligands also: for instance, one can imagine the number of ligand structures to match the number of skeletal substitution sites, but with different subsets of the ligand structures being identical, and so exhibiting a symmetry under interchange of the two. One may imagine that the permutation group for the ligands is naught but a product of over the ith set n ( i ) of n , equivalent ligands. For example symmetric groups Sn(,) one may be interested in placing one H-atom, two F-atoms and three C1-atoms onto the ethane skeleton, with a symmetry group taken as St,] X St2,3)X S{4,5,61. Generally the ligands may themselves exhibit a type of skeletal symmetry - for instance for bidentate ligands, they may or may not be symmetric under interchange of the two ends of the ligand, and in interchanging two equivalent bidentate ligands, one needs to simultaneously interchange both ends. For instance, for an octahedral coordination complex with two Cl ligands and two (bidentate) ethylenediamine ligands, the symmetry group of the ligands would be

(where I is the identity and we use the on-line notation for a permutation, e.g. so that (abc)(de) indicates a permutation carrying index c -+ b, b a, a c, d -+ e and e -+ d ) . In general there need not be too much formal mathematical distinction between skeleton and ligand sets - both may consist of disjoint pieces with equivalences between separate pieces, and within any one piece there may be multiple poiiits of attachment with internal symmetries entailing permutation of the attachment point labels. With disjoint pieces there may be chirality changing permutations which rather than changing any one piece into itself instead change one into another, and in addition would act on all pieces (whether skeletal or ligand pieces) simultaneously. One general approach would take the skeleton to be the atoms and the ligands to be the bonds to be used to -+

-+

62

Chemical Modelling: Applications and Theory, Vblume 2

interconnect the atoms. Anyway, given two sets S and L (a skeleton and ligand set) along with symmetries for both, the classical chemical isomer problem is to enumerate the equivalence classes of substitution or addition patterns under the equivalence mediated by the symmetry groups. The isomer enumeration problem as formulated here is a little different and a little more general than is usually considered. Often (as with Polya8y9)the ligands’ internal structure is dismissed; they are presumed to be unidentate, and different types of ligands are identified to different colors. Then instead of correspondences, one may speak in terms of mappings from the skeletal set of sites to the colors, whence a particular value (a color) may be taken several times. The approach indicated here seems to allow a greater generality of view, though not all the formal problems are so neatly solved to date. Even with unstructured ligands there are refinements and decorations to the basic problem so formulated. One may subclassify isomers according to the different symmetries of the substitution pattern; see, e.g., refs. 36 and 37. (Often this substitution-pattern symmetry is that of an intersection between skeletal and ligand symmetries.) One may subclassify isomers according to different substructures in the substituent pattern. For example, in dealing with alkane isomers, a subclassification according ~ ~subclassificato longest subchain (i.e. graph diameter) might be c ~ n s i d e r e d ,or tion according to number of primary, secondary, tertiary and quaternary carbons might be entertained. In the hndamental isomer enumeration problem with skeleton and ligands respectively being atoms and bonds, ‘isomers’ consisting of disconnected pieces can arise, so that it is relevant to make a subclassification into connectedness classes (i.e. according to the degree of interconnection). Another related type of chemical problem would be to enumerate the number of rearrangement processes conceivable for moving ligands around on a given skeleton. In this problem the ligands are viewed as ‘passive’, so that both the sets S and L are viewed as skeletal structures, and the considered rearrangement processes are the equivalence classes of permutations, which are viewed as moving whatever ligand is initially present at a skeletal site to another site. With both skeletons in such a process being the same (connected) structure, the equivalence classes are what has been termed a polytopal rearrangement, such as has been con~idered,~~,~’ or for rearrangements on a trigonal bipyramidal skeleton (or an octahedral skeleton, or other polyhedral skeleton). With the two skeletons different, and potentially disconnected, one faces a rearrangement enumeration problem for general chemical reactions. Evidently the enumeration problem of equivalence classes under group action is quite general, with Polya’s foundational work8y9marking a turning point in the consideration of the problem. Thus perhaps it is reasonable to briefly describe Polya’s work, where we have a skeleton with a set S of attachment or substitution sites and a set L of ligands viewed as colors (to be applied to the sites of attachment). The sites of S are acted upon permutatively by the elements of a symmetry group G for the skeleton. The application of the colors (or ligands) of L are viewed to correspond to a mapping f from S into L, with different mappings f and g being equivalent if there is a permutation n E G such that g = fn (i.e. the result of application of g is equal to the result of application of


63

n followed by the application of f t o this intermediate result, or also equivalently, g is the fimctional composition of n with f).Then an equivalence class of these

mappings corresponds to an isomer, and the number of different classes are to be enumerated. The numbers of sites of a given color (under such a mapping) correspond to the numbers of different sites occupied by the corresponding ligand, so that we seek equivalence-class counts #(n) with n a vector whose ath component nu gives the number of sites of color a. Polya's solution for this is in terms of a polynomial (called the cycle index) associated with the group G,

where [GI is the order of the group, and c,(x) is the number of cycles of size i in the disjoint cycle decomposition of n E G. The variables ziare expressed in terms of a set of color-correspondent dummy variables t, thus:

Then Polya's renowned Haupt Satz gives the various #(n) as the coefficients of a E L tl100-page) original quite lucid article, along with a survey

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Chemical Modelling: Applications and Theoy, Volume 2

(by Read) of the literature up through the mid-1980s. The 1991 book by Fujita3' reviews his own elaborated formulation, especially as regards the subclassification with regard to substitution-pattern symmetry. The book4* by TrinajstiC et aE. describes the isomer enumeration methodology for substitutional isomers on a fixed skeleton, and it treats especially nicely the case of (acyclic) alkanes, with extensive numerical tabulations. The recent book43 by S. El-Basil surveys the general theory focusing on the circumstance of symmetry subclassification, proceeding in a slow yet quite detailed palatable manner. The overall mathematical area is often referred to as Polya theoty though much was independently alternatively formulated by Redfield,1c'2 and again there have been other numerous elaborations, reformulations, and extensions. 2.2 Linear Recursive Methods - Kekule Structure Counting. - A rather general class of enumeration problems can be done in a recursive manner, which becomes most especially simplified when the recursion is the same at each stage, corresponding to different local regions of the structure. Thus such approaches are rather readily applicable to the treatment of polymers, crystalline solids, or perhaps other structures of high point-group symmetry. The well-known rotational isomeric model of polymer statistics (e.g. as in ref. 44) entails such linearrecursions for a variety of properties, such as mean square radius of gyration. Onsager's famed solution45 of the two-dimensional Ising model (on the square lattice) can be viewed as the evaluation via a linear recursive technique of a generating hnction for the enumeration of suitable subgraphs of the lattice - and in this case the exact solution appears quite non-trivial. The enumeration of Kekule structures in conjugated polymers, or in other highly symmetric molecules, such as buckminsterfullerene, provides another example of a linear recursive quantity. More generally there are other possible enumerations, e.g. of different subgraph types which may be used as graph invariants in various quantitative structure-property relationships (QSPR) and structure-activity relationships (QSAR). Beyond this, the general subgraph enumeration has other applications, as to statistical mechanics, say as involved with the different Mayer diagrams,23 or with an enumerative generating fwnction simply being a statistical mechanical partition function. Correlated electronic-structure wave-functions also give rise to a similar generating function problem.35 Again in the bulk of these subgraph enumeration problems equivalence under group action does not play a role and the difficulties dealt with in Polya enumeration theory do not arise, while entirely different problems connected with the subgraph relation arise. Often the subgraph enumeration problem can be solved (at least formally, and perhaps practically) in terms of linear recursions. This broad class of recursive enumeration problems appears to be quite ubiquitous, and in principle soluble by standard linear-algebraic techniques. As a concrete example, linear recursions have been extensively developed for the case of enumerating Kekule structures. A (molecular) graph might be denoted G, and a subgraph identified as a Kekule structure K if it has the same number of vertices every one of which has exactly one incident edge in K . For instance, for naphthalene (Figure 1) one h d s three Kekule structures each in correspondence

65

2: Enumeration in Chemistw

Figure 1 Kekule structures of naphthalene

with one of the possible (neighbor) n-bonding patterns. The greater the number K ( G ) of such Kekule structures, the greater the resonance stabilization (other things being equal - and in particular for molecular sizes being equal). It seems that the e a r l i e ~ t ’ and ~ . ~ ~now quite extensively ~ o n s i d e r e d ’ ~systematic ~’~ method for Kekule structure enumeration is based on a recursion to smaller graphs. Let e be any edge of graph G, let G - e be the graph obtained from G by deleting e, and let G 8 e denote the graph obtained by deleting e and all edges incident to e. Then there is a simple recursion

K(G)= K(G - e)

+ K ( G 8e)

which may be readily implemented in a computer program. But also it may be advantageously manipulated in different ways for different special types of graphs. It is noteworthy that if the recursion is iterated with a choice for a sequence of edges so as to disconnect the resultant graphs into disconnected fragments, then the fragments are separately treatable. For polymer graphs it can be USed15-17,24,4&48to yield recursions on Kekule-structure counts for chains of different lengths, and perhaps the bulk of Cyvin and co-workers’ books’”’* is given over to the development of quite explicit formulas as a function of length for a fair number of different particular polymer strips. Even for non-regular polymer graphs and especially catacondensed species there are elegant results, e.g. described in Chapter 6 of ref. 17. This latter scheme of Gordon and D a v i ~ o n ’ ~ has a neat pictorial presentation which might be illustrated for a catacondensed polyhex chain: first, given the displayed graph, one begins to write in a sequence of numbers in the hexagons starting from one end, with a 2 in the first hexagon and a 1 adjacent to it (in a 0th hexagon); second, each number in subsequent hexagons then is the sum of that immediately preceding and the 1st preceding number around a ‘kink’ in the chain; and finally the number appearing in the last hexagon is K(G). For the example shown in Figure 2, K ( G ) = 25. A natural extension to branched catacondensed species is also known.49 There is another neat scheme (based on the linear recursion) applicable for hand computation on polyhex benzenoids of up to a dozen or so hexagonal rings. This John-Sachs scheme is based on a one-to-one correspondence between Kekule structures and sets of mutually self-avoiding directed walks on the graph, and indeed this correspondence was (in a special context) utilized” in a statistical mechanical context in modelling collections of partly disordered polymer chains.

66


25 Figure 2

The Gordon-Davison scheme” for enumeration of Kekule structures in catacondensed benzenoids

The correspondence between a Kekule structure and a set of three mutually selfavoiding directed walks (from peak to valley) is illustrated in Figure 3, where the diagonal-oriented double bonds and (bold-face) vertical single bonds are the steps of the walks. Granted this correspondence, John and Sachs” developed a neat algorithm to give the Kekule-structure count as the determinant of a small matrix W whose elements count the number of possible directed walks from a position on one side of the polyhex graph G to each position on the other side (independently of any other such walks). Also see ref. 52. For example the path enumerations from either peak to either valley of the benzenoid structure below are readily obtained as indicated in Figure 4, and then there follows a Kekulestructure count as shown in eqn. (3):

P

V

P

P

v

v

Figure 3 A set of disjoint paths connecting peaks (P) with valleys (V) in a polyhex. The Kekule structure in correspondence with these paths is also indicated

Figure 4 A ‘Pascal-triangle’ scheme” for enumeration of paths joining the peak with the valleys

2: Enumeration in Chemisty

K ( G) = det( W ) = det

67

(i

:>=14

(3)

The consideration of the sets of mutually self-avoiding walkers has a wider impact in revealing a fundamental invariant (the number of walkers) which turns out to be important in transfer-matrix solutions, as applied to polymer graphs extending even to the two-dimensional limit53,54(as discussed in the next subsection). Also this invariant, which may also be described as an order, has physical irnplication~.~~~~~ Overall the general recursion of eqn. ( 2 ) is applicable beyond the case of Kekule structures here elaborated for illustration. The related so-called conjugated-circuit method20*2',s7,58 turns out to have quite neat (related) linear recurs ion^.^^,^^ Generally many sub-graph enumeration problems turn out to be of a linear recursive nature.

2.3 Transfer Matrix Methods. - The linear recursions of the preceding subsection can be alternatively framed in an especially elegant form for polymer graphs, of a rather general type, though here we focus on those polymers which are regular, with the same monomer unit repeated. The Kekule-structure count K L for a polymer chain of length L monomers can59y60quite generally be cast into the form of a trace

where T is a transfer matrix characteristic of the monomer unit and p is a matrix which encodes the character of the boundary conditions, i.e. of the polymer chain ends. Basically one may view T to give the various ways Kekule structures can propagate from one pattern at the boundary of one monomer unit to another pattern at the subsequent boundary of the next monomer unit. For instance, for a polyphene chain, e.g. of 14 hexagons in length, as shown in Figure 5, the chain may be divided into monomer cells of the form shown in Figure 6. There are just two patterns for the placement of double bonds at the boundary

Figure 5 The polyphene chain

Figure 6 The monomer repeating in the polyphene chain from Figure 5. Note that the monomer is not only repeated, but also flipped upside-down

68

Chemical Modelling: Applicalions and Theory, Volume 2

Figure 7 Propagation patterns of the two ways of connecting adjacent monomers in the polyphene chain from Figure 5. The patterns in the left column propagate into those in the right column as indicated by arrows (note that these are flipped upside-down and that the patterns in the left column switched their positions in the right column)

of a unit cell, these two patterns (consistent with types of ends indicated above) being as shown in Figure 7. Of these two patterns, the first may be propagated in two possible manners to the succeeding boundary, while the second may be propagated in but one manner. Thence, for this case, we have the transfer matrix

For narrow chains of say a couple hexagons width there are but a few such patterns, so that T is of a small size (say no more than 4 X 4), and upon diagonalization of Tone then easily raises T to the requisite power and a quite neat expression results for arbitrary length L. As strip width w increases (as measured in terms of the number of bonds crossing a monomer boundary), the size of T increases exponentially with w, but the technique is still readily applicable for widths w up to 12. A relation to linear recursions such as of Subsection 2.2 may be indicated. Denote the characteristic polynomial of an n X n transfer matrix T by

-

n

six"-'

p ( x ) = det(x1- T ) = i=O

Then (by the Hamilton-Cayley theorem) one has

i= 1

whence one can make a substitution (when L a n )


69

But in fact this is just a linear recursion as in the preceding subsection. For our example polyphene structure one evidently obtains K L + ,= KL + KL-1. Notably the recursion is largely independent of the chain ends, which then make their influence with the initial values upon which the recursion is based. A further means by which to deal with linear recursions and transfer-matrix approaches is by way of generating functions. One introduces generating functions with stage-l counts appearing as coefficients of a dummy variable z raised to the power L. For example, for Kekule structure counting,

Then, granted that the transfer matrix satisfies its own characteristic polynomial, one obtains a linear relation for the generating function. For example, for the polyphene chain, where the characteristic polynomial is x2 - x - 1, so that T 2 = T + I (where I is the identity matrix), we obtain

La2

=

KO

+ K , z+ ( z +

2')

F(2) - KO z

(91

Thence one obtains the generating function as a rational polynomial, here F ( z ) = [KO

+(K,

-

K") - 4 / 4 1 - 2

-

z2)

(10)

and this can then be expanded in powers of the dummy variable to obtain explicit expressions for the desired coefficients. The intimate contact with conjugated circuits theory also may be indicated. This theory concerns a formulation for the resonance energy of molecular structures G which support at least one (fully paired) Kekule structure. Within a Kekule structure K on G, a conjugated n-circuit is defined to be a cycle (in G) which exhibits alternating pairing in K around the cycle. Then the resonance energy is

where the sum is over not too large size n of (even-n) cycles in G, the y n are parameters, and a n K ( G )is the number of conjugated n-circuits as summed over all Kekule structures K of G. For regular polymers there is a modification to the transfer-matrix method for KL so as to also determine the a,&, and this is well described elsewhere.61But also something especially 'pretty' happens, granted the

70


recursion relation for Kekule structure counts of the form in eqn. (7). With the definition of a (squared) polynomial 2n

q(x) = [ p ( x ) l 2= -

C bix2"-1 i=O

and denotation of the conjugated n-circuit count for a length-L chain as a n K L , one finds these conjugated circuit counts satisfy a recursion

i= I

For example polyphene, q(x) = (x2 - x - 1)2 and

Notably the coefficients b, are independent of n (e.g. so that the numerator in the resonance energy expression obeys the same recursion). The distinction of the different conjugated n-circuit counts comes in the initial conditions. Again all this methodology extends to quite general enumerations. It applies for the Ising model (which may be viewed in essence to be an enumeration of evendegree subgraphs of the parent lattice graph), and it applies to many other statistical mechanical lattice models, as early emphasized by E. Montro11.62Indeed Onsager's famous solution4' to the square-planar-lattice Ising model is essentially just a solution of the transfer-matrix eigenproblem, which turns out to be challenging for the extended lattice, because of the dimension of the matrix approaching infinity. In any event the methodology extends to a great variety of graph-theoretic enumerations on polymer graphs. - Enumeration by explicit generation of all objects being counted is commonly done only when the counts are small or when no better method is (yet) available. The latter situation usually occurs when the underlying structures (which are being counted or on which the counting is being performed) are so irregular that no efficient recursive or grouptheoretical methods are applicable. The examples are counting of fullerene isomers,63fullerene caps,@benzenoid isomer^,^^,^^^' all connected subgraphs of a general graph68t69etc. Usually with such brute-force methods the only shortcut is provided by the divide-and-conquer strategy, which however still relies on exhaustive generation on the parts obtained by division. A frequent difficulty in these methods, e.g. when applied to generation of isomers, is an isomorphism of the generated objects, whence this requires additional efforts for diminishing redundancy of the generation algorithm and for recognition of isomorphic objects. In general, though such divide-and-conquer strategies are quite computer intensive, they still have significant advantages over more straightforward approaches. Both sorts of approaches often require computer time increasing

2.4 Exhaustive Generation (Brute Force) Methods.


71

exponentially with system size, though with the divide-and-conquer approach the exponential dependence may be in terms of the subsystem sizes, and thence immensely more efficient at a given size. They typically have a bound on system size which is then only weakly dependent on the computation power available, so that little improvement (in system size) occurs with new generations of computers.

2.5 Other Methods. - An occasional sort of approach is to change one enumeration problem into another. For instance, rather than enumerating Kekule structures, one can consider superpositions of pairs of the structures. These superpositions are still spanning subgraphs, now with components which either are isolated edges or even cycles. The number of such superposition graphs is just the square of the number of Kekule structures, if each cycle in the superposition graphs is identified with a factor of 2 somehow (so that if such a cycle arises , it in essence also from the superposition of Kekule structures K , and K ~ then arises from the superposition of K b and K,, in the reverse order), But in fact such superposition graphs are neatly identified to permutations on G, with the two directions around a cycle rather naturally giving the desired factor of 2 for each cycle. And sums over all the permutations arise in taking the permanent of the adjacency matrix A , the permanent of a matrix being like the determinant, but without the parity of the permutations entering into the sum. Thus

as was noted sometime ago by Percus.'" Permanents are generally difficult to compute, whereas determinants are much more convenient, but have a phase problem associated with the parity of the permutations. That is, the determinant of A adds or subtracts superposition graphs, so that one might naturally wonder whether this could be rectified if one were to adjust the signs on the elements of the adjacency matrix. And notably Kasteleyn" has found a neat way to solve this phase problem so long as G is planar (in a graph-theoretic sense). This powerful scheme results in a signed adjacency matrix S(G) with rows and columns that are labelled by the sites of G and with elements that are all 0 except those Sah= f l with a and b being adjacent sites in G. Then det(S( G)) = f[K( G)]' The signs are such that: first, S ( G ) is antisymmetric; second, if for an embedding of G in the plane one follows the edges of S around any even face (i.e. minimal ring) of this embedding, then the number of times a minus sign is encountered is odd. That is, if one proceeds around a ring of sites i,, iz, . . ., in then

Kasteleyn" describes how this odd orientation is readily achievable for any planar graph. For instance, if one inserts arrows on edges of G so that an arrow from a

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to b indicates Sub= +1 while Sba= -1, then an example of one such odd orientation is shown in Figure 8. For the special case of polyhex benzenoid structures, the determinantal formula for K(G) holds except that A(G) appears in place of S(G), as was earlier noted of Kasteleyn's by Dewar and L~nguet-Higgins.~~ Also there is a m~dification~~ scheme by which to deal efficiently with the conjugated-circuit-count problem: one inverts the matrix S(G) and computes determinants for submatrices associated with each cycle for which conjugated-circuit counts are sought. That is, the mean conjugated circuit count for a particular cycle C of a graph G is given as

where M c indicates just the submatrix of M for the rows and columns associated to the sites in C. The method has been applied74to several thousands of fullerenes (many with more than a million Kekule structures), and also7' to a couple of dozen different planar carbon network graphs, most with some five- and sevenmembered rings. Really very few significantly different other schemes for computation of Kekule structures or conjugated-circuit counts seem to have been explored to any extent, presumably because the preceding schemes have been so successful. Dewar and Longuet-Higgins7* and H e r n d ~ ndescribe ~~ an enumeration method based on the identification of non-bonding MOs for radical fragments of the graph under consideration. has considered a Monte Carlo scheme, but the range of sizes of benzenoids originally intended for this treatment are really quite easily treated exactly by the John-Sachs scheme. Still the Monte Carlo scheme is more evidently extendable to the treatment of the higher-level VB-theoretic models. There are some other less common schemes, e.g. as in ref. 78. Various special recursions are discussed in ref. 17. Moreover, the general idea of mapping one kind of enumeration to another seems to be of fairly general occurrence, though its manifestation typically seems to be quite different in different circumstances. Often there seems to be a correspondence between Ising-model enumerations and Kekule-structure enumerations. For instance, as follows from our brief note in subsection 1.3, for the hexagonal lattice P the Ising model solution can be viewed as involving enumerations of subgraphs with every vertex having degree 0 or 2. Then one can identifl Kekule structures on a corresponding lattice %* where each vertex of P

Figure 8 An example of Kasteleyn 5'' odd orientation of a planar polycycle


Figure 9

Hexagonal lattice % and the correspondent lattice % *. The subgraph of % with only even degree vertices corresponds to a unique Kekule structure (perfect matching) in % * as indicated in Figure 10

A-A Figure 10

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A-A

The correspondence between vertices with degrees 2 and 0, in subgraphs of %, and Kekule structure patterns in % *

is replaced by three vertices in %* as indicated in Figure 9. Then each Ising-type graph on % corresponds to a unique Kekule structure on %* as is indicated in Figure 10. Thus Ising-type graph enumeration on % is equivalent to Kekule-structure enumeration on and indeed such correspondences have sometimes been But again correspondences between different types of enumerations seem to be quite frequent. The correspondence of Kekule structures to sets of mutually self-avoiding walks as noted in passing in Section 2.2 in connection with the John-Sachs method of enumeration provides yet another example of such a correspondence.

**,

3 Current Results Here we survey with some critical commentary recent enumerative work from 1999 through May 2001. We divide up the discussion into different broad categories, separating off the work on fullerene isomer problems from other isomer work, in part because of the activity in dealing with fullerenes, and in part because, to deal best with fullerenes, Polya theory has played a lesser role. 3.1 Isomers: Enumeration and Generation. - During the last two years there seems to have been a degree of activity in the area of isomer characterization, more so with articles devoted to methodology development. There have been

74


some focusing on isomer enumeration methods. A few of them are classical enumerations for a simple fixed skeleton, with a selection of univalent ligands to be there attached. Lam,8o following up on earlier similar work,81 enumerates the numbers of isomers of alkyl-substituted cyclopropanes - the symmetry is not too high and as a consequence the development may be made without reference to the powerful Polya-theoretic mathematical machinery. Knopfmacher and Warlimont,82 following earlier work,83 use much more esoteric mathematical machinery (concerning semi-groups of polynomial-like structures) to develop methods to determine the asymptotic (large-“) form of isomer counts for N-atom alkanes. Polya4’ in fact developed asymptotic results, but Knopfmacher and Warlimont are interested in extensions, particularly to obtain asymptotics for the numbers of ‘generalized isomers’, viewed as sets of molecules such that different sets as a whole have the same elemental composition. Their main result is a theorem on the asymptotic number of ways in which a mixture of molecules may be realized when the total number of particular kind of atoms is fixed, and the asymptotic results for ordinary single-molecule isomers is already known. Examples of application to mixtures of alkanes, alkenes, substituted alkanes and achiral alkanes and alkanols are given. Baraldi and V a n o ~ s i ,also ~ ~ following up on earlier formal work,85 use (in a fairly conventional manner) the general Polya-theoretic machinery to enumerate substitutional isomers for several cyclic or polyhedral skeletons. They conclude with enumerations for icosahedral-symmetry skeletons, both for an icosahedron and for a truncated icosahedron, such as have become of some degree of popularity (as in ref. 86) over the last decade or so because of the ‘elegantly beautiful’ truncated-icosahedral structure of buckminsterfullerene. J. S Z U C S(refining ~~ earlier work of Kirby and Pollak88)generates elegant nearanalytic number-theoretic enumeration formulae for bucky-tori structural isomers (such structures being viewable as finite graphite fragments with cyclic boundary conditions), with special emphasis on asymptotics. Though one can imagine carbon tori with five- and seven-membered rings included (perhaps to contribute’’ to curvature strain relief), it has been argued” that the experimentally observed” (quite large) carbon tori do not have such non-benzenoid rings (at least arranged in any systematic fashion). For molecular polyhedral skeletons, it is well-known that there are naught but one or two topologically reasonable embeddings (with the two arising if the polyhedron is intrinsically chiral). But if for bucky-tori the topological embedding in 3-space is attended to (beyond just the molecular graph), then the same graph may be embedded in different topological manners. For example, a rectangular graphitic sheet may be rolled up in one direction or the other first, as indicated in Figure 11, though either way it is done, the same bonds become connected at the adjoining boundaries (and thence the same graph results). But more than this, even for a fixed order of rolling, one can after the first joining still twist the resultant open tube at the boundary before joining the last boundary, as indicated by the cyclic arrow in the Figure. If the twist is through a multiple of 2n radians, the same graph again results, though the embeddings are generally topologically distinct. Thence the theoretical number of such topologically distinct embeddings grows to infinity (with different multiples


75

Figure 11 Two diflerent ways of embedding the toroidal graph into a toroidal surface

of 2x),but steric hindrance restricts those which might be plausibly realized, and the labelling of such topological equivalence classes is Over the last decade and more S. Fujita has been singularly active in the development of isomer-enumeration methodology and its application. This effort has continued with a variety of hrther articles. One topic addressed has been93-97 that involving isomer enumeration with non-rigid skeletons. Examples are given involving skeletons such as that of tetramethylallene or of dimethylacetylene, with the methyl groups able to undergo internal rotation, while the H-atoms in these methyl groups provide the points of substitution. This involves taking a permutation group expanded over that corresponding to the classical point group, and using this in the Polya-theoretic apparatus. That is, the group is chosen to include permutations corresponding to the internal rotations, or inversions (as at the Natom of amines) or pseudo-rotations. Perhaps most of Fujita's work over the years has concerned the theory of subsymmetry classification (of the different isomers which arise with different substitution patterns in a skeleton), and he has now further d e v e l ~ p e d this ~ ~ ,theory, ~ ~ and made application^^^>'^ of it to a number of different circumstances. In yet another sort of circumstance Fujita addresses'oO-'02 the enumeration of isomers when the ligands themselves may be chiral, and illustrative applications are made. Here the crucial theoretical point is that permutations (which represent reflections and other improper rotations) need to be recognized as acting on both skeletal positions and the ligands. Further Fujita a d d r e s s e ~ ' ~ several ~ - ' ~ ~ interesting chirality characterization questions, concerning 'prochirality', 'stereogenic', 'prostereogenic', 'holotopic', 'hemitopic' etc. An-

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Chemical Modelling: Applications and Theoiy, blume 2

swers are proposed in terms of Fujita's mathematical constructs, and illustrative applications are made. S . El-Ba~il'~'builds from Fujita's work concerning the subclassification of substitutional isomers according to symmetry group of the substitution pattern. He uses so-called Cuyley-like diagrams as pictorial representations of the actions of group elements on the cosets associated to the various subgroup symmetries, and illustrative applications are made. VR. Rosenfeld''* extends Polya theory in two directions: to deal with sets of transformations on skeletal sites such that the set forms just a monoid rather than a full group; and to deal with symmetry subclassification. This latter extension evidently is proposed as an alternative to other approaches, such as Fujita's approach involving unit induced cycle indices and mark tables, though one can also recognize the mark table in Rosenfeld's normalizer scheme. Yet further it may be mentioned that S. El-Basil is to guestedit a special issue of the journal Match dedicated to the symmetry subclassification problem for isomer classes. This should appear in 2002. A few articles'09-''' have come from a German-Austrian group (of van Almsick, Dolhaine and Honig) utilizing the classical Polya-theoretic ideas to present substitutional isomer counts for a selection of skeletons, so as to illustrate a general isomer-enumeration program the group has developed. Their article is of an introductory nature insofar as Polya the#xy is concerned, with the main purpose being to indicate the framework and formatting for their software. Indeed especially one of the co-authors here (H. Dolhaine) has long pursued such a general program, which now is available at: http ://www-orgc.tu-graz.athoegroup.

Special efficiency in generating libraries of isomer structures is addressed. Another suite of programs due to Kerber's group may be found at:

http://www.mathe2.uni-bayreuth.de/axel/symneu.engl. html and the background for this is discussed in Kerber's m~nograph.'~ There are a couple of brief in the chemical literature considering this approach and the s o h a r e . Both the book and the software are adapted to doing a great variety of different things, besides enumeration also including possibilities for generation, and especially the book gives a wealth of mathematical material, which it seems has only meagerly been utilized to date in chemistry. Further in connection with the journal MutCh (from where a number of our references for this subsection come) there is an on-line address:

http://www.mathe2.uni-bayreuth.de/match/online/links, which offers isomer enumeration in a user-friendly format (without necessary reference to formal Polya theory) along with isomer generation (for the first 1000 isomers). The general MOLGEN program to generate isomeric structures, and some related characteristics, is illustratively briefly discussed in ref. 114. A formal


77

theoretical development of several aspects of this program are discussed in extended detail by Griiner."' It is available at: http://www.mathe2.uni-bayreuth.de/molgen4/ Hopefully such publicly available resources should find notable use by a wide community of chemists. Bytautas and Klein have written a set of articles with enumerations of different acyclic hydrocarbon structures: for alkanes;"6-"7 for fully conjugated polyenes;"* and for all hydrocarbons'19-'2' regardless of the degree of unsaturation. Here the alkane enumerations which are included in refs. 116 and 1 17 repeat or extend earlier work, e.g. as reviewed in Trinajstic et aZ.42though in the present articles there are additional considerations, beyond just isomer enumeration. The conjugated polyene enumerations considerably elaborate earlier work'22 so as now to distinguish radical and non-radical structures - i.e. so as to pay attention to the placement of Jt-bonds. The fist article on all acyclic hydrocarbons develops focuses on enumeration, developing the enumeration technology for all acyclic hydrocarbon graphs with arbitrary numbers of double and triple bonds, dealing with the (large-molecule) asymptotics of the isomer counts and applying the methodology for isomers with up to 26 carbons. For this case of all acyclic hydrocarbons, the work is developed with reference to a formula periodic table, as indicated in Figure 12. In this Figure the abscissa is half the number m of H-atoms, while the ordinate is the number n of H-atoms, so that at coordinate (m,n) one finds CnHZm, and the ordinary alkanes are found on the far right diagonal, the alkenes on the next diagonal in, both alkynes and alkadienes on the third diagonal from the right, etc. Moreover, beyond the classical enumerations, these authors seek in the bulk of their work' 16to extend much the same mathematical methodology to compute isomer-class averaged values for different graph invariants, including: atom-type counts, graph diameter, Wiener number, and second moments for atom-type

Figure 12

The formula periodic table of acyclic hydrocarbons

78


counts (so as to enable determination of associated standard deviations, and cross variances, for atom-type counts). Indeed in ref. 123 results for the Wiener number for alkanes of up to 90 carbons are computed (and asymptotic behaviors considered, not only for all isomers but for other equivalence partitionings of various tree structures). Yet further, the mean graph invariants are utilized along with sub-structural cluster expansions to give estimates for isomer-class averaged values of a few molecular properties, including: heat of formation, magnetic susceptibility, and index of refraction. The techniques thence allow such treatment for all of isomer classes even with enormous numbers of isomers, e.g. acyclics*19121and for just the alkanes.'I7 For the case of all acyclic hydrocarbons the results are presentedI2' graphically as 'property overlap plots', such consisting of contours superimposed on the formula periodic table (of Figure 12). An example of a 'property overlap plot' for the heat of formation is found in Figure 13. In this later figure the lines identify constant mean-AHf contours and the arrows normal to the contours indicate magnitudes of standard deviations (for AHf) at these positions, with the scale of the standard deviations set to correspond to the scale associated to the difference between the contour lines. In most of these papers attention is paid to asymptotics, not only for counts but also for the mean values of the various computed graph invariants. One of the articles on alkanes"' also illustrates a rather general method for selecting, from even a very large isomer class, particular structures exhibiting extreme values for the property estimates. It is proposed that the variety of extensions of standard enumerative methodologies considered offers a potential use in screening large classes of structures, in a sort of combinatoric chemistry.

m

n

Figure 13 The contour plot of heat of formations for acyclic hydrocarbons C,H,,. The lines orthogonal to the contours represent standard deviations


79

There has been some interest in an inverse isomer enumeration problem, of characterizing the symmetry group of the underlying skeleton from the numbers of isomers with different degrees of substitution. Historically this played some role in deducing molecular structures, e.g. in seeking a 6-position skeleton for benzene such that there are 1, 3, 3 , 3, 1 and 1 chloro-benzenes with respective numbers 1, 2, 3, 4, 5 and 6 of chlorine atoms substituted. Some rather general results were obtained (in 1930) by Lunn and Senior,'24 later (in 1985) by Ha~selbarth'~~ and now by V.V. Iliev.'26-'28 In particular Iliev develops a characterization via substitutional reaction processes; with the addition of one more substituent of a given ligand to one isomer with n of these ligands already in place, it may happen that only certain isomers with n 1 of these ligands arise. For instance, chlorination of p-dichlorobenzene gives rise to only one (of the three) trichlorobenzenes (while chlorination of o-dichlorobenzene gives rise to two trichlorobenzenes, and m-dichlorobenzene to all three). At least in favorable cases from the isomer counts (perhaps along with the substitutional reaction diagram just mentioned), the symmetry of the skeleton may be determined. Further Iliev provides explicit example applications for the skeletons of cyclopropane'28 and of ethane.12' However, in the case of benzene it has been emphasized129that, in considering the regular hexagon and the trigonal prism, the problem of determining the skeleton is indeterminant, no matter the number of different (independent monodentate) ligands one considers as substitutents, and indeed the problem remains indeterminant even with some additional information about reaction diagrams. That is, granted solely the isomer counts (and reactiondiagram information), both regular hexagonal and trigonal prismatic skeletons for benzene are acceptable. It may be noted that there are 'chemical' problems of different sorts for many of the enumerations which have been made over the years. First, in dealing with the alkanes there is a problem of steric hindrance which typically is entirely ignored. For instance, for alkane-chain conformations, such as treated by Tasi et a1.'30,'3'and others,'32 for the longer chains some of the enumerated conformations (thought of as walks on a lattice) end up walking over themselves. Moreover, in the field of 'polymer statistics' (concerned not only with conformer enumeration, but also mean spatial extent of these conformers) it is generally accepted (e.g. as in ref. 133) that this leads to quantitative differences in predictions as to the mean spatial extent of long-chain polymers (as well as a quantitative difference in the number of allowed conformations). For branched alkanes there are'34 some conceivable structures for which there exist no self-avoiding embedding on the relevant diamond lattice (even if the spatial requirements of the Hatoms are disregarded), so that one can imagine that the Polya enumeration counts should be reduced. Indeed this fact of steric crowding is135crucial in understanding the experimentally realized termination of dendrimer growth. Another problem arises in the treatment1l6of the acyclic polyenes simply as tree structures solely with a limitation on degree (on the H-deleted skeleton to a degree d 3), for then many of the structures turn out to be radicaloid - indeed, as revealed in a 'corrected' enumerationl10the great bulk of the otherwise counted poZyenoids are radicaloid, and even polyradicaloid. Yet further in computing the substitutional

+

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Chemical Modelling: Applications and Theoly, Volume 2

isomers of buckminsterfullerene there is a similar problem, in that many of the enumerated structures do not support a (fully paired) Kekule structure (and thence again are imagined to be radicaloid, and unstable). In some cases of enumeration or generation of isomers there is a 'reverse' problem, with some programs counting different Kekule structures of a single isomer as distinct isomers. This might be among the reasons that existing programs for generation of isomers in some cases do not agree in the numbers of produced isomers.'36 Another circumstance concerns enumerations with bidentate ligands - in this case for many such bidentate species, ligation is restricted to occur at adjacent sites of the skeleton (e.g. ethylenediamine would not ligate to trans positions in an octahedral skeleton), while the (extended) Polya-theoretic tools do not account for this. Evidently, even to deal systematically with just enumeration there are a number of 'chemical' problems which remain mathematically challenging. In a fair number of articles now many such problems are only overcome in a relatively brute-force explicitly constructional approach. Contreras et al.137have reported an extension of their program CAMGEC,13' which generates all isomers from a given molecular formula. The extended version, named US-CAMGEC, is intended to generate the geometrical and stereisomers induced by cumulative double bonds. Although quite comprehensive, the program does not yet account for chirality of atoms in cycles. The authors proposed an addition to Cab-Ingold-Prelog rules, but in several comment^'^^-'^^ that followed it was shown to be excessive and contradictory to the existing rules. In a follow-up of his previous paper,143Le Bret'36 has reported a comparison of his program Galvastructures with other existing programs for generation of isomers. The program Galvastructures is unique in using a genetic algorithm for the generation of isomers. Although slower than other programs, especially if all isomers are needed, its advantage is a relatively simple algorithm on which it is based. There were also discussed some practical difficulties in applying the common fingerprint methods for recognition of isomorphic structures, and a new fingerprint quantity, based on integer numbers, is proposed. L ~ k o v i t s ' ~is~ ,trying ' ~ ~ to devise an efficient algorithm for generation of all trees. By using properties of the Morgan labelling,'46 it is easy to generate all the, so-called, Morgan trees. The problem is that to any given isomer corresponds many different Morgan trees, and more so as the number of atoms increases. Thus if one is interested only in isomers, the code becomes highly redundant. However, each isomer has a unique Morgan tree which is used as the canonical code of the isomer and if the generation could be limited to canonical Morgan trees only, the redundancy would be eliminated. Lukovits seeks to formulate a set of semantic rules by which non-canonical Morgan trees could be detected and skipped earlier in the isomer generation. In his recent paper'47 two simple codes of adjacency matrix are formulated and discussed. An algorithm for generation of all boundary sequences that encompass a planar cubic map consisting of only pentagons or only hexagons was described by Deza et al.14* The algorithm works by producing all sequences satisfying the necessary condition on difference between the numbers of divalent and trivalent boundary vertices. The obtained sequences are subsequently checked for consistency by

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deconstruction - that is, by reducing the number of included rings and/or by splitting into smaller sequences which are separately checked in the same way unless the trivial case or some other special case is obtained. An additional, but simpler and faster, algorithm for generating sequences that correspond to planar polyhexes (those embedded into a regular hexagonal lattice) is also described. Generation and enumeration of benzenoid isomers and fbsenes is an everlasting subject considered by many authors. Caporossi et aZ.'49have counted the numbers of perifused and catafused polyhexes with up to 20 hexagons, and symmetry subcategorization and associated counts are also provided. Chyzak et a1.66 have applied holonomic guessing for estimation of numbers of not yet counted benzenoid isomers: those with 24 and 25 hexagons. The method relies on the built-in procedures of Maple and Mathematica by which a reccurrence relation for a given (presumably holonomic) sequence of numbers can be derived. The predicted number for benzenoids with 24 hexagons agrees with the most recent list of the numbers of benzenoid isomers and fbsenes reported by Brinkmann et aZ.67 The list contains exact numbers of fusenes with up to 26 hexagons and the number of benzenoids with up to 24 hexagons. Details of the algorithm will be published separately. An algorithm for generation and enumeration of polycyclic chains was described by Brinkmann et aZ.15' Polycyclic chain is a graph composed of rings of arbitrary size connected so that its inner dual is a path. Inner dual was also used for condensed representation of the polycyclic chain. A table with 26 representative counts for different combinations of ring sizes and their numbers is also given. 3.2 Fullerenes and Related Objects. - Apart from diversity of their own structural isomers, fullerenes bring additional variety with the derivatives obtained, e.g. by reactions of addition. Due to an almost uniform chemical character of all carbon atoms, the ligand atoms could be added in a wide range of numbers and patterns. Possible characteristics of such a reaction have been studied by Fowler et aZ.lsl in the model addition of hydrogen onto C24 fullerene. Even with so small a fullerene molecule (the smallest one with hexagonal rings), having also a symmetry, the number of all possible isomers is too large for a systematic and complete study. After abandoning isomers with an odd number of hydrogen atoms, since they imply an open shell in the remaining n-system, the number of isomers (including 0-24 hydrogen atoms) dropped to 352786. If the isomers containing odd disconnected components in the remaining conjugated nsystem (which implies a radical character too) are separated out, the number of isomers to be studied reduces to 63663 which was considered as acceptable. Nevertheless, due to an efficient method for calculation of electronic structure (density functional based tight-binding method), they were able to calculate energy and perform geometry optimization for all isomers with an even number of hydrogen atoms. The energies of the isomers without explicit radical character (containing no odd component in the conjugated n-system) were distributed at the low end of the cumulative distribution, slightly overlapping with radicaloid species. The most stable isomers for a given number of added hydrogens were

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Chemical Modelling: Applications and neory, Volume 2

always among those with no odd components. The analysis of the most stable isomers when increasing the number of hydrogens indicated that the hydrogencarbon bond energy does not change much up to 12 hydrogens, whereafter it decreases. It confirmed Kroto and W a l t ~ n ' s ' suggestion ~~ that fullerenes could behave as superatoms, exhibiting some sort of preferred coordination that might be interpreted as a preferred valency. In the case of C24, this number turned out to be 12. These twelve hydrogen atoms were bonded along the unique cycle dividing the molecule into two equivalent halves with the bare carbon atoms connected in two hexagonal rings. To check a possible specific effect of hydrogen atom, used as the model ligand, the isostructural most stable isomers with fluorine instead of hydrogen were also evaluated. Again, there was a marked decrease in binding energy at 12 added fluorine atoms showing thus a certain insensitivity to the bonded species. The sequence of the most stable isomers with increasing number of hydrogen atoms is consistent, with no rearrangement of bonded hydrogen atoms, and thus provides a possible pathway of the addition mechanism. A similar problem has been examined in another paper by Fowler et aE.'53in which there was studied an addition of bromine to the experimentally isolated isomers of c 6 0 , C70,C76and c84. As the bromine atom is much bigger than hydrogen or fluorine atoms, steric interactions between bromine atoms bonded to adjacent carbon atoms make such isomers less stable in comparison to those with no proximal bromines. This allows one to search for the most stable isomer among only those with no adjacent bromine atoms. The number of isomers to be checked in more detail is thus efficiently reduced, especially if only the isomers with maximum numbers of bromine atoms are examined. For example, the number of all isomers of C60Br24amounts to % 3 X loi4, while those with no adjacent bromines there is only 1085. Further reduction is possible if the isomers in which bare carbon atoms do not have a closed-shell electronic structure (as predicted by the Hiickel model) are also left out. In this case the number of C60Br24isomers reduces to a single one. The adducts of C70, c76, and c84 with the maximum number of bromine atoms are determined to be C70Br26(10 isomers), C76Br28(36 isomers), C84Br32(seven isomers and four isomers for two different c84, respectively). Their energies, calculated at the semiempirical level, are also given. An addition of bulky groups requires more space around each coordinated carbon atom. This may be formalized by a generalized requirement that all added groups must be separated by at least d carbon atoms. Again, for a given molecule and distance d, there may exist many different addition patterns which may not be easy to find. Enumeration and classification of such patterns, named d-codes for a given distance d, has been undertaken in the paper by de la Vaissiere et a1.Is4for classes of Platonic, Archimedean, face and medial duals of Archimedean polyhedra, as well as for general prism, antiprism and several chemical deltahedra corresponding to carboranes. &Codes were generated by the program Dense Clique, made by Hansen and Mladenovi~.'~~ The results are presented in tables containing the numbers and sizes of d-codes for all possible d, classified by symmetry and packing properties. Face-regular polyhedra, considered in the paper by Brinkmann and Deza,lS6are defined as polyhedra with similar surroundings of equally sized rings. The similar

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surrounding means that all n-gonal rings have equal sets of n adjacent rings (with no order implied and rings characterized only by their size), for each n. Written by mathematicians, the paper gives several lists of face-regular polyhedra with constant vertex degrees. The lists are characterized by the size of the maximal ring and the vertex degree, and for some selected cases they are claimed to be complete. Bifaced face-regular polyhedra with constant degree are fully characterized by four theorems. So far the chemical counterparts of regular-faced polyhedra exist in a small number of cases but this may change in the future. Related icosahedral-symmetry fulleroids are studied by Friedrichs and Deza. 15’ Fulleroids have all vertices of degree 3 , but sizes of faces other than 5 (and 6) are allowed. Those smaller polyhedral fu’leroids with icosahedral symmetry and no more than one size face other than 6 are comprehensively identified. The combinatorial structural characterization of icosahedral fullerenes is addressed by Quinn et al.,”* and shown to be of use in constructions for large such cages. These techniques presumably extend to the icosahedral symmetry fulleroids. Nanotubes have been theoretically studied by many authors (much of which is reviewed, e.g. in ref. 159). However, these studies generally have been concerned only with infinite nanotubes without ends. Brinkmann et ~ 1 have . ~studied ~ nanotube caps, that is, the possible ways of ending the tubes, with fullerenic structures. The tube is characterized by the vector ( n , m) in the hexagonal lattice connecting two hexagons that overlap each other when rolled into a tube. As there is no unique way to determine a boundary between the tube and the cap, the authors made their own choice by taking the hexagons lying on the two components of the vector (n, m) as the boundary. The patches corresponding to a given boundary were generated by the divide-and-conquer method: every patch can be uniquely divided into two special subpatches by using a Petrie path. The algorithm constructs all possible special subpatches which are subsequently combined into a patch with the given boundary code, The produced patches are uniquely coded in order to recognize and abandon isomorphic pairs. This is performed for all combinations of n and rn, with n + m d 25, and with n + m d 30 for patches with isolated pentagons. The patches, whose number exponentially increase with the diameter of the tube, are tabulated by the n and m values and symmetry group. And further the types of asymptotic forms of behaviors for different sorts of ends (regardless of whether the ends are fuilerenic or not) are characterized.160 The papers of Fowler and RogersI6’ and of Fowler et a1.162do not deal with enumeration but rather with coding. However, as enumeration is tightly interlaced with coding these papers are also interesting to workers in the field. The first code for fullerenes was devised by Fowler and Manolopoulos et al.’63It consists of a path going spirally from one face through all other faces. Although not all fullerenes have such a spiral, the smallest known counterexample has 380 vertices1@and so its use for smaller fullerenes seems to be safe. Particularly for fullerenes with C5 or C, symmetry it has been proven that such spiral In the present paper the authors derived the relation between the Goldberg representation of icosahedral fullerenes and their spiral codes. This relation has been extended also to analogous polyhedra derived from octahedron.

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The study of higher genus fulleroids, particularly with translational symmetry, has been a topic of some interest over the last decade or so. Terrones and T e r r ~ n e s identify '~~ some particularly favorable finite structures of genus up to a dozen or so. King167also presents some combinatonal group-theoretic aspects of the extended translationally symmetric high-genus such structures.

3.3 Counts of Resonance Structures and Related Items. - Resonance-theoretic based enumerations seem to have been somewhat less studied during the last two years, though a decade or two ago there were tremendous numbers of papers. For the case of Kekule-structure enumeration the methods we deem more powerful or elegant have been briefly indicated in Sections 3.2 and 3.3. Perhaps the current relative quietness of the field indicates that now developed methods are near optimum. Still there has been some work. Si16' considers the count of Kekule structures of a special subset of coronoids (which generally are benzenoid structures with a single 'hole' region covering two or more hexagons of area). Si establishes a determinantal formula like that of Dewar and L~nguet-Higgins~~ for a special subset of coronoids, and notes that the J o h n - S a ~ h s ~formula ' ~ ~ ~ (of Section 2.3) which involves the determinant of a much smaller matrix then also applies for these same coronoids. For more general coronoids there is'69 an extension of the John-Sachs formula, though this entails the evaluation of two or more John-Sachs determinants, and of course the method of Kasteleyn7' applies. In a series of ~ a p e r s ' ~ ( 'Dias - ' ~ ~has derived expressions for resonance-structure counts of several classes of mono- and diradical benzenoid hydrocarbons. The considered classes belong to polymer graphs which were extensively studied some time ago,53,59,60,175 though the considered ends of the oligomeric chains may be new. The recursions are apparently obtained by 'examination', so that formal derivations of them are not given. The recursions could be more formally achieved by application of the transfer-matrix method for matching polynomials of open polymer graphs (also called fascia graph^),'^^^'^^ though a slight extension is needed to obtain the first or second derivative of the matching polynomial evaluated at zero. That is, the first derivative of the matching polynomial at zero produces the number of resonance structures for monoradicals, and analogously the k-th derivative (divided by k!) gives the resonance-structure count for k-fold polyradicals. For the cases considered the unpaired electrons seem typically to be localized near the chain ends judging from the more numerous valence structures, and some qualitative chemical consequences are considered. Cash and H e r n d ~ n " ~described a program for calculation of a matching polynomial. The program is based on a successive removal of edges and application of the well known recursion relation for the matching polynomials. The edges are removed either in a sequence given by the user or automatically based on the heuristic choice. In order to keep all figures of the matching numbers, the program runs within Mathernatica. Zhang and Zhang17" have strengthened Gutman's result'" on the extremal values of Hosoya and Merrifield-Simons indices for hexagonal chains. These two indices are defined as sums of the absolute values of coefficients of the matching

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and independence polynomials, respectively. In the present paper it was shown that not only their sum but each coefficient of the two polynomials has extremal values for linear and zigzag chains. The linear chain has minimal number of matchings and maximal number of independent sets, and vice versa holds for the zigzag chain. The so-called algebraic structure count is closely related to Kekule structures but in many cases is more difficult to evaluate. It is defined as the absolute difference of the numbers of two parities (positive and negative) of Kekule structures. This algebraic structure count coincides with K( G) for benzenoids, but for more general alternants it has been advocated as a more reliable measure of aromaticity than K( G). Closed expressions for several special classes have been derived by Gutman et al.'81-'84Recently Graovac et al.Ig5 have formulated a general method for polymers, and elaborated it in detail for a case with two bonds between adjacent monomers. The method uses the fact that the determinant of the adjacency matrix is equal to a square of the algebraic structure count. For a given graph the determinant of the adjacency matrix can be recursively calculated from the determinants of certain subgraphs. Systematic application of such a recursion was efficiently organized by using the transfer matrix technique. The method is illustrated by three classes of phenylenes and two types of acenylenes. In the paper by Dias'*(' electronic structures of several types of subspectral graphs and infinite polymer strips were considered. The common point of enumeration and the subject of this paper are recursions for the characteristic polynomials of one-dimensional polymer graphs. Methods for deriving characteristic polynomial in these cases were first formulated by Polansky and Tyuty~lkov'*~ by using cyclic symmetry and representative graphs. Later it was formulated in terms of the transfer matrices for polymer graphs with singly connected monomers, and for both open and closed ends as well as for any type of the starting or ending part.Ig8 Some of these methods seem to be rederived in the present paper. Lin and Fan'89 have described an algorithm for finding all so-called 'linearly independent' and 'minimal linearly independent' conjugated circuits in benzenoid hydrocarbons. A different resolution into different-shaped conjugated circuits has also been advocated elsewhere.190In another paper"l Lin has used a simpler approach to count only the few smallest conjugated circuits.

3.4 Walks, Connected Subgraphs and Vertices at a Given Distance. - Counts of random walks are used for characterization of graphs and for definition of various molecular descriptors. These counts can be easily obtained from adjacency matrix powers and the recursion provided by the Hamilton-Cayley theorem. Basic mathematical properties of random walk counts were reviewed by Gutman et al.I9* In this paper the graphs with extremal walk counts were determined, and the relations between the structure, graph spectra and walk counts were discussed. The spectral moments (that is the numbers of self-returning walks) of phenylenes and their line-graphs were studied by MarkoviC et al.'93The authors are especially interested in expressing the lower moments by certain

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structural details, thus establishing a formal connection between the structure and moments or related molecular descriptors. Formulae for up to the ninth moments of phenylene line-graphs were given in terms of five obvious structural details and compared with similar formulae obtained by E ~ t r a d athat ' ~ ~ involved different and less obvious structural counts. A correlation between spectral moments of phenylenes and their line graphs was pointed out. Rucker and Rucker68 have reported an algorithm for the enumeration of all connected subgraphs of a given graph, whose number serves as a molecular descriptor for measuring a complexity of graphs (and molecules represented by them). The algorithm explicitly generates all connected subgraphs by a depthfirst path-tracing procedure. One might try to devise an algorithm by using a recurrence relation relating the number of connected subgraphs in a graph and in the subgraphs (some of them being 'rooted') obtained by removal of an edge. However, such a code would be useless when one wants to know the number of isomorphism classes of connected subgraphs, which is also used for estimation of molecular complexity, but taking into account its symmetry. This was the subject of the next Rucker and Rucker paper,69 in which the program for generation of connected subgraphs was combined with calculation of few graph invariants: numbers of vertices and edges, Balaban index, and the extremal eigenvalues of the distance matrix. These invariants were used for (approximate) recognition of isomorphism, so that all subgraphs with the same values of these invariants were considered as isomorphic. The program was generalized for edge- and vertex-colored graphs which are used for representation of heteroatomic molecules and those with multiple bonds. As a spin-off, there were found new examples of graphs with the same Balaban index and of isospectral colored graphs. The Hosoya polynomial of a graph G is defined as a generating function for the numbers of vertex pairs at varying distance expressed by the exponent of the dummy variable. There is no general recursion to express the Hosoya polynomial of the given graph over its subgraphs, as is possible for e.g. matching and characteristic polynomials. Gutman et ~ 1 . lhave ~ ~ studied several classes of hexagonal chains and formulated (inhomogeneous) recursions for their Hosoya polynomials. These recursions were solved by use of Mathernatica. Explicit formulas for the Hosoya polynomials were given for members of the considered classes with the number of repeating fragments as a parameter.

3.5 Other Enumerations.

- Evaluations of permanents have been pursued by Cash'96 (such permanents being involved in several different enumerations, including that of Kekule structures as mentioned in Section 2.5). He finds efficient means for their evaluation for matrices of up to 80 rows and columns, at least if the matrices have some sparsity. Tasi et aZ.'30,'31 make an enumeration of conformers of normal-alkane chains. Such does not require use of Polya-theoretic machinery and has been considered several times p r e v i ~ u s l y 'over ~ ~ the years. A difference in the present enumeration (using relatively straightforward mathematics to give a largely analytic enumera-


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tion) is that in addition to the familiar gauche and trans structures, an evidently theoretically predicted x-conformational structure is also included. Nitta19’ considers some ‘polymer statistical’ results for the variety of chains of branched acyclic molecules (such as alkanes). Tissandier et al.198 have studied all possible hydrogen bond arrangements in the water hexamer cage. Although enumeration of hydrogen bond arrangements was recognized as an interesting graph-theoretical problem long ag0,”3~’’ only recently2” has a simple procedure for generation of all arrangements been outlined. Yet for larger clusters a more efficient program might be necessary which also could analyse the arrangements (as already attempted202)and the possibilities for their interconversions. Vismar and Laurenco203consider an analysis of a molecular graph into various types of structures. This includes the enumeration and generation of all cycles, in addition to sets of independent cycles. As a further development Dietz et aL204 consider discrete decorations of a molecular graph so as to provide additional information about geometric structure, say as regards cis- and trans-stuctures, or as regards different enantiomorphs (including different distereomers). The enumeration or generation of the associated discrete mathematical structures then relates to the enumerations or generation of the different chemical isomers, not mediated by ordinary graphs. In fact there are already descriptions of geometric isomerization already implicit in Polya theory, as reviewed in Sections 2.1 and 3.1, though the (permutation-group-theoretical) representations implicit there are generally somewhat different. Xu and Johnson2o5consider the classification of molecules into equivalence classes identified by substructures associated with molecular ‘pseudographs’. Here the pseudographs represent homeomorphism classes of graphs (wherein degree-2 vertices are deleted), and again the Morgan extended neighbor idea146 is utilized. Further there is a great deal of work involving so-called topological indices, which might also be described as molecular graph invariants. Such indices often are integer valued and then count something. For instance the so-called Wiener number of a graph may be viewed as the count of the total number of steps in a set of minimal-length paths one between each pair of distinct vertices of the graph. Various walk counts and the Hosoya index as mentioned in Section 3.4 are further examples of (perhaps less popular) topological indices which enumerate something, and are often so described. The Wiener number however is usually not described as an enumeration, but as the sum over all shortest-path distances. There are numerous other integer-valued topological indices which can be viewed as enumerations, including the Platt index, the Gordon-Scantlebury index, the Zagreb group indices, centric indices, the Szeged index, the hyper-Wiener index, etc. And further there are a fair number of topological indices which take rational-number values but for which the numerator and denominator of the rational number both can be viewed as enumerations. There has in fact been a fair degree of work on such topological indices, but we have not attempted to review it fully. It is appropriate to note though that there are recent which review the field.

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4 Conclusion

It is seen that there is a great deal of activity in chemical enumeration. Perhaps in the classical area of isomer enumeration there is even a surprising amount of activity. This may be due to a recognition that it is desirable to treat whole sets of molecules beyond just individual molecules one at a time as has been the dominant focus in quantum chemistry and perhaps even the vast bulk of chemistry over the last several decades. Indeed in general chemistry, and especially medicinal chemistry, this recognition is involved in the intense interest in combinatorial libraries. Much of this area of combinatorial chemistry is viewed to be purely concerned with experimental methods of synthesizing large sets ( i e . ensembles) of compounds and their ensuant testing for their properties or bio-activities. But then perhaps the current theoretical activity is a quest for some sort of theoretical analogue approaches to comparably deal with virtual ( i e . theoretical or computer generated) ensembles of molecules to examine. As witnessed in Section 3.4 there has been much successhl work in developing libraries of molecular structures, especially in Brinkmann’s work. And the member structures of such libraries may be examined structure by structure, as exemplified in work by Fowler and colleagues. But beyond individual examination of each member, there are other systematic possibilities, which are beginning to be examined and which may ultimately prove quite useful. If a property is (approximately) expressed in terms of sufficiently nice graph invariants (as by e.g. cluster expansion in terms of local substructural counts), then searching through graphs, where the graph approximant is near optimal, could be addressed in a way that avoids structure by structure examination. There also seems still to be much room for theoretical methodological developments. We have noticed ( e . g . as in Section 2.5) correspondences with seemingly different types of enumerations. And thence there is a suggestion of a general classification of interconnections, with some sort of set of ‘canonical’ enumerations. Indeed the point of P- or NP-complete algorithms (as for enumerations) made in the mathematical theory of computation presumably focuses on some aspect of such correspondences, and in statistical mechanics the idea of ‘universality classes’ presumably relates to this also. But the mathematical theory relates to maximum computational time for a general case (e.g. enumeration for a particular type of subgraph on a general graph) and the statisticalmechanical ideas relate to asymptotic behaviors, whereas it seems that often there is a much more explicit correspondence between types of enumerations. Perhaps much more of such relations will become apparent in hture work. Overall there seems to have been a degree of activity in enumeration, with indications of a number of further developments to come.

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111. M. van Almsick, H. Dolhaine and H. Honig, Comm. Math. Comput. Sci., 2001, 43, 153. 112. A. Kerber, R. Laue, and T. Wieland, Discrete Mathematics for Combinatorial Chemistry, in Discrete Mathematical Chemistv, DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 5 1, eds. P. Hansen, I?W. Fowler and M. Zheng, American Mathematical Society, Providence, RI, 2000, pp. 225-234. 113. A. Kerber and A. Kohnwer, Comm. Math. Comput. Chem., 1998, 38, 163. 114. T. Griiner, A. Kerber, R. Laue and H. Meyer, Comm. Math. Comput. Chem., 1999, 39, 135. 115. T. Griiner, Comm. Math. Comput. Chem., 1999, 39, 39. 116. L. Bytautas and D.J. Klein, J Chem. In$ Comput. Sci., 1998, 38, 1063. 117. L. Bytautas and D.J. Klein, J Chem. In$ Comput. Sci., 2000, 39, 803. 118. L. Bytautas and D.J. Klein, Theor: Chem. Acc., 1999, 101, 371. 119. L. Bytautas and D.J. Klein, Croat. Chem. Acta, 2000, 73, 33 1. 120. L. Bytautas and D.J. Klein, Phys. Chem. Chem. Phys., 1999, 1, 5565. 121. L. Bytautas, D.J. Klein and T.G. Schmalz, New J Chem., 2000, 24, 329. 122. S.J. Cyvin, J. Brunvoll and B.N. Cyvin, J Mol. Struct. (Theochem), 1995, 357, 255; C. Yeh, J Chem. It$ Comput. Sci., 1995, 35, 912; S.J. Cyvin, J. Brunvoll, E. Brendsdal, B.N. Cyvin and E.K. Lloyd, J Chem. Znf: Comput. Sci. 1995, 35, 743; C. Yeh, J Phys. Chem., 1996, 100, 15800. 123. L. Bytautas and D.J. Klein, J Chem. Zn$ Comput. Sci., 2000, 40, 471. 124. A.C. Lunn and J.K. Senior, J Phys. Chem., 1929,33, 1027. 125. W. Hasselbarth, J Comput. Chem., 1987, 8, 700. 126. VV Iliev, Comm. Math. Comput. Chem., 1999, 40, 153. 127. VV: Iliev, Comm. Math. Comput. Chem., 2001, 43, 67. 128. V:V Iliev, Comm. Math. Comput. Chem., 2001, 43, 79. 129, D.J. Klein and L. Bytautas, Comm. Math. Comput. Chem., 2000, 42, 261. 130. G. Tasi and F. Mizukami, J Math. Chem., 1999, 25, 55. 131. G. Tasi, F. Mizukami, J. Csontos, W. Gyorffy and I. Palinko, J Math. Chem., 2000, 27, 191. 132. E. Funck, Zeit. Elektrochem., 1958, 62, 901; A.T. Balaban, Rev. Roum. Chim., 1976, 21, 1049; J. Brunvoll, B.N. Cyvin, E. Brendsdal, and S.J. Cyvin, Computers Chem. 1995, 19, 379; S.J. Cyvin, J. Brunvoll, B.N. Cyvin, and E. Brendsdal, Adv. Mol. Struct. Res. 1995, 2, 213. 133. E.g. I?G. deGennes, Scaling Concepts in Polymer Phvsics, Cornell University Press, Ithaca, New York, 1979. 134. D.J. Klein, J Chem. Phys., 1981, 75, 5186. 135. E.g. J.M.J. Frechet, Science, 1994, 263, 1710; D. Tzalis and Y. Tor, Tetrahedron Lett., 1996,37, 8293; F. Zeng and S.C. Zimmennan, Chem. Rev., 1997,97, 1681. 136. C. Le Bret, Comm. Math. Comput. Chem., 2000, 41, 79. 137. M.L. Contreras, G.M. Trevisiol, J. Alvarez, G. h a s and R. Rozas, J Chem. In$ Comput. Sci., 1999, 39, 475. 138. M.L. Contreras, R. Rozas and R. Valdivia, J Chem. Znj.’ Comput. Sci., 1994, 34, 610. 139. I? Mata, J Chem. In$ Comput. Sci., 1999, 39, 11 17. 140. J. Brecher, . IChem. In$ Comput. Sci., 1999, 39, 1226. 141. M.L. Contreras, G.M. Trevisiol, J. Alvarez, G . Arias and R. Rozas, J Chem. In$ Comput. Sci., 1999, 39, 1228. 142. P. Mata, J Chem. Zn$ Comput. Sci., 2000, 40, 1072. 143. C. Le Bret, J Chem. In$ Comput. Sci., 1996, 36, 678. 144. I. Lukovits, .IChem. In$ Comput. Sci., 1999, 39, 563.

2: Enumeration in Chemistg 145. 146. 147. 148. 149.

150. 151. 152. 153. 154. 155. 156. 157.

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M. Deza, PW. Fowler and V. Grishukhin, J Chem. Znf Comput. Sci., 2001,41, 300. G. Caporossi, P. Hansen and M. Zheng, Enumerations of Fusenes to h = 20, in Discrete Mathematical Chemistq DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 51, eds. P Hansen, PW. Fowler and M. Zheng, American Mathematical Society, Providence, RI, 2000, pp. 63 -78. G. Brinkmann, A.A. Dobrynin and A. Krause, Comm. Math. Comput. Chem., 2000, 41, 137. PW. Fowler, T. Heine and A. Troisi, Chem. Ph-vs. Lett., 1999, 312, 77. H.W. Kroto and D.R.M. Walton, Chem. Phys. Lett., 1993, 214, 353. PW. Fowler, K.M. Rogers, K.R. Somers and A. Troisi, J Chem. Soc. Perkin Trans. 2, 1999, 2023. B. de la Vaissiere, P.W. Fowler and M. Deza, J Chem. Znf Comput. Sci., 2001, 41, 376. P Hansen and N. Mladenovic, YUGOR, Yugoslav J Operations Rex, 1992, 2, 3. G. Brinkmann and M. Deza, J Chem. Znj.'Comput. Sci., 2000, 40, 530. O.D. Friedrichs and M. Deza, More Icosahedral Symmetry Fulleroids, in Discrete Mathematical Chemistv, DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 51, eds. P. Hansen, P. W. Fowler and M. Zheng, American Mathematical Society, Providence, RI, 2000, pp. 97- 1 16. C.M. Quinn, D.B. Redmond and P.W. Fowler, Group and Graph Theoretical Perspectives on the Structures of Large Icosahedral Cages, in Discrete Mathematical Chemistv, DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 51, eds. P Hansen, PW. Fowler and M. Zheng, American Mathematical Society, Providence, RI,2000, pp. 293-303. M.S. Dresselhaus, G. Dresselhaus and P.C. Eklund, Science ofFullerenes and Carbon Nanotubes, Academic Press, New York, 1996. H.-Y. Zhu, D.J. Klein, T.G. Schmalz, A. Rubio and N.H. March, J Phys. Chem. Solids, 1998, 59, 4 17. P.W. Fowler and K.M. Rogers, J Chem. h f . ' Comput. Sci., 2001, 41, 108. P.W. Fowler, T. Pisanslu, A. Graovac and J. Zerovnik, A Generalized Ring Spiral Algorithm for Coding Fullerenes and other Cubic Polyhedra, in Discrete Mathematical Chemistry, DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 5 1, eds. P. Hansen, P.W. Fowler and M. Zheng, American Mathematical Society, Providence, RI, 2000, pp. 175- 188. D.E. Manolopoulos, J.C. May and S.E. Down, Chem. Phys. Lett., 1991, 181, 105. D.E. Manolopoulos and PW. Fowler, Chem. Phys. Lett., 1993, 204, 1. P.W. Fowler, D.E. Manolopoulos, D.B. Redmond and R.P. Ryan, Chem. Phys. Lett., 1993,202, 371. H. Terrones and M. Terrones, Geometry and Energetics of High Genus Fullerenes and Nanotubes, in Discrete Mathematical Chemisby, DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 5 1, eds. I? Hansen, P W. Fowler and M. Zheng, American Mathematical Society, Providence, RI, 2000, pp. 333-342. R.B. King, Carbon Networks on Cubic Infinite Periodic Minimal Surfaces, in Discrete Mathematical Chemistry, DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 51, eds. P. Hansen, PW. Fowler and M. Zheng, American Mathematical Society, Providence, RI,2000, pp. 235-248. C.R. Si, Comm. Math. Comput. Chem., 2000, 41, 45.

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95

206. Topological Indices and Related Descriptors in QSAR and QSPR, ed. J.E. Devillers and A.T. Balaban, Gordon and Breach Science Publishers, NY, 1999. 207. R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Mannheim, 2000. 208. QSAWQSPR Studies by Molecular Descriptors, ed. M.V. Diudea, Nova Science Publishers, NY, 2000.

3 Density Functional Theory BY MICHAEL SPRINGBORG

1 Introduction In the course of developing the understanding of the interplay between composition and structure of materials on the one side and their properties on the other side, electronic-structure calculations continue to play an important role. Such studies have the advantage that well-defined systems are examined and that both existing and non-existing systems can be studied, whereby the composition structure - property relations can be investigated in great detail. On the other hand, for computational reasons there are limitations on the systems that can be examined. Therefore, theoretical studies of materials properties are most useful when they are combined with complementary experimental studies, and in that case the electronic-structure calculations most often deliver useful information when interpreting, explaining, and extending the experimental studies. It should therefore not surprise that very many research papers contain results of theoretical studies of the electronic and structural properties of materials. First of all, currently applied electronic-structure methods have been developed for calculating the total energy as a function of structure, thereby giving information on structure and relative energies of, e.g., different isomers and transition states. Also chemical reactions are, in principle, accessible with such methods. As a consequence, a very large part of the research papers containing results of electronic-structure calculations discuss structural and energetical properties of the systems of interest. In the first report in this series’ we gave a number of examples of such studies and, in principle, the present report could contain an update of the list of applications of electronic-structure methods (in particular of methods based on the density-functional formalism) to specific systems. However, due to the very large number of such studies such an overview would at most just give a feeling for the present state of this class of calculations. Instead, in the present report we have chosen to focus on a smaller number of special subjects where the density-functional methods have not yet been so fully developed that the methods are routinely applied. Thereby we hope to give an impression of the directions in which density-hnctional methods are being developed further at the moment. But it can simultaneously not be strongly enough emphasized that there are very many systems and properties for which density-functional calculations

+

Chemical Modelling: Applications and Theory, Volume 2 0The Royal Society of Chemistry, 2002

3: Density Functional Theoly

97

routinely yield accurate and useful information, although these studies will not be the subject of this report. One may split current electronic-structure methods into two classes, i.e. the wave function-based and the density-based ones. In our previous report' we discussed the similarities and differences between the two approaches, but since some of the recent developments within density-functional theory have been inspired from the wavefunction-based (i.e. Hartree-Fock) methods we shall in Section 2 briefly review the two approaches. Much effort is at the moment being devoted to the development of accurate, albeit still approximate, density functionals, and some of these are discussed in Section 3. Although computational power keeps on growing, there is still a very large class of materials that is out of reach with current accurate density-functional methods and where one has to introduce further approximations in order to make the calculations tractable. Some of these methods are discussed in Section 4. One of the problems with the more complex systems is their size. As an alternative to the methods of Section 4 some efforts have during the last almost 10 years been invested in developing methods for which the computational methods scale approximately linearly with the size of the system. These so-called order-N methods are described in Section 5. One specific class of systems for which the computational demands very easily become huge is that of catalysts. In particular, the study of chemical reactions on surfaces of crystalline materials is complicated, but since these are of enormous technological importance we shall in Section 6 review the present state of such studies. But also simpler chemical reactions, like those between two smaller, finite molecules, are far from trivial to investigate and it would be very useful if simpler tools could be developed that would permit simple estimates of how and when chemical reactions will take place. Some of the quantities that are being considered useful in this context will be discussed in Section 7. Of somewhat specialized interest is the quantum treatment of other particles, i.e. systems for which not only the electrons but also the nuclei (this is first of all the case for the lightest nuclei, the protons) have to be treated quantum-mechanically. A few examples will be presented in Section 8. In the last sections we shall focus on some of the limitations of current density-functional methods as well as on some suggestions to remove these. As the first example, Section 9, we discuss the fact that, when taking an electron very far away from the remaining part of the system, it experiences, according to most of the current approximate density functionals, a physically wrong potential. A possible solution to this problem (as well as to some other problems) is to use the so-called exact-exchange methods, which we will discuss in Section 10. Originally, density-functional theory was developed as a theory for the static ground state of a given system, and, accordingly, excited states were not accessible with this theory. With the introduction of time-dependent density-functional theory this has changed, as will be shown in Section 11. As a final example we consider the presence of macroscopic electrostatic fields which may lead to inaccuracies in the results when applying the current approximate density functionals, as we will discuss in Section 12. Finally, Section 13 contains a brief summary.

98


2 Basic Principles In some of the subsequent sections we shall discuss different recent developments of density-functional theory. Some of the fundamentals of electronic-structure calculations that we presented in our first report' will be very useful for that discussion, and we shall therefore repeat parts from that. We shall stress that these fimdamental considerations can be found in many modern textbooks on methods of electronic-structure calculations (see, e.g., ref. 2). We consider a molecule with M nuclei and N electrons. The positions of the nuclei are denoted gl,i2, . . . , iM and those of the electrons 6, 6 , . . . , FN. Moreover, we use Hartree atomic units and set accordingly me = [el = 4m0 = fi = 1. The mass and charge of the kth nucleus are then M k and z k , respectively. The combined coordinate Ti denotes the position and spin coordinate of the ith electron. In the absence of external interactions and relativistic effects, the Hamilton operator for this system can then be written as a sum of five terms,

i.e. as the kinetic-energy operator for the nuclei, that for the electrons, and the three potential-energy operators for the nucleus-nucleus, the electron-electron, and the electron-nucleus interactions, i.e.

H, =

N l -C-V' 2 .

The time-independent Schrodinger equation becomes then

Except for some few special cases (we shall consider some of those in Section 8) one resorts to the Born-Oppenheimer approximation in order to solve Eq. (3).

99

3: Density Functional Theory

Then, H h is ignored and, furthermore, the wavefunction Y that depends functionally on both electronic and nuclear coordinates is written as a product of two fbnctions,

Here, Y, is the wavehction for the nuclei (which in many cases is irrelevant) and Yeis the electronic wavefunction. Ye depends functionally on the electronic coordinates but also parametrically on the nuclear coordinates (implying that different structures have different electronic wavefunctions). Y is calculated fiom the electronic Schrodinger equation H,Y,

=

E,Y,

(5)

with H,

=

H k , 4- H,, 4- He,,,

Moreover, the total energy is E = E , -k H,,,,.

(7)

Within the so-called wavehnction-based methods, Eq. (5) is most often solved by first approximating Ye as a single Slater determinant. Thereby correlation effects are by definition ignored. (Parts of) these may, however, be added subsequently either directly or via perturbation theory. The N single-particle functions Ql,Q2,. , . , 4 1 of ~ the Slater determinant are calculated by solving the Hartree- Fock single-particle equations

with

Here,

and

100


Then,

It is clear that the calculation of E,, or the total energy, within the HartreeFock approach requires the calculation of a very complex object, the electronic wavehnction Ye? that depends on the coordinates of all N electrons (neglecting the addition parametric dependence on the nuclear coordinates). On the other hand, experiments probe properties of the system of interest that depend most often on only the electron density in the three-dimensional position space, implying that Yecontains very much redundant information. Hohenberg and K o h 3 (see also refs. 1 and 4) showed that - in principle - any ground-state property, including E,, could be calculated once the electron density p(?) was known. Thus, Ee is a functional of p(F),

but, unfortunately, the precise form of this functional is unknown. Kohn and Sham' showed that the problem of calculating E , [ p ( J ) ] of Eq. (13) can be cast into that of solving a set of single-particle equations,

with 1 ieff = - - v2+ V&( F).

2

xi&

In Eq. (11), Cijiis a multiplicative operator, but (i.e. the exchange operator) is a 'true' operator. Moreover, correlation effects are completely neglected within the Hartree-Fock approach. In contrast to this, Veff of Eq. (15) is a purely multiplicative operator, although it contains - as in the Hartree-Fock approach - both the external Coulomb potential from the nuclei, the Coulomb potential from the electrons, and the exchange interactions. Moreover, it contains also all correlation effects. Both in the Hartree-Fock and in the Kohn-Sham approach the total electron density is given as a sum over the N energetically lowest orbitals, i.e. in the Kohn-Sham case as

3: Densig Functional Theory

and a similar expression holds for the Hartree-Fock case with ?+bl replaced by Schematically, one may formulate the theorem of Hohenberg and Kohn as

101

#l.

i.e., from the electron density one may calculate the electronic energy. The approach of Kohn and Sham amounts to introducing an intermediate step (the calculation of Veff), i.e.

However, just as the precise form of E,[p( r')] of Eq. (13) is unknown, so is that of the first relation of Eq. (18), and one has to resort to approximation. In the next section we shall present some of the most common approximations, and in later sections we shall discuss some recent developments aimed at solving some of the problems related to approximating Veff.

3 Functionals Veff of Eq. (15) contains the external potential (i.e. most often the Coulomb potential from the nuclei), the Coulomb potential from the electrons, and a remainder,

When the external potential is solely the Coulomb potential of the nuclei, Vex, becomes

Moreover,

The remaining exchange-correlation potential Vxc(F) depends on the electron density, but the precise functional form is unknown. Accordingly, this is the quantity that is approximated and it may not surprise that many approximations have been proposed during the years (see, e.g., refs. 1, 6-8 and references therein).

102


As a first approximation V,, in the point Fis assumed to depend on the electron density in exactly that same point but in no other point, i.e. V,, becomes a function of the electron density. This amounts to the so-called local-density approximation (LDA). [For the sake of completeness we add that for spinpolarized system all functionals may be generalized to depend not only on the electron density

but also on the spin density

Here, p r F ) and pl?) are the electron density for the spin-up and spin-down electrons, respectively. We shall, however, not consider that generalization any further.] As a next step-one may introduce some non-locality by letting the functionals depend also on IVp(7)I and V*p(F), or rather on the scaled quantities

Including dependences on s leads to the so-called generalized gradient approximations (GGAs), whereas the inclusion of q, too, results in so-called metageneralized gradient approximations (MGGAs or meta-GGAs). The term meta-GGAs is, however, often also used for functionals that include dependences on local kinetic-energy densities. It may be observed that - fi-om a formal point of view - the solutions v i to the single-particle, so-called KohnSham equations (13) also are functionals of the electron density and, therefore, E , may include dependences on those. The quantity N

i

z(F) = ~ + v ~ i ( F ) 1 2 z= 1

2

gives, when integrated over the whole space, the total kinetic energy and is, moreover, everywhere non-negative. It can accordingly be considered a local kinetic-energy density and functionals (meta-GGAs) that depend explicitly on z have been derived. We would, however, like to stress that the definition of a kinetic-energy density is not unique and other definitions may be more useful (see, e.g., ref. 9). By definition, V,, contains all exchange and correlation effects. Since the Kohn-Sham orbitals are functionals of the electron density, one may also attempt to calculate exchange andor correlation effects directly with the help of these

3: Densip Functional Theory

103

orbitals. The hybrid methods (see, e.g., refs. 1, 6-8 and references therein) for which a part of the exchange effects are calculated using the Hartree-Fock expression [see Eq. (1 2)] and the remaining part is calculated using, e.g., a GGA expression represent one class of such approaches. Another class is formed by the so-called exact-exchange methods to be discussed below in Section 10. But before turning to those we shall discuss some specialized developments aimed at studying systems that are relatively complex.

4 Semi-empirical Methods Although density-functional methods as outlined in the preceding sections often are considered powerful and although the available computational power has increased enormously during the last decades, the class of systems that can be treated with such methods is limited. When the number of atoms increases above roughly 50 these methods run into severe problems and if one still attempts to use them, one most often has to make approximations that may or may not influence the results in some uncontrollable and undesirable way. For instance, only certain high-symmetry structures may be considered or the basis set may be limited. In order to overcome these problems, approximate methods constructed explicitly for certain classes of systems may be very usehl. In this section we shall explicitly study two such methods, i.e. a tight-binding density-functional method and the embedded-atom method. Within an LDA or GGA the total energy E is written as

Here, cxc is related to the exchange-correlation potential Vxc(F) through

By assuming that the Kohn-Sham equations (14) are solved exactly, E may also be written as

Within a tight-binding approximation it is assumed that E,,

contains inter-


104

actions that are of only short range and that these can be written as pair potentials,

(for more details, see, e.g., refs. 10, 11 and references therein). As one example we discuss briefly the method of Seifert et a1.I2-l6In this method, the repulsive potential Erepis written as a sum of pair potentials that each only depends on the types of and distances between the two atoms. They are given in terms of, e.g., powers of the interatomic distances up to a certain cut-off distance after which they are supposed to vanish identically. Their precise form is determined by fitting the expression of Eqs. (28) and (29) to parameter-free density-functional results on diatomic molecules. In order to calculate the so-called band-structure term, it is assumed that the orbitals of the system of interest can be written as a linear combination of atomic orbitals,

cLlci,

i

Im

4

where R represents the atom and (1, m) the angular dependence of the atomic orbital. The matrix elements

are calculated once and for all for the diatomics as a function of distance between the atoms and are used in setting up the secular equation from which the singleparticle energies ei of the system can be calculated. This method ignores long-range Coulomb potentials due to charge transfers. In a more recent development, Elstner et aZ.I7 included these whereby Mulliken populations were used as estimates for atomic charges. However, sitice the Mulliken populations define the Coulomb potential which in turn define the Hamilton operator and thereby the precise form of the orbitals and consequently the Mulliken populations, this approach requires that the equations are solved self-consistently, and it becomes accordingly computationally more involved. Such methods are most useful when studying large systems for which the computational requirements are too large for parameter-free methods. However, they should work equally well for smaller systems which then allow an assessment of their accuracy. Accordingly, we show in Table 1 the calculated hydrogenation-reaction energies for some small systems in comparison with results from parameter-free density-functional calculations and from experiments. Here, the tight-binding results were obtained with the method that includes the

105

3: Density Functional Theor),

Table 1 Hydrogenation-reaction energies (in kcal/mol) for some small organic molecules from semi-empirical density-functional calculations (DFTB) in comparison with parameterfree density-functional calculations (DFr) and experiment (Exp.). From ref 16 Reaction

CH3CH3 + H2 -+ 2CH4 CH3NH2 H2 CH, NH3 CHJOH H2 + CH, H20 NHzNHz + H2 + 2NH3 2H20 HOOH H2 CH2CH2+ 2H2 2CH4 CH2NH 2H2 -+ CH4 + NHq CH20 2H2 -+ CH4 H20 NHNH + 2H2 + 2NH3 CH2 + 3H2 4 2CH4 HCN + 3Hz -+ CH4 + NH3 CO + 3HI -+ CH4 + H2O N2 + 3H2 + 2NH3

+ + + + +

-+

+ +

-+

---$

+

DFTB

DFT

Exp.

20 23 32 30 101 71 66 65 56 124 88 83 37

18 24 28 43 80 67 67 67 89 131 102 93 71

19 26 30 48 86 57 64 59 68 105 76 63 37

Coulomb potentials as outlined above. The table demonstrates clearly that accurate results are obtained. Structural parameters are also accurate with this approach, most often independently of whether the Coulomb potentials are included or not. A special case is, however, formamide, for which the bonds have both ionic and covalent characters. Therefore, the results of Table 2 show a clear difference between the results obtained with the inclusion of the Coulomb potentials and those obtained without these potentials. This example is, however, atypical. Having established that tight-binding density-functional methods are accurate

Table 2 Optimized geometrical parameters for formamide from different methods: semi-empirical density-functional calculations with self-consistent charge (SCC-DFTB), tight-binding density-functional calculations (DFTB), parameter-free density-functional calculations (OFT), and experiment (Exp.). Bond lengths are given in and bond angles in degrees. From ref: 16 SCC-DFTB

DFTB

DFT

Exp.

1.224 1.382 0.996 1,131 125.5

1.296 1.296 1.003 1.130 127.0

1.223 1.358 1.022 1.122 124.5

1.193 1.376 1.002 1.102 123.8

~~

c=o C-N N-H C-H 0-C-N

106


we shall present some few examples of their applications to systems for which applying parameter-free methods would be difficult. In Figure 1 we show three different structures for C2F nanotubuli as obtained with such methods. Through f'unctionalization of the sidewalls of pure carbon nanotubes with fluorine atoms Seifert et aZ.'* showed that the electronic properties of these systems could be tailored. Since such functionalization has been obtained experimentally the results of Seifert et aZ. are important in the process of developing the field of carbon nanotubes into that of applied sciences. Another set of materials that have attracted much interest during the last years is that of nanoparticles (or clusters or colloidals) with between some 10 and some 1000 atoms. Quantum (finite-size) effects make the properties of these depend sensitively on the size of the particles. CdS is a popular example of these systems that moreover has the interesting property that for the macroscopic crystalline material the wurtzite and the zinc blende structures are very close to being

Figure 1 Different structures of CzF nanotubes either along (leff panels) or perpendicular (right panels) to the tube direction. Three- and fourfold coordinated carbon atoms are drawn as dark and light grey spheres, respectively, whereas the fluorine atoms are indicated by the grey sticks (Reproduced with permission fiom Appl. Phys. Lett., 77, 1313; 0 2000 American Institute of Physics)

107


energetically degenerate. In Figure 2 we show the calculated energy per CdS pair for finite Cd,S, particles with structures derived from either the zinc blende or the wurtzite structure. We see that the relative stability of the two structures is strongly size-dependent. Furthermore, by comparing with the energy gap between the occupied and unoccupied orbitals we also observe the interesting result that the stabler systems have a larger energy gap. As a final example we return to the carbon nanotubes but focus this time on dynamical properties, i.e. on how the ends of these close. Figure 3 shows the time evolution of this process and it is here seen how the caps are formed rapidly. The authors of that work2' could, furthermore, show that the addition of boron could largely prevent the closing of the nanotubes thereby assisting their growth. The tight-binding methods perform best for systems with strong covalent bonds between close neighbours. In that case both structural and electronic properties can be obtained with good accuracy. A different situation is encountered for metallic systems for which the electrons are delocalized. Then it is no longer a good approximation to assume that the electronic interactions are restricted to t

,

0

l

.

l

,

l

#

10 20 30

,

40

,

l

,

l

,

l

,

l

,

l

,

50 60 70 80 90 100

No. of CdS Pairs

0

10 20 30 40 50 60 70 80 90 100 No. of CdS Pairs

Figure 2

Relative total energy per CdS pair (dashed curve) and the energy gap between the occupied and unoccupied orbitals (solid curve) for3nite C d J , clusters as a function of n ,for clusters derived from either the wurtzite (upper panel) or the zinc blende (lower panel) cnistal structure (From ref. 19).

108


Figure3 Different stages of the closure of a carbon nanotube at 2500K affer approximately 2, 5, 8, 11, 15 and 18ps. In each case the structure is shown both from above and from the side (Reproduced with permission from J Chem. Phys., 113, 3814; 0 2000 American Institute of Physics)

near neighbours. On the other hand, the effects of placing a metal atom in a metal is first of all those of adding some extra, slowly varying electron density from the surrounding atoms inside the region of the atom of interest. One may thus assume that the effects of embedding the atom of interest largely are those of changing

3: Densihi Functional Theop

109

the energy of the atom due to the changed electron density. This is the main assumption behind the embedded-atom method. Daw and Baskes'' (see also ref. 22 and references therein) proposed accordingly to consider every atom in a metallic system as if it was an impurity in a host system consisting of the rest of the atoms. The binding energy of the whole system ( i . e . the total energy of the system minus that of the isolated atoms) is then written as

Here, F, is the energy required to embed atom i into the background electron The remaining parts of the binding energy are approximated density p, at site if. through pair potentials. The host electron density p f is approximated by a superposition of the spherically averaged electron densities of all atoms surrounding atom i,

PI

=

(33)

where p,( I i,- i,1) is the electron density of atom j at the position of the nucleus of atom i. The embedded-atom method as well as the related effective-medium method have their roots in the density-functional theory (see, e.g., ref. 23) but invoke some further approximations (that the functions F, and @, are only atom-specific but otherwise universal and that only the electron density at the sites of the nuclei need to be considered). The fwnctions F, and @,, have to be known in order to perform a calculation. They are most often determined by fitting to various experimental and/or theoretical information. Compared with the more conventional density-functional methods (including the tight-binding method of above) the embedded-atom method does not include any explicit description of the electrons and, thus, orbitals and their energies are not calculated. On the other hand, the great advantage is the efficiency which makes the study of also quite complex systems possible. As a single example of the application of this method we shall discuss the properties of Ni clusters. For simple metals, one often applies the spherical jellium model when studying the properties of finite clusters (see, e.g., refs. 24, 25 and references therein). This amounts to studying only the valence electrons whereas the core electrons and the nuclei are treated as a spherical, homogeneous background charge density whose size is determined from the number of atoms of the cluster. Such studies have been able to explain the occurrence of so-called magic numbers, i.e. clusters of certain sizes that are particularly stable. Ni can not be considered a simple metal, and it may even be suspected that the 3d electrons of Ni with their short range make the embedded-atom method also not suitable. In Figure 4 we show the calculated total energy per atom as a fimction of


110 4.0

I

I

1

1

n

2

3.7

v

E

,o m

8 3.4 n h

P

EQ,

3.1

N=2-100

-

c n t

.W

2.8

2.5

1.5

2.2

2.9

3.6

4.3

5.0

N113

Figure 4 Binding energy (the negative of the total enera) per Ni atom for NiN clusters as a function of N’13 (From ref. 26).

number of atoms N for 2 N d Particularly stable structures are found for N = 13, 19, 23, 27, 29, 43, 46, 49, 55, . . . , and in Figure 5 we show the structures of three of those. This Figure shows clearly that geometrical effects are important in determining the structures of particularly high stability. Finally, Table 3 compares our calculated structural parameters for Ni2 and Ni13 with those of experiment and of parameter-free density-functional calculation^.^^ There are some smaller discrepancies (in particular, we find that the Ni13 cluster has a

Figure 5

Optimized structures for NiI3,NiI9,and Niss clusters (From ref. 26).


111

Table 3 Calculated structural and energetical parameters for Ni clusters in comparison with experiment. EA marks results from embedded-atom calculations, DFT results from parameter-free density-functional calculations, and Exp. experimental results. l l e properties are the bond length in Niz (in the distances from the central atom to the outer atoms in Ni13(in i),and the difference in total energy per atom between Ni13and Ni2 (in eV). From ref 26

A),

Propere

EA

DFT

Exp.

NiZ bond length Ni distance

2.13 2.36 h 0.0 1.525

2.17 2.41 f 0.3 1.558

2.20

Relative energy difference

higher symmetry than what is found in the other calculations) but the overall agreement is good.

5 Order-N Methods In the preceding section we discussed approximate methods that were designed to make the calculation of the properties of complex materials possible. One set of methods, the tight-binding methods, were useful for systems with covalent bonds and localized electrons, whereas the other set, the embedded-atom or effectivemedium methods, were applicable to metallic systems with delocalized electrons. During recent years other sets of methods have been developed that are expected to be useful for complex materials but that are, in principle, exact. The basic idea behind these methods is that electronic interactions are largely short-ranged so that the major part of such interactions can be truncated after some cut-off distance. This means that the calculation of the matrix elements for these interactions will scale essentially linearly with the size of the system. Moreover, by also seeking computational methods for solving the secular equation that also scale roughly linearly with system size, one ends up with a so-called order-N method. In the present section we shall briefly outline the basic ideas behind the orderN methods as well as some related methods, i.e. the divide-and-conquer method and the elongation method. For more details the reader is referred to refs. 28, 29 and references therein. Within the conventional density-functional formalism one seeks an expression for the total electronic energy as a functional of the electron density of Eq. (16). The electron density of Eq. (16) can be considered the diagonal elements of the first-order reduced density matrix

I=

I

112


In the order-N methods this density matrix is the central quantity. It is known that

and often it is a good approximation to assume that the limit is reached already for small IF, - 61. Then one can write the density matrix in terms of a basis set of localized basis functions,

where Ki,i2are the sought expansion coefficients that vanish if the basis functions

xi, and xiz are sufficiently well separated.

The next step is to formulate the total electronic energy in terms of the density matrix. For the Coulomb and exchange-correlation parts this is easily done as they just depend on the diagonal parts of the density matrix, i.e. on the electron density of Eq. (16). Finally, the kinetic energy can be written as

The expansion coefficients K,,,, can then - in principle - be determined by minimizing the total electronic energy together with the constraint that the electron density gives the correct number of electrons,

However, density matrices are known to be idempotent, i.e.,

and it is not automatically given that the minimization procedure will lead to a density matrix satisfying this constraint. It has therefore been proposed3' to apply the purification

on any given density matrix and then replace p ( l ) by P(l) in the expression for the total energy. The purification leads to a density matrix that more closely obeys Eq. (39) but also to a more complicated expression for the total energy.


113

Finally, the minimization of the total energy with respect to the density-matrix expansion coefficients K,,12 can be done through a conjugate-gradient technique, which scales essentially as N instead of by solving a matrix eigenvalue problem (which would scale as N 3 ) . These techniques become even more powerful if they are combined with parameterized methods like those discussed in the preceding section. In that case also dynamical properties for complex materials can be studied. As examples of such applications we shall here briefly mention two studies. In the first study, Canning et ~ 1 . ~studied ' the deposition of c 2 g fullerene molecules on a semiconductor surface [i.e. the (1 11) surface of diamond]. The calculations were done by considering a film consisting of 12 layers of carbon atoms (simulating the surface and bulk regions of diamond) with periodic boundary conditions in the two directions of the surface. Each layer consisted of 256 atoms and on one side of those in total 78 Czg fullerene molecules were projected one by one. Of these, 50 molecules remained on the surface giving a final number of 4472 atoms per repeated unit. In Figure 6 we show a snapshot of the final structure of the molecular-dynamics simulation. Closest to the surface we recognize a quite compact layer of molecules and a more careful analysis of the results reveals that the c 2 g molecules are largely intact thus forming a superatom thin film on the diamond surface. As the other example we mention the work of Lewis et ~ l .who , ~studied ~ cyclic with rn = 1-4. The peptide structures of the form cyclo[(~-Ala-Glu-~-Ala-Gln),] calculated structures of those are shown in Figure 7, and by comparing the total energy per atom for the different values of m it was suggested that the system with rn = 1 has much more built-in strain than the other structures and, therefore, was considerably less stable than the other ones that, in turn, were essentially energetically degenerate. Experimentally, it has become possible to produce tubular structures of these cyclopeptides and also such ones were studied by Lewis et u E . ~ One ~ example of these is shown in Figure 8. The interior of these tubes may be used in transporting molecular systems, e.g. glucose as shown in Figure 9. The calculations by Lewis et aE.32on this system showed that at T = 0 the glucose molecule prefers to be placed inside the cyclopeptide ring structure rather than between two such ring structures and that the glucose molecule experiences some pressure from the cyclopeptide thus reducing its size whereas the structure of the cyclopeptide is largely unchanged. Giving the glucose molecule an initial velocity corresponding to T = 300 K does not lead to any transport of the molecule through the channel, but instead it quickly gets stuck at the walls of the tube. The fact that the diameter of this system is too small to allow transport for the glucose molecule is in accord with experimental observations. There are other methods based on the observation that most electronic interactions are of relatively short range and, therefore, that (parts of) the computational efforts may be made to scale essentially linearly with the size of the system above a certain smallest size. Among those we shall briefly mention two. The divide-and-conquer approach of Yang33,34is one such method. The main

114

Chemical Modelling: Applications and Theory, Yolume 2

Figure 6 Snapshot of the 50 C2, fullerene molecules deposited on the diamond (111) surface (Reproduced with permission from Phys. Rev. Lett., 78, 4442; 0 1997 American Physical Society)

idea behind it is to split the system of interest into smaller fragments that to some extent can be treated individually. Mathematically, a set of partitioning functions is introduced,

where a represents the different parts. At most points Tone of the functions pa will be dominating over all the others. For each subsystem one introduces furthermore a set of localized basis functions, with i = 1, . . . , M,. The Kohr-Sham operator of Eq. (15) can now be projected onto the space of these localized basis functions, giving

{x:},

3: Density Functional Theoy Cyclo [(D-Ala-Glu-D-Ala-Gln),]

115 Cyclo [(D-Ala-Glu-D-Ala-GIn)J

Figure 7 Geometries of the isolated ring structures cycle[('-Ala-Glu-D-Ala-Gln) ,,,I for m=I-4 (Reproduced with permission from J Phys. Chem. B, 101, 10576; 0 1997 American Chemical Society)

with

This leads to a set of single-particle Kohn-Sham equations for each subsystem, each in matrix form being

Since the Kohn- Sham operator heff contains information on the complete system, there will be 'cross-talk' between the different subsystems. However, the time-consuming part of diagonalizing a large matrix eigenvalue problem has been

Chemical Modelling: Applications and 7'heory, Volume 2

116

f a

Figure 8 A cyclopeptide nanotube constructed from c~vclo[(D-A~a-Gh-D-Ala-Gln)2] (Reproduced with permission from J Phys. Chem. B, 101, 10576; 0 1997 American Chemical Society)

reduced to that of diagonalizing a number of smaller eigenvalue problems, each having the size of the number of basis functions of the single subsystem. For large systems this can lead to a significant computer-time saving. Also the elongation method of Imamura et aZ.35--38 is based on the short range of the electronic interactions. It is a method with which polymeric systems can be studied. Most often such systems are studied by considering either finite systems whose length is increased until the property of interest is converged as a function of chain length or infinite periodic systems. In the first case one rapidly encounters the computational limits when the systems become larger thus setting - in some cases severe - limitations on the applicability of this approach. In the other case one is restricted to systems with perfect periodicity. With the

3: Densig) Functionul Theon)

Figure 9

117

Geometry of' the cyclopeptide nunotuhe cvclo[(o-Ala-Glu-D-Ala-Gln)4] with a glucose molecule (Reproduced with permission from J Phvs. Chem. B, 101, 10576; 1997 American Chemical Society)

elongation method one gradually increases the chain length by adding one unit (e.g. monomer) after the other. At any state the electronic orbitals are transformed into a set of orbitals that are localized either far apart from the region where the units are added (the so-called A-region) or close to that region ( i e . the so-called B-region). When adding an additional unit, only those orbitals that are localized to the B-region will be perturbed by the additional unit and, accordingly, only those need to be recalculated. From a certain size the length of the B-region will be constant and, thus, a larger part of the calculation will scale roughly linearly with the system size. A further advantage of the method is that the added units need not be identical, i.e. aperiodic systems can also be treated, as can systems with i m p ~ r i t i e s , ~Originally, '.~~ the method was developed for semiempirical or ab initio Hartree-Fock calculation^,^^.^^ but later it was also formulated for densityfunctional In their first density-functional studies based on this method, Aoki et considered chains of weakly interacting units, i.e. an aperiodic chain of Hz units38 as well as chains of hydrogen-bonded units.4" As an illustration of the effectiveness of their approach we discuss here briefly their results for a chain of water molecules, ( H 2 0 ) n . In Figure 10 we show the computer-time consumption as a function of n for three different studies based on the elongation method (they differ in the definition of the initial unit) in comparison with the time for a standard density-hnctional calculation where the full system is treated for any n. Figure 11 shows the interaction energy defined as

a1.38340

118


.E"

O 0 t

5

0

10

15

20

n Figure 10 CPU time used in studying a (H20), cluster as a function of n with either a standard density-jiunctional method (upper curve) or the elongation method (the three lower curves). For the elongation method three different initial structures were chosen leading to the three different curves. The data are from ref: 40

Ln t-

I

LJ-d&dJ 0

5

I0

15

20

n Figure 11

The interaction energy for a (H,O), cluster as a function of n with either a standard density-jiunctional method or the elongation method. The interaction energy is here defined as the total energy of the cluster of n - 1 units plus that of a single unit minus that of n units. The upper curve is from one of the sets of elongation-method calculations. The data are from ref 40


119

where Eto,(rn)is the total energy for a system consisting of rn units. The figures show clearly that the elongation method gives results that are as accurate as those of conventional density-functional calculations at a much lower computational cost.

6 Heterogeneous Catalysis Although studies of the properties of isolated systems - these being molecular, solid, liquid, . . . - are important, another also very important part of chemistry is concerned with the interactions, most notably chemical reactions, between different systems. From a theoretical point of view, the simplest reactions are those occurring in the gas phase between two otherwise isolated molecules that preferably should be not very large. In that case, theoretical studies can be used in identifiing reaction paths and transition structures as well as the energetics of a reaction (including answering the question whether a reaction is energetically possible). Some examples of such studies based on density-functional calculations were presented in our previous report.' From a theoretical point of view the situation is considerably more complex when the chemical reactions involve significantly different length scales. This includes heterogenous catalysis where molecules interact on the surface of some solid. The solid may be idealized as being infinite and periodic, but the periodicity is broken in the direction perpendicular to the surface at the surface where the chemical reactions are taking place. Moreover, it is almost never the case that the adsorbed molecules form some regular periodic structure on the surface so also in the directions parallel to the surface the periodicity is broken. In order to overcome these problems, simplified systems are studied theoretically. One such corresponds to approximating the semi-infinite solid through a finite cluster and then studying the interactions between this and the reactants. In this approximation a number of bonds that are present in the infinite solid have been cut and the resulting dangling bonds have therefore to be saturated through, e.g. hydrogen atoms. Nevertheless, finite-size effects as well as effects due to the saturated bonds may obscure the results of such calculations. Some of these problems may be removed by studying infinite, periodic structures. Then, the semi-infinite crystal is modelled as a film or slab of finite thickness that is periodic in the two directions parallel to the film, Periodicity may also be introduced in the third direction, perpendicular to the film, although one then has to be very careful avoiding interaction between the different films. Reacting molecules or atoms will then have to be introduced periodically, too, and this may lead to the fbrther complication that different, but equivalent, molecules may interact if the size of the repeated unit is not chosen sufficiently large. As a first example of such studies we discuss briefly the work of Alavi et aZ.4' on the CO 0 + C 0 2 reaction on the Pt( 11 1) surface. Figure 12 shows the initial structure as well as the size of the repeated unit in the plane of the surface. The semi-infinite crystal itself was modelled through a film containing three

+

120

Figure 12


+

The initial geometry for the CO 0 --+ CO, reaction on the Pt(ll1) surface looked from above. The unit shown in thejigure is the periodically repeated unit (Reproduced with permission from Phvs. Rev. Lett., 80, 3650; 0 1998 American Physical Society)

layers of Pt atoms, and in the calculations only the topmost layer of Pt atoms was allowed to relax. The initial configuration was chosen according to the experimental observations about how CO molecules and 0 atoms are adsorbed on the Pt( 1 1 1) surface. In Figure 13 we show a series of snapshots of how the CO molecule approaches the 0 atom on the surface ultimately leading to the formation of the CO, molecule, and in Figure 14 the corresponding changes in

Figure 13 Snapshots of the reaction pathwa-v from the initial state (a) to thejinal state C 0 2 reaction on the Pt(ll1) surface. The (h) for the reaction CO + 0 large spheres represent Pt atoms, whereas the smaller darker ones are 0 and the smaller lighter ones are C atoms (Reproduced with permission from Phys. Rev. Lett., 80, 3650; 0 1998 American Physical Society) ---f

121


I 0.5 0.0 1.O

Figure 14

1.5 2.0 2.5 3.0 3.5 C-qa) separationlAA

4.0

Variation in the total energy.for the reaction CO + 0 CO, reaction on the Pt(ll1) surface. The squares show results from LDA calculations and the circles those from GGA calculations (Reproduced with permission from Phvs. Rev. Lett., 80, 3650; Q 1998 American Physical Society) --f

the total energy are shown both as calculated using a local-density approximation and using a generalized-gradient approximation. Notice that in this figure the initial structure in Figures 12 and 13(a) corresponds to the large C - 0 separation. In the study of the CO 0 -+ C 0 2 reaction on the Pt( 11 1) surface the reaction path was identified, and it was found that one CO molecules moves in the direction of an 0 atom on an essentially flat surface. In reality the situation may be somewhat more complicated, including that the reactants move more or less randomly on the surface before reacting, that the surface may contain steps or other types of defects, that also other types of molecules may be present on the surface, and that also the surface itself may not be inert, Examples of theoretical studies where these issues have been addressed shall now be briefly mentioned. Persson et aZ.42studied theoretically the Eley-Rideal reaction in which an incident species reacts directly with an adsorbate to form a product molecule that promptly leaves the surface. As an example of this reaction they studied the H + H + H2 reaction on the Cu( 11 1) surface. The calculational approach was very similar to that discussed above for the CO 0 + C 0 2 reaction on Pt( 111). They studied different configurations of two H atoms on the Cu( 1 11) surface, shown in Figure 15, and calculated the total energy as a function of the structural parameters also shown in the figure. Moreover, they studied also an isolated H2 molecule. In Figure 16 we show the calculated total energy as a function of interatomic distance for the isolated H2 molecule. As a side remark we mention the difference between the results for the spin-unpolarized and the spin-polarized calculations that show up for the larger interatomic distances when the two atoms are essentially non-interacting. By comparing the results of Figure 16 with those of Figure 17 for the H2 molecule on the Cu(1 11) surface it is very clear that the surface has strong effects on the properties of the H2 system, and also that there is a strong dependence on where and how the molecule is placed on the surface.

+

+

122


w

c8

Figure 15 The four different configurations that were considered in the study of the H -+ H -+ H2 reaction on the Cu(ll1) surface viewed both from the side and from above (Reproduced with permission from J Chem. Phys., 110, 2240; 0 1999 American Institute of Physics)

Subsequently, Persson et aZ.42used the results like those of Figures 16 and 17 in deriving a simpler model potential containing various parameters whose values were determined by fitting to the calculated total energies. This potential could then be used in model studies of the dynamics of H atoms on the Cu( 111) surface, which, however, was not done by the authors. Hammer44 studied whether the chemical reactivity of a surface depends on the atomic structure of the surface or only on its chemical composition. Specifically, he studied the dissociation of NO on the Ru(0001) surface. He considered the four different structures shown in Figure 18, i.e. a flat surface and three surfaces where steps that are one atom high have been introduced. The calculations showed, indeed, that the dissociation barrier was strongly reduced on a corrugated surface compared with the one for a flat surface, i.e. he observed reductions from 1.28 eV on the flat surface to values as low as 0.15 and 0.17 eV on some of the corrugated surfaces. Analysing the results, Hammer found that this reduction could be explained through differences in the bonding behaviour of the dissociation products, i.e. the chemisorbed N and 0 atoms. When these can easily bond to the atoms of the substrate, the energy barrier is lowest. And this is obviously the case when the NO molecule is placed next to a step on a corrugated surface. Also the presence of other molecules on the surface may influence the chemical reactions on the surface. In another study, Hammer4’ studied how the presence of N, 0, or H on the Ru(0001) surface would influence the dissociation of N2 on the


123

Figure 16 Calculated potential energies jbr an isolated H2 molecule as a function of interatomic distance. The closed and open circles are results from spinunpolarized and spin-polarized GGA calculations, respectively, the solid curve is aJit to a Morse potential, and the dashed curve shows the highly accurate results of Kolos and Wolniewiczfrom re$ 43 (Reproduced with permission from J Chem. Phys., 110, 2240; 0 1999 American Institute of Physics)

surface. As in the other study, the surface was modelled through a film of finite thickness and periodicity was assumed in all three directions. The film had a thickness of five layers and the repeated unit in the directions parallel to the film contained 4 X 4 Ru atoms in the topmost layer. For each of those 4 X 4 Ru atoms one N2 molecule was deposited on the surface and its dissociation was studied. Moreover, also 4, 8 or 12 N, 0 or H atoms per unit were also placed on the surface and it was explored whether their presence would modifjr the reaction energy for the N2 N N dissociation. These numbers of X = N, 0 or H correspond to coverages of Ox = 0, 25, 50 and 75%, respectively. Figure 19 shows the calculated reaction energies and energy barriers as functions of the coverage Ox. It is seen that both quantities depend strongly both on the coverage and on the type of the additional adsorbants. As discussed by Hammer, parts of the dependence can be understood as being due to differences in the numbers of nearest neighbours of the N atoms, but in order to understand the differences between X = H, 0, and N one needs to include effects from the electronic structures of the different adsorbates. These studies have considered isolated surfaces interacting with single molecules or atoms. Although also the presence of other species on the surface

-

+


124 1.o 0.5

0.0

5 Y

h

N

-0.5

-1 .o

W

>*

-1.5

-2.0 -2.5

4.0

tt..

0.0

.

.

.I., 0.5

4

I

*

I

1.0

I

I

.

4

1 - 1

13

.

L

.L. 1

2.5

...d 3.0

3.5

Figure 17 As Figure 15 but for the H2 molecule on various sites of the Cu(1ll) surface (Reproduced with permission from J Chem. Phys., 110, 2240; 0 1999 American Institute of Physics)

Figure 18 Top view of the four different periodic structures used in modelling the NO dissociation on Jrat and corrugated Ru(0001) surfaces. Ru atoms are represented by large circles and small circles represent chemisorption sites. Lighter and darker Ru circles represent uppermost and second-uppermost layers of Ru atoms, respectively (Reproduced with permission from Phys. Rev. Lett., 83, 3681; 0 1999 American Physical Society)

125

3: Density Functional Theonj

-

Figure 19 Calculated reaction energy (rep) and energy barrier (right) f o r the N2 N N dissociation on a Ru(0001) surface precovered with H (lower curves), 0 (middle curves), or N (upper curves) as a function of the coverage (Reproduced from ref. 45)

+

was included, no attempt was made to account for the fact that the surfaces themselves may change due to the environment. An attempt to study such effects was recently presented by Wang et al.46 who studied the stability of the a-A1,0,(000 1 ) surface (corundum) in an atmosphere containing both oxygen and hydrogen. In the (0001) direction, corundum can be considered as formed by layers of oxygen atoms with A1 atoms in between (Figure 20). Whereas the 0 layers are planar, those of A1 are not and in the stacking direction the A1 atoms take two different positions. This leads to a stacking sequence like (-Al-Al-03-)x, or in

Figure 20 Side and top view of the Al-O,-Al-R surface of corundum in which the sites 1, 2, and 3 indicate three sites which could be occupied by the topmost A1 atoms. Large spheres represent oxygen atoms and small spheres aluminium atoms. Also the size of the repeated unit used in the calculations is shown (Reproduced with permission from Phys. Rev. Lett., 84, 3650; 2000 American Physical Society)

io

126


the notation of Wang et al. R-A1-AlLO-R, where R denotes the continuing sequence in the bulk. At the (0001) surface the stacking can be terminated in different ways, i.e. as Al-0,-A1-R (this is the termination shown in Figure 20 and this is the termination found in theoretical studies of the isolated surface see ref. 46 and references therein), as A1-A1-O3-R or as 0,-A1-Al-R (here, the leftmost atoms are those closest to the surface). For the last surface, one may actually also consider oxygen-deficient ones of the type 0,-Al-A1-R or OI-Al-Al-R, and in the presence of hydrogen also hydrogen atoms may be deposited on the surface. The system was studied by considering a periodic structure with a finite number of 0 and A1 layers (six, and between ten and fourteen, respectively) separated by a vacuum region of 10 In the directions parallel to the surface, a periodic structure as that of Figure 20 was used. The relative stability of the different systems was studied through Gibbs free surface energy

A.

where AGGvib is the vibrational contribution Q. Moreover, the summation runs over all types of atoms, N j is the number of the ith type and p i ( T , P ) the chemical potential at temperature T and pressure P. For the latter, the authors used

where pp is the chemical potential at 0 K and where A p i ( T , P ) was taken from thermochemical tables. For bulk aluminium and corundum the chemical potentials were taken from the density-functional calculations, whereas for H2, 02,and H 2 0 experimental values were used. The chemical potential of oxygen can be varied by varying the pressure of oxygen. Therefore, the results will depend on the chemical potential of oxygen. This, in turn, can be varied between the value for the maximum concentration of O2 (which corresponds to O2 condensing on the surface and which is the zero in Figure 21) and a minimum value below which A1 will condensate on the surface. Figure 21 shows the results. Without hydrogen the A1-O3-A1-R structure is clearly the most stable one, independent of oxygen concentration. This result is in agreement with earlier theoretical predictions for the free surface when effects of the environment were not taken into account. Including hydrogen, however, changes the situation so that the oxygen-terminated surface with additional hydrogen (i.e. the H3-O3-A1-R structure of Figure 21) is the stabler one almost independent of the oxygen concentration. This result is accordingly at variance with those of the other theoretical studies on the fiee, isolated corundum surface but, as discussed by Wang et al.,46in agreement with experimental observations. In total, therefore, this study shows how important it can be to include all effects in theoretical studies of materials properties.

127

3: Density Functional Theoy

500

400 N

5

2E

300

W

)r

p

200

a>

100

2a, 0

I , v)

-100

-200 -7

-6

-5

-4

-3

-2

-1

0

1

Oxygen chemical potential (eV) Figure 21 Surface energies of different terminations of the (0001) surface of corundum as ,functions of the chemical potential of oxygen. The shaded regions indicate the range where hydrogen on the surface is in equilibrium with H2 and H 2 0 (Reproduced with permission from Phys. Rev. Lett., 84, 3650; 0 2000 American Physical Society)

7 Descriptions of Chemical Reactions

The results of the preceding section show that it is a very far from trivial endeavour to obtain detailed information on how molecular species interact with each other, and the results may depend sensitively on the structural and compositional aspects of all involved parts. In that section this was made clear for the case of reactions on surfaces of crystalline materials. It should therefore not be surprising that the description and study of chemical reactions between isolated molecules in the gas phase also may become complicated as soon as the molecules contain just a little more than very few atoms (for a few examples, see, e.g., our previous report'). Nevertheless, it is often believed that a major part of the way two molecules interact is dictated by properties of the individual molecules. There exist therefore different attempts to identify the important quantities of the individual molecules that are responsible for their reaction properties as well as to develop computational methods with which they can be calculated. Hardness and softness are two such quantities that originally were introduced

128


by Pearson4' (see also ref. 48) and later were put on a formalistically more correct footing by Parr et ~ 1 who . based ~ ~ the discussion on density-functional theory. The quantities were briefly discussed in our previous report but shall here for the sake of completeness be introduced again. Subsequently, some more recent developments will be discussed. The chemical potential for the electrons is defined as

3E,

dE

p=3N=3N7 i.e. it tells how the total energy changes when the number of electrons is changed. Such a change can occur when two molecules start interacting and ultimate take part in a chemical reaction. When the two molecules start interacting, electrons will flow from one to the other until the chemical potentials for the electrons of the two molecules are identical. Accordingly, we need to study the changes in the chemical potential which can be calculated from

where the first term comes from the change in the number of electrons and the second from the change in the potential. Except for a factor of 2, the first term on the right-hand side is essentially the chemical hardness

and the second term is related to the Fukui function,50

The second identity shows that the Fukui function tells how the electron density is modified when changing the total number of electrons. Neglecting relaxation effects, the Fukui function is as a first approximation the density of the LUMO or of the HOMO, depending on whether electrons are added or removed. Correspondingly, one has to distinguish between two different Fukui functions,

3: Density Functional Theon;

129

Changing the number of electrons N means changing the electron density locally, and one may accordingly define a local hardness through

d2E

v(T) = -

p( J ')dT ' .

(53)

The local softness is defined through

and is the reverse of the local hardness in the sense that

J ?)r( S(

?)d? = 1.

(55)

Also for the Fukui function one may use the fact that changing N means changing p ( T ) locally and, accordingly, define a local function,

The hardness, softness and Fukui function quantify the responses of a given system to a change, i.e. to changes in the number of electrons, in the electron density, or in the external potential. They are, moreover, the linear responses. Another quantity that also describes the linear response to some perturbation is the static polarizability a that is the linear change in the total energy due to an applied electrostatic field, E=

E(O)

+

-

aVp, E,,,,+

.

-

(57)

Here, Po)is the total energy of the system without the external field, @ is the dipole moment of the system and is the DC field vector. The static polarizability is an experimental observable and it would, therefore, be interesting if this quantity would yield information not only on the system's response to external electrostatic fields but also on the system's behaviour in a chemical reaction. Accordingly, there exist some theoretical studies devoted to identiQing relations first of all between the softness and the static polarizability (see, e.g., refs. 5 1-54 and references therein). One of these studies is due to Chattaraj et aLS4who studied an electrocyclic cis-butadiene reaction. In this reaction, one C-C reaction, i.e. the cyclobutene bond is broken and the groups bonded to the breaking bond are rotated. One may now consider two possibilities, i.e. either these groups rotate in the same direction (both clockwise or both counterclockwise) or they rotate in opposite directions

zDc

---+


130

during the process. The first process is called conrotatory and the second disrotatory. In their study, Chattaraj et al. explored whether there was a correlation between the hardness, the polarizability and the total-energy variation for the mode chosen by the reaction cyclobutene --+ cis-butadiene. They used both Hartree-Fock-based methods and density-functional methods. In Table 4 the results from the latter are collected. Here, the average polarization ( a ) is defined as

The results confirm the different predictions based on total-energy, hardness, and polarizability arguments, i.e. when two (or more) different reaction pathways between reactants and products exist, the one of lower energy, higher hardness and lower polarizability will be preferred. Although the total energy, the hardness and the polarizability may be useful concepts in analysing chemical reactions, it is desirable to explore whether other quantities would yield other information that could be of use. Some of those quantities were studied by Chattaraj et al.55in another work. In order to identify chemical bonds, the electron localization function (ELF)56

[ + (32] -1

L(J)= 1

(59)

with

3 Dh(F) = -[3Jt’p(7)]”’ 10

has turned out to be an interesting quantity. Although it does not directly give

Table 4 Total energy (in a.u.), point-group symmetry, mean value of the polarizability of Eq. (58) (in a.u.), and hardness (in eV) for the cyclobutene cis-butadiene reaction via either the conrotatory or the disrotatory transition state. The results are from ref 54 --$

~~

Point group

Energy

(4

11

Reactant

c 2 v

- 155.9940

Product Transition state Transition state

C2V

-156.0120 - 155.9340 -155.8650

47.70 54.70 56.80 71.20

3.36 2.48 2.35 0.91

Species

Cyclobutene cis-Butadiene Conrotatory Disrotatory

C2 C,

3: Dens@ Functional Theory

131

information on chemical reactions, these latter involve the formation and breaking of chemical bonds and, therefore, the ELF may also be relevant for chemical reactions. Also kinetic-energy densities have been proposed as descriptors to be used for chemical reactions. The definition of a local kinetic-energy density is, however, not unique (see, e.g., ref. 9). Some proposals, considered by Chattaraj et are

t2(

1

7 ) = t , (F) - 2 V 2 p (F)

f3(F) = I \ ( ? )

-

1 -V2p(F). 8

For any of those, t( F), one may define a local temperature through

2 t(F) O(F) = --

3

kB

and a local entropy density through

These quantities may be compared with the Fukui function which - according to the discussion above - is a descriptor of chemical reactivity. Chattaraj et al. studied the different quantities for the water molecule and some lighter atoms. They found that the local temperature and the local entropy are closely related to the ELF and as such most likely only of secondary importance as descriptors for chemical reactions although in some cases also similarities with the Fukui function were observed. Above, we mentioned that - when neglecting relaxation effects - the Fukui function simply is the electron density of either the HOMO or the LUMO. When are written as a sum over atom-centered the Koh-Sham wavehnctions I#,(?) basis functions.

132


(2 labels the atoms, k any other parameter of the basis functions), the same procedure as used in the definition of Mulliken gross populations,

can be applied for the HOMO andor LUMO and thereby so-called regional Fukui functions can be defined.57 Fuentealba et ~ 1studied . ~ whether ~ these functions (actually, they are numbers and not functions) were good descriptors of chemical reactivities. First they considered some smaller test systems for which they compared the regional Fukui functions with atomic populations that also have been used in analysing chemical reactivity. Among those systems was formaldehyde, OCH2, and Table 5 contains a

Table 5 Regional Fukui functions for nucleophilic (+) and electrophilic (-) attacks for formaldehyde. The results are from ref 58 and are based on density-functional calculations Atom

Phase

ft

fa

C 0 H C 0 H

gas gas gas aq. aq. aq.

0.635 0.358 0.003 0.647

0.03 1 0.724 0.124 0.039 0.7 13 0.124

0.346

0.003

3: Densip Functional Theon

133

part of the results for this molecule. It is known that formaldehyde has two active sites, one for nucleophilic attacks (the carbon atom) and one for electrophilic attacks (the oxygen atom). Thus, the regional Fukui function f ' (with representing the LUMO) should be particularly large for the carbon atom, and the regional Fukui function f - (with - representing the HOMO) should be particularly large for the oxygen atom. That it in fact is so, is verified by Table 5. Furthermore, it is also seen that embedding the molecule into a solvent (treated as an infinite, continuous dielectricum with a dielectric constant as for water) hardly changes the chemical reactivities. Subsequently, Fuentealba et a1.58 studied the regional Fukui hnctions for different sites of aniline and some of its substituents (see Figure 22). Aniline may be protonated at the nitrogen atom of the amino group and also at the aromatic ring para to the NH2 substituent. The presence of other substituents may change

+

ANILINE

Figure 22 Aniline and some of its substituents (Reproduced with permission from J Chem. Phys., 113, 2544; American Institute of Physics)

0 2000


134

the protonation sites, and the authors suggested that the regional Fukui functions could be useful in identifying these sites. Table 6 contains some of their results for the molecules of Figure 22, which also gives the numbering of the atoms used in the table. The larger value of f for the nitrogen sites suggest that this is the site of protonation but the fact that for m-OMeC6H4NH2 this function has a comparable value for the C5 atom suggests that this site is also of importance for the protonation - in agreement with experimental findings. Another approach for studying chemical reactions theoretically - also based on the Fukui function - was presented by Clark et al.59 They suggested that a chemical reaction was accompanied by some charge transfer and that this meant that a chemical reaction was facilitated when the Fukui function f +(F) of the electron-receiving system as well as the Fukui function f -(F) of the electrondonating system simultaneously were large, or, otherwise stated, that the Fukui function overlap integral

Imp=

[/.;(F) f L(?)dF'

Table 6 Regional Fukui functions for nucleophilic (+) and electrophilic (-) attacks for different sites of the different molecules of Figure 22 where also the numbering of the atoms is given. The results are from re$ 58 and are based on density-functional calculations Molecule

Atom

fk

fa

C6H5NH2

N c2 N C6 N C6 N C6 N

0.002 0.006 0.012 0.06 1 0.002 0.006 0.002 0.006 0.0 13 0.129 0.065 0.002 0.007 0.001 0.00 1 0.005 0.005 0.002 0.006 0.000

0.343 0.22 1 0.330 0.240 0.339 0.230 0.332 0.232 0.289 0.202 0.26 1 0.315 0.177 0.306 0.096 0.317 0.21 1 0.250 0.147 0.141

m-FC6H4NH2 m-ClC6H4NH2 rn-MeC6H4NH2 m-OMeC6H4NH2

c1

c5 N C6 N Cl N C6 N C6 0

3: Density Functional Theoiy

135

was large. Here, A and B denote the electron-donating and electron-receiving system, respectively. This integral depends on the relative position and orientation of the two systems and for configurations where it is large there will be a large tendency to transfer electrons from the nucleophile to the electrophile. Accordingly, they suggested that such configurations would be relevant when attempting to identify transition-state structures. The approach was tested on different electrophilic aromatic substitution reactions and was found to be promising.

8 Quantum Treatment of Other Particles Quantum effects are most important for the lightest particles, and, correspondingly, they are often included solely for the electrons when studying molecular systems, whereas the nuclei are considered so heavy that a classical treatment is sufficiently accurate. This is the physical basis for the Born-Oppenheimer approximation discussed in Section 2. There are, however, cases where a quantum treatment of the nuclei can become important. Such cases occur first of all for the lightest nuclei, i.e. most notably for hydrogen atoms, but also for systems where small energy barriers between different isomers can lead to tunneling effects. In the preceding sections we have discussed how quantum effects of the electrons are treated first of all within density-functional theory, but we emphasize that, from a conceptual point of view, treatments based on the Hartree-Fock approximation are fairly similar. There exist some few methods where the quantum treatment of the electrons is extended by a quantum treatment of (some of) the nuclei (see, e.g., refs. 60-62) but whereas the electrons still are treated within the 'standard' electronic-structure approaches, the path-integral method of Feynman is used for the nuclei. The basic ideas behind these will be briefly outlined here followed by some few examples of their implications. For a quantum many-body system with Hamilton operator H the statistical mechanics can be formulated in terms of the reduced density matrix

where

is the inverse temperature. The trace of y " ) is the partitioning function 2. p(')(&, &, p) can also be represented in terms of path integrals,

where k(r) represents the configuration of the system as a function of an imaginary time z. The paths are restricted to those beginning at 2, and ending at

136


i2.4D[R(r)]is the differential element for all paths. Finally, the Euclidean action S [ R ( z ) ]for a certain path is defined as

where the first term of the integrand is the kinetic energy and the second term is the potential energy. Moreover, the dot represents derivative with respect to time. In a practical calculation the time and path integrations are discretized (for details, see refs. 60-62) leading to the following expression for the partition function for M nuclei

2=

Here, E o ( i ) is the electronic ground-state energy for the electrons when the nuclei are at the positions i = (il, i2, . . . , ZM)(this term can, e.g., be calculated using a density-functional method), P is the number of time steps, u’p = $, M I is the mass of the Zth nucleus and the index s represents the time step. Once the reduced density matrix (or the partitioning function) has been calculated, observables can be calculated from Eqn. (72) (p. 139).

Figure 23 Superimposed configurations for the CzH,’ molecule. A and B show results for T = 5 K, C results for T = 3000 K. In A and C the nuclei are treated classically, in B quantum-mechanically (Reproduced with permission from Science, 271, 179; 0 1996 American Association for the Advancement of Science)

137


CLASSICAL (100K)

Figure 24

QUANTUM (100K)

Distribution function of the relative proton positions between the two neighbouring oxygen atoms for ice for a classical treatment (left panels) and a quantum-mechanical treatment (right panels) of the nuclei. The difSerent panels correspond to diflerent pressures (or molar volumes) (Reproduced with permission from Classical and Quantum Dynamics in Condensed Phases, eds. B.J. Berne, G. Ciccotti and D.F. Coker, p. 359; 0 I998 World Scientific Publishing)


138

CLASSICAL

QUANTUM

I

2 v44s c m ’ h

2

1

.

1

.

1

t

.

V15.41 aa’h

V

..--1.0

-0.5

0.0

0.5

1.0

6 (A) Figure 25 Relative free energy along the one-dimensional proton transfer path 6 shown in Figure 24 (Reproduced with permission from Classical and Quantum Dynamics in Condensed Phases, eds. B.J. Berne, G. Ciccotti and D.F. Coker, p. 359; 0 1998 World Scientific Publishing)

139

3: Density Functional Theoiy

(72)

under the assumption that 6 is diagonal in the coordinate representation. C,H,+ is a so-called floppy molecule. It is characterized by an essentially linear H-C-C-H backbone with the third (bridging) proton forming a nearly equilateral triangle with the two carbon atoms. Since this last proton may easily move around the C-C bond, quantum effects of this may be important. Marx and Parrinel10~~ therefore studied this system using a method as just described. Figure 23 shows pictorially their main conclusions. The classical distribution at 5 K is essentially the planar structure with one proton bridging between the two C nuclei. At 3000 K some thermal broadening is seen. For the quantum treatment of the nuclei at 5 K a structure similar to that of the classical treatment is found, but the protons can tunnel between different positions. 2.0

1

.

f C

e

3

1.0

Y

n

B

0.0

I 5

e;

1.0

Y

0.0 4.0

5.0

6.0

7.0

Figure 26 Pair correlation function g ( r ) for Li,. The upper panel shows the classical results for temperatures 50 K (solid line) 100 K (dashed line) and 200 K (dotted-dashed line), whereas the lower panel shows the quantum-mechanical (dashed line) and the classical (solid line) results for T = 50 K (Reproduced with permission from J. Chem. Phvs., 108, 8848; 1998 American Institute of Physics)

(c

140


In a further study these authors studied the motion of protons along the hydrogen bonds between different water molecules in ice@ as well as how extra protons bond to the water molecules.6s They found6’ that the extra proton in HS02+behaved in an essentially classical way, whereas for H302- quantum effects were important. For the proton motion along the hydrogen bonds between different water molecules we show some representative results in Figures 24 and 25. Figure 24 shows the distribution function of the relative proton position between the two oxygen atoms for as functions of the distance between the two 0 atoms (ROaob) and the displacement away from the midpoint position between the two 0 atoms (6). The different panels correspond to different pressures (i. e. different molar volumes) and the left-hand part shows the results when the protons are treated classically, whereas they are treated quantum-mechanically in the right-hand part. The much more delocalized behaviour of the quantum particles, which is due to tunneling effects, is clearly recognized when comparing the two parts. This tunneling is also seen in the results for the free energy shown in Figure 25. In fact, this Figure shows even more clearly the significant differences between the classical and quantum-mechanical treatment of the protons. A related study was reported by Mei et a1.66who studied proton transport along a chain of water molecules. Also they found clear signals of quantum effects of the protons. Instead of dwelling further on this study, we shall turn to another one where not protons but Li atoms were treated quantum-mechanically, i.e. a slightly heavier atom.62 Also here quantum effects were found to be important, in particular at lower temperatures (below, say, 50 K). This is examplified in Figure 26, which shows the pair correlation function g ( r ) for Li4. The differences between the classical and the quantum-mechanical results should be obvious.

9 Problems with l/lcPotentials Let us consider the Hartree-Fock approximation as discussed in the introduction. In the Hartree-Fock equations (8) the Fock operator (9) operating on the wavefunction @ k leads to one term being

This term is the so-called self-interaction which within the Hartree-Fock approximation vanishes (as it should). However, with current approximate density functionals the exchange interactions are approximated and the term of Eq. (73) does not vanish. This leads to some artifacts of which we here shall focus on one, i.e. the long-range behaviour of the exchange potential. In order to understand it we consider a finite system like a molecule or an atom. Using that in an exact theory the term of Eq. (73) is identically 0 and using Eq. (1 1) we have

3: DensiQ Functional Theor?/

141

x' and are combined position and spin coordinates but for our purpose only the position-space dependence is important. When Q k is bound to the system of interest, we have

1

- @ L ( x ' )for I'

Y --+ 3c.

(75)

-

These arguments show that the exchange part of the self-interaction potential co but do not consider the remaining parts of the approaches 1 / r for r exchange potential. It may, however, be shown" 7 1 that when including these also, the above result holds. On the other hand, with the current approximate density functionals the exchange potential is approximated through some function of the electron density and its gradients in the point of interest. This means that the potential far away fi-om the system of interest will depend essentially on the density of the most extended orbital, i.e. of the highest occupied molecular orbital (HOMO). Far away from the system this orbital decays exponentially with a decay constant determined by its energy. Thus, for many approximate density functionals the long-range behaviour of the exchange potential is rather exponential instead of I / Y , which can be a severe problem when studying charge transfers, excitations, or ionizations. One may add any constant to the potential since this will just lead to an overall up- or downwards shift of all orbital energies. It has therefore been suggested72 (see also refs. 73-75) that the exchange-correlation potential should obey

where k is a system-dependent positive constant. For some systems, k is expected to be related to the first ionization potential I and the electron affinity A through the hardness ( I -- A ) / 2 (this hardness is a finite-difference approximation to the integral of the local hardness q defined in Section 8) or to the I and the energy of the HOMO (see, e.g., refs. 72, 73 and 7 5 ) . Tozer and Handy76 have recently proposed some new approximate exchangecorrelation functionals that contained a number of parameters, including different k for different systems (and even allowing for different values for k for a and p spin). The values of these parameters were determined by fitting to the exchangecorrelation potential obtained by using the procedure of Zhao et al.77 (the theory behind this potential will not be discussed further here) as well as to various total exchange-correlation energies for a given set of atomic and molecular systems. For the present purpose the most important result is that they also obtained values for k for these systems. The values of k , given in Table 7, show that the constant shifts are considerable

142


Table 7 The calculated values of the parameters k for a and p spin for d@erent atomic and molecular systems compared with the hardness ( I - A)/2. All quantities are in eVand are from ref 76

C N NH2 CH Li CH2 0

F 0 2

H BH CH4

co

F2 H2 H20 HF LiH N2 Ne

4.6 5.7 4.6 4.4 3.3 4.6 6.3 6.5 5.2 6.5 4.4 4.9 4.4 5.2 5.7 5.4 6.3 4.6 4.4 7.6

5.2 6.0 4.4 4.4 14.7 4.9 6.3 6.5 4.9 4.4 4.9 4.4 5.2 5.7 5.4 6.3 4.6 4.4 7.6

4.9 7.3 5.2 4.6 2.4 4.9 6.0

7.1 5.7 6.5 10.1 7.9 6.3 8.7 9.5 11.2

9.0

and amount to several e\! This is actually the typical error when comparing the density-functional results for the energy of the highest occupied orbital with the first ionization potential although these should, when applying an exact density functional, be identical (see, e.g., ref. 69). Moreover, the shifts are strongly system dependent which is somewhat unfortunate since this makes it less obvious how to determine its value for any general system. Finally, the Table shows that the correlation between the values of k and those of the hardness is not very strong.

10 Exact-exchange Methods In Section 2 we discussed the theorem of Hohenberg and Kohn that states that once the electron density of the ground state is known any other ground-state property, including the total electronic energy E,, can be calculated ‘somehow’. Schematically, this procedure was depicted in Eq. (17). The approach of Kohn and Sham introduced an intermediate step, i.e. from the electron density one calculates first some effective potential and from that one obtains E,, cJ Eq. (18). The Kohn-Sham approach leads actually also to a set of Kohn-Sham orbitals

3: Density Functional Theor?,

143

vi(

{ F ) } that, consequently, also are some hnctionals of the electron density. Therefore, one can also apply these in the calculation of E, and still have an expression where E, is some functional of the electron density p(F). Thus, the schemes of Eqs. (17) and (18) may be generalized to

Within the Kohn-Sham approach, E , is given as

where h l is the operator for the kinetic energy and for the electrostatic potential from the nuclei [see Eq. (lo)], Vc is the Coulomb potential from the electrons [see Eq. (21)] and cxc is the exchange-correlation energy density per particle that is related to the exchange-correlation potential through the functional derivative

Here, Ex, is the total exchange-correlation energy that may be split into an exchange and a correlation part,

Ex, = Ex

+ E, =

s

s

c.r(T)p(T)dF+ cc(F)p(F)dF.

(80)

In the preceding parts of this report we have discussed studies based on approximations to both t, and cc. In many of these approximations, each part by itself (i. e. exchange and correlation separately) are not optimally approximated but their sum benefits from a favourable error cancellation so that Ex, is more accurate than Ex and E,, separately. This is, e.g., the case for (most of) the LDAs and GGAs. One may, however, replace the approximate treatment of the exchange energy through the exact expression,

The only difference from the Hartree-Fock expression, Eq. (12), is that in Eq. (81) the Kohr-Sham orbitals { v l }and not the Hartree-Fock orbitals enter. Within the exact-exchange (EXX) methods the exchange energy is calculated through Eq. (81) and, accordingly, only the correlation energy is treated approximately. Naively, one might think that this leads to the Hartree-Fock equations, Eq. (8), with the only differences being that the single-particle operator i1 of Eq. (10) contains an extra term from the correlation potential and that an

144

Chemical Modelling: Applications and Theoq: Volume 2

equivalent extra term shows up in the expression for the total electronic energy, E,, of Eq. (12). However, the Hartree-Fock equations are obtained by minimizing E, as a functional of the Hartree-Fock orbitals { G k } whereas within the densityfunctional formalism we minimize E , as a functional of the electron density p, i.e. we need to determine the exchange potential resulting from Ex of Eq. (81),

It has been shown7*-*’that V, satisfies the following equation

In this equation, a denotes occupied orbitals and s unoccupied orbitals and

t,

and

cs their single-particle energies. Moreover,

and

In a practical calculation, the Kohr-Sham orbitals { V j } are expanded in some basis set and also V, can be expanded in a set of basis functions. Then, Eq. (83) becomes a matrix equation which can be solved leading to the exchange potential Vx in terms of its basis functions. The EXX approach has several advantages over the standard density-functional approaches we have studied so far: The self-interaction, discussed in the previous section, is absent and the problems related to the asymptotic 1/r behaviour of the exchange potential, i.e. the subject of Section 9, are removed. As we shall see below, this leads first of all to a significantly improved agreement between ionization potentials and the orbital energies. On the other hand, the calculations are significantly more complex, and the field is still in its infancy so that only few systems have been treated and the corresponding computer codes have not reached a mature state of optimal efficiency. Moreover, since now only correlation effects are treated approximatively, the above-mentioned cancellation of errors when approximating both exchange and correlation effects can no longer be exploited. Therefore, the future perspectives of this approach may depend crucially on new approximate correlation functionals. Here, we shall briefly discuss three recent studies using exact-exchange methods. In all of them, smaller atomic and molecular systems were examined


145

and the results compared with those obtained with other, Hartree-Fock- or density-functional-based, methods. In one of these, Ivanov et al.** studied the orbital energies for H 2 0 , NZ,and CO for the orbitals closest to the Fermi level. The results were compared with those of Hartree-Fock and of local-spin-density calculations and are reproduced in Table 8. This table shows clearly how the correct description of the long-range 1/r potential and the removal of the self-interaction leads to a much better description of the energies of the occupied orbitals. In fact, these become fairly close to those of the Hartree-Fock calculations which may not surprise. The Table shows also that the energies of the unoccupied orbitals are lower than those of both Hartree-Fock and of the local-spin-density calculations. The energy gap between the occupied and the unoccupied orbitals is largest in the Hartree-Fock calculations and smallest in the local-spin-density calculations, which actually represents well-known deficiencies of such approaches: this energy gap is significantly overestimated within the Hartree-Fock approximation and underestimated within the local-density or generalized-gradient approximation. Thus, it may be hoped that the results of the exact-exchange calculations imply that such calculations yield good approximations to this energy gap. GOrlingg3studied atomization energies and ionization potentials for a number of smaller molecules and used thereby both an LDA and a GGA for the

Table 8 Energies (in ev) of the highest (HOMO), second-highest (HOMO-I), and third-highest (HOMO-2) occupied orbital as well as of the lowest (LUMO)and second-lowest (LUMO+I)unoccupied orbital for H20, N,, and CO from exact-exchange ( E Z ) , Hartree-Fock (HF), and local-spindensity (LSDA)calculations in comparison with the experimental first ionization potentials (IP). *For N,, the ordering of the HOMO and the HOMO-I is not correctly reproduced by the Hartree-Fock calculations. The results are from re$ 82 System

Orbital

EXX

HF

LSDA

H2O

LUMO+ 1 LUMO HOMO HOMO- 1 HOMO-2 LUMO+ 1 LUMO HOMO HOMO- 1 HOMO-2 LUMO+ 1 LUMO HOMO HOMO- 1 HOMO-2

-3.3 1 -5.35 -13.71 -15.74 - 19.28 -0.26 - 7.91 -17.16 -18.12 -20.23 -2.19 -7.23 - 15.03 - 17.99 -20.57

5.43 3.58 - 13.76 - 15.77 - 19.29 10.77 4.35 -17.23* - 16.67* -21.18 7.02 3.72 - 15.07 - 17.40 -2 1.88

2.17 0.08 -6.99 -9.05 - 12.93 6.30 -2.07 - 10.30 -1 1.81 - 13.35 3.42 -2.13 -9.05 - 12.06 -14.13

N2

co

IP

- 12.62

- 15.58

-

14.01

146


approximate description of correlation effects. The results were compared with those of LDA and GGA calculations where also the exchange effects were approximated and are reproduced in Tables 9 and 10. The ionization potentials of Table 10 confirm the conclusions of the discussion above, i.e. the removal of the self-interaction and the correct long-range behaviour

Table 9 Atomization energies (in kcal molt') for different molecular systems calculated with local-density (LDA) or generalized-gradient (GGA) approximation for either only correlation (EXX-LDA and EXX-GGA) or correlation and exchange (LDA and GGA) effects in comparison with experimental values (Exp.). The results are from re$ 83 System H2 LiH OH FH Liz LiF

co N2 0 2

F2 NH3 HzO CH4 CN-

EXX-LDA 115 62 96 124 21 120 209 167 55 -28 282 210 419 237

EXX-GGA 104 52 94 123 17 116 217 171 68 -2 1 275 207 41 1 246

LDA

GGA

Exp.

113 61 124 164 24 163 298 266 174 80 337 266 462

105 53 110 142 20 144 268 242 143 53 302 234 420

109 58 107 141 24 139 259 229 121 39 297 232 419 240

Table 10 Ionization potentials (in e V) for different molecular systems calculated with locaI-density (LDA) or genera 1ized-gradient (GGA) approximation for either only correlation ( E n - L D A and EXX-GGA) or correlation and exchange (LDA and GGA) effects in comparison with experimental values (Exp.). The results are from re$ 83 System

EXX-LDA

EXX-GGA

LDA

GGA

Exp.

17.5 16.3 18.9 6.0 15.6 16.9 14.8 15.9 13.0 15.1 15.4

17.0 15.8 18.4 5.8 15.1 16.4 14.3 15.4 12.5 14.6 15.0

10.3 8.2 9.8 3.2 9.1 10.4 6.9 9.7 6.3 7.4 9.5

10.4 8.0 9.6 3.2 9.1 10.3 6.8 9.5 6.2 7.2 9.5

15.5 13.0 16.0 5.0 14.0 15.6 12.1 15.7 10.2 12.6 12.6

147


of the exchange potential leads to significantly more accurate ionization potentials than otherwise are obtained with density-functional calculations. On the other hand, the error cancellation when approximating both exchange and correlation effects means that the atomization energies are more accurate with the current GGAs compared with any current exact-exchange method. Since LDA calculations are known to predict much too large bond energies (typically from some tenths to some eV per covalent bond), the atomization energies of the LDA calculations in Table 9 are in general too large and the errors become comparable with those of the exact-exchange calculations. Gorling considered actually also the case that correlation effects were completely neglected and in agreement with the well-known findings of standard Hartree-Fock calculations (that predict too small bond energies) these calculations (not shown in the Table) led to too small atomization energies. Finally, V e ~ e t halso ~ ~studied the total energy as well as the orbital energies from exact-exchange density-functional calculations for some small molecules and compared the results with those of Hartree-Fock calculations. In addition, Veseth noticed that since the self-interaction error is absent and the asymptotic behaviour of the exchange potential is correctly reproduced within the exact-exchange approach, the orbitals that are obtained with such calculations may represent physical objects that can be used in studying other quantities of interest, including excitation energies. As a consequence of these arguments, Veseth explored whether limited configuration-interaction calculations based on the so-obtained KohnSham orbitals could be used in obtaining accurate excitation energies. As one

Table 11 Vertical excitation energies (in eV) for N2. KS and HF denote limited conJguration-interaction calculations based on either the Kohn -Sham (KS) or Hartree-Fock (HF) orbitals, whereas CCSD and SOPPA represent results of sophisticated configurntion-interactionmethods. The final state and the electronic excitation are also shown and the results are compared with experimental values (Exp.). The results are from ref 84 ~

State

Excitation

KS

HF

CCSD

SOPPA

Exp.

A3X;

In, lx, In, + In, 30, + In, In, -+ In, In, + In, lx, + ln, 20, -+ In, 30, -+ 4a, 3 0 , --+40, 30, --+ 2n,, 30, 30, In, 40, 3 0 , -+40,

7.61 8.19 8.92 9.58 9.93 10.10 10.70 11.54 12.20 12.61 13.22 12.89 13.73 14.57

7.25 8.78 8.19 10.69 9.13 9.13 9.60 12.56 13.36 13.92 13.35 13.85 14.49 14.28

7.56 8.05 8.93 9.27 9.86 10.09 10.54 11.19 11.75 12.20 12.84 12.82 13.61 14.31

7.91 7.87 9.01 9.32 9.99 10.05 10.54 11.10

7.75 8.04 8.88 9.3 1 9.67 9.92 10.27 11.19 12.0 12.2 12.90 12.98 13.24 14.25

B3n, W3Au

a'n,

Bt3ZU a"E; w'A,

c3nu E3E: a"'2; C'I-I, C"C:

b'l7,

b"Ei

In, 30,

+

-+

-+

---f

12.11 12.30 13.68 14.30

148


example we show in Table 11 the results for the N2 molecule in comparison with similar results based on the Hartree-Fock orbitals and results of much more sophisticated configuration-interaction calculations as well as experimental results. As suggested above, the results do in fact show a considerably better agreement between the Kohn-Sham results and experiment than between the Hartree-Fock results and experiment, and only a slightly worse agreement than between the results of the much more sophisticated calculations and experiment. This result is very interesting since it suggests that the orbitals obtained within the EXX scheme provide very good approximations to experimental observables.

11 Time-dependent Density-functional Theory In the previous section we saw that the fact that exact-exchange methods remove the unphysical self-interaction from which most other density-functional methods suffer and have the correct asymptotic behaviour means that the orbital energies provide accurate approximations to experimental excitation energies. However, it should be stressed that this finding is lacking mathematical rigour so that it at most can be very useful, but there is no guarantee that it is applicable for all systems or that improving the quality of a calculation automatically leads to improved agreement between theory and experiment. Instead, the time-dependent density-functional theory, first formulated by Runge and Grossg5(see also refs. 86 and 87), provides a mathematically exact scheme with which excitation energies can be calculated within density-functional theory. We shall here briefly describe its main foundations and subsequently give a few recent examples of its applications. Although the theory was formulated almost two decades ago, it is only within the last few years that it has become of practical use for a larger class of systems. The theorems of Hohenberg and Kohn combined with the approach of Kohn and Sham (i. e. the foundations for the previous parts of this report) apply to the stationary case where there are no time-dependent interactions. Excitations involve, on the other hand, the response of the system of interest to some external time-dependent perturbation (for instance an electromagnetic field) and one has accordingly to extend the density-functional theory of Hohenberg, Kohn, and Sham to the time-dependent case. However, since the Hohenberg-Kohn theorems are derived by applying the variational principle for which there, in the general non-stationary case, does not exist an analogue, it is non-trivial to derive a general time-dependent density-functional theory. To derive such was the accomplishment of Runge and Runge and Gross proved that under certain conditions the total electron density for a general system subject to some time-dependent interaction can be written as

i.e. an expression similar to the stationary case.

149

3: ilensity Functional Theory

The single-particle orbitals are calculated from time-dependent Schrodinger- or Kohn- Sham-like equations,

where the effective potential is the sum of the external potential (e.g. the electrostatic potential of the nuclei plus an electromagnetic potential), the Coulomb potential of the electrons, and the remaining time-dependent exchangecorrelation potential,

This expression is very similar to the standard Kohn-Sham expression (except for an extra time-dependence) with, however, the important difference that the last term contains not an energy but an action. For a general quantum-mechanical system with the wavefunction Y ( t ) (the dependence on all other coordinates is not shown) the action is defined as A

= f ( Y ( t ) / Z3- I(,

fi(t)lY(t))dt.

at

A x , is the exchange-correlation part of the action. In the general case, A as well as A,, depends on the complete history of the system but a considerable simplification is obtained when considering only the linear response of the system to a time-dependent perturbation. This corresponds to writing

A(t) = A0 + A f i ( t ) ,

(90)

where the only time-dependence is contained in A f i which, moreover, is assumed to be much smaller than Po.By letting p o ( r ' ) be the electron density without the perturbation A H , the exchange-correlation potential can in this case be written as

where 6 p is the change in the electron density due to the perturbation and f x cis the so-called exchange-correlation kernel evaluated for the density po,

The exchange-correlation kernel is thus the functional derivative of the exchange-correlation potential which in turn can be expressed in terms of the functional derivative of the exchange-correlation energy density ex,. All these

150


quantities are time-dependent and their precise forms are unknown as for the time-independent case. Accordingly, as in the latter case one has also in the timedependent case to resort to approximations and here much less work in developing accurate approximations has been performed compared with the timeindependent case. The simplest approximation, which also is the one that has been used most often, is the adiabatic local-density approximation (ALDA). In the static case, the local-density approximation amounts to letting E~~ be a function of the electron density in the point of interest, and in the ALDA this approximation is kept and any time-dependence is ignored. Accordingly, f F [ p ] ( r ' , t, F', t ' ) = d(F- F')d(t - t ' )

I

dv~cDAIP'l dP' p'=p -

(93)

In the frequency domain one may write the effective potential as

where we still restrict ourselves to linear responses and have neglected the case w = 0. Then Vext(F,w ) does not contain the Coulomb potential from the nuclei. One may also write 6 p in terms of the response function

x,

where, in the static case, the response function can be expressed in terms of the Kohn-Sham orbitals

Here, f j and f k are the occupation numbers of the orbitals and E~ and E k their energies. 7 is an infinitesimal positive energy. By solving Eqs. (95) and (96) self-consistently, one obtains the linear response of the electron density to the external perturbation, which could, e.g., be an electromagnetic field. From this, various quantities like polarizability etc. can be extracted. The response function xs of Eq. (96) has poles at the orbital energy differences f(c, - c k ) (where one of the orbitals q, and q kis occupied and the other empty). Therefore, these poles represent first approximations to the excitation energies. Better estimates can be obtained by identifying the poles of the true response function. However, the resulting equations are very involved and have been solved for only few systems like, e.g., closed-shell atoms where the spherical symmetry and closed shells lead to important simplifications. In order to demonstrate the


151

effects of going beyond simply using the orbital eigenvalues of the Kohn-Sham equations we show in Figure 27 results for the He atom. In this case it is possible to calculate the exact Kohn-Sham density-functional potential ( i e . the selfinteraction is removed and the potential has the proper asymptotic behaviour), but, as seen in the Figure, the orbital eigenvalues do still suffer from some inaccuracies which to a large extent are removed through the ALDA. Improving the density hctionals (these calculations are marked TDOEP in the figure) may, but need not, lead to even further improvements. In order to obtain better approximations to the electronic excitation energies compared with the simple orbital-energy differences but without having to solve very complicated equations, one may use an approach presented by Jamorski et aZ.@They considered the linear response to a perturbation w ( t ) that was assumed to be turned on slowly at some time in the distant part. In frequency domain the linear change of the electron density is

Continuum

-

0.90

n v)

- 0.85

.as

-g Y

W

A

-

0.80

M

8

d

8 *a .-d

2P

V

2s

1

- 0.75

Triplet

KS

ALDA

.,-...___

A -

TDOEP x-only

TDOEP SIC

a

-----_-_-

Exact

-

0.70

Figure 27 Excitation energies for the He atom obtained with digerent appmaches. KS marks the orbital eigenvalues for the exact Kohn-Sham potential, ALDA the results with the adiabatic local-density approximation, and for the TDOEP approaches digerent orbital-dependent finctionals have been used. Both the ALDA and the TDOEP results have been obtained using the time-dependent densiy-functional theory (Reproduced with permission from Int. J Quant. Chem., 80, 534; 0 2000 John Wiley and Son. Inc.)

152


i.j

where the response of the density matrix is

) element for the extra field which Here, wg(w)= ( ~ i ~ w ( w is) ~the~ Jmatrix would have been the only contribution if the electrons were non-interacting. But since the electrons interact through Coulomb, exchange and correlation effects, an extra term describing the linear response of the self-consistent field to the perturbation has to be included. This term contains the quantities

where an adiabatic approximation has been made for the exchange-correlation kernel. Jamorski et aL8' showed that the excitation energies oIcould be calculated from the eigenvalue equations

with 0 0 = €, - &;

and with the coefficients Fb related to the oscillator strengths. Jamorski et aLS8 showed moreover that their approach also allowed for the calculation of polarizability. In a later paper, Casida et ~ 1 used . this ~ ~formalism to calculate the excitation energies of some smaller molecules (N2, CO, CH2, and C2H4). In Table 12 we have collected their results for N2 and in Table 13 those for CO. Those for N2 can be compared directly with those of Table 11 obtained with an exact-exchange method. The results of both tables show that the time-dependent densityfunctional methods give results that are almost as accurate as those of the sophisticated correlation methods (like coupled-cluster, configuration-interaction, multiple-configuration, or polarization-propagator methods) and considerably

E3Zi

c3n,

wlAu

a"2;

Bl3Z;

--f

--f

l n u -+ lx, 30, -+ In, In, + In, 30, ln, lx, -+ In, ln, In, In, -+ In, 20, -+ In, 30, + 40,

A32: B3H, W3Au

alng

Excitation

State

7.85 7.54 8.82 9.05 9.63 9.63 10.22 10.36 10.29

TDLDA

7.29 7.14 8.32 8.68 9.18 9.18 9.82 10.06 12.32

TDGGA

7.56 8.05 8.93 9.28 9.87 10.09 10.54 11.19 11.75

MRCCSD

7.91 7.87 8.93 9.32 9.96 10.02 10.51 11.05

SOPPA

7.64 8.17 8.86 9.60 10.07 10.40 10.76 11.43

MRTDHF

3.47, 3.46 7.62, 7.62 5.80, 5.86 9.76, 9.77 7.94, 7.94 7.94, 7.94 8.75, 8.78 11.26, 11.28

TDHF

6.23, 6.25 7.99, 7.94 7.33, 7.35 10.02, 9.94 8.50, 8.51 8.50, 8.51 9.06, 9.09 11.74

crs

7.75 8.04 8.88 9.31 9.67 9.92 10.27 11.19 12.0

Exp.

Table 12 Vertical excitation energies (in e V ) for N2. TDLDA and TDGGA represent results from time-dependent density-functional calculations with either a local-density (LDA) or a generalized-gradient (GGA) approximation, whereas TDHF are similar 3 results @om time-dependent Hartree- Fock calculations. MRCCSD, SOPPA, MRTDHF are results from sophisticated 3 configuration-interaction calculations, and CIS are less sophisticated configuration-interaction calculations. The final state and the electronic excitation are also shown and the results are compared with experimental values (Exp.). The results are from ref: 89

2.

2

3

2

b

?

d3A e3Z-

n

-

50 4 2n l n ---t 2 n 50 2n In 3 2n In + 2n

a3II ar3Zt

A'

Excitation

State

5.96 8.37 8.18 9.17 9.84

TDLDA

5.58 8.36 7.98 9.17 9.86

TDGGA

6.32 8.26 8.79 9.18 9.82

MRCC

6.02 8.02 8.53 8.96 9.64

SOPPA

5.35, 5.30 6.33, 6.36 8.89, 8.79 7.90, 7.93 9.35, 9.38

TDHF

5.88 7.82 9.07 8.78 9.74

CIS

6.32 8.5 1 8.51 9.36 9.88

Exp.

Table 13 As Table 12, but for CO. In addition, MRCC are results from sophisticated configuration-interaction calculations. The results are from re$ 89

c

tJl P

155


more accurate than those of the time-dependent Hartree-Fock calculations. Moreover, the local-density calculations are more accurate than the generalizedgradient calculations, where a functional of van Leeuwen and Baerendsgo was used. This functional has been explicitly constructed to give a correct asymptotic behaviour. In another work, Stener et aL9' solved Eqs. (95) and (96) self-consistently for some closed-shell atoms. From the frequency-dependent changes in the electron density they calculated the dynamic polarizability

from which the photoionization cross section can be calculated, 4nw

a ( w ) = -Im[a(w)]. C

They were interested in the performance of different density-functional approaches. In all cases they used the ALDA for the exchange-correlation kernel, but for the response function xs of Eq. (96) they used different density functionals. Since most of the current density functionals give a wrong ionization threshold (cf the discussion of the preceding section), they rigidly shifted the single-particle energies so that the experimental ionization potential was obtained. a ( w ) has peaks at resonances corresponding to specific excitations. In the neighbourhood of the resonances a as a function of w can be modelled through the analytic expression

with

This expression contains various (so-called Fano) parameters that are specific for the resonance: the peak position E R , its width r, the shape factor q, the background non-resonant cross section ao,and the mixing parameter p2. Table 14 contains a part of the results of the study of Stener et ~ 1 . The ~ ' table shows that the exact-exchange method without adjusting the orbital energies leads to reasonably accurate results with, however, some shifts of the resonance positions. This may not be surprising when taking the discussion of the preceding section into account. Moreover, adding LDA correlation to an exact-exchange calculation does not necessary improve the results, whereas the GGA calculations (using the functional of van Leeuwen and BaerendSgO which has the correct asymptotic behaviour) give somewhat less accurate results.

2s -+ 3 p

2s + 4 p

3s -+ 4 p

4s -+ 5p

Ne

Ne

Ar

Kr

exact KS GGA EXX+LDA EXX EXX* Exp. exact KS GGA EXX+LDA EXX EXX* Exp. GGA EXXfLDA EXX EXX* Exp. GGA EXX+LDA EXX EXX* Exp.

45.544 46.224 45.168 45.438 43.170 45.546 47.123 47.352 47.034 47.093 45.365 47.121 26.888 26.225 26.522 27.196 26.606 25.199 24,662 24.959 24.474 24.992

16.4 14.2 22.0 17.9 28.9 13 5 .O 3.8 5.5 5.5 5.0 4.5 183.8 202.4 157.3 162.5 76 129.7 135.1 111.1 109.0 22.8

-3.07 -3.61 -3.41 -3.18 - 1.98 -1.6 -3.34 -3.50 -3.76 -3.35 -3.52 -1.6 -0.15 -0.06 -0.14 -0.09 -0.249 0.144 0. I98 0.176 0.13 -0.5 14 27.5 1 24.88 29.56 3 1.03 27.9

8.25 8.18 7.75 8.09 8.71 8.6 8.06 7.97 7.53 7.89 8.46 8.0 30.88 27.38 3 1.76 3 1.25

0.538 0.528 0.501 0.70 0.900 0.896 0.899 0.900 0.890 0.883 0.874 0.883 0.885 0.778

0.544 0.5 19 0.541 0.547 0.499 0.70 0.535 0.523

Table 14 Fano parameters for dwerent ionization resonances for some closed-shell atoms. Except for the calculations marked E m * , the Kohn-Sham single-particle eigenvalues have in all cases been shifted to match the ionization potential. Exact KS represents results with the exact Kohn-Sham potential, GGA those with a gradient-corrected density functional, EXX exactexchange without correlation, EXX+LDA the same but with the inclusion of correlation effects with a local-density approximation, and Exp. experimental results. R e results are from re$ 91

2

2

3 3

9

g

%

2Q

2.

R

z

k

5

g

$

?i. r?,

0 s .

c VI o\

3: Dens@ Functional Theory

157

Table 15 Vertical excitation energies (in e V) for different molecular systems as calculated with the parameterized density-functional method (TD-DFTB) and with a parameter-free density-functional method (TD-DFRT) in comparison with experimental values (Exp.). The results are.from ref 92 Molecule

State

Excitation

0.08

TD-DFTB

TD-DFRT

Exp.

5.47 7.8 1 4.47 4.92 4.47 5.39 5.81 7.01 7.03 4.04 4.04 2.42 2.42

4.16 7.44 3.71 4.36 4.37 5.28 5.94 6.48 6.69 2.74 3.37 1.53 2.14

4.40 7.65 4.10 4.84 4.5 1 5 .OO 6.45 7.23 7.23 2.99 3.56 2.38 2.73

I

I

I

I

1

0

1

2

3

4

1

1

0.06

0.04 0.02 0 -0.02 -0.04

-0.06 -0.08 5

6

7

8

E (W Figure 28 Circular dichmism of C76. The solid curve shows theoretical results and the dashed one experimental results (Reproduced with permission from Phys. Rev. A , 60, 1271; 0 1999 American Physical Society)

158

Chemical Modelling: Applications and Theoq Volume 2

Niehaus et ~ 1 implemented . ~ ~ the method of Jamorski et ~ 1 discussed . ~ ~ above in the parameterized density-fbnctional program that we discussed in Section 4 in order to calculate excitation energies. They explored the performance of their method on a number of smaller molecular systems and compared the results with those of parameter-free density-functional calculations and with experimental values. Some representative results are shown in Table 15, where it is seen that their method performs well also in comparison with the parameter-free calculations. The Table also shows, however, that a perfect agreement with experimental values is by no means automatically obtained, independent of the theoretical approach that is being used. Finally, Yabana and Bertschg3 calculated various properties related to optical activity for some chiral molecular systems using time-dependent density-functional theory. We stress that it is very far from trivial to obtain even a fair agreement with experimental results, so the results of Figure 28 for the circular dichroism of CT6are promising.

12 Polarizability and Hyperpolarizability In the preceding two sections we have discussed different ways in which a molecular system may respond to an electromagnetic field, first of all through excitation or ionization. Other responses include that the particles (electrons and nuclei) change positions due to these fields. This may, e.g., be quantified through the dipole moment, i.e.

where ,MY' is the ith component of the permanent dipole moment in the absence of the external field, a is the polarizability, p, y, . . . are hyperpolarizabilities and I? is the electromagnetic field vector. As an alternative to Eq. (106) one may also express the total energy in terms of the polarizabilities and hyperpolarizabilities through

I=x, y,z

In the static case, the frequencies of all field components as well as of the dipole moment vanish, but in the dynamic case they may be non-zero, obeying, however,

159

3: Density Functional Theovy

with

Within orbital-based methods like the Hartree-Fock method one may apply standard (time-independent or time-dependent) perturbation theory and thereby calculate the polarizabilities and hyperpolarizabilities. The resulting perturbationtheoretic expressions have been given by Genkin and M e d n i ~ Within . ~ ~ densityfunctional methods the simplest approach amounts to ignoring the formal problems related with using the Kohn-Sham orbitals as if they were electronic orbitals and, accordingly, proceed as in the Hartree-Fock case. In practical calculations both approaches suffer, however, from severe problems, including that the absorption threshold will be underestimated when using the Kohn-Sham orbitals and their energies and overestimated when using the Hartree-Fock orbitals and their energies. A formally more correct approach is to use time-dependent density-functional theory. In their approach, Jamorski et a1.88showed that the average polarizability 1 G(w) = - [ a x x ( - w ; w )

3

+ a,(-w;

w)

+ a,(-w;

w)]

could be calculated directly from the excitation energies (w,) and oscillator strengths cf,),

They used N2 as a test system and obtained the results of Figure 29. Thereby they studied the importance of different terms in the coupling matrix K , . of Eq. (99), so that within the RPA only the Coulomb term (the first integral on the right-hand side) is included, within TDLDAx exchange but not correlation effects are included in the exchange-correlation kernel, and within the TDLDAxc, finally, also the correlation effects are included. The results show an overall good qualitative agreement with experimental results and also quantitatively an accuracy of some few % can be obtained. Also van Gisbergen et al.95 applied the time-dependent density-functional theory in calculating frequency-dependent hyperpolarizabilities of some small molecules. They considered different density functionals and compared their results with those of previous theoretical studies and of experiment. In Tables 16 and 17 we reproduce their results for N2 and CO. The molecules were assumed placed along the x axis, and the authors considered also the averaged hyperpolarizabilities given by Eq. (1 12).

160


0

1

2

3

4

5

6

7

8

Figure 29 The average polarizability of Nz as a function of frequency using different approaches based on the time-dependent densiiy-functional theory in comparison with experimental results (EXPT) (Reproduced with permission from J Chem. Phys., 104, 5134; 0 1996 American Institute of Physics)

Table 16 Results for hyperpolarizabilities (in a.u.) for N2. TDHF and MBPT(2) are results from time-dependent Hartree- Fock and perturbation-theory calculations, respectively, whereas CCSD and CCSD(r) are coupledcluster results. Exp. denotes experimental results, and LDA, GGA, and LB94 are results from time-dependent density-functional calculations with different dens@ functionals. For a description of the quantities, see the text. The results are from ref: 95 Quantity

GGA LB94

TDHF

MBPT

CCSD

CCSDV) Exp.

LDA

780 1220 340 930 1200 1100

810, 782 1220, 1180 350, 313 950, 903 1200, 1212 1100, 1041

860 1290 370 1010 1300 1100

1100 1200 610 1800 2000 1000 450 510 260 1300 1400 740

THG EFISH

660 790 270 730 937 822

EOKE

756

1000

1000, 947

1100

Y U X X

Y Yx.rz2. ZZZZ

Yll

1295 f 206 1057.6 f 6.4, 1500 1800 1030 f 12 1 4 3 0 f 160 1400 1500

840 770

161


Table 17 As Table 16 but for CO. The results are from ref 95 Quantity TDHF

MBPT CCSD

Exp.

26.0 6.6, 7.8 23.5, 26.6 SHG 27.0, 29.9 f 3.2 30.1 f 0.6 OREOPE 2 1.9 23.6 24.1 24.6 ynn 920, 1173 1380 1360, 1239 1470, 1357 yzuz 1200, 1173 1740 1740, 1616 1880, 1758 yuzz 360, 348 520 510,460 540, 500 YIi 1020,992 1500 1480, 1353 1590, 1475 THG 1484 2200 2200 2300 1720 k 48 EFISH 1211 1800 1800 1900, 1790 f 90 EOKE 1071 1600 1600 1700

BZZZ

BW PI1

28.2, 31.47 25.6 26.1, 28.9 3.5, 4.89 6.0 6.1, 7.3 21.1 22.6 23.0, 26.1 24.1 25.9 26.4

CCSDfl)

LDA

GGA

LB94

33.9 34.4 24.8 8.43 8.74 4.80 30.5 31.1 20.6 36.6 37.8 23.4 32.3 33.1 21.5 2000 2200 1000 2700 3000 1400 830 390 750 2200 2400 1100 2900

3200

1400

2400

2700

1200

Furthermore, they examined the performance of different density functionals, including a local-density approximation and a generalized-gradient approximation as well as the functional of van Leeuwen and BaerendP that has been constructed to have the correct asymptotic behaviour. Moreover, they considered different frequency-dependent processes, including third-harmonic generation [THG, corresponding to y( -30.1; cu, w , a)], electric-field-induced second harmonic generation (EFISH, y(-2w; cu, w , O ) ] , electro-optic Kerr effect [EOKO, y(-cu; cu, 0, O ) ] , and optical rectification [OR, B(0; cu, -w)]. For the dynamical properties (cu # 0) they considered light with a wavelength of 694.3 nm. The results show systematic overestimates for the LDA and GGA calculations. For the asymptotic corrected functional of van Leeuwen and Baerends a general trend is less clearly recognized. In total this means that one has to be very cautious when using such calculations in a predictive manner and it emphasizes the need for improved density-fhctionals to be used in time-dependent densityfunctional theory. On the other hand, van Gisbergen et al.96showed in an earlier work that despite the possible inaccuracies of the calculated hyperpolarizabilities the results could provide a useful extension of experimental studies. They calculated the dynamic hyperpolarizability of CM for which the experimental studies had provided values differing by as much as 10 orders of magnitude. The theoretical results were able to reduce the possible values to a small interval.


162

In more recent work, Schipper et aZ.97 presented a new exchange-correlation potential that was constructed to give a correct asymptotic behaviour (as in the potential of van Leeuwen and Baerends”) and also an accurate description in the regions closer to the nuclei. The resulting potential (SAOP, for Statistical Average of different model Orbital Potentials) became orbital dependent and was subsequently tested on some small molecular systems for which both excitation energies and polarizabilities and hyperpolarizability were calculated. It was indeed found that an improved agreement with experimental results was obtained. In opposition to these prospective results are some recent studies on larger systems. Champagne et aL9’ studied the polarizabilities and hyperpolarizabilities of finite oligomers of a polyacetylene chain, i.e. of CH2(CH),CH2, for different values of n. They found that all conventional and most nonconventional density functionals fail. Specifically, the correlation correction to the polarizability was either too small or in the wrong direction, the hyperpolarizability was significantly overestimated, the dependence on n was too strong (more for y, but also for a), and the dependence on the so-called BLA parameter was not correctly reproduced. Polyacetylene contains a zigzag backbone of carbon atoms and the bond lengths between those alternate. The BLA parameter is the difference in the lengths of the shorter and the longer bonds. This study was extended by van Gisbergen et aZ.99 who made a systematic investigation of the n dependence of the static polarizability a and the static hyperpolarizability y (for brevity we have set GI = a , and y = y- with x being parallel to the polymer axis). The results are shown in Figure 30. Since it was known that Hartree-Fock calculations provided results that were in qualitative and quantitative agreement with those of significantly more advanced studies, the density-functional results were compared with Hartree-Fock results. Figure 30 shows that there is not even a qualitative agreement between the results of the

1000

20

30 40 #C atoms

50

Figure 30 Polarizability (a) and hyperpolarizability (y) forJinite polyacetylene oligomers C,H,,, as a function of n both from Hartree-Fock (HF) and densityfunctional (LDA) calculations relative to the Hartree-Fock values for n = 20 (Reproduced with permission from Phys. Rev. Lett., 83, 694; 0 1999 American Physical Society)

3: Density Functional i?heo?y

163

two types of calculations. In agreement with the statements above, the densityfunctional calculations predict a much stronger dependence on system size than is the case for the Harkee-Fock calculations. In somewhat earlier theoretical work, Gonze et aZ.'@-' argued that for an infinite system in an electric field a linear term will arise in the exact exchangecorrelation potential. This term cannot be captured by a functional that depends only on the (periodic) electron density (note, for an infinite periodic system in an external field, the electron density is periodic, too) and, accordingly, an extra dependence on the polarization needs to be included in a more precise density hctional. van Gisbergen extended the study on the polyacetylene oligomers by also studying two simpler model systems, a linear chain of H atoms and a linear chain of H2 molecules, for which more accurate calculations could be performed. They showed that by applying a more accurate exchange-correlation potential (the precise definition of this is beyond the scope of this report) improved polarizabilities and hyperpolarizabilities could be obtained although there still were deficiencies. They concluded, accordingly, that the theoretical treatment of a0 16

I

---_

1

I

-

1

-

0 -......

.I.........

.... - ............ -.. ..L ......

.

Figure 31 Real and imaginary part of the calculated dielectric function (solid curve) in comparison with experimental results (dashed curve) for diamond (Reproduced with permission from .IChem. Phys., 112, 6517; 0 2000 American Institute of Physics)

164


extended systems in external electric field with current density-functional methods may be problematic. That such calculations not always lead to erroneous results was demonstrated by the last work we shall discuss in this section. Kootstra et a1.l'' calculated the dielectric responses (i. e. essentially the polarizabilities) for some non-metallic crystalline systems using time-dependent density-functional theory. They found a good agreement with experimental results, as, e.g., shown in Figure 31 for the case of diamond.

13 Conclusions Density-functional theory is a very active field and the number of papers reporting results of calculations based on this theory is enormous. In many cases these theoretical studies are used in supplementing experimental work, in others to obtain alternative theoretical information compared to that obtained with more traditional wave function-based methods, and in yet others to explore new systems and/or properties that are not or have not been accessed with other approaches. Due to this huge amount of density-functional studies on specific systems we felt that a report simply listing all these studies at best would be useless. Instead we have in this report discussed some of the issues that currently are the topic of new theoretical developments. However, first it should be stressed that within density-functional theory, in its most widespread formulation, it provides a method for calculating static groundstate properties of a given atomic, molecular, crystalline, liquid, . . . system. This is the field where density-functional theory has found most applications and where the experience has shown that the results in by far the most cases are accurate and useful. Typically, structural, energetical, and vibrational properties are as accurate as those obtained with Hartree-Fock methods augmented with perturbational inclusion of correlation effects at the MP2 level although there are exceptions (both for the case that the density-functional calculations are inaccurate and for the case that the Hartree-Fock-based calculations fail). In this report we have not attempted to review all the studies supporting this conclusion. Therefore, it is very important at this point to emphasize the importance and accuracy of such studies. Such studies are, in principle, parameter-free and although the calculations, in general, are faster than MP2 calculations the classes of systems that can be treated in a routine way are limited to those of high symmetry and/or containing a small number of atoms. Many systems of relevance in nature, technology, or experiment do, however, not fall into those catagories and in order to treat these classes of materials one has to use special methods. Two such methods, that introduce various approximations and therefore become computationally more efficient, were discussed in Section 4. We found that through parametrization of density-functional results on small molecules an efficient tight-binding method suitable for systems with localized electrons could be obtained. Alternatively, for delocalized electrons the precise description of the electron density could


165

be abandoned and large metal-atom-containing systems accordingly be treated. Their computational efficiency also allowed for a largely unbiased structureoptimization of larger systems of lower symmetry. It should be stressed that structure-optimization is a severe problem whenever the number of atoms exceeds, say, 10 and will, without intelligent selection of the structures to be studied, easily require that more than 1O7 electronic-structure calculations are carried through. Therefore, very efficient methods for the calculation of the total energy for a given structure are required. When the number of structures to be considered is smaller (e.g. when some information about the structure is available fi-om other sources) but the number of atoms still is large, the order-N methods of Section 5 may be useful. They are, in principle, parameter-free and are, accordingly, computationally heavier than those just mentioned. On the other hand, for large systems (but not for smaller ones) they are more efficient than the traditional methods. Some of the systems where the symmetry is low and the number of atoms relatively high are those occurring in the field of heterogeneous catalysis. The combination of a surface of a (in the ideal case, semi-infinite) crystal and one or more finite molecules (maybe even together with the presence of some medium) makes the study of the chemical reactions on the surfaces of crystals very complicated. In Section 6 we saw that such studies are becoming possible but still only for the simplest reactions on fairly idealized surfaces. Moreover, the calculations explore mainly energetical aspects and not kinetic aspects of the reactions. This means also that the time scales that can be treated with quantummechanical molecular-dynamics simulations are much too small (10-'5-10-*2s) compared with those that often are relevant for chemical reactions. In order to simplify and rationalize the studies of chemical reactions it would be useful to identify the central properties of any molecule that determine whether and how it would react with any other molecule. Some approaches in this direction were discussed in Section 7 but we saw that this field is not yet mature and, although some trends are emerging (including those based on the HSAB principle and on the Fukui function), the approaches can still not be used in a quantitative manner. The common density-functional methods are methods for calculating the electronic ground-state properties. There are, however, other systems and/or properties where extensions of these methods could be highly relevant, In Section 8 we saw that a quantum treatment of some light nuclei (most notably, of protons) could lead to qualitatively different results than when treating these classically. Thus, first of all when studying hydrogen bonds quantum-mechanical effects of the protons could very well be important. Another, maybe even more important, field is that of excitations and ionizations. Here, the facts that the conventional density-functional methods are ground-state methods and that the most common approximate density functionals suffer from a non-physical self-interaction of the electrons and from a wrong asymptotic behaviour (Section 9) lead to problems when attempting to study excitations and ionizations. The exact-exchange methods remove some of the problems of the approximate density functionals (Section 10) and were found to

166


produce orbitals that appeared to be relevant in studying such processes (although the mathematical justification for this is lacking) at the expense of increased computational demands. Alternatively, the time-dependent density-functional theory (Section 11) is a mathematically well-founded theory for studying excitations and ionization processes and was found to give promising results for small test systems - once again at the expense of increased computational demands. A special problem where the time-dependent density-functional theory could be usefwl is that of calculating the polarizability and hyperpolarizability (Section 12). It turned out that although accurate results could be achieved for smaller molecules (partly, however, requiring a careful choice of the approximate density functional), severe problems could turn up (but did not always) when considering extended systems. It might mean that the current density functionals are lacking an explicit dependence on the polarization, but further studies are needed in order to clarifl this point.

Acknowledgments The author is very grateful to Fonds der Chemischen Industrie for very generous support.

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4 Numerical Methods for the Solution of lD, 2D and 3D Differential Equations Arising in Chemical Problems BY T.E. SIMOS

1 Introduction Many mathematical models of chemical applications are expressed via onedimensional, two-dimensional or three-dimensional differential equations. l y 2 A well known example is the Schrodinger type differential equations. For example we have mathematical models of chemical applications which are expressed via the one-dimensional Schrodinger type differential equations

. [

v"(x> =

l(1

+ 1) + V ( x ) - El y(x)

where one boundary condition is y(0) = 0 and the other is specified at x = 00 and is dependent on the properties of the problem. In equation (1) the fbnction W ( x ) = l(E + l)/x2 + Y ( x ) is denoted as the effective potential, for which W ( x ) -+ 0 as x -+ 00, and E is a number denoting the energy. We have also mathematical models of chemical applications which are expressed via the two-dimensional Schodinger type differential equations 1 au 2dx2

1 du 2ay2

+ V ( x , y)u = Eu

u(x, f o o ) = 0,

-oo < x < +oo

u ( f o o , y ) = 0,

-0c < y < +oo

where E is the energy eigenvalue, V ( x , y ) the potential and u(x, y ) the wave function. Finally one can find mathematical models of chemical applications which are expressed via the three-dimensional Schrodinger type differential equations which Chemical Modelling: Applications and Theory, Volume 2 0The Royal Society of Chemistry, 2002

4: Numerical Methods for the Solution of ID,2 0 and 3 0 Diflerential Equations

171

have form analogous of the two-dimensional Schrodinger type differential equations mentioned above. In the present review we shall present recent advances in the development of numerical methods for the numerical solution of one-dimensional, two-dimensional or three-dimensional differential equations. In each case we shall present remarks or improve the developed methods. In Section 2 we shall investigate a new class of methods named symplectic integrators. We shall emphasize adapted and exponentially fitted symplectic integrators, i.e. methods with coefficients dependent on the frequency of the problem (i.e. on the quantity Jl( I( I l)/x2) V ( x ) - El). In the same paragraph numerical tests for these methods are presented on well known problems with oscillating or periodical solution. We note here that in Section 2 a new family of exponentially fitted linear multistep methods is developed [these methods have the property of symplecticness (see Section 2.6)]. In Section 3 we shall study a new class of methods named dissipative methods. These methods are nonsymmetric and they haven’t an interval of periodicity. We shall make some comments on recent papers in the subject and then present the dissipative methods developed in the literature. Finally two cases of exponentially fitted and trigonometrically fitted methods are developed. Numerical illustrations for the methods developed in previous sections are presented in Section 4. In Section 5 we make comments on the new developed numerical methods with constant coefficients and on the new developed numerical methods with coefficients dependent on the frequency of the problem. Numerical illustrations for the methods presented in Section 5 are presented in Section 6. General comments on the application of the above methods to one-dimensional, two-dimensional and three-dimensional problems are presented in Section 7. In Appendix A a computer algebra program for the construction of the exponentially fitted method and the trigonometrically fitted method produced in Section 2 is presented. Finally in Appendix B a computer algebra program for the construction of the exponentially fitted method and the trigonometrically fitted method produced in Section 3 is also presented.

+

+

2 Adapted, Exponentially Fitted and Trigonometrically Fitted Symplectic Integrators In this section, we describe the adapted symplectic integrators, a new procedure for the efficient solution of Hamiltonian problems that has been introduced in 2001 by Aguiar, Simos and T ~ c i n o . ~ The new procedure is based on the combination of the trigonometric fitting technique and the symplecticness conditions. Hamiltonian problems can be written in the form:


172

where N denotes the number of degrees of freedom. It is noted that the dot denotes differentiation with respect to time. Consider the problem, q” + w2q = f(q),

(3)

where f(4) is the gradient of a potential scalar. This problem can be transformed into the system of first order equations

+

The following (rn 1)-stage modified Runge-Kutta-Nystrom method has been introduced3 in order to solve numerically the above problem:

where

+

The free coefficients C, D, Pi, bj, Ci, Dj, ai,j, i, j = l(1)rn 1, A, B are parameters that are defined in order for the method to be symplectic, adapted and to have the maximum algebraic order. The symplecticness conditions come from the requirement,

Based on the above relation Aguiar, Simos and Tocino3 have found the equations which satisfy (8). As Aguiar, Simos and Tocino have provea3 in order that the Runge-KuttaNystrom method defined by (5)-(7) integrates exactly equation (3) with f(4) = 0 the coefficients A , B, C, D and C,,Di of the method must be given by:

gisin( giwh) Di = , giwh

(9) Ci = cos(giwh).

4: Numerical Methods for the Solution of 1 0 , 2 0 and 3 0 Differential Equations

173

2.1 Case m = 0. - In the case m = 0, solving the above relations, Aguiar, Simos and Tocino3 have found the following family of Runge-Kutta-Nystrom methods with order two:

Pn+l

= cos(wh)p,

-

wsin(wh)q,

+ hblf , ,

where

J; = f(cos(wh)qn

+sin(wh) W

Pn)

*

+

and bl = 1 O(w2h2). The local truncation error in the fbnction y(x) is of order O(h3) and the local truncation error in the function y ' ( x ) [ie. derivative of y(x)] is of order O(h3) (see ref. 3 for more details).

2.2 Case m = 1. - In the case rn = 1 and based on the above relations and on the symplecticness conditions, Aguiar, Simos and Tocino3 have found the following family of Runge-Kutta-Nystrom methods with order two:

where .fl

f 2

= f(C2qfl

= .f(qfl),

+ hD2pfl+

h2a2,,fd,

where the parameters of this method are given3 by:

174

Chemical Modelling: Applications and Theoy, Volume 2

2 v2 v4 v6 307~' " = O' g2 =7+810+68040+6123600+21824510~000

167v" 109vl 2 39720608928000 - 649973600640000 -

5729723~'~ 537049~'~ 1148600848370976000000- 598585 1278367600640000

F = C2 -k a2,v2,PI =

D2v(1 - cos(v))

b2= -

blv - sin(v) VF

7

a2,1 =

bl =

F(- 1

+ cos(v)) + D2vsin(v)

+

9

D2v2

C2(2- 2 cos(v) - v sin(v)) cos(v) v sin(v))

v2( - 1

(17)

7

V2

),

v 2F

+ F( -(v) + sin(v))

+

*

where v = wh. The local truncation error in the function y(x) is of order O(h4)and the local truncation error in the function y'(x) [i.e. derivative of y(x)] is of order O(h3). Definition 1. A method is called classical if it has constant coefficients and it can be obtained from an exponentially or trigonometrically or adapted method for w + 0. When w -+ 0, the corresponding classical method obtained by the method (13)-( 14) is given by (see ref. 3 for more details):

CLASSICAL METHOD

where

2.3 Runge-Kutta-Nystrom Method with FSAL Property. - We give the following definition. Definition2. A Runge-Kutta-Nystrom

method is said to have the FSAL

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Differential Equations

175

property if the last stage of the current step of integration of the method is used also as first stage of the next step of integration. For the Runge-Kutta-Nystrom method described by equations ( 13)-( 17), Aguiar, Simos and Tocino3 have produced the corresponding method that has the FSAL property. This method is given by:

g1 = 0,

g2 = 1,

1 sin(v)

P1 = y y ,

p2

=o, 1 sin(v) a2,1= -___ 2 v '

1 bl = - COS(V), al,2= 0, 2

where v = wh and al,l, a2,2,b2 are free parameters. In order to maximize the algebraic order of the method Aguiar, Simos and Tocino3 chose b2 = (1/2)cos(v), al,l = 0 and a2,2= 0. Based on the above coefficients the following RungeKutta-Nystrom is ~btained:~ qnt1 = cos(v)qn

pn+l = cos(v)p,

-

sin(v) p n

+h

w sin(v)q,

+ h2sin(v) 2v ~

f l ,

h cos(v) +2

( f l

+f 2 ) ,

(23) (24)

where

sin(v) h

sin(v)

The local truncation error in the function y(x) is of order O(h3) and the local truncation error in the function y ' ( x ) [i.e. derivative of y(x)] is of order O(h3) (see ref. 3 for more details). For w -+ 0, the above method is equivalent to the classical symlpectic second algebraic order Runge-Kutta-Nystrom method mentioned in the paper of Calvo and Sa~u-Serna.~

2.4 Trigonometrically Fitted Symplectic Integrators. - Consider the Hamiltonian problems which are given by equation (3). We introduce the following n-stage modified Runge-Kutta-Nystrom method:

176


METHOD TFSI

where

In order for the above method to satisfy the symplecticness conditions (8) and the functions cos(vx), sin(vx) the following system of equations is obtained:

p1g3 -

b1g2

+ c,b1g, = 0

+ p1w4a1,0sin(wco)

sin(w) = g 2 w - p1w2 sin(wco) - PIw3c1

+

sin(w) = bowcos(wco) bl w - bl ~

~cos(wco) a ~

cos(w) = g3- bowsin(wco) - bl c1w2

+ bl ~

,

~sin(wco) a ~

~

,

~

where w = v h and h is the step size of integration. One of the family of solutions of the above system of equations is given by: co = 0,

1 2

a1fJ = - ,

c1 = 1 bl

= 0,

p1 = 0

cos( w) g1= 1 + cos(2w)’ bo = g2,

sin(w) g2

=

W

cos(w) - cos(3w) cos(2w))

= 2wy1

+

(3 1) 9

~

g3 =cos(w)

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Drferential Equations

177

For small values of w the above formulae are subject to heavy cancellations. In this case we use the following Taylor series expansions: g1= 1+-w2+-w4+-w6+1 5 61 2 24 720

50521 +-3628800

wl o

+

1 g 2 = 1--w2+-w 1 6 120

-

1

39916800

WI0

540553 w 1 2 199360981 w14+ . . . 95800320 87178291200 +

1 w6+- 1 5040 362880 w8

4 --

1 + 6227020800

1 1 1 g 3 = 1--w2+-w4--w6+2 24 720 1 +

Po=

47900 1600 1 6

277 w8 8064

w12 -

31 360

w12 -

1 w’4+ . . . 1307674368000

1 w8 - 1 40320 362880 w l o

1 w’4+ . . . 87178291200

173 2526 1 w6 +5040 1814400 w8

1+-w2+-w4--

675691 w l o 9968049 1 1211969509 w 1 4 + . . . 1 19750400 43589145600 w12 1307674368000 +

+

+

2.5 Exponentially Fitted Symplectic Integrators. - Consider the Hamiltonian problems which are given by equation (3). We consider the n-stage modified Runge-Kutta-Nystrom method given by equations (27)-(29). In order for the above method to satisfy the symplecticness conditions (8) and the function exp(vx) the following system of equations is obtained:

+ g2w+ Bow2exp(wco)+ Blw’ + Blw’cl + w4a1,0exp(wco) exp(w) = g3 + bowexp(wco)+ blw + blW2c1+ blw3al,oexp(wco)

exp(w) = g,

One of the family of the solutions of the above system of equations is given by:

178

Chemical Modelling: Applications and Theory, Volume 2 co = 0,

1 al,o= 2’

bo =

b1 =

CI =

1,

g, = 1,

g2 = 1,

g3 = 1

PI = 0 (34)

exp(w) - 1 - w W2

(-1 +exp(-w)

9

P o = bo

+ w)

W2

For small values of w the above formulae are subject to heavy cancellations. In this case we use the following Taylor series expansions: 1 1 1 1 1 b - -+-w+-w~+-w~+-W~+‘-2 6 24 120 720 1 +-40320

W6+-

1 5040 w 5

1 w7 +w8+ w9 362880 3628800 39916800

1 1 w’2+ . . . Wl1+ W1O 87178291200 6227020800 47900 1600

+

+

b

1 24

- -1- - W 1 + - W ~ - - W ~ 1+ - W ~ - - 1W ~

‘-2

6 1 +-40320

w6

120

1 -w7+-

362880

720 1

3628800

1 5040

w 8-

) (3 , 5,

1 39916800 w 9

1 1 1 w10 w12- . . . 6227020800wll 87178291200 479001600 +

2.6 Exponentially Fitted and Trigonometrically Fitted Symplectic Linear Symmetric Multistep Methods. - The linear mulstistep methods are very important since they are also symplectic (see Sam-Serna2* and Sam-Serna et aZ.29).We study here the exponentially fitted and trigonometrically fitted linear multistep methods. Consider the following nine-step linear symmetric multistep method:

2.6.1 First Family of Methods - Case to = -2. - We require that the above family of methods should integrate exactly any linear combination of the hctions:

4: Numerical Methods for the Solution of 10,2 0 and 3 0 Digerential Equations

Case I

{ I , x, x2, x3, x4, x5, x6, x’, exp(*wx)}

Case I1

{ 1, x, x2, x3, x4, 2,exp(fwx), xexp(fwx)}

179

(37) Case I11

{ 1, x, x2, x3, exp(fwx), x exp(fwx), x2 exp(&wx)}

Case J Y

{ 1, x, exp(fwx), xexp(fwx), x2 exp(kwx), x3 exp(+wx)}

In order to construct a method of the form (36) that integrates exactly the functions (37), we require that the method (36) integrates exactly the functions:

and then put

Case I

w0 = w1 = w2 = 0 and w 3= w

Case I1

wo = wl = 0 and w 2 = w 3 = w

Case I11

wo = 0

Case IV

and w1 = w2 = w3 = w

wo = W] = w2 = w3 = w

The method (36) integrates exactly the functions 1, x. Demanding that (36) integrates (38) exactly, we obtain the following system of equations for the coefficients bili = 0(1)3:

+2 COS~(~V,) = ~:(2boC O S ~ ( ~+ V261 , ) C O S ~ ( ~+ Y2b2 , ) C O S ( Y , ) + b3)

4 C O S ~ ( ~-Y4, )C O S ~ ( ~Y2, )cosh(v,)

where v, = w,hlj = O( 1)3. Solving the above system for biJi= 0(1)3 we obtain:

(40)

180


THOD TAMMI - C

bo = [327v2cosh(2~)- 8 0 8 cosh(v) ~ ~

w

+ 6 0 1 ~ 96~ cosh(2~)- 48 cosh(4~) -

+ 48 cosh(v) + 96 cosh(3v)]/ +

[288v2cosh(2~) 4 8 0 ~ 7~ 2 0 cosh(v) ~ ~ - 48v2 C O S ~ ( ~ Y ) ]

+

+

bl = [ 1 0 9 cosh(3~) ~ ~ 1 9 ~cosh(v) ' - 96 cosh(4~) - 192 C O S ~ ( ~ Y112v2 )

+ 96 cosh(v) + 192 cosh(3v)]/ [-96v2 cosh(2v) - 160v2+ 240v2cosh(v) + 16v2cosh(3v)l

b2 = [-808v2 cosh(3~)- 57v2 cosh(2~)- 9 3 5 ~ 1440 ~ cosh(3~)

- 720 cosh(v)

+ 720 C O S ~ ( +~ Y1440 ) COS~(~Y)]/ +

[ - 2 8 8 ~cosh(2~) ~ - 4 8 0 ~ 7~ 2 0 cosh(v) ~ ~

b3 = [935v2cosh(v)

+ 48v2 C O S ~ ( ~ V ) ]

+ 480 C O S ~+( ~6 )0 1 cosh(3~) ~ ~ + 960 cosh(3~)

- 480 cosh(4v) - 960 cosh(2v)

+

-

336v2cosh(2v)]/

[-144v2 cosh(2~)- 2 4 0 ~ 3~ 6 0 cosh(v) ~ ~

+ 24v2C O S ~ ( ~ V ) ]

where v = wh. For small values of v the above formulae are subject to heavy cancellations. In this case we use the following Taylor series expansions:

4: iVumerica1 Methods for the Solution of 10, 2D and 3 0 Differential Equations

17671 12096

45767 +-725760

ho = -

y2

+

164627 y 4 47900160

9190171 + 3201186852864000

y'0

-

20 16

120960

+ ...

y'4

164627 - ____

520367 7983360 y4 - 264 1766400y 6

-

9 19017 1 666292 1 do 533531142144000 567677 1352412160

-

2866814089 y'J + . . . 34060628114472960000

+-

-

-

76873 v8 14944849920

+

20483 45767 v2 b2 = 4032 48384

164627 +-3193344

9 19017 1 y'0 213412456857600

-

520367 y4

y'4

12629 3024

45767 36288

v6+

+ 1056706560

76873 y 8 5977939968

666292 1 2270708540964864 '12

2866814089 + 13624251245789184000 b3z

520367 v 6 + 76873 y8 15850598400 89669099520

666292 1 340606281 14472960'I2

2866814089 + 204363768686837760000 3937 45767 bl = _ _ - -y2

+

+ ...

164627 520367 76873 y 8 2 3 9 ~ 0 v84 - 792529920 y 6 - 4483454976

_ _ _ - _ _ _ y2 - ____

-

6662921 9190171 do+ 160059342643200 1703031405723648'12

-

2866814089 y'4 + . . 10218188434341888000

The Local Truncation Error (LTE) of the above method is given by:

181

182


-

METHOD LMMII CASE 11

+ 480 sinh(4v)v 660 cosh(2v) + 125v3sinh(3v) 1 6 8 snh(5v) ~ + 600 cosh(3~)+ 144 cosh(5~) 24 cosh(6v) + 744 sinh(2v)v + 385v3 sinh(v)

bo = [- 144 - 444 sinh(v)v

+ 408 cosh(v) + 24v sinh(6v)

-

-

-

- 380v3 sinh(2v) - 756 sinh(3v) - 384 cosh(4v)]/[324v3 sinh(3v)

+ 504v3sinh(v) + 12v3sinh(5v) 96v3sinh(4v) - 576v3sinh(2v)l b, = [84 + 438 sinh(v)v 612 sinh(4v)v + 696 cosh(2v) + 285v3sinh(3v) 372 cosh(v) + 174v sinh(5v) 708 cosh(3v) - 84 cosh(5v) - 6v sinh(7v) - 24 cosh(6v) + 12 cosh(7v) 840 sinh(2v)v + 145v3sinh(v) - 250v3 sinh(2v) + 954 sinh(3v)v + 396 cosh(4v) - 125v3sinh(4v)ll [ 162v3sinh(3v) + 252v3 sinh(v) + 6v3sinh(5v) -

-

-

-

-

- 48v3 sinh(4v) - 288v3 sin(2v)l

b2

+ 492 sinh(v)v + 576 sinh(4v)v 168cosh(2v) - 30v3sinh(3v) - 696 cosh(v) + 120v sinh(5v) + 264 cosh(3v) - 624 cosh(5v)

[624

- 21 6v sinh(6v)

-

+ 48v sinh(7v) + 408 cosh(6v)

+ 24 sinh(2v)v + 625v3sinh(v)

-

-

(44)

96 cosh(7v)

580v3sinh(2v) - 828 sinh(3v)v

+ 288 cosh(4v) + 125v3sinh(5v)]/[324v3 sinh(3v) + 504v3sinh(v) + 12v3sinh(5v) - 96v3sinh(4v) 576v3sinh(2v)l b3 = [-648 - 924 sinh(v)v + 168 sinh(4v)v 624 cosh(2v) + 145v3sinh(3v) + 1032 cosh(v) - 300v sinh(5v) + 552 cosh(3v) + 648 cosh(5v) + 192v sinh(6v) - 36v sinh(7v) 336 cosh(6v) + 72 cosh(7v) + 912 sinh(2v)v - 40v3sinh(v) + 20v3sinh(2v) - 324 sinh(3v)v 696 cosh(4v) - 95v3sinh(5v) + 10v3sinh(4v)]/[162v3 sinh(3v) + 252v3sinh(v) + 6v3sinh(5v) 48v3sinh(4v) - 288v3sinh(2v)l -

-

-

-

-


4: Numerical Methods for the Solution of ID, 2D and 3 0 Differential Equations

bo =

~

17671 12096

96865 +-362880 45767 v 2 + 19160064

17608099 + 123122571264000

y’0

+

b1

b2 =

21971953 26 1534873600v 6

82561 v8 448345497600

1184824691 v12 75690284698828800

45767 1491199 32 1593093 189532561 v 8 60480 v 2 - 15966720v4 - 43589145600 v 6 - 532069747200

-

28082396599 460 150601 y’0 v12 38109367296000 113535427048243200

-

v14+ . . 16468224911 2058609391534080000

20483 4032

~

+

1864866399161 v14+ . . . 1 124000727777607680000 3937 2016

--

-

v4+

2943449 +-45767 v2 +------16128 3548160

10190684747 + 9484998082560

y10 -

v4

+

107557349v6 5074066909 v 8 792529920 348713164800 +

5994017812967 v , 2 103214024589312000

549960074207 + 240171095678976000

v 1 4 +.

12629 3024

b3 --

183

45767 y2 - 9837221 153204313 12096 7983360 v 4 - 653837184 v 6 ~

-

20347993339 y10 - 8744186458121 77 10518441984000‘I2 9700566220800

-

133502728560739 v 1 4 + . 28100018194440192000

2356782689 v 8 - 87178291200


184


+ 102v2sinh(2v) 81v2sinh(v) 45v2sinh(3v) 57 sinh(4v) + 39v2sinh(5v) + 65v cosh(4v) 24v2 sinh(4v) + 27 sinh(3v) - 39 sinh(v) + 24 sinh(2v) + 3 sinh(7v) 69v cosh(5v) + 3w2sinh(7v) + 30vcosh(6v) 18 sinh(6v) 81v + 15vcosh(3v) + 15v4sinh(v)

bo = [45 sinh(5v) - 18v2sinh(6v)

-

-

-

-

-

-

-

+ 155v cosh(v) - 110v cosh(2v) 5v cosh(7v)]/ [-2v4 sinh(3v) + 36v4sinh(v) + 12v4sinh(4v) - 27v4sinh(2v) 6v4sinh(5v) + v4 sinh(6v)l

- 5v4 sinh(3v)

-

-

+ 8v2sinh(6v) + 24v2sinh(2v) + 12v2sinh(v) + 6 sinh(8v) 16v2sinh(3v) 8v cosh(8v) + 30 sinh(4v) + 24v2 sinh(5v) + 30v4 sinh(2v) 15v4sinh(4v) + lOvcosh(4v) 30v2 sinh(4v) - 12 sinh(3v) - 30 sinh(v) + 24 sinh(2v) 30 sinh(7v) + 26v cosh(5v) + 3v2 sinh(8v) - 12v2sinh(7v) 60v cosh(6v) + 60v sinh(6v) + 174v 6v cosh(3v) 250v cosh(v)

bl = [-60 sinh(5v)

-

-

-

-

-

-

-

+ 76v cosh(2v) + 38v cosh(7v)]/[2v4 sinh(3v) -

12v4sinh(4v)

-

-

36v4sinh(v)

+ 27v4sinh(2v)l (47)

b2 = [-45 sinh(5v) + v 2 sinh(9v) + 15v2sinh(6v) - 21v2sinh(2v)

+ 99v2sinh(v) - 6 sinh(8v) - 33v2sinh(3v) + 2v cosh(8v) + 9 sinh(4v) + 15v2sinh(5v) 15v4sinh(5v) + 33v cosh(4v) - 12v2sinh(4v) + 9 sinh(3v) - 33 sinh(v) + 15 sinh(2v) 12 sinh(7v) + 9v cosh(5v) - 12v2sinh(7v) - 63v cosh(6v) - 3v cosh(9v) + 45 sin(v) + 3 sinh(9v) + 141v + 33v cosh(3v) + 75v4sinh(v) 167v cosh(v) 17v cosh(2v) + 32vcosh(7v)J/[-2v4 sinh(3v) + 36v4sinh(v) + 12v4sinh(4v) 27v4sinh(2v) 6v4sinh(5v) + v4 sinh(6v)l -

-

-

-

-

-

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Diferential Equations

185

b3 = [ 120 sinh(5v) + 2v2 sinh(9v) - 22v2sinh(6v) + 144v2sinh(2v)

+ 12v2sinh(v) -

- 24 sinh(8v) -

124v2sinh(3v)

+ 20v cosh(8v)

156 sinh(4v) + 60v2sinh(5v) + 75v4sinh(2v) - 5v4 cosh(6v)

+ 176vcosh(4v) 12v2sinh(4v) + 96 sinh(3v) - 84 sinh(v) + 30 sinh(2v) + 42 sinh(7v) - 172v cosh(5v) - 6v2sinh(8v) + 6v2sinh(7v) + 54v cosh(6v) 6v cosh(9v) - 66 sinh(6v) + 6 sinh(9v) -

30v4sinh(4v)

-

-

-

+

2 2 8 ~ 1 0 8cosh(3~) ~ + 4 7 6 cosh(v) ~ - 4 0 6 cosh(2~) ~ - 2 2 C~O S ~ ( ~ Y ) ] / [2v4sinh(3v) - 36v4sinh(v) - 12v4sinh(4v)

+ 27v4sinh(2v)

+ 6v4sinh(5v) - v4 sinh(6v)J where v = wh. For small values of v the above formulae are subject to heavy cancellations. In this case we use the following Taylor series expansions:

bo =

22 153 45767 v 2 +456 1920v4 24 1920

17671 -+12096

+

41092123 7321421 v 8 130767436800v 6 - 348713164800

5642643317 y10 v,2 210863655707 2134124568576000 681212562289459200 364884558191 y'4 + ... + 10035720783728640000 3937 2016

b 1 --

b2

45767 v2 - 5 1408821 35318011 y 8 8607 40320 39424v4 - 27243216OOv6- 34871316480

-

3348191339 y10 5610471 1163 v , 2 118562476032000 43667471941632000

-

1538176483573 v14 31222242438266880000

20483 4032 -

2943449 +-45767 v 2 +16128 3548160

+

v4

. ..

+

107557349v 6 792529920

+

10190684747 y10 - 5994017812967 v 1 2 9484998082560 103214024589312000 549960074207

+ 240 171095678976000

v14

+

...

5074066909 v 8 348713164800

186


b3 -- - 12629 ~ 3024

45767 9837221 153204313 2356782689 v 8 y2 12096 7983360 v4 - 653837184 v 6 - 87178291200 ~

- 20347993339 y10

9700566220800 -

-

8744186458121 v 1 2 77410518441984000

133502728560739 v 1 4 28100018194440192000

+

...


THOD LMMIV- CASE IV

+ 12 sinh(7v) - 12 sinh(5v) - 48 sinh(2v) + 84 sinh(v) - 36 sinh(3v) - 36v - 60v3 + 60 sinh(4v) + 90v3cosh(2v) + 62v2 sinh(v)

bo = [-24 sinh(6v)

+ 8 1cosh(5~) ~ - 2 7 cosh(7~) ~ - 2 7 cosh(v) ~ - 2 7 cosh(3~) ~ + 126v2sinh(3v) - 36v3cosh(4v) + 144v cosh(2v) + 36v cosh(6v) + 60v3cosh(5v)- 108v3cosh(3v) + 6v3cosh(6v) - 144v cosh(4v) + 60v3cosh(v) - 22v2 sinh(6v) + 26v3 sinh(7v) - 12v3cosh(7v)

-

110v2sinh(5v) - 200v2 sinh(2v)

+ 45v5cosh(2~)

-3

+ 1 1 5 ~sinh(4v)]/[-18v5 ’

cosh(4v)

0 ~+ ’ 3 ~ C’ O S ~ ( ~ V ) ]

+ 36 sinh(5v) + 48 sinh(2v) - 48 sinh(v) + 12 sinh(3v) + 12 sinh(8v) - 48v + 50v3 - 48 sinh(4v) - 54v3cosh(2v) - 244v2 sinh(v)

bl = [-24 sinh(7v)

+

+

- 8 7 cosh(5~) ~ 3 0 cosh(7~) ~ 8 4 cosh(v) ~ -27~~0sh(3~) - 22v2 sinh(3v) - 12v3cosh(4v)

- 24v

cosh(2v)

+ 48v cosh(6v)

+ 22v3cosh(6v) + 48v cosh(4v) - 60v3sinh(6v) - 12v2sinh(7v) + 50v2sinh(5v) + 184v2sinh(2v) - 4v2sinh(4v) + 19v2sinh(8v) - 2 4 cosh(8~) ~ - 6v3C O S ~ ( ~ V ) ] / cosh(4~) [-~V~

+ 15v5cosh(2v) - 10v5+ v5cosh(6v)l

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Diflerential Equations

187

+ 60 sinh(5v) - 12 sinh(9v) + 84 sinh(2v) 132 sinh(v) + 36 sinh(3v) + 24 sinh(8v) 60v - 14v2sinh(9v) + 4v3cosh(9v) + 21v cosh(9v) + 10v3 108 sinh(4v) - 15v3cosh(2v) 194v2sinh(v) - 165vcosh(5v) + 6vcosh(7v) + 159vcosh(v) - 2 l v cosh(3v) - 306v2sinh(3v) + 6v3cosh(4v) - 186v cosh(2v) + 6v cosh(6v) 60v3cosh(5v) + 140v3cosh(3v) - v3cosh(6v) + 264v cosh(4v) 84v3cosh(v) v 2 sinh(6v) + 4v2sinh(7v) + 170v2sinh(5v) + 42 1v2sinh(2v) 131v2sinh(4v) + 6v2sinh(8v)

b2 =

-

[ 12 sinh(6v) - 24 sinh(7v)

-

-

-

-

-

-

-

-

- 24vcosh(8v)]/[6v5 cosh(4v) - 15v5cosh(2v)

b3 = [36 sinh(6v)

+ 10v5 - v 5 cosh(6v)l

+ 156 sinh(7v) - 276 sinh(5v) + 24 sinh(9v) - 336 sinh(2v)

+ 324v3cosh( 1Ov) 84 sinh(3v) + 18v cosh( 1Ov) - 1lv2sinh( 1Ov) + 3v3cosh( 1Ov) 96 sinh(8v) + 432v + 4v2sinh(9v) 18w cosh(9v) 378v3 + 336 sinh(4v) - 12 sinh(l0v) + 420v3 cosh(2v) + 1614v2sinh(v) + 6 7 5 ~ ~ 0 s h (-5 ~1 )7 1 ~ ~ 0 ~ h ( 747vcosh(v) 7~) +2 6 1 ~ ~ 0 ~ h ( 3 ~ ) + 3 10v2sinh(3v) + 60v3cosh(4v) + 216v cosh(2v) 378v cosh(6v) -

-

-

-

-

-

-

-

135v3 cosh(6v) - 4 3 2 cosh(4v) ~ + 441v2sinh(6~)+ 50v2 sinh(7~) 298v2 sinh(5v) - 1340v2sinh(2v) - 124v2sinh(4v) - 1 0 0 ~ sinh(8v) ’

+ 144v cosh(8v) + 30v3cosh(8v)]/[ 18v’ cosh(4v) -

4%’ cosh(2v) + 30v5 - 3v5cosh(6v)I

where Y = wh. For small values of v the above formulae are subject to heavy cancellations. In this case we use the following Taylor series expansions: 17671 45767 bo = 12096 181440v 2

135959 14453093 90901339 v 8 47900 160 v 4 + 16345929600 v 6 - 896690995200

3513993676211 1564247467 y’0 v,2 106706228428800 1703031405723648000 31447292193 4077316614312701 y’4 y16 + . . . 1094805903679488000 103408066955539906560000

188


3937 b1 -- ---2016

45767 y2 30240

-

3156581 2 1796097 2365857293 y 8 7983360 y 4 - 681080400y 6 - 1046139494400

~

102137141 y 1 0 - 3 198002983423 y12 17784371404800 283838567620608000

+

1774367017771 y ’ 4 229698448358107 y16 1277273554292736000 1148978521728221184000

+

... (5 1)

20483 45767 b2 = v2 4032 12096

+-

3549253 y 4 2280960

+-

1385811370311

+

36881797y 6 99066240

106905916402097

-I-35568742809600’Io

+

957 14204623 v8 2092278988800

v12

i- 567677135241216000

553568118810067 2724564546449 y ’ 4 y16 + . . . -I-23223 1555325952000 313357778653151232000 12629 3024

b3 --

45767 27865393 557684327 9072 ” - 11975040 y4 - 8 17296480 y 6

- 575696865983

y10 -

26676557107200 -

-

235111157089 v8 1569209241600

73845973877087 y 1 2 32750603956224000

28484706610277 y 1 4 - 264201425989315289 v,6 174173666494464000 25852016738884976640000

+

.. .


We note here that a method of the form (36) which integrates the functions of the Case I (i.e. the methods LMMI) has been constructed in a different way by Simos and Aguiar.s We also note that the corresponding trigonometrically fitted case for the methods LMMI-LMMIV mentioned above is obtained by the substitution of v = iv.

2.6.2 Second Family of Methods - Case to # 2. - We require that the family of methods (36) should integrate exactly any linear combination of the functions: Case V

In order to construct a method of the form (36) that integrates exactly the functions (53), we require that the method (36) integrates exactly the functions:

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Dzflerential Equations

189

and then put Case V

wo = WI = w2 = w3 = w4 = w

(55)

Demanding that (36) integrates (54) exactly, we obtain the following system of equations for the coefficients ao, bi(i= O( 1)3:

+

+2 COS~(~Y,) = v:(2bo C O S ~ ( ~+ V 2bl , ) cosh(2vj) + 2b2 C O S ( V ~+ ) b3)

4 C O S ~ ( ~ Y2, )to C O S ~ ( ~-V2, )cosh(v,)

(56)

where v j = w j h ( j= 0(1)4. Solving the above system for ao, bi(i = O( 1)3 we obtain:

+ 4284v2 cosh(4v) - 936v sinh(3v) - 290 sinh(v)v3 + 120v sinh(5v) + 2040 sinh(v)v + 375v2cosh(v) + 4728v3sinh(2v) + 24v sinh(7v) + 50v3sinh(5v) - 2v3 sinh(7v) + 18v3sinh(3v) + 864v sinh(2v) 472v3 sinh(8v) 567v2cosh(3vj + 120vsinh( 1Ov) 20 16v sinh(4v) + 1944v sinh(6v) - 8 16v sinh(8v) + 50v3sinh(l0v) 4488v3sinh(4v) 3 v 2 cosh(7v) + 195v2cosh(5v) - 1 0 5 ~cosh(l0v) ’ - 2979v2 cosh(6v) + 888v2 cosh(8v) + 1962v3sinh(6v) + 60 cosh(7v) 1 8 0 +~ 540 ~ cosh(3~) 6 cosh(l0v) + 2160 cosh(2~) - 180 cosh(6v) + 240 cosh(8v) + 1512v4 12v4 cosh( 1Ov) + 120v4 cosh(8v) 5 4 0 cosh(6~) ~ ~ + 1 4 4 0 cosh(4~) ~~ 2 5 2 0 cosh(2~) ~~ - 300 cosh(v)

to = [- 1440 - 1908v2cosh(2v)

-

-

-

-

-

-

-

-

-

- 300 cosh(5v)

-

-

720 cosh(4v)]/[6v3 sinh(9v) - 72v3sinh(7v)

+ 18OOv sinh(5v) 540 cosh(v) 1377v2cosh(v) + 450v3 sinh(5v) + 3186 sinh(v)v3 + 3672 sinh(v)v - 60 cosh(9v) 1662v3sinh(3v) - 33v2cosh(9v) + 360 cosh(7v) - 576v sinh(7v) + 72v sinh(9v) + 342v2cosh(7v) 1395v2cosh(5v) + 1140 cosh(3v) - 900 cosh(5~) + 2 4 6 3 C~ O~ S ~ ( ~ V ) ] - 3096v sinh(3v)

-

-

-

-

190

Chemical Modelling: Applications and meory, Volume 2

b, = [-324 - 2 9 3 7 cosh(v) ~~

+ 1 8 7 8 cosh(2~) ~~

-4

7 1 0 cosh(4~) ~~

+ 870 sinh(v)v5 + 684 sinh(v)v3

- 432v sinh(3v)

- 150v5sinh(5v)

+ 720v sinh(5v)

-

- 144v sinh(7v)

+ 540v3sinh(5v) - 36v3sinh(7v) - 3348v3sinh(3v)

- 288v sinh(2v)

+ 4095v3sinh(8v) + 2178v2cosh(3v) - 180v sinh(l0v)

1296 sinh(v)v - 1242v2cosh(v) - 33 12v3sinh(2v)

+ 1440vsinh(4v) - 1764v sinh(6v) + 900vsinh(8v) - 675v3sinh(l0v) + 9720v3sinh(4v) + 114v2cosh(7v) - 1050v2cosh(5v) - 54v3sinh(3v)

+ 435v2cosh( 1Ov) + 4887v2cosh(6v) 2355v2cosh(8v) - 9279v3sinh(6v) + 72 cosh(7v) + 7 5 6 0 cosh(2v) ~~ 4320~ +~1 6 2 0 cosh(6v) ~~ -

-

-

360v6 cosh(8v)

+ 36v6cosh( 1Ov)

- 4 3 4 4 sinh(2v) ~~

135v2 - 2 1Ovs sinh( 1Ov)

+ 8016v5sinh(4v) - 216 cosh(3v)

+ 36 cosh( 1Ov) + 504 cosh(2v) - 180 cosh(8v) - 2826v4 -

-

-

5 9 9 4 ~ sinh(6v) ’

-4

536~~

+ 324 cosh(6v)

+ 1 8 3 6 ~sinh(8v) ’ + 529v4cosh(l0v)

+

3 8 3 0 cosh(8~) ~~ 1 0 1 0 7 cosh(6~) ~~ - 1 2 0 4 0 cosh(4~) ~~

+ 8492v4cosh(2v) + 6v5sinh(7v) + 216 cosh(v) - 19v4cosh(7v)

+ 67v4cosh(5~)+ 1 7 3 7 cosh(3~) ~~ - 72 cosh(5~)- 360 C O S ~ ( ~ V ) ] / [- 1620v4cosh(v) - 4 131v6 cosh(v)

+ 5400 sinh(5v) + 9558 sinh(v)v7

+ 11016 sinh(v)v5

- 9288v5 sinh(3v)

+ 18v7sinh(9v)

99v6cosh(9v) - 4986v7 sinh(3v)

-

- 172812 sinh(7v)

- 180v4cosh(9v) - 216v7 sinh(7v)

+ 1350v7sinh(5v)

+ 2 16v5sinh(9v) + 1080v4cosh(7v) - 2700v4 cosh(5v)

+ 3420v4cosh(3v) + 1 0 2 6 cosh(7v) ~~ + 7 3 8 9 cosh(3v) ~~

-4

1 8 5 cosh(5v)l ~~

191


b, = [ -72 -

-

1406v4cosh(v)

+ 48 cosh(9v) + 52v sinh( 11v) - 120v2cosh(2v) + 1200v5sinh(5v) - 228 sinh(v)v5

152v2cosh(4v) - 576v sinh(3v)

+ 1592 sinh(v)v’ + 260v sinh(5v) + 352 sinh(v)v

-

616v2cosh(v)

+ 1440v3sinh(2v) + 176vsinh(7v) + 1975v3sinh(5v) + 4v3sinh(7v) + 480v sinh(2v)

-

3 5 0 4 sinh(3v) ~~

-

64v sinh(4v) - 160v sinh(6v) + 32v sinh(8v)

- 236v2 cosh(7v)

-

-

32v2sinh(8v)

+ 1344v2cosh(3v)

+ 256v3sinh(4v)

775v2 cosh(5v) - 1628v5sinh(3v)

+ 120v2cosh(6v)

+ 2v2cosh(8v) + 160v3sinh(6v) - 56v4cosh( 1lv) - 48 cosh(7v)

+ 107v3sinh( 1I v) + 150v2

+ 2 7 4 cosh(9~) ~ ~

-

-

48 cosh(3v) + 12v5 sinh( 11v)

+

+ 128v4cosh(4v) - 480v4cosh(2v) + 48 cosh(v)

-

+

24 cosh(8~) 1 2 7 2 ~ ~ 8v4cosh(8~)- 1 6 0 cosh(6~) ~ ~ 12 cosh( 1lv) - 48v5 sinh(7v)

-

101v2cosh( 1 lv) - 76v5sinh(9v)

+

+ 1 16v4cosh(7v)

+

-

2302 cosh(5~) 3 0 8 6 cosh(3~) ~~ 12 cosh(5~)+ 3 8 4 cosh(9~) ~ ~

-

192v sinh(9v) - 4 3 2 ~ sinh(9v) ’

- 459v6 cosh(v)

+ 96 cosh(4v)]/[ - 180v4cosh(v)

+ 600v5sinh(5v) + 1062 sinh(v)v7 + 1224 sinh(v)v5

- 1032v5sinh(3v) - 20v4 cosh(9v)

-

1lv6 cosh(9v) - 554v7sinh(3v)

+ 24v5 sinh(9v) + 120v4cosh(7v)

-

24v7sinh(7v)

+ 2v7sinh(9v)

+ 150v7sinh(5v) - 192v5sinh(7v) -

+ 1 14v6cosh(7v) + 821v6cosh(3v)

300v4cosh(5v)

-

+ 380v4cosh(3v)

465v6cosh(5v)I (57)

192


b2 = [-156 - 1 9 7 7 cosh(v) ~~ + 2 4 ~ 0 s h ( 9 -~ )1 0 9 4 cosh(2~) ~~ - 1307v2cosh(4v) - 756 sinh(v)v3

- 496v sinh(3v) -

+ 400v sinh(5v)

-

1 5 0 ~sinh(5v) ’ - 1062 sinh(v)v5

816 sinh(v)v - 402v2 cosh(v)

+ 2456v3sinh(2v) + 64v sinh(7v) - 460v3 sinh(5v) - 64v3sinh(7v) - 2684v3 sinh(3v) + 64v sinh(2v) + 1 4 6 5 sinh(8v) ~~ + 774v2cosh(3v) + 40v sinh( 1Ov) + 836v sinh(4v) - 1064v sinh(6v) + 380v sinh(8v) 10v3sinh(l0v) + 231 lv3sinh(4v) - 4v2cosh(7v) - 350v2cosh(5v) + 554v2sinh(3v) 10v2cosh( 1Ov) + 2352v4cosh(6v) 1045v2cosh(8v) -

-

+ 48 cosh(7v) + 1035v2 + 3 3 8 4 sin(2v) ~~

- 3454v3sinh(6v) -

-

588v5sinh(4v) - 168cosh(3v) - 24 cosh( 1Ov) + 7v4cosh(9v)

+ 168 cosh(2v)

-

1128v5sinh(6v)

+ 144 cosh(6v) - 60 cosh(8v)

+ 12 cosh( 12v) + 2106v4 + 300v5sinh(8v) - 61v3sinh( 12v) + 4v4cash( 1 0 ~ ) 1 0 3 0 cosh(8~) ~~ + 2 9 3 2 cosh(6~) ~~ - 2 6 2 cosh(4~) ~ ~ - 3056v4cosh(2v) + 24v5 sinh(7v) + 2 16 cosh(v) - 4v5sinh( 12v) -

- 2v5 sinh(9v)

+ 26v4cosh( 12v) - 26v4 cosh(7v) + 565v4cosh(5v)

- 489v4 cosh(3v) -

+ 4v3sinh(9v)

-

120 cosh(5v) - 18 v 2 cosh(9v) - 16v sinh(9v)

44v sinh( 12v) - 84 cosh(4v)

[- 180v4cosh(v) - 459v6 cosh(v)

+ 1224 sinh(v)v5

-

+ 2v7sinh(9v)

1lv6cosh(9v)

-

- 1 9 2 ~ sinh(7v) ’

+ 69v2cosh( 12v)]/

+ 600v5sinh(5v) + 1062 sinh(v)v7

1 0 3 2 sinh(3v) ~~ - 20v4 cosh(9v) - 24v7 sinh(7v) -

5 5 4 ~ sinh(3v) ’

+ 150v7sinh(5v)

+ 24v5 sinh(9v) + 120v4cosh(7v) - 300v4cosh(5v)

+ 380v4cosh(3v) + 114v6cosh(7v) + 821v6cosh(3v) - 465v6 cosh(5v)I

4: Numerical Methods for the Solution of 1 0 , 2 0 and 3 0 Diflereential Equations

b3 =

193

+ 1 9 3 8 4 cosh(v) ~~ - 972 cosh(9v) - 864v sinh(1 lv) + 2388v2cosh(2v) + 2688v2cosh(4v) + 1 1 8 4 4sinh(3v) ~ 1 9 4 4 0 sinh(5v) ~~ + 25320 sinh(v)v5 + 6552 sinh(v)v3 - 5040v sinh(5v) 6048 sinh(v)v + 1 2 5 0 4 cosh(v) ~~ 32832~ sinh(2v) ~ - 4284v sinh(7v) 30060~ sinh(5v) ~ - 2601v3 sinh(7v) + 5 1 4 7 1 sinh(3v) ~~ 1 0 3 6 8 sinh(2v) ~ + 288v3sinh(8v) - 295 1lv2cosh(3v) + 576v sinh(4v) + 3456v sinh(6v) 288v sinh(8v) - 1 0 9 4 4 sinh(4v) ~~ + 3969v2cosh(7v) + 1 8 9 0 0 cosh(5v) ~~ + 2 1 7 6 2 ~sinh(3v) ~ - 30v2cosh(l0v) 2358v2cosh(6v) 168v2cosh( 8v) 5 184v3sinh(6v) + 742v4cosh( 11v) + 972 cosh(7v) - 1584v3sinh( 1lv) 2520v2cosh(3v) + 72 cosh( 1Ov) 120v5sinh(l1v) 5391v4cosh(9v) 108vsinh(l3v) + 144cosh(2v) 216 cosh(6~)+ 576 cosh(8~) 2 6 2 0 8 cash( ~ ~ 1 0 ~ ) 32v4 cosh(8~) + 2952v4cosh(6v) 3712v4cosh(4v) + 3952v4cosh(2v) - 6v5sinh(l3v) + 144 cosh( 1lv) - 594d sinh(7v) 1 152cosh(v) - 1 17v3sinh( 13v) + 1632v2cosh(1 lv) + 1350v2sinh(9v) + 1071v4cosh(7v) + 4 1 8 5 0 cosh(5~) ~~ -49059~ cosh(3~) ~ + 3 6 ~ 0 ~ h ( 1+3 ~43v4 ) cosh(l3~) 7641v2cosh(9v) + 40681, sinh(9v) + 8703v3 sinh(9v) + 147v2cosh( 1311) 2304 cosh(4v)]/[-1620v4 cosh(v) 4 1 3 1 cosh(v) ~~ + 5400v5sinh(5v) + 9558 s1nh(v)v7 + 11016 sinh(v)v5 - 9288v5sinh(3v) - 180v4cosh(9v) 2 1 6 ~ sinh(7v) ’ + 18v’ sinh(9v) 99v6cosh(9v) - 4 9 8 6 ~ ’sinh(3v) + 1 3 5 0 ~sinh(5v) ’ 1 7 2 8 sinh(7v) ~~ + 216v5sinh(9v) + 1080v4cosh(7v) 2 7 0 0 cosh(5~) ~~ + 3 4 2 0 cosh(3~) ~~ + 1 0 2 6 cosh(7~) ~~ + 7 3 8 9 cosh(3v) ~~ - 4 1 8 5 cosh(5v)l ~~ -

[ 1728

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-



194

t - - 2 - - - -45767 -y100 -

1451520

17671 + - 45767 bo = v2-12096 145152

2389 9377033 34164329 v 8 2395008v4 +498161664OV6 - 114124308480

-

6173666447789 y10 3201186852864000

-

94160670614060173 v14 1348800873333129216000 3937 2016

b 1 --

b2 =

164627 y12 - 1978139 v14 19160064 1268047872

-

78047984217869 v12 291948240981196800

45767v2 --71237 109848019 24192 114048 v4 - 2490808320v 6

-

2268523525897 V'O - 539411389537379 v12 66691392768000 4865804016352800

-

2190469739609771 v 1 4 910122046783488000

20483 4032

___

2001803 v 2 ++-228835 48384 798336

v4

+

-

999525767 v 8 209227898880

71501777 40856602993 v 8 90574848 v6 -I- 4 18455797760

-

10197353667901 y10 - 1094269095668321 v 1 2 213412456857600 97316080327065600

-

13204586585862157 v 1 4 89920058222208614400

b3 -- - 12629 3024

228835 4507043 372263873 15054000099 v 8 ~ y2 36288 1 197504 v4 - 249080832 v6 - 3 13841848320 ~

-

457772474221 y10 - 185791672599151 2500927228800 4293356485017600

-

128871120244918297 v 1 4 13488008733331292160

,2

The Local Truncation Error (LTE) of the above method is given by: LTELMMy

=45767 ~ l o ( y j l l o ) - 5v2yJ18'-

725760

lOv6y'n4' + 5v*y?' - v'Oy,)

(59)

In Appendix A a Maple program for the production of the above formulae (41)(42), (44)-(45), (47)-(48), (50)-(51) and (57)-(58) is presented. We also note that the corresponding trigonometrically fitted case for the method LMMV mentioned above is obtained by the substitution of v = iv. 2.6.3. Stability Analysis. - In the last decade there has been great interest in the numerical solution of special second order periodic initial-value problems (see refs. 6, 7 and references therein)

4: Numerical Methods for the Solution of 10, 2 0 and 3 0 Differential Equations

195

In order to investigate the periodic stability properties of numerical methods for solving the initial-value problem (1) Lambert and Watson8 introduced the scalar test equation

and the interval of periodicity. Based on the theory developed in ref. 8, when a linear symmetric multistep method k

k

j=O

/=O

is applied to the scalar test equation (61), a difference of the form

is obtained, where H = qh, h is the step length and y n is the computed approximation to y(xo + nh), In = 0, 1, 2, . . .. The general solution of the above difference equation is given by:

where Qj, j = l(1)k are the distinct roots of the polynomial

where p and o are polynomials given by:

We note here that the roots of the polynomial (65) are perturbations of the roots of p. We denote as Ql and Q2 the perturbations of the principal roots of p. Based on Lambert and Watson,8 when a symmetric multistep method is applied to the scalar test equation y ” = - q 2 y a difference equation (63) is obtained. The characteristic equation associated with (63) is given by:

196


The roots of the characteristic polynomial (65) are denoted as Qili = l(1)k. We have the following definitions.

Definition3. Following Lambert and Watson’ we say that the numerical method (62) has an interval of periodicity (0, Hi), if, for all H2 E (0, Hi), Qili= l(l)k satisfy:

Definition 4. (see ref. 8). The method (4) is P-stable if its interval of periodicity is equal to (0, 00). For the families of symmetric linear multistep methods developed in this review we have that the polynomials p and a are given by (66) where k = 8 and a. = as = 1,

a, = al = to,

a2 = a6 = 2,

a3 = a, = -1,

a4 = 0

(69)

where to is given by (57)-(58) and the coefficients:

are given by: Method LMMI- Case I: Equations (41)-(42) Method LMMII- Case II: Equations (44) - (45) Method LMMIII- Case HI: Equations (4 7) - (48) Method LMMIV- Case IF Equations (50)-(51) Method LMMV- Case K Equations (57)-(58) In Figures 1-5 and based on the above theory and on the coefficients given above we present the stability polynomial for the Linear Multistep Methods LMMI - LMMV. In Table 1 and based on the above mentioned Figures we present the interval of periodicity for the Linear Multistep Methods LMMI - LMMV developed in this review.

2.7 Numerical Examples. - In this section, we apply the symplectic methods presented above to three well-known problems. The first two are inhomogeneous non-autonomus Hamiltonian problems and the last a homogeneous Hamiltonian problem but with two frequencies. 2.7.1 Inhomogeneous Equation. - We consider the following problem, y ” = -lOOy+99sinx,

y(0) = 1,

whose analytical solution is y(x) = cos 1Ox

y’(0) = 11,

+ sin 1Ox + sin x.

(71)


197

Stability Polynomial of LMMI

Figure 1 Stability Polynomial for LMMI

Stability Polynomial of LMMll

Figure 2 Stability Polynomial for LMMII

Equation (71) has been solved numerically for 0 6 x 1000 using the well known Runge-Kutta-Nystrom eighth algebraic order method" (which is indicated as Method MI), the classical second algebraic order symplectic RungeKutta-Nystrom method [see equations (3 1)-(32)] with w -+0 (which is indicated as Method MII), the classical second algebraic order symplectic Runge-KuttaNystrom method presented in ref. 4 (which is indicated as Method MIII), the new trigonometrically fitted second algebraic order symplectic Runge-Kutta-Nystrom method TFSI developed in this review (which is indicated as Method MIV), the new FSAL trigonometrically fitted method S13 [see equations (23)-(26)] (which

198

Chemical Modelling: Applications and Theory, Volume 2 Stability Polynomial of LMMlIl

Figure 3 Stability Polynomial for LMMIII

Stability Polynomial of LMMIV

Figure 4 Stability Polynomial for LMMIV

is indicated as Method MV), the other new trigonometrically fitted symplectic Runge-Kutta-Nystrom S12 [see formulae (13)-( 17)] (which is indicated as Method MVI), the classical linear multistep symmetric eighth algebraic order (which is indicated as Method MVII) and method developed by Quinlan et the linear symmetric multistep method LMMI [see equations (41)-(43)] (which is indicated as Method MVIII). For this problem, w = 10. In Figure 6, we present the absolute maximum error aL97"

4: Numerical Methods for the Solution of 10,2D and 3 0 Differential Equations

199

Stability Polynomial of LMMV

Figure 5 Stability Polynomialfor LMMV

Table 1 Properties of Symmetric Linear Multistep Methods Method

Algebraic order

Quinlan and Tremaine (Cla~sical)~" LMMI' LMMII LMMIII LMMIV LMMV

8 8 8 8

8 8

Interval of periodicity (0, H i )

(0, 0.57) (0, 0.90) (0, 1.22) (0, 7.36) (0, 3.48) (0, 3.64)

*Based on the Definition 1

for the number of function evaluations which are equal to NFE X 1000. The nonexistence of a value of Errmaxfor some of the Methods MI-MVIII indicates that for this number of function evaluations the values of Errmaxare unaccepted, i.e. positive. For this problem the frequency is equal to w = 10. 2.7.2 Dufin k Equation. tion,

-

We consider the non-linear undamped Duffin's equa-

y"

+ y + y 3 = B cos(wx),

(73)

where B = 0.002 and o = 1.01. The analytical solution of the above equation is given by

Chemical Modelling: Applications and Theoq blume 2

200 0

2

x 1000

E

L

w

4

6 l

~

Err,,,for

l

~

l

~

l

several values of NFE x 1000

MVll MVlll

L

Lu

-1 2

1 0

500

1000

1500

2000

2500

NFE x I000

Figure 6 Errmmfor the number of function evaluations which are equal to NFE Inhomogeneous equation.

X

1000.

~

4: Numerical Methods for the Solution of 10, 2 0 and 3 0 Diflerential Equations

20 1

where A l = 0.200179477536,

A3 = 0.246946143 X

A5 = 0.304016 X

A , = 0.374 X

(75)

Equation (73) has been solved numerically for 0 6 x 6 1000 using the methods mentioned above (see Section 2.7.1). For this problem the initial condition is of the form

In Figure 7, we present the absolute maximum error Err- [see equation (72)] for the number of hnction evaluations, which are equal to NFE X 1000. The nonexistence of a value of Errma for some of the Methods MI-MVIII indicates that for this number of hnction evaluations the values of Errmx are unaccepted, i.e. positive. For this problem the frequency is equal to MI = 1.

2.7.3 An Orbit Problem Studied by Stiefel and BettisI2-l5.- We consider the following ‘almost’ periodic orbit problem studied by Stiefel and bet ti^,'*-'^

z” + z

= 0.001e’”,

z(0) = 1,

z’(0) = 0.99951,

z E C,

(77)

whose analytical solution is given by:

+ iv(x), u, Y E R, u(x) = cosx + 0.0005~ sinx, u, Y E R, z(x) = u(x)

(78)

~ ( x= ) xin x - 0.0005~ cos x.

The solution of equation (77) represents motion on a perturbation of a circular orbit in the complex plane. We write equation (77) in the equivalent form, u”

+ u = 0.001 cosx,

u(0) = 1,

u’(0) = 0,

Y”

+ v = 0.001 sinx,

v(0) = 0,

v’(0)= 0.9995.

(79)

The equivalent system of equations (79) has been solved numerically for 0 d x d 1000 using the above mentioned methods (see Section 2.7.1). In Figure 8, we present the absolute maximum error Errmm(see equation (72)) for the number of function evaluations, which are equal to NFE X 1000. The non-

202

Chemical Modelling: Applications and Theory, Volume 2 0

Errma,forseveral values of N F E x 1000 MI MI1 Mlll MIV

__t_

* +

2

X

kE

-4

W

6

1 -8 200

400

600

N F E x 1000

Err,,,for

several values of N F E x 1000 MV __6_ MVI MVII

2

__t_

X

t' -4 W

6

0

400

200

600

N F E x 1000

Figure 7 Errma for the number of function evaluations which are equal to NFE X 1000. Dufin b equation.

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Drflerential Equations

203

0

2

.

t'

-4

w

6

a

1 '

I

I

I 200

0

I 400

I

3 600

NFE x I 0 0 0

0

4

Err,.,for

a

several values of NFE

+MV +MVI

+MVll

E

L

w

MVlll -12

-16

-20

1 0

200

400

600

NFE x 1000

Figure 8 Err,, for the number of function evaluations which are equal to NFE Stiefel and Bettis problem.

X

1000.


204

existence of a value of Errma for some of the Methods MI-MVIII indicates that for this number of h c t i o n evaluations the values of Errma are unaccepted, i.e. positive. For this problem the frequency is equal to w = 1.

3 Dissipative Methods For the numerical solution of the problems described in Section 1 a new class of methods has been developed the last three years. These methods are called ‘dissipative methods’.

Definition 5. A method is called dissipative when it is non-symmetric. Remark 1. A non-symmetric (i.e. dissipative) multistep method has empty interval of periodicity. For the numerical solution of the type of the problems described in Section 1, the most important properties are the following: (i) (ii) (iii) (iv) (v) (vi)

algebraic order of the method, the interval of periodicity of the method, The minimization of the phase-lag of the method, the symmetry of the method, the exponential fitting and, in special cases, the adaptive properties such as Bessel and Neumann fitting.

More details of the above are given in refs. 6 and 16. The development of methods with these properties is an open problem. The methods for the solution of the problems described in Section 1 are divided into two categories based on Simos.6.’6The categories are: (1) Methods with constant coefficients and (2) Methods with coefficients dependent on the frequency of the problem. The most important properties for the construction of methods of the first category are the properties (i)-(iv) mentioned above while the most important properties for the construction of methods of the second category are (i), (ii), (iv) and (v) or (vi) mentioned above. For the numerical solution of the problems described in Section 1 much For complete reference to the research has been done in the last two methods developed for the solution of the problems described in Section 1 (with constant coefficients and with coefficients dependent on the frequency of the problem) see refs. 6, 16, 23 and references therein. It is well known that the most finite difference methods developed in the literature for the numerical solution of the problems described in Section 1 belong to the class of multistep and hybrid techniques. Papageorgiou, Tsitouras and F a m e l i ~in~ ~a very recent paper have


205

developed a new class of Runge-Kutta-Nystrom type methods which are not symmetric, i.e. are dissipative. The methods developed by Papageorgiou et al. are multistep but have the logic of Runge-Kutta-Nystrom methods since they have stages. Some inaccurate remarks on the construction of methods for this type of problems are made in the paper of Papageorgiou et al. as follows. (1) The methods of the second category (i.e. the methods with coefficients dependent on the frequency of the problem) are not very useful since in most of the cases the frequency of the problem is not known. For most of the problems of physics and chemistry this remark is not correct. In most of these cases the frequencies of the problems are known or can be determined very effectively (for example Schrodinger equation, Celestial Mechanics, Hamiltonian problems etc. For example, in satellite orbits the frequency is known with the accuracy of the J 2 coefficient while in the radial, for example, Schrodinger equation the frequency of the problem is given by: Frequency =

Ji

+ V(x)

-

El.

See for more details refs. 6, 16 and references therein. (2) The methods of the second category are not useful when the problem has many frequencies.

From the theory of exponentially fitted methods (see refs. 6 and 23) it is very well known that the exponentially fitted methods developed in the literature have been developed for many frequencies and in the specific papers some results for special cases are given, i.e. some results for special values of multifrequencies are given in each paper (see all the literature of exponentially fitted methods and for example see the review paper,6 the dissertationI6 and finally the very old paper of Stiefel and BettisI2 and the paper of rap ti^^^). In these papers one can see that the exponentially fitted methods which have been developed in the literature were constructed in the beginning for multifrequency cases and then, for the specific case of each paper (e.g. for Orbital Problems or for the Schrodinger equation), the frequencies are defined and the specific methods are produced. We must note here that in the literature there are exponentially fitted methods developed for four frequencies (see the paper by Cash et alS2')and also exponentially fitted methods developed for up to five frequencies [see the method (36) developed in this review and the Cases I-V]. ( 3 ) The requirement of non empty interval of periodicity is not necessary since the minimal phase-lag property seems more important. This remark of Papageorgiou et a1.24is based on a remark of SimosI6 in which

206


he has mentioned that the phase-lag property is more important than the nonempty interval of periodicity. But Simos has proved16 that the non-empty interval of periodicity and the minimal phase-lag properties are necessary conditions for the development of the methods for the numerical solution of problems with oscillating solution. The meaning of the remark of Simos is that the two properties are necessary but if one requires optimization of a property then the preferable property is the optimization of the phase-lag under the condition that the property of the non-empty interval of periodicity is satisfied. The developments of methods without interval of periodicity leads to non-efficient methods for the solution of the problems described in Section 1. Dissipative methods (see Avdelas et aL2’)have been constructed for the solution of specific problems and in that case a special strategy for the application (see ref. 23) is required. The authors of ref. 24 don’t follow this specific procedure since they don’t study the solution of the Schrodinger equation. Another disadvantage of the methods developed in ref. 24 is that these methods additionally with the empty interval of periodicity are also non-symmetric. This is important since symmetry is equivalent to syrnplecticness (see Section 2.6). So, these methods are inefficient for the solution for Hamiltonian problems i.e. for example for the numerical solution of all Schrodinger type equations.

3.1 Phase-lag of Non-symmetric (Dissipative) Two-step Methods. - In this section we study the numerical solution of the problem:

In order to examine the stability properties of methods for solving the initialvalue problem described by equation (Sl), Lambert and Watson* have introduced the scalar test equation

and its interval of periodicity. When we apply a non-symmetric two-step method to the scalar test equation (82) we obtain the following difference equation:

where s = wh, h is the step length, Q(s) and C(s) are polynomials in s and y n is the computed approximation to y(nh), n = 0, 1, 2, . . .. The characteristic equation associated with equation (83) is given by: z2

+ Q(s)+ C(s) = 0

(84)

Theorem 1. A method which has the characteristic equation (84) has an interval of periodicity (0, Hi), if for all s2 E (0, H i ) , IQ(s)I < 1 and C(s) = 1.

4: Numerical Methods for the Solution of 10, 2 0 and 3 0 Dlfferential Equations

207

Definition 6.' The method with characteristic equation given by (84) is P-stable if its interval of periodicity is (0, 00).

Consider that we approximate the equation (82) by the method given by equation (83). Substitution of y = eJwxinto equation (83) and taking the real part gives a relation cos(s) = -(Q(s)/(l c)),whereas taking the imaginary part gives C = 1 (which when is satisfied then we have the non-dissipative case, i.e. we have the symmetric case). So, for the dissipative case we can see that a measure of the efficiency is how accurately the above conditions are satisfied. Based on this we make the following definition.

+

Definition 7. For any method corresponding to the characteristic equation (84) the quantity:

is called the dispersion or the phase error or the phase-lag of the method. If t = O(sP+') as s -+ 0 the order of phase-lag is p . The quantity

is called dissipation. If u = O(s') the order of dissipation is r.

From Definition 7 and based on an analogous remark of Coleman (1 989), we have the following remark.

Remark 2. If the order of dispersion is 2 r then: t = cs2'

+ O(s2r+3)* cos(s) + 1 +QW C(S)

= cos(s) - cos(s

~

-

(87)

+ O(s2'+4)

t ) = csZrt2

where t is the phase-lag of the method.

3.2 Dissipative Methods Developed in the Literature. - For the numerical solution of the second-order periodic problem (8 1) the following dissipative methods have been developed in the literature during the last two years. Simos and Williams3' have considered the two fiee parameter ( y o and vl) family of explicit fifth algebraic order methods for problem (81):

208

Chemical Modelling: Applications and Theoy, Yolume 2

where fn,-l = fn . Requiring maximal algebraic order for each approximation and for the final method Simos and Williams3' have obtained the following coefficients:

1 a. = --,

2

a4 = -1,

1 20

3 a l = -,

2

a5 = 2,

b5 = 0,

1 c,=-,

co=--=c 4'

5 bl = 16

1 bo = 16,

30

b,

1

= - = b7 = b8

3

8 15

c2=-=c

The local truncation error (LTE) of the above family of methods is given by

For v1 = 151/43008 (and vo free) a method with phase-lag of order eight is produced. For v1 = 151/43008 and vo = 25829/773120 a method with phase-lag of order ten is developed. T s i t o ~ r a s has ~ ~ developed the following family of dissipative two-step Numerov-type methods:

4: Numerical Methods for the Solution of ID,2 0 and 3 0 Differential Equations f n

209

= f ( x n , Yn),

In order that the above family has algebraic order six, the following specific method is produced:

7 d2, = --, 1 44

5 d22= --, 48

1 a2]=-,

36

For the above specific values of the parameters the Local Truncation Error is given by:

LTE=--

hs y‘,“’ O(h9) 120960

+

T s i t o u r a ~has ~ ~ developed the following family of dissipative two-step Numerovtype methods of algebraic order eight:

Chemical Modelling: Applications and Theoty, klume 2

210

%+I

= 2Yn-

+

Yn+l - 2 ~ n

h'

-

+ 10fn

Y n - 1 +E(fn+l

Yn-1

= h2

[

107

-Un+l 30870

+fn-l),

+fn-1)

283 1 7560

+-fn

80384

-

-

+-1065015 (fn+3/4 + f n-3/41

265625 +-1136016 (f + in-2l5)] . n+2/5

We note here that there is an error in the formula given in ref. 33 which we have corrected here. In order to have phase-lag of order 22, dissipation order 10 and algebraic order eight Tsitowra~~~ has chosen the set of parameters given by Table 2. In order to have phase-lag of order 18, dissipation order 14 and algebraic order eight T s i t ~ u r a shas ~ ~chosen the set of parameters given by Table 3. For more details about the Local Truncation Error see ref. 33. sir no^^^ has obtained the first family of dissipative Numerov-type methods for the solution of (81). This method has the form:

4: Numerical Methods for the Solution oj’lD, 2 0 and 3 0 Differential Equations

21 1

Table 2 Coeficients of the eighth algebraic order method of T s i t 0 u t - a ~with ~~ phase-lag of order 22 and dissipation order 10 gl = -0.327866 18933175 g3 = 1 .O 1484856799525 dl = 0.001285807291666666 d3 = -0.0146647 1354166666 d5 = -0.08424479166666666 ~2 =z 0.5856314873314576 ~4 = 0.01576042590075025 c6 = 0.457 1 19005406094 k2 = -0.02431 196195189366 k4 = 0.003684031277186183 k6 = 7.81360527358 x rl = 0.008179617736005171 r3 = 0.1 195199889374067 r5 = 1.899787462877609 r7 = -0.59 1666384816221 1 SI = -1.377668289974674 s3 = - 1.377668289974674 $6 = -9.148885251510144

g2 12.2484471359905 g4 = -2.560429514654 dz = 0.01064453125 d4 = -0.006770833333333334 C I = 0.01569149760700887 cj = -0.1 112076707490621 ~5 = -0.3067447454962486 k , -0.0004766854383 15481 I k3 = -0.003494149915634853 k5 = -0.08164166925930805 k7 = -0.01375957252563939 r2 = 0.2358016708666944 r4 = 0.05955674442839562 r6 = -0.04240495495488 127 r8 = - 1.408774145075008 ~2 = 1 1.65373547923761 s4 = 5.199685801866009 1

Table 3 Coeficients of the eighth algebraic order method of Tsitourasj3 with phase-lag of order I 8 and dissipation order I 4 g1 g3 dl d3 d5 ~2 ~4

=

-0.3275064808539252

= 1.013769442561776

= 0.001285807291666666 = -0.01466471354166666

= -0.08424479 166666666 = 0.5857066073301425 = 0.01573037790127627

c6 = 0.4572563676894037 k2 = -0.02431531295954992 k4 = 0.003683230812710925 k6 = 2.454519362719256 x rl = 0.00820226463177012 r3 = 0.1195813195859239 r5 = 1.900670459112769 r7 = -0.5919828216271334 SI = - 1.378402397366143 s3 = - 1.378402397366143 s6 = -9.148885251510144

g2 = 2.246288885123552 g4 = -2.557551846831402 d2 = 0.01064453125 d, = -0.006770833333333334 C I = 0.01569686332120066 ~3 = -0.11 12452307484046 CS = -0.3068949854936 185 k , = -0.0004776893915544161 k3 = -0.003491709816257519 k5 = -0.08163282637645776 k7 = -0.01376814678825401 r2 = 0.2357706404459578 r4 = 0.0596717574254971 r, = -0.04249926804562959 rx = -1.409414351529155 S? = 11.65973068960128 s4 = 5.202409446680736

212


(94)

where fn,-l = f n . The Local Truncation Error is given by:

where Fn = a f /ax, FL = dF’/dx. Applying the theory developed in Section 3.1 we find the values of parameters mentioned in Table 4 in order for the method to have minimal phase-lag. Papageorgiou et ~ 1 have . ~ developed ~ the following dissipative two-step Numerov-type method:

+

h2(d41f

n-1

+ d42f + a4lf + a42fb + a43fc)), n

n

4: lVumerical Methods for the Solution of ID, 2 0 and 3 0 Differential Equations

0 0 0 rg

2 E: rg \o 00

Ccl

0

m

" \o

N

0 0 0 0

\o 0 -3 v)

N N v) N 00

0

I

0 0 0 0

I

8 00 m N o\ b m

rg

m

IA IA

2

" 0 0 0 0

0

oooc

0

0

0 0 0 0

0

0

b d

(r,

c1

s

O

I

I

cs

Where PLl = -

4 1 ~ ~ ~ 37s2* PL2 = 3 1642868488395211407360000’ 4496002911 1 104307200000’ s24 S2b PL4 = PL3 = 567595389733741264896000000’ 479110255818407495073792000000

Phase-Lag

33 30752 96 1 536355 3973 1530256 1051 282700 1285 232462 - 116231 32558400 1 232560 PL 1

37 43960 157 114180 173 89280 155 58058 319 85032 3543 639400 _ _ 3197 __ 895356 1 447678 PL2

0

0

0

0

0

0 41 60528 1261 1 160320 1036 691713 137 68360 1709 63 1904 217 57594 9599 1730310 57677 -~ 1615 1520 _~ 1 807576 PL3

Method VIII

Method VII

Method VI

Table 4 (continued)

9 16168 202 1 2297640 19147 16086616 1163 747252 37 18160 227 83384 1489 394320 1643 295989 -~98663 27627600 1 -1381380 PL4

Method IX

2 P

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Digerential Equations

215

In order for the above method to have algebraic order 6, phase-lag order 14 and dissipation order 9 Papageorgiou et aZ.24have obtained the parameters presented in Table 5 . 3.3 Generator of Dissipative Numerov-type Methods. - Recently Avdelas and sir no^^^ have considered the following generator of dissipative two-step Numerovtype methods:

where b is the number of the family of methods and

Pn,o = y n . The free

Table 5 The coeficients of method of Papageorgiou et al. with algebraic order 6, phase-lag order I4 and dissipation order 9 dll = -0.01 198958741218540 d21 = -0.1354926086240548 a21= 0.1056425232832385 d32 = -0.1244946227062173 a32 = 0.008375035141675025 d42 = 0.1793 101099560068 a42 = -0.00771298419462741 I C I = 1.853745004884331 ~3 = 0.2575963849069488 W I = -0.01095654182197717 bl = 0.003328481791861325 b3 = 1.253365756591692

dl I = 0.8033023565368236 dzl = 3.570963452815986 dil = 0.005343514535940652 a31= -0.03293580148977421 d41 = -0.05260980968085666 a41= 0.03279785282508096 a43= 0.0006 109986184401625 C: = -2.207808474569488 ~4 -0.2448438326576166 W: = - 1 S353305 18304029 bl 10.001 185580954875260 b4 = 1.288407240787577


216

parameters ab-k+l I k = 1( 1)b are chosen in order for the phase-lag of the family of methods to be minimal. We also note that based on the analysis of Avdelas and Simos (see ref. 35 for more details) the number of stages is given by

N=b+7

(98)

The Local Truncation Error is given by: 1

(1

+ 16w)y',6)Fn

1

1 (1 + 16w - 1 9 2 ~ a ~ - . k + ~ ) y j f ) F , ,F~ 2880

--

(99)

Applying the theory developed in Section 3.1 we find that for the values w=--

23 336'

Si- 1

,

i = l(1)b

(100)

-;

where s - ~= and s,li = 0, 1, . . . are the coefficients of the Taylor series expansion of a well known function, the phase-lag of the family of methods is O( H 2 N )= O(H2b+14). 3.4 Exponentially Fitted Dissipative Numerov-type Methods. - Simos and Williams36 have developed an exponentially fitted fourth algebraic order method which has the form: 1

Yu = -2- ( - Y n - i

yc = -yn-1

+

h2

+ 3Yn) + ,,Ufl-l + 5 f n )

h2 + 2yn +T(3'fn-4

+ f n

+

+fu

+fb)

+

where x, = x, ah, x b = x, bh and x, = x, ch. Simos and Williams36 have chosen a = 1/2, b = - 1/2 and c = 1. The method (101) integrates exactly y = 1, x. In order that the method (101) integrates exactly any linear combination of the functions x2, x3, x4, x5, exp(fvx) the following system of equations is obtained:

4: Numerical Methods for the Solution of 10, 2 0 and 3 0 Diflerential Equations

2co + CI

COW

217

+ 2c2 = 1

13824 + 6912w2+ 576w4 + 5w6ePw + 25w6 c,w2 6912

+

+ 5w4ePw+ 25w4+ zcow4e-w- 2 cosh w + 2 = 0 + c2w 4608 + 576w22304 where w = vh. It is obvious that we have a system of three equations for the four unknowns co, c l , c2 and z to integrate any linear combination of the functions [ l , x, x2, x3, x4, x5, evx}. So, there is one free parameter. Simos and Williams36 have chosen z = 0 for computational simplicity. Solving the above system of equations the coefficients of the method are obtained. For small values of Y the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The local truncation error is O(h6). For the trigonometrically fitted case a replacement of w by iw in the third equation of the system of equations (102) must be done. Taking real and imaginary parts gives two relations for z. The real part gives:

+ 1728w4(4cozcos w + 4Co +

+ 6912w2(-2co -

CI

~ 2 )

+ 13824(1 - cos W)= 0

- 2~2)

while the imaginary part gives

If we eliminate z between the above relations we obtain a linear relation between the civalues: -

+

( 2 5 ~' 5 7 6 ~ 6~ 9 1 2 ~ ~13824w2)co+ 6 9 1 2 ~ ~ ~ 1

+ ( 7 5 -~ 1~ 7 2 8 +~ ~13824w2)c2+ 13824(~0~ w

-

1) = 0

(103)

The values of the coefficients co, c1 and c2 are obtained by the solution of the system of the two equations of the system of equations (102) and (103). After the determination of the coefficients co, cl and c2 the coefficient z can be defined from the equations given before equation (103). The finite difference method will now integrate exactly any combination of the function set: { 1, x, x2,x 3 , x4, x 5 , cos ux, sin ux}. For small values of Y the above formulae are subject to heavy cancellations. In this case Taylor series expansions must be used. The local truncation error is O(h6).

218


sir no^^^ has developed an exponentially fitted sixth algebraic order method which has the form:

+

+

+

where x, = x n ah, xb = xn bh and x, = x n ch. sir no^^^ has chosen a = 1/2, b = - 1/2 and c = 1. The method (101) integrates exactly y = 1, x. In order that the method (104) integrates exactly any linear combination of the functions x2, x3, x4, x5, exp(fvx) the following system of equations is obtained (for the case of q =

5):

2c0

+ CI + 2c2 = 1

e -2

+ e-w - (2c0 + cI + 2c2)w2- w4

c2

+ co + -29 coe-w+ coze-w

where w = vh. Solving the above system of equations the coefficients of the method are obtained. For small values of v the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The local truncation error is O(h8). For the trigonometrically fitted case (and for free q ) we have the following equations (additionally to the first three equations of the system of equations ( 105)):

219

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Differential Equations cs,.

-

2cs

+ cs- + (2co + CI + 2c2)w2cs c2cs

+ 5cs) - -(cs36

[

+ w6 -(cs_ 5?6 ss,

-

2ss + ss-

CO

- cs)

+ (2co + c1 + 2c*)w2ss

c2ss

+ w66;[-(ss_

2 + coq + -cots+ cozcs- + -3 cocs 9

+ cog + -29 toss- + cozss- + -3 coss + 5ss)

-

co 36

-(ss-

-

ss)

+

where cs = cos(vx), ss = sin(vx), cs- = cos[v(x - h ) ] , cs, = cos[v(x h ) ] , ss = sin[v(x - h)] and ss, = sin[v(x + h ) ] . The values of the coefficients co, cl, c2, q and z are obtained by the solution of the system of the three equations of the system of equations (105) and the equations (106). The finite difference method will now integrate exactly any combination of the function set: { 1, x , x 2 , x3, x4, x s , cos vx, sin vx}. For small values of v the above formulae are subject to heavy cancellations. In this case Taylor series expansions must be used. The local truncation error is O(h8).

3.4.I New Exponentially Fitted Dissipative Two-step Method. Case I. the dissipative two-step method: 1 Yo = -(-Yn--l 2

-

Consider

h'

+ 3Yn) +Ecf"'+ 5 . f n )

In order for the above method to be exact for any linear combination of the functions:

{ 1, x , x 2 , x 3 , x4, x 5 , exp(fwx)} the following system of equations is obtained:


220

2 cosh(wh) - 2 = w2h2(2co

+ w6h6

+ c1 + 2c2) + w4h4

(A

c2(5

+ exp(

-wh))

+ 361 co( 1 -

- exp( -wh))

1 +w8h8co(5+ exp(-wh)). 864 The solution of the above system of equations gives the coefficients of the method. For small values of w the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The local truncation error is given by:

LTE=--

h8 (y',")- 12w2 h 2 y (6) ,) 120960

(110)

3.4.2 New Trigonometrically Fitted Dissipative Two-step Method. Case I. Consider the dissipative two-step method (104). In order for the above method to be exact for any linear combination of the fwnctions:

{I, x, x2, x3, x4, cos(fwx), sin(fwx)}

(1 11)

the following system of equations is obtained: 2co

+ + 2c2 = 1

44Co

c1

4CoZ

9C2 = 3

+

- ~3 4 5 6 2 ~ ~w2h2(432c2 2cos(wh) - 2 = L w 2 h 2 [ - 3 4 5 6 c o - 1 7 2 8 ~ 1728

+ 1 7 2 8 ~+0 384~0COS(wh) + ~ ~ ~ c ~ z c o s ( w ~ ) ) + w4h4(-3c2(5 + cos(wh)) - 48c0(l cos(wh)) + 2w6h6co(5+ cos(wh))]. -

1 w4h4 sin(wh)(384co + 192c0z- 3w2h2c2 1728

--

+ 48w2h2co+ 2w4h4co)= 0

(1 12)


22 1

The solution of the above system of equations gives the coefficients of the method. For small values of w the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The local truncation error is given by:

hu LTE = - _ _ _(-3~:) 362880

+ 36w2h2yj16'+ 13w4h4yy')

(1 13)

3.4.3 New Exponentially Fitted Dissipative Two-step Method. Case II. - Consider the dissipative two-step method (104). In order for the above method to be exact for any linear combination of the fUnct ions :

{ 1, x, x2, x3, x4, exp(fwx), xexp(fwx)}

(1 14)

the following system of equations is obtained: e(wh)

-

2 + e(-wh)

-

1 5 h2w2c,+2c2h2w2+-c2h4w4 +-h6w6c2 4 576

1 +-h6w6cze(-wh) 576

+ 2coh2w2+ coh4w4+ -29 h 4 ~ 4 ~ o e ( - w h ) 1 + -coh6w6 36

5 + 864

1 36

- - h6w6coe(-wh)-h8wuco

1 5 1 + 2h2w2cox+ h6w6cox+ h4w4cOx+ h6w6c2x+ - h4w4c2x 36 576 4 -

5 +h8w8cox+ 4h4w3co+ 864

~~'W'CZX

5 + 108 h8w7co+ -

h2W2C1X

5 +h6w5c2. 96

The solution of the above system of equations gives the coefficients of the

222


method. For small values of w the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The local truncation error is given by:

h7 LTE=-- 7560 w3y',4'

3.4.4 New Trigonometrically Fitted Dissipative Two-step Method. Case II. Consider the dissipative two-step method (104). In order for the above method to be exact for any linear combination of the functions:

{ 1, x, cos(fwx), sin(+wx), xcos(fwx), xsin(fwx)}

(1 17)

the following system of equations is obtained:

2 Cos(wh) - 2 =

-h2w2(- 1

1728

+

7 2 8 ~-~3 4 5 6 ~ ~4 3 2 h2w2 ~ ~ - 15c2h4w4

+ 1728~0h2w2

- 3c2h4w4cos(wh) - 3 4 5 6 ~ 0

+ 384~0h2 COS(wh) 48c0h4w4 + 48c0h4w4cos(wh) + 1Ocoh6w6 + 2c0h5w6cos(wh) + 192c0h 2 m 2cos( wh)) -

W'

0= -

1728

--

h4w4sin(wh)(-3c2h2w2

+ 384c0 + 48coh2w2+ 2c0h4w4+ 192c0z)

+

h2w2(-432c2h2xw2 3456c2x - 18c2h4w3sin(wh)

1728

+ 288coh4w3sin(wh) + 1536coh2wsin(wh)+ 16coh6w5sin(wh) + 1728clx + 3456c0x + 384c0h3w2 cos( wh) + 768c0h 2 m sin(wh) - locoh6w6x + 48c0h4w4x 1728c0h2w2x + 15c2h4w4x + 3h4c2w4cos( wh)x 48h4cow4cos(wh)x + 192coh3m2cos(wh) 384coh2w2cos(wh)x 192c0h 2 m 2cos(wh)x 2c0h6w6cos( wh)x + 2c0h7w6 cos( wh) + 48 h5cow4COS(wh) 3 h5 w4 COS(wh)) = ~X(COS( wh) 1) -

-

-

-

-

-

~2

-


1 h2~ ( 9 0 h4 ~ w4 2 - 6912~0h2w2 - 1728~2 h2w2- 80c0h6w6 1728

223

+ 3456~1

+ 18c2h4w4C O S ( W-~ )288coh4w4C O S ( W+~ )6 9 1 2 ~ +0 6912~2 1536coh2w2cos(wh) 16coh6w6cos(wh) + 384coh3w3sin(wh) - 768c0h 2 m 2cos(wh) + 288c0h4w4 + 3 h4c2w5 sin(wh)x 48h4cow5sin(wh)x + 192cOh3m3 sin(wh) 384coh2w3sin(wh)x 192cOh2m3 sin(wh)x 2coh6w7sin(wh)x + 2coh7w7sin(wh) + 48 h5cow 5sin(wh) 3 h5c2w5 sin(wh) = 2 sin(wh)h -

-

-

-

-

-

-

The solution of the above system of equations gives the coefficients of the method. For small values of w the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The local truncation error is given by:

LTE

==

hs ( - 15ylp)+ 126w2h2yj16’+ 227w4h4y:’ 18 14400

-~

+ 156w6h6yjf)) (1 19)

In Appendix B we present a Maple programme for the construction of the above exponentially fitted and trigonometrically fitted dissipative two-step methods (Cases I and 11). In Table 6 we present the basic characteristics for the dissipative methods with

Table 6 The basic characteristics for the dissipative methods with constant coeficients ~~~~~~

~

Method

A 0 ojthe method*

PLO of the method*

Simos and Williams3’ (Case I) Simos and Williams3’ (Case 11) Tsitouras” T s i t o u r a ~(Case ~ ~ I) T s i t o ~ r a s(Case ~ ~ 11) sir no^^^ (Case I) sir no^^^ (Case 11) sir no^^^ (Case 111) sir no^^^ (Case IV) Simos3‘ (Case V) sir no^^^ (Case VI) sir no^^^ (Case VII) sir no^'^ (Case VIII) sir no^^^ (Case IX) Papageorgiou et al.” Avdelas and sir no^^^

5 5 6 8 8 6 6 6

8 10 0 22 18 10 12 14 16 18 20 22 24 26 14 [8 - 00)

6 6 6 6 6 6 6 6

“ A 0 = Algebraic order of the method and PLO = Phase-lag order of the method.


224

Table 7 The basic characteristics for the dissipative methods with coeflcients dependent on the frequency of the problem Method

A0

IEF*

Simos and Williams36

4 6 6 6

p=5,m=0 p=5,m=0 p=5,m=0 p=4,m=1

sir no^^^ New exponentially fitted method (Case I) New exponentially fitted method (Case 11)

*IEF = Integrated exponential functions and p , m are based on formula (120).

constant coefficients that have been developed in the literature and which have been described above. In Table 7 we present the basic characteristics for the dissipative methods with coefficients dependent on the frequency of the problem that have been developed in the literature and which have been described above. We note here that the classification of the methods with coefficients dependent on the frequency of the problem is based on the set of functions:

4 Numerical Illustrations for Linear Multistep Methods and Dissipative Methods In this section we present some numerical results to illustrate the performance of the classes of methods described above. Consider the numerical integration of the Schrodinger equation (1) using the well-known Woods-Saxon potential which is given by V ( r ) = Vw(r)=

UO ~

(1

+ 2)

UOZ

-

[a(l

+ z)’]

with z = exp[(r - & ) / a ] , uo = -50, a = 0.6, and & = 7.0. In Figure 9 we give a graph of this potential. We note that this potential supports energies in the interval [-50, 10001. In the case of negative eigenenergies (i.e. when E < [-50, 01) we have the well-known bound-states problem while in the case of positive eigenenergies (i.e. when E E [l, lOOO]) we have the well-known resonance problem.

4.1 Resonance Problem. - In the asymptotic region the equation (1) effectively reduces to

x.>

+ (k2 - l(1 + 1)

y”(x)

for x greater than some value X.

y(x) = 0

4: Numerical Methods for the Solution of ID,2 0 and 3 0 Diflerential Equations

225

-50 -(O1

Figure 9

Plot of the Woods-Saxon potential

The above equation has linearly independent solutions kxj,(kx) and kxn ,( kx), where j , ( kx), nl( kx) are the spherical Bessel and Neumann function respectively. Thus the solution of equation (1) has the asymptotic form (when x .+00) y(x) ‘v Akxj,(kx) - Bnl(kx)N D[sin(kx - xZ/2)

+ tandl cos(kx

-

nZ/2)] (123)

where 6, is the phase shift which may be calculated from the formula (124) for xI 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 step size) with S(x) = kxj,(kx) and C(c) = kxnl(kx). Since the problem is treated as an initial-value problem, one needs yo and yI before starting a two-step method. From the initial condition, yo = 0. The value y1 is computed using the Runge-Kutta-Nystrom 12(10) method of Dormand et al.” With these starting values we evaluate at x1 of the asymptotic region the phase shift 6, from the above relation. 4.1.1 The Woods-Saxon Potential. - As a test for the accuracy of our methods we consider the numerical integration of the Schrodinger equation (1) with 1 = 0 in the well-known case where the potential V(r)is the Woods-Saxon one (121). One can investigate the problem considered here, following two procedures. The first procedure consists of finding the phase shift 6(E) = 6, for E E [ l , lOOO]. The second procedure consists of finding those for E, for E E [ 1, lOOO], at which 6 equals n/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 by Blatt3’

226

Chemical Modelling: Applications and Theory, Volume 2 0

1

2

L

g

-3

w

4

Error versus number of function evaluations = N F E x 100. 5

6

0

8

4

16

12

NFE x 100

1

L

g Lu

-

E ri evaluations = NFE x 100. MV MVI MVll MVlll

+

I

I 0

+ I

1 4

1

1 8

1

1 12

1

1 16

NFE x 100

Figure 10 Errors Err for the number of function evaluations which are equal to of the lowest eigenenergy Eo = 53.588872.


227

The boundary conditions for this problem are: y(0) = 0 and y(x)

N

cos[dEX] for large x

The domain of numerical integration is [0, 151. For comparison purpose in our numerical illustration we use the following methods: (1) The explicit version of the Numerov method which has better stability properties than Numerov’s method since it has an interval of periodicity equal to (0, 12) while Numerov’s method has an interval of periodicity equal to (0, 6). This method has been produced by Chawla3*and is indicated as method MI. (2) The method developed by Tsitoura~,~* which is indicated as method MII. (3) The method developed by Simos and William~,~’ which is indicated as method MIII. (4)The method developed by Papageorgiou et uZ.,*~ which is indicated as method MIV. (5) The method developed by T s i t ~ u r a s which , ~ ~ is indicated as method Mv; (6) The method developed by Sin10s~~ (Case IX), which is indicated as method MVI. ( 7 ) The exponentially fitted method developed by S i m o ~ ,which ~ ~ is indicated as method MVII. (8) The exponentially fitted method developed by Simos and A g ~ i a (see r ~ ~Appendix C), which is indicated as method MVIII (Case I). (9) The exponentially fitted method developed by Simos and A g ~ i a r which , ~ ~ is indicated as method MIX (Case 11). (10) The exponentially fitted method developed in this review (Sections 3.4.1 and 3.4.2), which is indicated as method 0 -

4-

L

g

a-

W

-12

-

-

Error versus number of function evaluations NFE x 100. __f__ MIX MX __t_ MXI

+

-16

I

0

I 4

1

I 8

NFE x 100

Figure 10 (continued)

1

I 12

I

1 16


228

MX. (11) The exponentially fitted method developed in this review (Sections 3.4.3 and 3.4.4), which is indicated as method MXI. The numerical results obtained for the above methods, with the same number of function evaluations, which are equal to NFE X 100 (in order to have the same computational cost) were compared with the analytic solution of the WoodsSaxon potential resonance problem, rounded to six decimal places. Figure 10 shows the errors

of the lowest eigenenergy Eo = 53.588872. Figure 11 shows the errors

of the highest eigenenergy E3 = 989.701916. The performance of the exponentially fitted and trigonometrically fitted methods is dependent on the choice of the fitting parameter w. For the purpose of obtaining OUT numerical results it is appropriate to choose w in the way suggested by Ixaru and R i ~ e a . ~That ' is, we choose:

=

{

(-50 (-E)1/2

for x E (0, 6.5) for x E (6.5, 15)

For a discussion of the reasons for choosing the values 50 and 6.5 and the extent to which the results obtained depend on these values see Ixaru and R i ~ e a . ~ '

4.1.2 Modified Woods-Saxon Potential. - We have applied the above methods for the numerical solution of the resonance problem of Schrodinger equation (1) with 1 = 0 in the well-known case where the potential is given by: V(x)= Yw(X)

+D

X

where V , is the Woods-Saxon potential [equation (1 17)]. For the purpose of our numerical experiments we use the same parameters as in Ixaru and R i ~ e a , ~i.e. ' D = 20, 1 = 2. Since V ( X )is singular at the origin, we use the special strategy of Ixaru and R i ~ e a . ~We ' start the integration from a point E > 0 and the initial values Y ( E ) and Y ( E h) for the integration scheme are obtained using a perturbative method (see 1xa1-u~~). As in Ixaru and Rizea4' we use the value E = 1/4 for our numerical experiments. For the purpose of obtaining our numerical results it is appropriate to choose w in the way suggested by Ixaru and R i ~ e a . ~That ' is, we choose:

+


229

1 A

L

0

L

Lu

-

Error versus number of function evaluations = NFE x 100. MVI

+

MVlll

1

I1

I

1 1

1

8

4

1

1

12

16

NFE x 100

0

2

4

6

Error versus number of function evaluations = NFE x 100. _t_ MIX __h_. MX

0

-10

I I 0

I

I 4

1

1 8

1

1 12

1

1 16

NFE x 100

Figure 11 Errors Err for the number of function evaluations which are equal to of the highest eigenenergy E3 = 989.701916.

230


[V(a~) 2

+

w=I

v(E)l for x E ( E ,

Val)

al)

for x E (a,, a2)

2 V(a31

for x E (a2,a 3 )

. V(15)

for x E (a3,15)

where ai, i = I( 1)3 are fully defined in Ixaru and R i ~ e a . ~ ’ Since the results for the above potential are exactly of the same form as in Figures 10 and 11 is not necessary to present them. 4.2 The Bound-states Problem. - For negative energies we solve the so-called bound-states problem, i.e. the equation (1) with I = 0 and boundary conditions given by

In order to solve this problem numerically we use a strategy which has been proposed by Cooley4’ and has been improved by Blatt.37 This strategy involves integrating forward from the point x = 0, backward fiom the point xb = 15 and matching up the solution at some internal point in the range of integration. As initial conditions for the backward integration we take (see Cash et aZ.26): y(xb)= exp(-

d q )and y(xb

-

h ) = exp[ -1 -d

where h is the step length of integration of the numerical method. The true solutions to the Woods-Saxon bound-states problem were obtained to nine decimal places using the analytic solution and the numerical results obtained for the methods mentioned above were compared to this true solution. Since the results for the above potential are exactly of the same form as in Figures 10 and 11 it is not necessary to present them. We have also applied the above methods to other potentials like the parabolic potential :43 0.5(x - a)2 for - 00 < x S 2a OS[x - (2i - l)a]’ for 2(i - 1)a < x d 2ia(2 L i d N OS[x - (2N - l)a]’ for 2(N - 1)a < x < 00

-

1

where N is the number of minima. We have also applied the methods mentioned above to the two-dimensional Schrodinger equation (1a) mentioned in Section 1 and also we have applied the methods mentioned above to the three-dimensional potentials. The procedure for the application of the above mentioned methods to two- and three-dimensional problems is fully described in ref. 43. For all the problems mentioned above the results have the same form as described in Figures 10 and 11.

4: Numerical Methods for the Solution of ID, 2 0 and 3 0 Dzfferential Equations

23 1

4.3 Remarks and Conclusions. - Based on the above we have the following remarks: 1. For all the problems the exponentially fitted and trigonometrically fitted

2.

3.

4.

5.

methods are much more efficient than the other methods with the same basic characteristics for the same cost. From exponentially fitted and trigonometrically fitted methods the most efficient are the linear symmetric multistep methods. This is because these methods have two additional properties [symmetry (ie. symplecticness) and non-empty interval of periodicity] to the dissipative methods and one additional property (non-empty interval of periodicity) to the symplectic methods. From linear symmetric multistep exponentially fitted and trigonometrically fitted methods the most efficient are the methods that integrate exactly as much as possible functions of the form (120) (for Y E C). As a conclusion we can say that symmetric (non-dissipative) methods are more efficient because they have non-empty interval of periodicity and because they are symplectic (in the case of linear symmetric multistep methods). We note also that symmetric linear multistep methods are very simple in programming and have very low computational cost (only one function evaluation per step). The only way to produce efficient dissipative (i.e. non-symmetric) methods for the numerical solution of problems described in Section 1 is the exponentially fitted and the trigonometrically fitted version of these methods. The above methods are based on the well known exponential fitting procedure, fist introduced by Ly~he.~O

5 New Developments on Numerical Methods with Constant Coefficients and on the Methods with Coefficients Dependent on the Frequency of the Problem 5.1 Methods with Constant Coefficients (Generators of Numerical Methods). Avdelas et al.44have introduced a generator of hybrid implicit eighth algebraic order method of the form:

GENERATOR I

232


where b denotes the family of methods defined by the user. We observe that the above family of methods contains free parameters wi. The local truncation error of the final families of methods is of order O(h'') (see ref. 14 for details), i.e. is of eighth algebraic order. In order that the above method has a minimal phase-lag the following system of equations must hold: In order to reach a minimal phase-lag the following implicit equations for the free parameters wi are obtained (for the proof see ref. 44):

where

4: Numerical Methods for the Solution of ID,2 0 and 3 0 Digerential Equations

mo = 1,

m4 =

~

1255 ml =24192’

14079 7169 ’

349 1 m2 =-2903040’

5605 m5 = ___ 57344 ’

m6 =

233

199 m3 = 34836480’

378 1 3440640

~

and s, is the half of the power of s. Avdelas and sir no^^^ have developed a generator of explicit eighth algebraic order hybrid methods of the form:

GENERATOR I1

234


where

We note that ab-k(k = l(1)b are free parameters, which can be chosen so that the important properties of an approximation method are satisfied. The computational cost of the method is determined via the number of total stages, which are N = b + 6. The Local Truncation Error of the method is of the form O(h") (see ref. 45 for more details), i.e. is of eighth algebraic order. In order that the above method has a minimal phase-lag the following formulae for the coefficients of the generator must hold (for proof see ref. 45): a l =- s 1 - --,S1 -2ao 2ao

a. =so, a2 =

s2

(-2)2aoal

-

-.-

s2

(-2)2s0

-2s0 s1

s2 -and

-2sl

generally

Avdelas et aE.46have introduced a generator of hybrid explicit eighth algebraic order method of the form:

GENERATOR I11

Yn+l

= 2Yn - Y n - 1

+

h2 ,,cm+l

+ lOE, + f n - , ) ,


h2 (-1527fn+1 786432

--

- 27414fn -

1289fn-1

-

31O88fn+1/2

235

+ 27420fn-I/2),

We observe that the above family of methods contains free parameters w i . The local truncation error of the final families of methods is of the form O(h'O) (see ref. 46 for more details) i.e. is of eighth algebraic order. In order that the above method has a minimal phase-lag the following formulae for the coefficients of the generator must hold (for the proof see ref. 46):

,t=b

n;:;''

where i = 0, 1, . . . , b and u', = 1 for i = 0. According to the desired approximation, we choose the value of b. Avdelas et aZ.47.48 have introduced a generator of hybrid explicit tenth algebraic order method of the form:

236


GENERATOR IV %+l

~

n

jn+l

= 2yn - yn-I

+ h2y:

=, ~n - Wih2[jS:+l

= 2yn - yn-1

2.~:

-

h2 +60

[Y:+l

+ ynN-13,

+ 26y: + y:-1 +

839

w7:+1/2

333 --y;+2240

+ 7:-1/2)1

17 24 192

II

n-l

4: Numerical Methods for the Solution oj’ID, 2 0 and 3 0 Differential Equations

101237

+ 566231040 122333 + 8847360

-

5507449 =,, 141557760 n’

-

713689 =,, 141557760 Y n - 1 / 2 ’

~

Yn+1’4

8847360 Y n - 1 ’ 4 456099 10485760”

-

981067 566231040

181229 +-8847360

where

713689 - 5507449 - 141557760Y n + 1 ’ 2 141557760

-11

Yn+1’4

i~

237

238


We note that wi are free parameters, which can be chosen so that the important properties of an approximation method are satisfied. The Local Truncation Error of the method is of the form O(hI2)(see refs. 47 and 48 for more details), i.e. is of tenth algebraic order. In order that the above method has a minimal phase-lag the following formulae for the coefficients of the generator must hold (for the proof see refs. 47 and 48):

j=b

GENERATOR V (OPTIMIZATION OF GENERATOR IV) This generator is based on the previous with the exception that the parameter wo remains free. The determination of the free parameter is based on the maximization of the interval of periodicity (see refs. 47 and 48 for details).

5.2 Methods with Coefficients Dependent on the Frequency of the Problem. In this section recent developments on exponentialy fitted methods and Bessel and Neumann-fitted methods are presented.

5.2.I Exponentially Fitted Hybrid Methods. - Kalogiratou and sir no^^^ have considered the following method:

They have defined the coefficients of the above method in order the method integrates exactly any linear combination of the functions { 1, x, x2, x3, exp(kwx)} or { 1, x, x2, x3, cos(fwx), sin(fwx)} and to be P-stable. The Local Truncation Error of the method is of the form O(h6)(see ref. 49 for more details), i.e. is of fourth algebraic order. More recently Avdelas et aLsohave developed the following family of methods:

4: Numerical Methods for the Solution of ID, 2D and 3 0 Differential Equations Y n

=y,

-

a0 h 2 C f n i i

-

2fn

239

+j-n-11,

The above methods for appropriate values of the free parameters a, a j , b,li = 0, 1, is of algebraic order four (see ref. 50 for details). Avdelas et aLS0have defined the parameters of the method in order for: i. The methods to be P-stable and ii. The methods to integrate exactly any linear combination of the functions (1 20) with: p=5,rn=0,

p=3,rn=1,

p = 1,rn=2, andp=O,rn=3

Simos and Williams" have considered the following method:

where, for example, pi = f(xn, yn).We require that the method to be exact for any linear combination of the functions { 1, x, x2,x3, exp(fwx)} or { 1, x, x2, x3, cos(fwx), sin(fwx)} and also to be P-stable. The Local Truncation Error of the method is of the form O ( h 8 )(see ref. 51 for more details), i.e. is of sixth algebraic order. Aguiar and Simoss2have considered the following method:

240

Chemical Modelling: Applications and T h e o q Volume 2

1

h2

Yn+1/2

= -2( v n

Yn-1/2

= -(yn

Yn+l

1 2

= 2y,

-

+ Yn+d - &c+1 + 10j7:+1/2 + y,),

+Yn-I)

-

h2 -(yL 96

yn-I - h2[qoy:+l

+ 10y:-1/2 + y,),

+ q1y: + qoY:-l + 92(X++1/2+

X-I/2)

We require that the method to be exact for any linear combination of the functions { 1, x, x2, x3, x4, x5, x6, x', exp(*wx)) or { 1, x, x2, x3, x4, x5, x6, x7, cos(kwx), sin(fwx)} and also to be P-stable. The Local Truncation Error of the method is of the form O(h'O) (see ref. 52 for more details), i.e. is of eighth algebraic order. 5.2.2 Bessel-jtted and Neumann-jtted Methods. - sir no^^^ has considered the following second algebraic order explicit method: Yn+l

Yn+l

- 2%

= 2yn

+ Yn-1 =

- Yn-I

h2(b0fn+l

+ h2fn + + boffl-1)

(134)

blfn

SimosS3has defined the free parameters in order for the method to be exact for the functions J , = kr,jl(krq), Y, = krqyj(krq),q = n - l(1)n 1, where j/(kr), yl(kr) are the spherical Bessel and Neumann functions respectively. He also considered the fourth algebraic order explicit method:

+


24 1

Again the free parameters of the method are defined in order for the method to be exact for the spherical Bessel and Neumann functions respectively. sir no^^^ has considered the following eighth algebraic order explicit method:


242

Again the free parameters of the method wii = 0, 1 I are defined in order for the method to be exact for the spherical Bessel and Neumann functions respectively. SimosS5has constructed the following implicit eighth algebraic order method:

h2

+ -[-9(~:+~ 5 12

+ Y:-~)

-

171(y:

+ J&)],

The free parameters of the method a j i = 0, 1 I are defined in order the method to be exact for the spherical Bessel and Neumann functions respectively. In Table 8 we present the basic characteristics of the above described hybrid methods. We note that for the classification of the exponentially fitted and trigonometrically fitted methods is based on the set of functions (120).

4: Numerical Methods for the Solution of 10, 2 0 and 3 0 Differential Equations

243

Table 8 The basic characteristics for the P-stable exponentially jtted methods and Bessel and Neumann-jtted methods Method

A0

Form”

IEF

Kalogiratou et Avdelas et al.” Avdelas et al.” Avdelas et ~ 1 . ~ ’ Avdelas et aLso Simos et al.” Aguiar et a1.52 SirnoP Simoss3

4 4

p=3,m=0 p=5,m=0 p=3,m=1 p=l,m=2 p=O,m=3 p=5,m=0 p=7,m=0

sir no^^^

8 8

PET PET PET PET PET PET PET BNE BNE BNE BNI

4 4 4

6 8 2 4

SirnoP

~

~

~~~~~~~~~

*PET = P-stable exponentially fitted and trigonometrically fitted method. BNE = explicit Bessel-fitted and Neumann-fitted method. BNI = implicit Bessel-fitted and Neumann-fitted method.

5.3 Runge-Kutta Exponentially Fitted Methods. - Williams and sir no^^^ have considered the four-stage explicit Runge-Kutta method presented in Table 9, where the parameters of the method are given by:

1

a2!= -, 2 a42= 1

-

a31= 0, 24k4,

1

a32= -, 2

a4]= 0,

a43= 24k4.

Williams and SirnosS6have determined the free parameter k4 in order that the

Table 9 Thefour-stage explicit exponentiallyjtted and trigonometricallyJitted RungeKutta method of Williams and Simos 0 c2 c3

a32

c4

a42

I

a43

b4

244


method integrates exactly any linear combination of the functions { 1, x, x2, x3, x4, exp(*wx)}. sir no^^^ has constructed a four-stage fourth algebraic order explicit RungeKutta exponentially fitted and trigonometrically fitted method of the form presented in Table 9. The parameters of the method are determined in order that the method integrates exactly any linear combination of the functions { 1, x , x2, x3, exp(fwx)} and { 1, x, x 2 , x3, cos(fwx), sin(Awx)}. In the same paper a three-stage second algebraic order explicit Runge-Kutta exponentially fitted and trigonometrically fitted method of the form presented in Table 10 is constructed. The parameters of the method are determined in order that the method integrates exactly any linear combination of the functions { 1, x, exp(Awx)} and { 1, x , cos(fwx), sin(fwx). Avdelas et al.58have constructed an embedded Runge-Kutta explicit exponentially fitted and trigonometrically fitted method of the form given in Table 11. The parameters of the method are given by: 1

c2=z1 a31= 0,

1 c 3 = 2- ’ a32

1 2

= -,

c4 = 1 , a41= 1,

1 a2]= -, 4 a42

=

-2,

a43= 2 .

Table 10 The four-stage fourth algebraic order explicit exponentially fitted and trigonometrically fitted Runge-Kutta method of Simos 0 c2 c3

a2 I a3 I

a32

Table 11 The embedded Runge-Kutta explicit exponentially fitted and trigonometricallyfitted of Avdelas et al.58

c3 c4

1 I

a3 I

a41

a32 a42

a43


245

The free parameters of the method are determined in order for the method to be exact for any linear combination of the functions { 1, x, x2, exp(fwx), { 1, x, x2,cos(fwx), sin(fwx)} and for the functions { 1, x, exp(fwx) and { 1, x, cos(fux), sin(fwx).

5.4 Modified Runge-Kutta Phase-fitted Methods. - Simos et al.59have considered a modified four-stage explicit Runge-Kutta method presented in Table 12. The parameters of the method are given by: 1

c7

-

=-

2'

a31 = 0,

C?

1 2

= -,

u32=

1 5,

c4

= 1,

a41 = 0,

1

LIZ[

=-

a42

2'

= 0,

a43= 1,

The free parameter g , is determined in order that the method has phase-lag of order infinity (see ref. 6 for more details on phase-lag analysis of Runge-Kutta methods). Simos6' has considered a modified seven-stage and six-stage explicit embedded Runge-Kutta method presented in Table 13. The parameters of the method are given by:

Table 12 The mod$ed four-stage explicit Rurtge-Kutta method of Simos et al.

phase-jtted

Table 13 The mod$ed seven-stage and six-stage explicit embedded phase-jtted Runge-Kutta method of Simos

246


3 a3, =-,

1 5

a21= - ,

40

19372

aS1= 6561 ’

a52

5179 57600 ’

=-

s2

44 45

a4,=-,

= 0,

s3

a42

64448

25360 2187 ’

46732 5247 ’

a63 = -

757 1 16695 ’

= ____

s4=-

56 15

32 9

= --,

a43=-,

212 729

a54= --,

a53= 6561 ’

= --

355 a62 = - 33 *

9017

a6]= -3168’

S]

9 40

a32 =-,

49 176’

aM=-

3 93 640 ’

s5=--

5103 18656’

a65=--

92097 339200 ’

187 2100‘

s6 =-

The free parameters b6 and s7 are determined in order that the method has phaselag of order infinity (see ref. 6 for more details on phase-lag analysis of RungeKutta methods). 5.5 Modified Runge-Kutta-Nystrom Phase-fitted Methods. - Simos et aL6’ have considered a modified four-stage explicit Runge-Kutta-Nystrom method presented in Table 14. The free parameter g3 is determined in order that the method has phase-lag of order infinity (see ref. 6 for more details on phase-lag analysis of Runge-Kutta-Nystrom methods).

Table 14 The mod$ed four-stage explicit phase-fitted Runge-Kutta-Nystrom method of Simos et al. 0 1/3 213 1

1 1

0

g3

113

1/18

13

219 0

3

116

3

-

-

10

40

1 60


247

6 Numerical Illustration on Variable-step Methods

Our error estimation policy is based on the error estimation described in ref. 6. 6.1 Coupled Differential Equations. - The close-coupling differential equations of Schrodinger have the form

for 1 G i d N and rn # i. The following boundary conditions hold (for the case in which all channels are open - see ref. 65 for details): yq = 0 at x

=0

where j l ( x ) and n r ( x ) are the spherical Bessel and Neumann functions, respectively. Based on the detailed analysis developed in ref. 65 we find that the asymptotic condition (140) may be written as:

Denoting, as in ref. 65, the entrance channel by the quantum numbers ( j , I ) , the exit channels by ( j ’ , Z’) and the total angular momentum by J = j + Z = j’ + 1’, we find that

where

E is the kinetic energy of the incident particle in the centre-of-mass system, I is the moment of inertia of the rotator and p is the reduced mass of the system. Based on the analysis of ref. 65, the potential Vmay be written as

and the coupling matrix element is given by


248

where the f 2 coefficients can be obtained from formulae given by Berstein et a1.,66 k j r j is a unit vector parallel to the wave vector k j r jand Pi,i = 0, 2 are Legendre polynomials (see ref. 66 for details). The boundary conditions may then be written as (see ref. 65): yg = 0 at x = 0 -

(2)

[(

31

”2sJ(jl; j’Z‘)exp i k j P j x- -

where the scattering S matrix is related to the K matrix of (140) by the relation

A numerical method is required for the calculation of the cross sections for rotational excitation of molecular hydrogen by impact of various heavy particles. This method will solve the above problem from the initial value to matching points. In our numerical test we choose the S matrix which is calculated using the following parameters:

3= 1000.0, h2

!r!= 2.351,

E

=

1.1

As is described in ref. 65, we take J = 6 and consider excitation of the rotator from the j = 0 state to levels up to j’ = 2, 4, 6, giving sets of four, nine and sixteen coupled differential equations, respectively. Following Bersteid7 and Allison65 the reduction of the interval [0, 0 0 3 to [0, xo] is obtained. The wavefunctions then vanish in this region and consequently the boundary condition (146) may be written as

For the numerical solution of this problem we have used (1) the embedded Runge-Kutta method of Avdelas et aE.58(which is indicated as method MI), (2) the embedded Runge-Kutta method of Simos5’ (which is indicated as method MII), (3) the embedded Runge-Kutta method of Simos60 (which is indicated as method MIII), (4) the variable-step Bessel- and Neumann-fitted method developed by 54im0.s~~ (which is indicated as method MIV), ( 5 ) the generator of eighth algebraic order methods developed by Avdelas et aE.46(which is indicated as method MY), (6) the generator of eighth algebraic order methods developed by Avdelas et (which is indicated as method MVI), (7) the variable-step Bessel ~

1

1

.

~

~

3

~

~

4: Numerical Methods for the Solution of ID,2 0 and 3 0 Differential Equations

249

and Neumann fitted method developed by sir no^^^ (which is indicated as method MVII), (8) the generator of eighth algebraic order methods developed by Avdelas et a1.44(which is indicated as method MVIII), (9) the variable-step Bessel- and Neumann-fitted method developed by Simos” (which is indicated as method MIX), ( 10) the exponentially fitted variable-step method developed by Konguetsof and sir no^^^ (which is indicated as method MX), (1 1) the generator of eighth algebraic order methods developed by Avdelas et aZ.47,48 (which is indicated as method MXI), (12) the optimized generator of eighth algebraic order methods developed by Avdelas et a1.47,48 (which is indicated as method MXII) and (13) the exponentialy fitted P-stable variable-step method developed by Aguiar and Simoss3 (which is indicated as method MXIII). In Table 15 we present the real time of computation required by the methods mentioned above to calculate the square of the modulus of the S matrix for sets of four, nine and sixteen coupled differential equations. In Table 15 N indicates the number of equations of the system of coupled differential equations.

7 General Comments

From the results presented in Table 15 we have the following remarks: i. The most efficient variable-step methods for the solution of coupled differential equations arising fiom the Schrodinger equation is the P-stable exponentialy fitted variable-step method developed by Aguiar and sir no^.^^ Another very efficient variable-step method is the variable-step Bessel- and Neumann-fitted method of sir no^.^^ Efficient variable-step methods for the solution of the above problem are also the variable-step Bessel- and Neumann-fitted method of Sirnos” and the variable-step exponentialy fitted method developed by Konguetsof and sir no^.^^ Finally efficient methods for the solution of the above problem are the generator and the optimized generator developed by Avdelas et aZ.47,48 ii. Based on the above we conclude that the following areas are interesting for further research on computational efficiency: 1. Bessel and Neumann fitting 2. P-stability and exponentially fitting and 3. Construction of generators of methods.

We note here that similar results are obtained for the numerical solution twodimensional and three-dimensional Schrodinger equations. For the application of exponentialy fitted methods an accurate estimation of the fiequency is required. Recently Vanden Berghe et ~ 1 . ~have ’ introduced a new method for the determination of the frequency of the problem. Vanden Berghe et aL7’have also constructed multistep exponentialy fitted methods for first order Ordinary Differential Equations. A new area for the solution of the problems studied in the review has been


250

Table 15 Real time of computation of the modulus of the S matrix for sets of 4, 9 and I 6 coupled differential equations

MI MI1 MI11 MIV

Mv MVI MVII MVIII

4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9

MIX MX

MXI MXII MXIII

16 4 9 16 4

9 16 4 9 16 4 9 16 4 9 16

0.43 3.54 19.33 0.39 3.48 19.31 0.22 2.56 13.22 0.19 1.35 7.1 1 0.10 1.04 7.96 0.06 0.38 2.71 0.07 0.54 2.05 0.05 0.92 7.2 1 0.08 0.55 2.20 0.05 0.70 5.71 0.06 0.75 6.13 0.04 0.63 5.32 0.03 0.2 1 1.72

*RTC = Real time of computation.

introduced this year by Chen et al.72This area is based on quadrature formulae. We must note here that there is a very powerful new area on this subject, which is based on exponentialy fitted and trigonometrically fitted quadrature formulae and on the symplectic integrators (it is known that matrices which are obtained from quadrature formulae are s y m p l e c t i ~ ) . ~ ~ , ~ ~

4: Numerical Methods for the Solution of ID,2 0 and 3 0 Differential Equations

25 1

Another powerful area is the construction of exponentialy fitted and trigonometrically fitted symplectic integrators (as we have seen in this review). Re~ently'~.'~ symplectic integrators have been applied for the numerical solution of the onedimensional and two-dimensional Schrodinger equations. We will investigate the above new areas in a future review.

Appendix A % In this programme the derivation of the methods LMMI-LMMV is presented. YOThe derivation is based on the theory developed in ref. 6 (see Section 2.1)

%

YOThe equations will be solved by an application of Cramer's rule. % From the theory of exponentially fitted methods it has been shown % that to avoid divisions by zero valued determinants we must apply YO1'Hospital's rule. Hence we find the appropriate derivatives of the YOdeterminants of the matrices. YO YOMETHODS LMMI - LMMIV

YO YOThe elements of the matrices for the solution of the system of equations YO(40) are determined. YO > parl: = -2*wA2*cos(3*w); > par2: = -2*wA2*cos(2*w); > par3: = -2*wA2*cos(w); > par4: = -w"2; > par5: = -2*cos(w) + ~ * c o s ( ~ * w~ ) *cos(~*w +)~ * c o s ( ~ * w ) ; > parl0: = subs(w = w0,parl); > parl 1: = subs(w = w1,parl); > parl2: = subs(w = w2,parl); > parl3: = subs(w = w3,parl); > par20: = subs(w = wO,par2); > par21: = subs(w = wl,par2); > par22: = subs(w = w2,par2); > par23: = subs(w = w3,par2); > par30: = subs(w = wO,par3); > par3 1: = subs(w = wl,par3); > par32: = subs(w = w2,par3); > par33: = subs(w = w3,par3); > par40: = subs(w = wO,par4); > par41: = subs(w = wl,par4); > par42: = subs(w = w2,par4); > par43: = subs(w = w3,par4); > par50: = subs(w = wO,par5); > par5 1: = subs(w = wl 'pa1-5);


252

> par52: = subs(w = w2,par5); > par53: = subs(w = w3,par5); > with(linalg); > > 'Method LMMI'; > % % The Denominator %

> matl: = array(l..4,1..4,[[par10,par20,par30,par4O],[parl 17par21,par31,par41], > [parl2,par22,par32,par42],[par13,par23,par33,par43]]); %

YOWe must find % the 6th derivative w.r.t w2, the 4th derivative w.r.t wl, % and the 2nd derivative w.r.t w0. YO > den: = det(mat1); > den: = diff(den,w0$2,~1$4,~2$6); YO % Substitutions based on the theory developed in ref. 6 %

> den: = subs(w0 = 0,wl

= O,w2 = O,w3 = w,den);

YO

YOCalculation of the exponentially fitted version YOof the denominator YO

> den-exp:

= simplify(subs(w =

-I*w,den));

% % Coefficient bo

YO

> mat2: = array( 1..4,1..4,[ [par5O,par2O,par3O,par40],[par51,par2 1,par3 1,par4 11, > [par52,par22,par32,par421,Cpar53 ,par23,par33,par43]]);

> db0: = det(mat2); > db0: = diff(dbO,wO$2,~1$4,~2$6); YO % Substitutions based on the theory developed in ref. 6 %

> db0: = subs(w0 = O,W1 = O , W = ~ O,W~ = w,dbO); > db0-exp: = simplify(subs(w = -I* w,dbO)); > b[O][trig_fitted]: = combine(dbO/den); % % Calculation of the exponentially fitted version % of the coefficient bo % > b[O] [exp-fitted] : = combine(dbO-exp/den-exp); %


253

YOCoefficient b, YO > mat3: = array( 1..4,1..4,[[parlO,par50,par30,par40],[parl l,par5l,par3 l,par41], > [par 12,par52,par32,par42],[par 13,par53,par33,par43]]); > dbl : = det(mat3); > db 1: = diff(db 1,w0$2,w 1$4,w2$6); YO YOSubstitutions based on the theory developed in ref. 6 YO > dbl: = subs(w0 = 0,wl = 0 , ~ = 2 O,W~ = W,dbl); > dbl-exp: = simpliQ(subs(w = -I*w,dbl)); > b[ l][trig_fitted]: = combine(dbl/den); YO YOCalculation of the exponentially fitted version % ofthe coefficient b, %

> b[ l][exp-fitted]: = combine(db 1-exp/den-exp); YO YOCoefficient b2 %

> mat4: = array( 1..4,1..4,[[parlO,par20,par50,par40],[parll,par2l,par5l,par41], > [par 12,par22,par52,par42],[par 13,par23,par53,par43]]); > db2: = det(mat4); > db2: = diff(db2,w0$2,w 1$4,w2$6); % % Substitutions based on the theory developed in ref. 6

YO

> db2: = subs(w0 = 0,wl = O , W = ~ O , W ~= w,db2); > db2-exp: = simpliQ(subs(w = -I*w,db2)); > b[2][trig_fitted]: = combine(db2/den); YO % Calculation of the exponentially fitted version % of the coefficient b2 %

> b[2J[exp-fitted]:

= combine(db2_exp/den_exp);

YO % Coefficient b3 %

> mat5: = array(l..4,1.,4,[[par10,par20,par30,par50],[parl l,par2l,par3 l,par51], > [par 12,par22,par32,par521,[par 13,par23,par33,par5311); > db3: = det(mat5); > db3: = diff(db3,~0$2,~1$4,~2$6); % % Substitutions based on the theory developed in ref. 6

YO

> db3: = subs(w0 = 0,wl

= O , W= ~ O,W~ = w,db3);

254


> db3-exp: = simplify(subs(w = -I*w,db3)); > b[3][trigfitted]: = combine(db3/den); % % Calculation of the exponentially fitted version % of the coefficient b3 %

> b[3] [exp-fitted]: = cornbine(db3_exp/den_exp); YO YOTaylor series expansion of the coefficients b[i], i = O( 1)3. YO YONote: exp-fitted: exponentially fitted method and % trig-fitted: trigonometrically-fitted method. YO > b[O][exp-fitted-taylor]: = convert(taylor(b[O][exp-fitted],~= 0,24),polynom); > b[ l][exp-fitted-taylor]: = convert(taylor(b[ l][exp-fitted],~ = 0,24),polynom); > b[2] [exp-fitted-taylor]: = convert(taylor(b[2][exp-fitted],~= 0,24),polynom); > b[3] [exp-fitted-taylor]: = convert(taylor(b[3][exp-fitted],~= 0,24),polynom); > b[O] [trig-fitted-taylor]: = convert(taylor(b[O][trigfitted],~= 0,24),polynom); > b[ l][trigfitted-taylor]: = convert(taylor(b[ l][trig_fitted],w = 0,24),polynom); > b[2][trig-fitted-taylor]: = convert(taylor(b[2][trig_fitted],w = 0,24),polynom); > b[3] [trigfitted-taylor]: = convert(taylor(b[3][trig_fitted],w = 0,24),polynom); > > 'Method LMMII'; > YO % The Denominator

YO %

YONote: As before with [exp-fitted] or -exp we denote the YOexponentially fitted versions of the coefficients, denominator % and numerators YO > mat 1: = array( 1..4,1..4,[[par 1O,par20,par3O,par40],[par 1 1,par2 1,par3 1,par4 11, > [par 12,par22,par32,par42],[par 13,par23,par33,par43]]); > den: = det(mat1); YO YOWe must find YOthe 1st derivative w.r.t w2, the 4th derivative w.r.t wl, YOand the 2nd derivative w.r.t w0. YO > den: = diff(den,w0$2,w 1$4,w2$1); > den: = subs(w0 = 0,wl = O,w2 = w,w3 = w,den); > den-exp: = simplify(subs(w = -I*w,den)); YO YOCoefficient b, %


255

> mat2: = array( 1..4,1..4,[[par50,par20,par30,par40],[par51,par2 1,par3 1,par41], > [par52,par22,par32,par42],[par53,par23,par33,par43]]); > dbO: = det(mat2); > db0: = diff(dbO,wO$2,~1$4,~2$1); > db0: = subs(w0 = 0,wl = O , W = ~ W , W=~ w,dbO); > db0-exp: = simplify(subs(w = -I*w,dbO)); > b[O][trigfitted]: = combine(dbO/den); > b[O] [exp-fitted]: = combine(dbO_exp/den-exp); YO % Coefficient 6,

YO

> mat3: = array( 1..4,l..4,[[parlO,par50,par30,par40],[parll,par5 l,par3 l,par41], > [parl2,par52,par32,par42],[parl3,par53,par33,par43]]); > dbl: = det(mat3); > dbl: = diff(dbl,wO$2,~1$4,~2$1); > dbl: = Subs(w0 = 0,wl = O , W ~= W , W=~ w,dbl); > dbl-exp: = simplify(subs(w = -I*w,dbl)); > b[ l][trig_fitted]: = combine(dbl/den); > b[ 11[exp-fitted] : = combine(db 1-exp/den-exp); %

YOCoefficient b, %

> mat4: = array( 1..4,1..4,[[parlO,par20,par50,par40],[parl l,par2l,par5l,par41], > [parl2,par22,par52,par42],[parl3,par23,par53,par43]]); > db2: = det(mat4); > db2: = diff(db2,w0$2,w 1$4,w2$1); > db2: = Subs(w0 = 0,wl = O , W = ~ W , W=~ w,db2); > db2-exp: = simplify(subs(w = -I*w,db2)); > b[2][trig_fitted]: = combine(db2lden); > b[2][exp_fitted]: = combine(db2_exp/den_exp); YO

YOCoefficient b3 %

> mat5: = array(l..4,1..4,[[parlO,par20,par30,par50],[parl l,par2l,par3 l,par51], > [par 12,par22,par32,par52],[par13,par23,par33,par53]]); > db3: = det(mat5); > db3: = diff(db3,~0$2,~1$4,~2$1); > db3: = Subs(w0 = 0,wl = 0 , ~ = 2 W , W= ~ w,db3); > db3-exp: = simplify(subs(w = -I*w,db3)); > b[3][trig_fitted]: = combine(db3lden); > b[3] [exp-fitted] : = combine(db3_exp/den_exp); % % Taylor series expansion of the coefficients b[i], i = 0(1)3. % % Note: exp-fitted: exponentially fitted method and % trig-fitted: trigonometrically-fitted method.

256


YO

> b[01[exp-fitted-taylor] : = convert(taylor(b[O][exp-fitted] ,w = 0,24),polynom); > b[ l][exp-fitted-taylor]: = convert(taylor(b[ l][exp-fitted],~ = 0,24),polynom); > b[2] [exp-fitted-taylor]: = convert(taylor(b[2][exp-fitted],w = 0,24),polynom); > b[3][exp-fitted-taylor]: = convert(taylor(b[3][exp-fitted],w = 0,24),polynom);

> b[O][trigfitted-taylor]:

= convert(taylor(b[O][trig-fitted],~= 0,24),polynom);

> b[ l][trig-fitted-taylor]: > b[2][trigfitted-taylor]: > b[3] [trig-fitted-taylor]: > > 'Method LMMIII'; >

= convert(taylor(b[ l][trig-fitted],~= 0,24),polynom); = convert(taylor(b[2][trig-fitted],w = 0,24),polynom); = convert(taylor(b[3] [trig-fitted],~= 0,24),polynom);

% % The Denominator % %

YONote: As before with [exp-fitted] or -exp we denote the % exponentially fitted versions of the coefficients, denominator

% and numerators

YO

> mat1 : = array(l..4,1..4,[[parlO,par20,par30,par40],[parl l,par2l,par3 l,par41], > [par 12,par22,par32,par42],[par 13,par23,par33,par43]]); > den: = det(mat1); % YOWe must find % the 2nd derivative w.r.t w2, the 1st derivative w.r.t w 1,

YOand the 2nd derivative w.r.t w0. %

> den: = diff(den,w0$2,w 1$1,w2$2); > den: = subs(w0 = 0,wl = w,w2 = w,w3 = w,den); > den-exp: = simplify(subs(w = -I*w,den)); YO % Coefficient bo %

> mat2: = array( 1..4,1,.4,[[par5O,par2O,par3O,par40],[par5l,par2l,par3 l,par41], > [par52,par22,par32,par42],[par5 3,par23,par33,par43]]); > db0: = det(mat2); > db0: = diff(dbO,w0$2,w1$1,w2$2); > db0: = subs(w0 = O,W1 = W , W=~ W , W=~ w,dbO); > db0-exp: = simplify(subs(w = -I*w,dbO)); > b[O][trig_fitted]: = combine(dbO/den); > b[O][exp-fitted] : = combine(db0-exp/den-exp); % % Coefficient bl %

> mat3: = array(l..4,1..4,[[par10,par50,par30,par40],[parl l,par5 l,par3 l,par41],

4: Numerical Methods for the Solution of 10, 2 0 and 3 0 Digerential Equations

257

> [par 12,par52,par32 ,par42],[par 13,par53,par33,par43]]); > dbl : = det(mat3); > db 1: = diff(db 1,w0$2,w 1$1,w2$2);

> dbl: = subs(w0 = 0,wl = W , W=~ W , W= ~ w,dbl); > dbl-exp: = simplify(subs(w = -I*w,dbl)); > b[ l][trigfitted]: = combine(dbl/den); > b[ l][exp-fitted]: = combine(db 1-exp/den-exp); % % Coefficient b2 %

> mat4: = array( 1..4,1..4,[[parlO,par20,par50,par40],[parll,par2l,par5 1,par41], > [parl2,par22,par52,par42],[parl3,par23,par53,par43]]); > db2: = det(mat4); > db2: = diff(db2,~0$2,wl$l,w2$2); > db2: = sUbs(w0 = 0,wl = W , W ~= W , W=~ w,db2); > db2-exp: = simplify(subs(w = -I*w,db2)); > b[ 21[tri~fitted] : = combine(db2/den); > b[2][exp_fitted]: = combine(db2_exp/den_exp); YO YOCoefficient b3 YO

> mat5: = array( 1..4,1..4,[[parlO,par20,par30,par50],[parl l,par2l,par3 l,par51], > [parl2,par22,par32,par52],[parl3,par23,par33,par53]]); > db3: = det(mat5); > db3: = diff(db3,~0$2,wl$l,w2$2); > db3: = subs(w0 = 0,wl = W , W=~ W , W=~ w,db3); > db3-exp: = simplify(subs(w = -I*w,db3)); > b[3][trig_fitted]: = combine(db3/den); > b[ 31[exp-fitted] : = cornbine(db3_exp/den_exp); YO YOTaylor series expansion of the coefficients b[i], i = O( 1)3. YO % Note: exp-fitted: exponentially fitted method and YOtrig-fitted: trigonometrically-fitted method.

YO

> b[ 01[exp-fitted-taylor] : = convert(taylor(b[01[exp-fitted],~= 0,24),polynom); > b[ l][exp-fitted-taylor]: = convert(taylor(b[ l][exp_fitted],w = 0,24),polynom); > b[2] [exp-fitted-taylor]: = convert(taylor(b[2][exp-fitted],~= 0,24),polynom); > b[3][exp_fitted_taylor]: = convert(taylor(b[3][exp-fitted],w = 0,24),polynom); > b[O][trig-fitted-taylor]: > b[ l][trigfitted-taylor]:

= convert(taylor(b[O][trig_fitted],w = 0,24),polynom); = convert(taylor(b[l][trig_fitted],w = 0,24),polynom); > b[21[tri~fitted-taylor] : = convert(taylor(b[21[trigfitted],w= 0,24),polynom); > b[3] [tri~fitted-taylor]: = convert(taylor(b[3][tri~fitted],w= 0,24),polynom);

> > 'Method LMMIV';

>

258


% % The Denominator % % % Note: As before with [exp-fitted] or -exp we denote the

% exponentially fitted versions of the coefficients, denominator % and numerators %

> matl: = array(l..4,1..4,[[par10,par20,par30,par40],[parl 1,par2l7par3l,par41], > [par 12,par22,par32,par42],[par 13,par23,par33,par43]]); > den: = det(mat1); % % We must find % the 3rd derivative w.r.t w2, the 2nd derivative w.r.t wl,

YOand the 1st derivative w.r.t w0. YO > den: = diff(den,wO$l ,w 1$2,w2$3); > den: = subs(w0 = w,wl = w,w2 = w,w3 = w,den); > den-exp: = simplify(subs(w = -I*w,den)); % % Coefficient bo % > mat2: = array( l..4,1..4,[[par5O,par20,par3O,par4O],[par5 l,par2l,par3 l,par41],

> [par52,par22,par32,par42],[par53,par23,par33,par43]]); > db0: = det(mat2);

> db0: = diff(dbO,wO$ 1,w 1$2,w2$3); > db0: = subS(w0 = w,wl = W , W=~ W , W=~ w,dbO); > db0-exp: = simplify(subs(w = -I*w,dbO)); > b[O][trig_fitted]: = combine(dbO/den); > b[O][exp-fitted]: = combine(db0-exp/den-exp); YO % Coefficient b, %

> mat3: = array( 1..4,l..4,[[parlO,par50,par30,par40],[parl l,par5l,par3 l,par41], > [par 12,par52,par32,par42],[par13,par53,par33,par43]]); > dbl: = det(mat3); > dbl: = diff(dbl,wO$l,w1$2,~2$3); > dbl: = sUbs(w0 = w,wl = W , W=~ W , W=~ w,dbl); > dbl-exp: = simplify(subs(w = -I*w,dbl)); > b[ l][trig_fitted]: = combine(db l/den); > b[ l][exp-fitted]: = combine(db 1-exp/den-exp); % % Coefficient b2 %

> mat4: = array( 1..4,1..4,[[parlO,par20,par50,par40],[parl 17par21,par51,par41], > [par12,par22,par52,par42],[par13,par23,par53,par43]]);

4: Numerical Methods for the Solution of 10,2 0 and 3 0 Drferential Equations

259

> db2: = det(mat4); > db2: = diff(db2,wO$l,w 1$2,w2$3); > db2: = subS(w0 = w,wl = W , W=~ W , W=~ w,db2); > db2-exp: = simplify(subs(w = -I * w,db2)); > b[2][trig_fitted]: = combine(db2/den); > b[2][exp_fitted]: = combine(db2_exp/den_exp); YO % Coefficient b3 %

> mat5: = array( 1..4,1..4,[[parlO,par20,par30,par50],[parll,par2l,par3 l,par5 I], > [parl2,par22,par32,par52],[parl3,par23,par33,par53]]); > db3: = det(mat5); > db3: = diff(db3,wO$l,w 1$2,w2$3); > db3: = subs(w0 = w,wl = W , W=~ W , W=~ w,db3); > db3-exp: = simplify(subs(w = -I*w,db3)); > b[3][trig-fitted]: = combine(db3/den); > b[3][exp_fitted]: = combine(db3_exp/den_exp); YO YOTaylor series expansion of the coefficients b[i], i = 0(1)3. YO YONote: exp-fitted: exponentially fitted method and YOtrig-fitted: trigonometrically-fitted method. YO > b[01[exp-fitted-taylor] : = convert(taylor(b[01[exp-fitted],~ = 0,24),polynom); > b[ l][exp-fitted-taylor]: = convert(taylor(b[ l][exp-fitted],~ = 0,24),polynom); > b[2] [exp-fitted-taylor]: = convert(taylor(b[2][exp-fitted],w = 0,24),polynom); > b[3][exp-fitted-taylor]: = convert(taylor(b[3][exp-fitted],w = 0,24),polynom); > b[O][tri~fitted-taylor]: = convert(taylor(b[O][trig_fitted],w = 0,24),polynom); > b[ l][trig_fitted-taylor]: = convert(taylor(b[ l][trig_fitted],w = 0,24),polynorn); > b[2] [trig-fitted-taylor]: = convert(taylor(b[2][trig+fitted],w = 0,24),polynom); > b[3] [trig-fitted-taylor]: = convert(taylor(b[3][trigJitted],w = O,%),polynom);

> > 'Method LMMV'; > %

YOThe elements of the matrices for the solution of the system of equations % (56) are determined. %

> par0: = - - ~ * C O S ( ~ * W ) ; > parl: = -2*wA2*cos(3*w); > par2: = -2*wA2*cos(2*w); > par3: = -2*wA2*cos(w); > par4: = -wA2; > par5: = -2*cos(w) + ~ * c o s ( ~ * w +)~ * C O S ( ~ " W ) ; > par00: = subs(w = w0,parO); > par01 : = subs(w = w 1,pad);

260

Chemical Modelling: Applications and T h e o q Vblume 2

> parO2: = subs(w = w2,parO); > parO3: = subs(w = w3,parO); > parO4: = subs(w = w4,parO); > parlo: = subs(w = w0,parl); > parll: = subs(w = w1,parl);

> parl2: = subs(w = w2,parl); > parl3: = subs(w = w3,parl); > parl4: = subs(w = w4,parl); > par20: = subs(w = wO,par2); > par2 1: = subs(w = wl ,par2); > par22: = subs(w = w2,par2); > par23: = subs(w = w3,par2); > par24: = subs(w = w4,par2); > par30: = subs(w = wO,par3); > par3 1: = subs(w = w 1,par3); > par32: = subs(w = w2,par3); > par33: = subs(w = w3,par3); > par34:

= subs(w = w4,par3);

> par40: = subs(w = wO,par4); > par41 : = subs(w = w 1,par4); > par42: = subs(w = w2,par4); > par43: = subs(w = w3,par4); > par44: = subs(w = w4,par4); > par50: = subs(w = wO,par5); > par5 1: = subs(w = w 1,par5);

> par52: = subs(w = w2,par5); > par53: = subs(w = w3,par5); > par54: = subs(w = w4,par5); > with(linalg); % % The Denominator

% %

YONote: As before with [exp-fitted] or -exp we denote the % exponentially fitted versions of the coefficients, denominator % and numerators %

> mat1 : = array( 1..5,1..5,[[par00,par10,par20,par30,par40],[par01,par1 1,par21, > par3 1,par4 11,[parO2,par12,par22,par32 ,par42],[par03,par 13,par23,par33,par43], > [par04,par 14,par24,par34,par44]]); > den: = det(mat1); % % We must find % the 4th derivative w.r.t w3, 3rd derivative w.r.t w2, the % 2nd derivative w.r.t wl, and the 1st derivative w.r.t w0. %


26 1

> den: = diff(den,wO$l,w 1$2,w2$3,w3$4); > den: = subs(w0 = w,wl = w,w2 = w,w3 = w,w4 = w,den); > den-exp: = simplify(subs(w = -I*w,den)); YO YOCoefficient a. YO > mat2: = array( 1..5,1..5,[[par5O,parlO,par20,par30,par40],[par5 1,parl l,par21, > par3 1,par4 1],[par52,par 12,par22,par32,par42],[par53,par 13,par23,par33,par43], > [par54,par 14,par24,par34,par44]]); > daO: = det(mat2); > daO: = diff(daO,wO$l ,w 1$2,w2$3,w3$4); > daO: = subs(w0 = w,wl = w,w2 = w,w3 = w,w4 = w,daO); > dd-exp: = simplify(subs(w = -I*w,daO)); > a[O][trig_fitted]: = combine(daO/den); > a[0][exp-fitted] : = combine(da0-exp/den-exp); YO YOCoefficient bo YO > mat3: = array( 1..5,1..5,[[parOO,par50,par20,par3O,par40],[par0 1,par5 1,par2 1, > par3 1,par4 17,[par02,par52,par22,par32,par42],[par03,par53,par23,par33,par43], > [par04,par54,par24,par34,par44]]); > db0: = det(mat3); > db0: = diff(dbO,wO$l,w 1$2,w2$3,w3$4); > db0: = subs(w0 = w,wl = W , W=~ W , W ~= W , W=~ w,dbO); > db0-exp: = simplify(subs(w = -I*w,dbO)); > b[O][trig_fitted]: = combine(dbO/den); > b[O][exp-fitted]: = combine(db0-exp/den-exp); YO % Coefficient b, YO > mat4: = array(l..5,1..5,[[par00,parlO,par50,par30,par40],[parOl,parl l,par51, > par3 1,par4 13 ,[parO2,par12,par52,par32,par42],[par03,par13,par53,par33,par43], > [par04,par 14,par54,par34,par44]]); > dbl: = det(mat4); > dbl: = diff(db1,wO$l,w1$2,w2$3,w3$4); > dbl: = subs(w0 = w,wl = W , W=~ W , W= ~ W , W=~ W,dbl); > dbl-exp: = simplify(subs(w = -I*w,dbl)); > b[ l][trig_fitted]: = combine(dbl/den); > b[ l][exp-fitted]: = combine(db1-exp/den-exp); %

YOCoefficient b2 YO

> mat5: = array( 1..5,1..5,[[par00,par10,par20,par50,par40],[parOl,parl l,par21, > par5 1,par4 l],[par02,parl2,par22,par52,par42],[par03,parl3,par23,par53,par43], > [par04,par 14,par24,par54,par44]]); > db2: = det(mat5);

262


> db2: = diff(db2,w0$1,w 1$2,w2$3,w3$4); > db2: = subs(w0 = w,wl = W , W=~ W , W=~ W , W=~ w,db2);

> db2-exp: = simplify(subs(w = -I*w,db2)); > b[2][trig_fitted]: = combine(db2/den); > b[2][exp-fitted]:

= combine(db2_exp/den_exp);

% % Coefficient b3 %

> mat6: = array(l..5,1..5,[[par00,par10,par20,par30,par5O],[par0l,parl l,par21, > par3 1,par5 l],[parO2,par12,par22,par32,par52],[par03,par13,par23,par33,par53], > [par04,parl4,par24,par34,par54]]); > db3: = det(mat6); > db3: = diff(db3,wO$l,w 1$2,w2$3,w3$4); > db3: = subs(w0 = w,wl = W , W=~ w,w3 = w,w4 = w,db3); > db3-exp: = simplify(subs(w = -I*w,db3)); > b[3][trig_fitted]: = combine(db3/den); > b[ 31[exp-fitted] : = combine(db3_exp/den_exp); YO

% Taylor series expansion of the coefficients a[O] and b[i], i = 0(1)3. % % Note: exp-fitted: exponentially fitted method and % trig-fitted: trigonometrically-fitted method. % > a[ 01[exp-fitted-taylor] : = convert(taylor(a[O][exp-fitted],~= 0,24),polynom);

> b[O] [exp-fitted-taylor]: = convert(taylor(b[O][exp-fitted],~= 0,24),polynom); > b[ l][exp-fitted-taylor]: = convert(taylor(b[ l][exp-fitted],~= 0,24),polynom); > b[2] [exp-fitted-taylor]: = convert(taylor(b[2][exp-fitted],~= 0,24),polynom); > b[3][exp-fitted-taylor]: = convert(taylor(b[3][exp-fitted],~= 0,24),polynom); > a[O][trig-fitted-taylor]: = convert(taylor(a[O][trig_fitted],w = 0,24),polynom); > b [01[trig-fitt ed-t ay1or]: = convert(tay1or(b[01[trig-fitt ed] ,w = 0,24),p01ynom) ; > b[ l][trig-fitted-taylor]: = convert(taylor(b[ l][trig_fitted],w = 0,24),polynom); > b[2][trig_fitted_taylor]: = convert(taylor(b[2][trig-fitted],~= 0,24),polynom); > b[31[trig-fitted-taylor] : = convert(taylor(b[31[trigfitted],w = 0,24),polynom); > Appendix B % % Exponentially Fitted Dissipative Two-step Method - Case I % % The system of equations (106) %

> eql: = 2*c[2] + c[1] + 2*c[0] = 1; > eq2: = 2*hA4= 88/3*hA4*c[0]+ 8/3*hA4*c[0]*z+ 6*hA4*c[2]; > eq3: = 0 = -80/3*hA5*c[0] - 40/3*hA5*c[0]*z;


263

> eq4: = exp(w*h) - 2 + l/exp(w*h) = hA2*w"2*c[l]+ 2*c[2]*h"2*wA2 + > 1/4*c[2]*h"4*wA4+ 5/576*hA6*w"6*c[2]+ > 1/576*hA6*wA6*c[2]/exp(w*h)+ 2*c[O]*hA2*w"2+ c[O]*h"4*wA4+ > 2/9*hA4*w"4* c[O]/exp(w*h) + 1/36* c[01* h" 6" w"6 1/ > 36* h"6*~"6*c[O]/exp(w*h) + 5/864* h"8* w"8*c[O] + 1/864*h"8*wA8"c[0]/ > exp(w* h) + 1/9*h 4* w"4* c [01* z/exp(w*h); % % The solution of the system of equations YO

> solut: = solve({eql,eq2,eq3,eq4),(c[O],c[l],c[2],z}); > assign(solut); > c[O]: = combine(c[O]); > c[ 11: = combine(c[ 13); > c[2]: = combine(c[2]); YO % The Taylor series expansions for the coefficients of the method

YO

> cot: = convert(series(c[O],w = 0,22),polynom); > cl t: = convert(series(c[ l],w = 0,22),polynom);

> c2t: = convert(series(c[2],w = 0,22),polynom); %

YOWe note that the solution of the above system gives z = -2. For this reason YOthere isn't need for the computation of the Taylor for the coefficient z. YO YO %

YOTrigonometrically Fitted Dissipative Two-step Method - Case I YO % The system of equations (109)

YO

> eql: = 2*c[2] + c[1] + 2*c[0] = 1; > eq2: = 2*hA4= 88/3*hA4*c[0]+ 8/3*hA4*c[O]*z+ 6*hA4*c[2]; > eq3: = 2*cos(w*h) - 2 = 1/1728*h"2*wA2*(-1728*c[l] 3456*c[2] + > 432*c[2]*hA2*w"2 - 15*c[2]*h"4* w"4 - 3 *c[2]*h"4* w"4*cos(w* h) > 3456*c[O] + 1728*c[0]*h"2*wA2+ 384*c[0]*h"2*w"2*cos(w*h) > 48* c [01* hA 4 * w" 4 + 4 8 * c [01* h"4 * w 4 * cos(w * h) + 10* c [01* h A 6 * wA 6 + > 2 * c [01* h A 6 * wA 6 * cos(w * h) + 192* c [01* h"2 * z * w"2* cos(w *h)); > eq4: = 0 = -1/1728*h"4*w"4*sin(w*h)*(-3*c[2]*h"2*w"2+ 384*c[O] + > 48*c[0]*hA2*w"2 + 2*c[O]*h"4*wA4+ 192*c[O]*z); -

-

% % The solution of the system of equations

YO > solut: = solve({eql ,eq2,eq3,eq4),{c[O],c[ l],c[2],z}); > assign(so1ut); > c[O]: = combine(c[O]); > c[ 11: = combine(c[ 13);

264


> c[2]: = combine(c[2]); > z: = combine(z); % % The Taylor series expansions for the coefficients of the method %

> cot: = convert(series(c[O],w = 0,22),polynom); > clt: = convert(series(c[ l],w = 0,22),polynom); > c2t: = convert(series(c[2],w = 0,22),polynom); > zzt: = convert(series(z,w = 0,22),polynom); % % % % Exponentially Fitted Dissipative Two-step Method - Case I1 % % The system of equations (1 12) %

> eql: = c[l] + 2*c[2] + ~ * c [ o = ] 1;

> eq2: = 2*hA4= 6*hA4*c[2]+ 88/3*hA4*c[0]+ 8/3*hA4*c[O]"z; > eq3: = exp(w*h) - 2 + l/exp(w*h) = hA2"w"2*c[l] + 2*c[2]*hA2*w"2 + > 1/4* c[2] * hA4*w"4 + 5/576*h" 6*w 6 * c [21 + 1/5 76" h" 6 * wA 6 * c [2]/ > exp(w*h) + 2*c[O]"h"2*wA2+ c[O]*hA4*w"4+ 2/9*h"4*wA4*c[0]/ > exp(w*h) + 1/36*c[0]*h"6*wA6 - 1/36*h"6*w"6*c[O]/exp(w*h)+ > 5/864*h" 8* w" 8* c [01 + 1/864* h" 8* w" 8* c [O]/exp(w *h) +

> 1/9* h A 4* w A 4* c[01* z/exp(w* h) ; > eq4: = exp(w*h)*x + exp(w*h)*h - 2*x + exp(-w*h)*x - exp(-w*h)*h = > 8/9*hA4*wA3* c[01* exp(-w*h) + 1/96* h" 6*w" 5* c[2]* exp(-w* h) + > 1/9 * hA4*wA4*c [01* z*exp( -w * h) * x + 1/ 108 * h" 8 * wA7* c[01* exp( -w * h) > 2/9* hA5 *wA4*c [01* exp(-w* h) - 1/36* h" 6" w" 6* c[01* exp(-w * h)* x + > 2/9* hA4*wA4*c [01* exp(-w* h)* x - 1/5 76" h 7* w 6" c[2] * exp(-w * h) A

A

-

> 1/864* h*9 * w" 8* c [01* exp(-w * h) - 1/6 * h" 6" wA 5* c [01* exp( -w * h) + > 1/864 * h" 8 * wA 8 * c [01* exp( -w * h) * x + 1/3 6*h "7* w 6* c[01* exp( w * h) + > 4/9* hA4*w"3 * c[O]* z* exp( -w* h) - 1/9 * h" 5* w"4* c[O]* z* exp(-w* h) + > 1/576*h"6*wA6*c[2]*exp(-w*h)*x + 2*hA2*w*c[l]+ 4*hA2*w*c[0]+ > 4*hA2*w*c[2]+ hA4*w"3*c[2]+ 1/6*h"6*~"5*c[O]+ > 2*h"2*wA2*c[0]*x + 1/36*h"6*wA6*c[0]*x+ h"4*wA4*c[0]*x + > 5/576"h"6*wA6*c[2]*x + 1/4*h"4*wA4"c[2]*x + 5/864*hA8*w"8*c[0]*x + > 4*hA4*w"3*c[O]+ 2*h"2*w"2*c[2]*x + 5/108*h"8*wA7*c[O]+ -

> h*2*wA2*c[1]*x+ 5/96*hA6*w"5*c[2];

% % The solution of the system of equations %

> solut: = solve({eql,eq2,eq3,eq4},{ c[O],c[l],c[2],z)); > assign(so1ut); > c[O]: = combine(c[O]); > c[ 11: = combine(c[ 11); > c[2]: = combine(c[2]);


265

> z: = combine(z); Y O

% The Taylor series expansions for the coefficients of the method %

> cot: = convert(series(c[O],w = 0,22),polynom);

> clt: = convert(series(c[ l],w = 0,22),polynom); > c2t: = convert(series(c[2],w = 0,22),polynom); > zzt: = convert(series(z,w = 0,22),polynom); % % % % Trigonometrically Fitted Dissipative Two-step Method - Case I1 Y O

% The system of equations (1 15)

%

> eql: = 2*cos(w*h) - 2 = 1/1728*h"2*wA2*(-1728*c[l]- 3456*c[2] + > 432*c[2]*hA2*w"2 - 15*c[2]*hA4*w"4- 3*c[2]*h"4*wA4*cos(w*h) > 3456*~[0]+ 1728*c[0]*hA2*w"2+ 3 8 4 * ~ [ 0 ] * h " 2 * ~ " 2 * ~ 0 ~-( ~ * h ) > 48*c[O]*h"4*wA4+ 48*c[0]*h"4*w"4*cos(w*h) + 1O*c[0]*hA6*w"6+ > 2*c[O]*h"6*~"6*cos(w*h) + 192*c[0]*h"2*z*wA2*cos(w*h)); > eq2: = 0 = -1/1728*h"4*wA4*sin(w*h)*(-3*c[2]*h"2*wA2+ 384*c[O] + > 48*c[0]*hA2*w"2 + 2*c[0]*hA4*w"4 + 192*c[O]*z); > eq3: = - 1/1728*h"2*~"2*(-432*~[2]*h"2*~*~"2 + 3456*c[2]*x > 18* c[2]* h"4 * w" 3* sin(w*h) + 288* c[01* h" 4* w"3* sin(w * h) + > 1536* c[01* hA2* w * sin(w* h) + 16*c [01* h" 6* w" 5* sin(w* h) + > 1728*c[1]*x + 3456*c[O]*x + 384*c[0]*h"3*w"2*cos(w*h)+ > 768*c[0]*hA2*z*w* sin(w*h) - 10*c[O]*h"6* wA6*x+ > 48 * c[01* h"4 * W" 4* x - 1728* c [01*hA 2 * w"2 * x + 15 * c [21* h"4*w"4 * x + > 3*hA4*c[2] *w"4* cos(w*h)*x - 48*h"4* c[O]*w"4* cos(w*h)*x + > 192 * c [01* h" 3* z* w"2 * cos(w*h) 384* c[O] * h"2 * w" 2 * cos(w * h) * x > 192* c [01* h A 2 * z* w A 2 * cos(w* h) * x - 2 * c [01* h"6 * w"6 * cos(w * h)* x + > 2* c[O]* h"7* wA6*cos(w*h) + 48* h"5 * c[O] * w"4* cos(w*h) > 3*h"5*~[2]*~"4*cos(w*h)) = ~*x*(cos(w*~ -) 1); > eq4: = 1/1728*h"2*w*(90*c[2]*hA4*w"4 - 6912*c[0]*hA2*w"2 > 1728*c[2]*h"2*wA2- 80*c[0]*hA6*w"6+ 3456*c[ 11 + > 18*c[2]*h"4*wA4*cos(w*h) - 288*c[O]*h"4* w"~*cos(w*~) + > 6912*~[0]+ 6912*~[2]- 1536*~[O]*h"2*~"2*~0~(~*h) > 16* c[O]* h" 6*w" 6* cos(w*h) + 384" c[01* h"3 * w" 3 * sin(w*h) > 768* c [01* h A 2 * z * wA 2 * cos(w* h) + 288* c[01* hA 4 * w"4 + > 3* h"4* c[2]* w" 5* sin(w*h)* x - 48 * h"4* c[01* w" 5* sin(w*h)* x + > 192" c[O]* h"3 * z*w" 3* sin(w*h) - 384* c[01* h"2* w" 3* sin(w*h)* x > 192*c[O]* hA2*z*w" 3* sin(w*h)* x - 2* c[O] * h" 6*w" 7" sin(w *h)* x + > 2 * c[01*h" 7* w" 7* sin(w* h) + 48 * h A 5* c [01* wA 5 * sin(w*h) > 3*hA5*c[2]*wA5*sin(w*h))= 2" sin(w*h)*h; -

-

% % The solution of the system of equations

266


%

> solut: = solve({eql,eq2,eq3,eq4),(c[O],c[l],c[2],z}); > assign(so1ut); > c[O]: = combine(c[O]); > c[ 11: = combine(c[ 13); > c[2]: = combine(c[2]); > z: = combine(z); %

YOThe Taylor series expansions for the coefficients of the method %

> cot: = convert(series(c[O],w = 0,22),polynom); > c It: = convert(series(c[ l],w = 0,22),polynom); > c2t: = convert(series(c[2],w = 0,22),polynom); % % We note that the solution of the above system gives z = -2 1/96 wA4*hA4. % For this reason there isn't need for the computation of the Taylor for the % coefficient z.

+

YO

Appendix C Simos and Aguia?' have developed the following symmetric linear multistep method:

Case I Requiring the above method to be exact for any linear combination of the h n ct ions :

Simos and A g ~ i a have r ~ ~ obtained the following system of equations:

2cas(iwh)

-

Icos(iwh) =

- 2w2h2 (b2 cos

(f wh) + bl cos (iwh) + bocos (iwh) )

267


Solving the above system of equations the coefficients of the methods are obtained. For small values of w the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The corresponding exponentially fitted method ( i e . a method which integrates exactly any linear combination of the functions: { 1, x, x2, x3, x4, x5, exp(&wx)}) is produced via the substitution: w = 4 9 . Case I1

Requiring the above method to be exact for any linear combination of the functions:

{ 1, x, cos(wx), sin(wx), xcos(wx), x sin(wx), x2cos(wx), x2 sin(wx)}

(A.4)

Simos and Aguia?’ have obtained the following system of equations: ICOS(5h)

-2cos(;wh)

=

+ bl cos(iwh) + bocos(iwh))

- 2w2h2(bzcos(iwh)

G1

7sin -wh h - 5 s i n

(i 1

-wh h =4h2wbocos(iwh) -5h3w2bosin(iwh)

+ 4h2wbIcos(iwh)

- 3h3w2blsin(iwh)

+4h2wb2cos(fwh)

-

= 4 h2bo cos

(2

wh) - 2 h2b, w2x2cos

+4h2b2cos(fwh)

(i (i

9 h4b,w2cos 2

--

+ 4h2b,cos

h3w2b2sin(~wh)

-

wh)

wh) -

(i

12h3blwsin(iwh)

- 2h2blw2x2cos

h4bow2cos

- 4h3b2wsin

wh)

(i

-

(i

(k

wh)

20h3bowsin(:wh)

wh)

-

21 h4bzw2cos

wh) - 2h2bow2x2cos

(i

($~

wh)

h )

268


Solving the above system of equations the coefficients of the methods are obtained. For small values of w the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The corresponding exponentially fitted method (i.e. a method which integrates exactly any linear combination of the functions: { 1, x, exp(fwx), x exp(kwx), x 2 exp(kwx)}) is produced via the substitution: w = -1q.

References 1. L.D. Landau and EM. Lifshitz, Quantum Mechanics, Pergamon, New York, 1965. 2. I. Prigogine, S. Rice (eds.), New Methods in Computational Quantum Mechanics, Advances in Chemical Physics, Vol. 93, John Wiley & Sons, 1997. 3. J. Vigo-Aguiar, T.E. Simos and A. Tocino, Int. J Modern Phys. C, 2001, 12, 225. 4. M.P. Calvo and J.M. Sam-Serna, SIAMJ Sci. Comput., 1993, 14, 1237. 5. T.E. Simos and J. Vigo-Aguiar, Comput. Phys. Commun., in press. 6. T.E. Simos, in Chemical Modelling - Applications and Theory, Vol. 1, ed. A. Hinchliffe, Specialist Periodical Reports, The Royal Society of Chemistry, Cambridge, 2000, p, 38. 7. J.P Coleman, in Proceedings of the First International Colloquium on Numerical Analysis, ed. D. Bainov and V Civachev, Bulgaria, 1992, p. 27. 8. J.D. Lambert and I.A. Watson, J Inst. Math. Appl., 1976, 18, 189. 9. G.D.Quinlan and S. Tremaine, Astron. J , 1990, 100, 1694. 10. J.R. Dormand, M.E.A. El-Mikkawy and P.J. Prince, IMA 1 Numer Anal., 1987,7,423. 1 1 . G.D. Quinlan, 1999, preprint (astro-pW9901136) 12. E. Stiefel and D.G. Bettis, Numer Math., 1969, 13, 154. 13. PJ. Van der Houwen and B.P Sommeijer, SIAMJ Numer. Anal., 1987, 24, 595. 14. J.P. Coleman and S.C. Duxbury, J Comput. Appl. Math., 2000, 126, 47. 15. S.C. Duxbury, Mixed Collocation Methodsfor y ” = f(x, y), PhD Thesis, University of Durham, 1999. 16. T.E. Simos, Doctoral Dissertation, National Technical University of Athens, Athens, Greece, 1990. 17. G. Avdelas and T.E. Simos, Comput. Math. Appl., 1996,31, 85. 18. G. Avdelas and T.E. Simos, J Comput. Appl. Math., 1996, 72, 345. 19. A. Konguetsof and T.E. Simos, J Comput. Methods Sci. and Eng., in press. 20. T.E. Simos, Int. J Quantum Chem., 1998, 68, 191. 21. T.E. Simos, Int. J Quantum Chem., 1995, 53,473. 22. T.E. Simos and G. Mousadis, Mol. Phys., 1994, 83, 1 145. 23. L.Gr. Ixaru, Numerical Methods for Differential Equations and Applications, Reidel Publishing Company, Dordrecht, Boston, Lancaster, 1984. 24. G. Papageorgiou, Ch. Tsitouras and I.Th. Famelis, Int. J Modern Phys. C, 2001, 12, 657. 25. A.D. Raptis, Bull. Greek Math. SOC.,1984, 25, 113. 26. J.R. Cash, A.D. Raptis and T.E. Simos, J Comp. Phys., 1990, 91, 413. 27. G . Avdelas and T.E. Simos, Phys. Rev. E, 2000, 62, 1375. 28. J. M. Sam-Serna, Acta Numerica, 1992, 1, 243. 29. J.M. Sam-Serna and M.P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994.


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75. X.-S. Liu, X.-Y. Liu, Z.-Y. Zhou, P.-Z. Ding and S.-F. Pan, Int. J Quantum Chem., 2000, 79, 343. 7 6 . X.-S. Liu, X.-Y. Liu, Z.-Y. Zhou, P-Z. Ding and S.-F. Pan, Znt. J Quantum Chem., 2000, 79, 343. 77. X.-S. Liu, L.-W. Su, X.-Y. Liu and P-Z. Ding, Int. J Quantum Chem., 2001, 83, 303.

5 Computer-aided Drug Design 2000-2001 BY RICHARD A. LEWIS

1 Introduction

The year starting from May 2000 has been a very fertile time in the field of molecular modglling, driven by an increased understanding of the big picture of drug discovery and the increasing availability of high-quality data to validate modelling techniques. This review is also not intended to be a catalogue of papers: the coverage would be cursory, and that task is far better done by modem electronic methods for literature alerts. In consequence, this review will be at best a subjective success, at worst, a skim through the rich variety of modelling papers that have appeared this year. As an emphasis has been placed on new work, rather than minor improvements to existing solutions, it is inevitable that some key papers will be missing. The emphasis has been placed on refereed, widely available journals, rather than on books or patents. This will cut down the number of papers reviewed, but it is the author’s hope that this approach may give a better insight into the directions in which modelling is moving. I have tried to abstract out some of the key points from the papers reviewed. Inevitably, this will lead to gross simplifications; the goal is to stimulate interest, and to encourage reading of the primary literature. The main themes of this review are: 3D-QSAR, pharmacophores, library design, ADME/Tox, Docking and Scoring, Cheminformatics and Structure-based drug design.

2 3D-QSAR

The production of high quality 3D-QSAR models is becoming more routine, as the tools for aligning structures, for generating descriptors and processing the statistics improve. It is now becoming possible to include more conformational flexibility into the initial alignments, which will reduce the need for rigid analogues. Models can also be recreated for local use, which increases the value of this method further. The alignment of molecules in their bioactive conformations is, despite a great deal of research in the area, still a significant challenge. The issues have been set out in a paper’ that uses three HIV-1 RT inhibitors. Three molecules are aligned Chemical Modelling: Applications and Theory, Volume 2 0The Royal Society of Chemistry, 2002

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rigidly, flexible, painvise and severally, and the results compared to the bioactive alignment determined from the X-ray structure of the complexes with HIV-1 RT. The conclusions were that painvise rigid alignment will generate the bioactive alignment, but it will not necessarily be scored highly compared to other alignments. The addition of flexibility does not always improve the ranking, but different rigid alignments may converge to the same optimised alignment. The use of multiple alignment does improve the situation, as might be expected, but care does need to be taken with charge redistribution after optimisation. Mills et aL2 describe an alignment method based on stochastic optimisation of distance-matrix overlays, called SLATE. A molecule is represented in terms of its hydrogen-bonding capabilities, using acceptor atoms and donor atom site points, and aromatic ring planes. In turn, this information is expressed as a distance matrix (which is independent of coordinate space). Two molecules are compared by pairwise comparison of their matrices, as the torsions in the molecules are varied to improve the fit. The final alignment is then converted back into coordinate space. Among the parameters that can be adjusted are the resolution of the torsion adjustments, and which bonds are included in the search, and which groups should be included in the hydrogen-bonding analysis. This allows one to apply existing SAR knowledge. The most promising application of this technique is not for overlaying single pairs, but when there are a number of molecules in the SAR that can be used to delineate the alignment more precisely. The program has been applied3 to derive a pharmacophore model for Histamine H3, which was validated by the design and synthesis of novel compounds with potent activity. Another method towards the alignment of molecules has been de~cribed;~ it uses Gaussian feature densities to compute the similarity of two molecules, and Monte Carlo sampling of conformers. The best fits are scored in terms of internal strain and molecular overlap. The method is validated using ligands abstracted from X-ray complexes. The method has not been compared rigorously with existing methods, so it is difficult to say whether it is an improvement. A genetic algorithm (GA) for superposing small molecules5 was also published this year; the chromosome code for torsional flexibility and for atom pairs matches, i.e. the method takes a pharmacophoric representation rather than using all atoms. This allows for a degree of partial matching, which is a very common situation when constructing overlays. The approach also performs Pareto optimality with respect to the size of the substructures matched, to move towards the best compromise to rms fit and the number of atoms matched. Being a genetic algorithm, the method can be applied to multiple molecules, and can be tailored to include knowledge about preferred conformations and so on. Pastor et aL6 have tried to get round the alignment issue by developing GridIndependent Descriptors (GRIND) for use in developing SARs. The first step is obtain the molecular interaction fields around the molecules, using a program like GRID. The key information is extracted by looking at positions of the minima and distances between then, so that the field is reduced down to about 100 key points. These points are then encoded autocorrelograms. These descriptors were then used to derive S A R s with @ values > 0.6. One great advantage of the method is that the key variable for the SAR can be interpreted in terms of the key

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5: Computer-aided Drug Design 2000-2001

points, leading to a pharmacophoric interpretation of the results. The drawback is that the current method is still dependent on the starting conformation of the structures. Another study’ used a homology model followed by DOCK to generate alignments that were validated by constructing a CoMFA model; the results were compared with a manual atom-based alignment. In the latter case, no predictive models could be found, but the DOCK alignment produces a model with @ 0.5, despite the observation that the DOCK scores were uncorrelated to the activities. In a neat refinement cycle, the CoMFA fields were analysed to look for inconsistencies in the homology model, illustrating a complete model building cycle. Guccione et a1.* have approached the alignment problem in 3D-QSAR problem in a novel and interesting manner. Rather than choose a single alignment rule, they used the structure of a template (refined with NMR constraints) and then flexibly superposed the other compounds; naturally, there are several ways in which the molecules can be aligned, so the five lowest energy alignments for each molecule were used in the HASL analysis. This improved the @ value fiom 0.75 (when just a single conformer alignment was used) to 0.92 (for the averaged multiple conformers). A similar philosophy occurs in methods for 4D-QSAR, as described by SantosFilho and Hopfinger.’ A conformer model for each compound in the SAR set is generated, and the Conformers are aligned using predefined alignment rules. The dependent variables for the QSAR are the frequencies of occupancy by

-

Table 1 Some representative high-quality 3D-QSAR models Target

Alignment method

QSAR method

@

Nicotinic a4b2I2 Adenosine A2AI3 Peripheral benzodiazepine

Manual Manual Active analogue Pharmacophorehomology model Field fit Field fit X-ray/dockmg Rigid fit

GRID/GOLPE CoMFA GRID/GOLPE

0.8 1 0.54 0.76 0.72

CoMFA CoMFA GRID/GOLPE CoMFA and GRID/GOLPE HASL ChemX Cerius2 GRID/Golpe CoMFA CoMFA CoMFA CoMSIA CoMFA

0.62 0.85 0.9 0.8

receptor^'^.'^ HIV-1 RT inhibitorsl6-I7

Estrogen18 Cytochrome P450 2A5 and 2A6I9 5HT 1a and Alpha 1 Adenosine A 1 Epothilones’’ Acetylcholine esterase2’ Dopamine D4 antagonists” HIV- I P r ~ t e a s e ~ ~ Cyp450 2C924 CDK2’ c0x-226

*’

FL096 3D query/ChemX Fit to most active analogue X-ray/doclung Fit to rigid analogue X-ray /docking Fit to rigid analogue Field Fit Fit to pharmacophore

0.86 0.78 0.6 0.9 0.73 0.68 0.8 0.74 0.7

274


pharmacophore centres in the conformers of a set of lattice cubes, as before. These frequencies are analysed by PLS and then by the GFA method to produce robust predictive S A R equations with a similar flavour to the models produced by Catalyst." The improvements are in the number of conformers examined (lOOOs), but need for the alignment rules could still be limiting. Another neat confluence of ideas occurs in the work of Lee and Briggs," who, in the course of constructing a 3D-QSAR model, also analyse the best 100 equations to deduce a most probable pharmacophore. A new method for constructing SARs based on the K nearest neighbours was described by Zheng and Tr~psha.~'This is based on the assumption that the activity of an unknown can be estimated by some form of interpolation of the activities of its K nearest neighbours. The distance metric is made up of standard molecular descriptors, and the subset of descriptors that make up the distance metric is established by simulated annealing on the objective function of maximising the value of @ in the training set. Despite its simplicity, the method produces predictive S A R s that have @ significantly above random, and when the authors applied the method to some literature datasets, they obtained better values than other standard methods. The interpolation method used was averaging, and the descriptors were 2D-based, but other functions and descriptors could easily be added.

3 Pharmacophores Pharmacophore analysis is starting to be used more widely, now that more software vendors are producing products that allow one to create and manipulate pharmacophores and their keys more easily. It is also fairly easy to use published models to guide further studies, for example, by providing the alignment rule for 3D-QSAR. Bradley et aL2* have described a new way of applying the pharmacophore method to lead discovery and evolution. The classical description of the pharmacophoric conditions necessary for activity has been as a single set of features separated by distances or other geometric criteria. Any uncertainty has been introduced using tolerances or ranges in the geometry. In this paper, an ensemble of four-point pharmacophores is used, selected on their ability to distinguish actives and inactives. The pharmacophores themselves are part of a general set, built up in PDQ style.29The extra fuzziness in the description deals with the issues of binding mode variance, and of partial matches. The method has been used to find al-adrenergic antagonists, whose 2D similarity to the initial lead is very low. A welcome development this year has been the work of Norinde?' on the refinement of Catalyst pharmacophore hypotheses using simplex optimisation. An initial pharmacophore can be optimised in terms of the positions of the features, using the error in the predicted activities as a scoring function. When this method is applied to hypotheses that contain excluded volumes, the usefulness of the refined hypotheses is greatly improved. This is because the optimisation resolves


275

Table 2 Some high quality pharmacophore models published this year Target

Method

Features

Tolerances

D2 /D4 antagonist32

Manual

4 features, 2(D4) or 3 (D2) 4 features 7 features

Geometry not given 0.5-2.0 NIA

516 features

0.5-1.6 0.5 0.3

5HT 1b/d33 Manual Catalyst/HipHop Mesangial cell proliferation inhibitors34 6, p and IC opioid a g ~ n i s t s ~ ~ MOLMOD L-Type Ca Channels36 Active Analogue A4b2 nicotinic'* Active Analogue E-selectin3' Homology model 5HT-338 Catalyst K-opioid~~~ Sybyl Search GABA(A)40 MOLMOD Cyp450 2C94' Catalyst

4 features 4 features 3 features 3 features 4 features 5 features 3 and 4 features

2.0 NIA I .o 0.5-1.5 N/A

any incorrect steric clashes caused by the (slight) mispositioning of the excluded volumes, giving a better picture of the receptor site. Another application of substituent-based conformational analysis is pharmacophore searching. In the work of Olender and R~senfeld,~' conformer libraries of substituents are created with the attachment point to a central scaffold as the reference. To search the virtual library with minimal enumeration, they first apply a series of filters to eliminate most of the possibilities. First conformers that clash when attached to the scaffold are removed, followed by substituents that do not contain any of the features specified in the pharmacophore queries. The next filter looks at pairs of substituents, to remove rotamers that cannot meet the distance constraints, followed by triples and so on. Using this hierarchy of filters, most rotamers/substituents can be eliminated quickly, leaving a small number to be enumerated and tested explicitly. A library based on five scaffolds with six substituents, each with 143 rotamers (equivalent to 1OI6 entries), was scanned with a six-point pharmacophore in a few hours.

4 Library Design Library design has been one of the main challenges for modelling over the last few years, partially because the technology of library synthesis has been evolving rapidly. Pickett et aL4*provided an elegant demonstration of how the incorporation of drug-like properties into library design can enhance the quality of the synthesised compounds in terms of providing better leads. The molecular descriptors used to describe diversity or similarity to a reference standard are 3D pharmacophores, and the drug-like properties are based around the degree of oral absorption (as measured by polar surface area). The selection of an efficient library design is done by a Monte Carlo algorithm. In the example described, a

276


follow-up library was designed with the goal of maintaining of improving the biological activity, while increasing the CACO-2 permeability. The second library contained more actives (85% compared to 60%), and the number of compounds with good permeability on the Caco-2 screen rose from 60% moderately absorbed to 80% well-absorbed. The querying of massive virtual libraries is a neat challenge for the cheminformatician, and Lobanov and A g r a f i ~ t i shave ~ ~ devised a strategy based on stochastic sampling and building block analysis. A large random sample of the virtual library is built, and assessed against the query, resulting in a ranking. The top M compounds are broken down into their building blocks, and the blocks reassembled in proper combinatorial fashion. These products are also evaluated against the query to produce the final selection on matches. The process is repeated several times, to improve the sampling and hence the final selection. The performance is dependent on the size of the sampling, but can be better by 1-2 orders of magnitude, with the quality of the final set being very comparable with the best set obtained by brute force. Two similar approaches towards optimising a library d e ~ i g nhave ~ ? ~been ~ put forward. As an example, consider a simple two-component reaction, in which the reagents can be arrayed as rows and columns of a matrix. The first step is to score the virtual library in some way (bioactivity, druggability), so that a smaller subset of potential actives can be flagged. The next stage is to design a library that maximises the number of actives in the library (the efficacy) and the number of reagents used to build the library (the efficiency). The algorithms are both greedy algorithms; in each iteration, the reagent that adds least to the efficacy and efficiency of the library is eliminated. The methods are both extremely fast, and allow the chemist to experiment with various parameters (such as the definition of an active) and weightings of efficacy and efficiency. Another way of implementing a greedy algorithm for library design is provided by Agrafiotis and L o b a n ~ v . ~ ~ The library is prescored as before, and instead of removing the worst reagent, the best sets of reagents are selected, scanning across the rows and columns of the matrix repeatedly until no better selections can be found. All these pieces of work are important contributions towards helping combinatorial chemists to use design tools as a standard part of their library generation process. have described a method for assessing diversity (and conseLeach et quently for monomer selection) based on how the candidate monomers occupy space when attached to a common core. The algorithm involved the attachment of the monomer to the predefined core, followed by conformational analysis. The space occupied by a monomer is computed by placing a 3D grid of cells around the core, and counting which cells are occupied by a monomer atom, and how frequently, during the conformer generation stage. The grids can be further classified depending on whether one wants to count hydrogen bonding interactions, or steric occupancy and so on. The shape and extent of the grids can also be modified, to take into account information known about how the core binds to an active site. The advantage over whole-molecule based pharmacophore analyses is that the cell signature is quicker to compute; the assumption is made, however, that the core binds in a similar manner in all cases. More traditional metrics can

5: Computer-aided Drug Design 2000- 200I

277

be added as secondary filters. As with other cell-based approaches, it is straightforward to determine sets of similar, complementary and diverse monomers from a query database. A work that is similar in spirit is the OSPPREYS approach of Martin and H~effel.~' Although this dense paper is ostensibly about addressing the limitations in pharmacophore analysis of the components of a combinatorial library, it describes a similar substituent-based method for assessing diversity by computing the pharmacophore profiles of the substituents, as opposed to the grid-based metric. Maximising the pharmacophoric diversity of the substituents should maximise the overall diversity too. The number of pharmacophoric centres in the description is also increased, as each substituent part of the whole molecule can contribute three centres. The conformational analysis is performed independently of the rest of the molecule (in contrast to the previous method), so that the pharmacophore fingerprint can be stored and reused in several analyses. Genetic Algorithms (GAS) have also been used to design truly combinatorial libraries. The strategy is simple: molecules are built from reagents using defined rules (e.g. a molecule could be made from an acid and an amine using a carboxamide condensation rule). Libraries are built from the combinatorial combination of the reagents. If the reagents are expressed as the bits on a chromosome, the choice of reagents making up the library can be optimised. Optimisation is performed by enumerating the molecules and scoring them by some means (an SAR equation). The fitness of the library is the mean of the fitness of the component molecules. Sheridan et have used this approach, extending it to use 3D scoring functions: however, they found that library-based design was too slow, and that finding the optimum molecules, rather than the optimum libraries, was a more efficient approach. When libraries are too large even for GAS to handle, there is the option of random ampl ling.^' By analysing a random sample of sufficient size, the properties of the whole library can be estimated with a predictable error. Also, by analysing the library components that pass a SAR test, one can eliminate those reagents that rarely occur in the successful molecules (in contrast to the MFA method, which slums from the top). In the example given, the library size was reduced from 10" to lo5. After this reduction, more rigorous methods can be applied. The topic of privileged motifs for combinatorial libraries is given new impetus by a paper by Hajduk et aE.,5' which uses an NMR protocol for identieing small molecule ligands binding to proteins. They found that the hit lists were considerably enriched with molecules that contained motifs like acids, biphenyls and diphenylmethanes. Moreover, they found that this was not just non-selective binding (as might be expected for hydrophobic groupings): distinct structure-activity relationships were observed. They conclude that combinatorial libraries and screening sets ought to contain more of these motifs, to improve hit rates. Another paper using the same strategy as the RECAP work looks at the scaffolds present in natural products as a potential source of building blocks for combinatorial chemistry.52They broke down the Bioscreen NP databases3 to find all the ring fragments that did not occur in the World Drug Index. The approach

278


was validated by comparison of the pharmacophore space of the two sets of compounds, which seemed to be quite similar. The final word on library design should go to Hann, Leach and Harper,54who have published an intriguing paper on molecular complexity and lead discovery. Their thesis, supported by a simple model of molecular recognition and an analysis of drugs and their corresponding leads, is that there is an optimal complexity for finding leads by high-throughput screening (HTS), and that the conversion of leads to drug involves the addition of more complexity. Their conclusion is that screening libraries should not be made more drug-like, but more lead-like. The optimal complexity argument is based on the probability of finding a hit in an HTS screen: a complex ligand will have reduced chance of binding successfully, but if it does it will have high affinity (hexapeptides are a good example). Less complex ligands will have a greater chance of being recognised, as they have fewer features that might cause rejection, but there is a greater chance that their binding will not be sufficiently strong to be detected. To compare leads to their cognate drugs, profiles of MW, ClogP etc. were plotted, using the WDI as reference. In all cases, the leads were more simple, allowing for elaboration to improve affinity, bioavailability etc. The irony is that combinatorial libraries are being made more complex, with more points of attachment to increase the number of compounds that can be made by a particular synthetic route.

5 ADMElTox

Recent years have seen a welcome shift in emphasis in medicinal chemistry away from considering binding affinity as the key parameter in driving forward an SAR, towards bringing in absorption, metabolism and biopharmaceutics in the process at an earlier stage. This in turn has spurred on modelling efforts in these fields. The field of cytochrome P450 modelling has been very active, with several pharmacophore and 3D-QSAR models being published. The progress in this area has been reviewed by Ekins et u1.,55-57who also looks at the current state of the art in modelling in absorption (and efflux pumps such as p-glycoprotein). The reviews contain a wealth of references back to the original datasets used to derive the models. Also the first homology model of the HERG channel5*has appeared, which will help us to understand the mechanism of another key determinant of drug toxicity. There has been more activity in the area of trying to discriminate drugs from Anzali et ~ 1 . ~claim ’ an 80% success non-drugs. Using the PASS rneth~dology,’~ rate. More interesting is their definition of what is a non-drug, driven as it is by a series of substructural features. It is not clear that the arbitrary definition of what is a non-drug, and the definitions of the atomic environments that PASS uses in . ~ the~ same challenge its classification, are truly independent. Muegge et ~ 1 attack using pharmacophore filters, which look for the presence of functional groups, e.g. amide, amine, alcohol, ketone, carboxylate etc. Using the counts of the pharmacophore points, they come up with some very simple rules that give 69%


279

discrimination success. Although this is lower than other methods in the field, the authors claim the advantage of interpretability of their rules. On a slight tangent, King et aZ. have used their ILP program W a d 2 to discriminate between carcinogenic and non-carcinogenic compounds. They found that the minimum size of substructure that provided a useful classification was > 7 atoms/bonds. It will be interesting to see if this work can be extended to the drughon-drug problem. Some new rules for describing drug-likeness have been published by Xu and S t e ~ e n s o nbased , ~ ~ on the non-peptide, orally bioavailable drugs listed in the CMC database. The reference drug-like structures are surveyed to produce frequency histograms of 25 descriptors, for example, the number of heavy atoms, the number of N-H groups, the number of cyclic building blocks and so on. The histograms are normalised to produce a scoring metric, Drug-Like Index or DLI, which for a query compound is a function of the product of the normalised frequency scores for the descriptors found in the query. This forces a strict Pareto optimality, in that if the query has a descriptor that has a score of zero its overall DLI score will also be zero. This approach is thus based on the assumption that the reference database from which the frequency histograms are derived covers drug-like space fully. However, it does not make the assumption that the drugs database and a chosen source of general compounds (often the ACD database) are disjoint. In fact, the authors show that 60% of compounds in the ACD have a reasonable DLI score. This method can be used to profile libraries. Llorens et aLM have used Ames test screening data to derive mutagenicity profiles. The compounds in the data based were described using Molconn-X and MACCS key descriptors, and significance tests were performed to try to find which descriptors were associated with mutagenicity. This was followed by recursive partitioning to derive a prediction for whether a query compound was likely to be a mutagen. Some of the key factors are the presence of nitro groups and the number of Hbond acceptors. The FW model gives 60-90% accuracy in its predictions. Another interesting article on drug-like properties has been written by Lipin~ki,~’in which he argues that ‘druggability’ screening should take place concurrently, or even before, screening for biological activity. This seems too polemic, as it does not allow for the role played by informative compounds that may not ever become drugs, but which help to elucidate mechanisms and define SARs. He does come up with the useful guideline that solubility levels of 50 mg 1-’ are required to get good bioavailability for a medium-permeability compound dosed at 1 mg kg-’ . He also notes the trend towards increasing log P in Pfizer clinical candidates, and explains this by the observation that many Pfizer leads come from HTS, and a common way of increasing potency is by the addition of suitable lipophilic functions. In Merck, where the leads have come more from rational drug design, the bias is towards explicit intermolecular interactions, so that the hydrogen-bond count has increased over time. Both strategies have led to compounds with higher molecular weight. In turn,the lively debate about how to estimate solubility continued. Although QSPR is not usually within the remit of molecular modellers, the crucial dependence of oral bioavailability on solubility (among several other factors) has made the estimation of this property a hot field within the pharmaceutical industry. The recent

280


publication by Jorgensen and has been critically assessed by Ran and Yalk~wsky,~~ whose earlier work in the field led to a simple equation relation solubility to melting point and C l o p . The Jorgensen equation involves simple quantities derived from a Monte Carlo simulation. Perhaps not surprisingly, Ran and Yalkowsky find that their own equation is more accurate by about 0.1 log units, but the caveat is that one still has to know the melting point, and both methods are only applicable to neutral species. Classical QSAR methods, based on 2D descriptors, are being applied widely to the field of ADME/Tox. It seems that the main stumbling block is not the modelling methodology, but the availability of high-quality data. Several equations for estimating physicochemical properties (solubility, partitioning) have been published by Abraham and co-workers.68The equations are all based around a set of six descriptors, and a fragment-based method for computing them has now been p ~ b l i s h e d .Yoshida ~~ and T o p l i ~ slooked ~ ~ at oral bioavailability in a set of 232 diverse drugs using classical QSAR, and found that log D6.5 was a strong factor influencing bioavailability. They produced a classification model that was 70% accurate in its predictions. The model is relatively simple, allowing one to see which factors in a molecule are most affecting its estimated bioavailability.

6 Docking and Scoring There have been several major papers published in this area, in which the difficult issues of the accuracy of docking and scoring methods are addressed. In an ideal case, we would want a docking program that could always generate the same pose of a ligand as observed in an X-ray complex, and a scoring function that would clearly rank this pose as the best. The OWFEG scoring method has been tested on several system^,'^ and it seems to perform better than several of the traditional empirical scoring metrics such as DOCK or ChemScore. Despite using a molecular dynamics paradigm, OWFEG is as fast as the traditional methods, once the initial scoring grid has been created. The receptor and the solvent are allowed to move during the simulation, so that both entropic and solvation effects are considered, giving a better approximation to the true AG. In a head-to-head trial, OWFEG did as well as or better than the other metrics both in terms of getting better correlations between score and observed binding, and in database enrichment. There are limitations to the approach, in terms of the charge model used, the treatment of ligand solvation, and large scale entropy changes but, that having been said, this approach does seem to be a big step forward in virtual screening scoring functions. Another development in the field of scoring fbnctions is the work of Jorgensen et al.,72who have used PCA to combine several known functions, following on from the consensus scoring approach described by Charifson et al.73 The difference is that the current analysis introduces different weights for the scoring functions, rather than treating them equally. The functions examined were Ludi, GOLD, DOCK, ChemScore, PMF, GRID, SCORE, FlexX, and the test set used was made up from X-rays of the matrix metalloprotein complexes. In the first part


281

of the paper, the scoring functions were analysed to select the best docking from a choice of alternative dockings generated using GOLD. This can be assessed by computing the rmsd to the known X-ray docking. A single-component model could be obtained, with R2 = 0.73, @ = 0.59. The second part deals with correlating score with observed affinity, for a given docking. Again, a onecomponent model could be obtained, with R2 = 0.68, @ = 0.67 on a general test of 60 X-ray complexes, compared to R2 = 0.56 for the best of the individual force fields. Another view of the scoring issue is provided by Bissantz et aZ.,74who examine seven scoring functions combined with three docking algorithms. Use of a single scoring function allowed one to pick 70% of the true hits hidden in a database of random compounds. Use of consensus functions improved this enrichment factor. However, the best scoring method varied with the target; to deal with this, the authors propose a training run to deduce the best protocol, before screening the entire database. In their hands, GOLD performed the best in most situations, PMF did well for polar sites, and DOCK for apolar sites. Muegge et aL7’ have continued to investigate further applications of the PMF scoring function, in particular by using it as the scoring function within DOCK. The combined method was studied by trying to reproduce the correct binding modes of 61 MMP-3 inhibitors. The results were compared to those obtained using DOCK with native scoring, and FlexX. It was found, perhaps not surprisingly, that DOCWPMF does much better than the other approaches, finding the correct binding mode 93% of the time (compared to 85% for the other approaches), and was the only approach to show significant correlation between binding affinity and predicted score (R2 = 0.49). Stahl and rare^'^ have also performed comparative studies of the common scoring functions used in docking and screening. They used FlexX to perform their dockings. Their conclusions in brief are that the FlexX scoring function performs best for complexes that have a significant number of intermolecular hydrogen-bonds, PMF does less well in sterically restricted sites and consensus scoring can exploit synergies between scoring methods. However, they found that if one knows a priori which scoring function is best suited for a particular situation, better performance can be obtained using this function alone rather than using a consensus which might dilute the hit rate. They also propose a new function, Screenscore, which gives more robust rankings when used with FlexX generated dockings. An entirely different and novel approach to docking and scoring has been taken by Vieth and based on the assumption that a congeneric series will have a common binding mode. The first step is to dock the ligands in the site, followed by a cluster analysis to look for the common binding modes. Any ligands not docking in a particular mode are then redocked in a constrained manner, so that all the structures are aligned to the chosen mode. An MD protocol is used, to allow for sidechain movements or errors in a homology model. At this point, there may still be several possible binding modes, so an QSAR approach is used to score the modes. The descriptors are the inter- and intra-molecular interaction energies, and the QSARs are built up using PLS with four-fold crossvalidation. In the

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Chemical Modelling: Applications and Tkeory, Volume 2

examples given, it was possible to deduce a binding mode that was consistent with the known X-ray binding mode, and to show that the QSAR equation had a high degree of confidence (@ = 0.66). This approach will be useful both in generating binding modes, and in identifying problems in homology models. Fradera et U Z . ’ ~ have used this same assumption of docking mode in their similarity-driven docking method: the score from a similarity calculation to a reference ligand (which defines the binding mode) is used either to modify the final scores from a DOCK4.0 run,or to modify the search paths during the run. Although molecular simulations do not fall with the scope of this review, some papers in this area do seem to have a more general impact. In addition, this is the appropriate place to pay tribute to the life and work of Peter Kollman [ 19442001],79 whose contribution to the whole area of molecular modelling has been simply immense. We mourn the passing of one of the most original and respected thinkers of this generation. Essex et aLS0have neatly demonstrated that there can be a very strong configurational dependence of binding energies computed using the kite-difference Poisson-Boltzmann equation. In one case, a rms shift of 0.6 in a small molecule causes a change of 6.9 kcalmol-’ in the hydration energy. This implies that averaged free energies over many configurations should be used in preference to single FDPB estimates. Another approach to estimating binding constants is to use the Linear Interaction Method; unfortunately, although the theory is precise, practice has been a different matter, with no one universal set of coefficients emerging. Wall et aZ.*l have looked at neuraminidase inhibitors to assess the LIE parameters derived by other workers, and to derive their own QSAR equations. The first finding was that the error in using published parameters was > 5 kcal mol-I. When other descriptors were included, a more predictive equation could be derived (@ = 0.6): one important point to note is that the descriptors are strongly correlated, so that MLR cannot be used. This leads to the conclusion that there is no general equation for LIE, but good casedependent coefficients can be derived. Wang and Wade looked at the same systems2 using the COMBINE methodology, which correlates selected interaction energy components with binding free energy. They derive models with @ = 0.78. From this study, the COMBINE model, with all its inherent assumptions, does better, but it seems that the LIE method is improving to comparable levels. The new algorithms behind DOCK 4.OS3have been published. The main change has been the introduction of ligand flexibility through the use of a guided conformational analysis strategy. Previously, one of the common ways of bringing in flexibility was by the use of Flexibases. In tests to reproduce known X-ray complexes, the crystallographic pose was always found, and was scored the best in seven out of 15 cases. Each docking takes about 15 cpu seconds on an RlOOOO processor. Tao and Lais4 have independently produced a new scoring h c t i o n , ScoreDock, that was incorporated into DOCK4. The function itself is made up of standard terms such as hydrogen-bonding, solvation, strain and so on, with the coefficients being derived by regression on a set of training X-ray complexes. There is the essential interplay between the docking and scoring algorithms to consider. A scoring function cannot perform well if the docking algorithm cannot produce the crystallographic poses as one of its solutions. In the cases where the

A


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proper pose was generated, ScoreDock does seem to do better than the default scoring functions available in DOCK. Verkhivker et aZ.85have looked in detail at three complexes for which most current protocols fail to identify the X-ray docking as the most favourable. They propose a hierarchical approach, in which docking space is sampled by a simplified PL energy function that is not so sensitive to local side-chain conformation, followed by a leader clustering of the conformers, The cluster leaders are then minimised using AMBER. The docking landscapes were also examined using AMBER directly, to explain why the more complex force field can run into sampling issues; the landscapes were either shallow with the global minimum poorly defined, or rugged, in which case the global minimum may not be sampled. The simplified force field overcame both issues, and seems to be a good way to seed more rigorous scoring functions. Another method for docking, based on the Mining Minima optimiser, has been published.86This uses ideas for the Global Underestimator method, tabu search, GA and poling to sample the docking configuration space and propose a global minimum. The scoring function used is based on Charmm, but contains ideas from the PL function of Verkhi~ker.~~ In a comparison with existing methods, this approach seems competitive but not a quantum leap forward. The inclusion of some of the other work on improved scoring functions might make a big difference. Further progress has been made in the area of protein sidechain prediction, with enhancements to the Dead End Elimination algorithm.87DEE can now be used for the placement of sidechains for nearly all sizes of protein, and therefore can be used to generate ensembles of active site conformations very quickly. Another strategy for introducing protein flexibility into docking protocols has been described by Broughton.88The conformational space of the receptor is sampled using an MD run of 50 ps, saving one conformation every 1 ps. The conformers are aligned using a predefined frame of reference (e.g.a ligand), and docking grids are created for each conformer. The grids are then combined by an averaging process at each point on the grids. This emphasises the interaction points that occur in many conformers, and down-plays the interactions due to interactions with mobile side-chains. The composite grid is used to guide the docking run. In the cases examined (DHFR and COX-2), the use of weighted grids improved performance markedly over standard (single conformation) grids, both in terms of cpu time and lead-finding efficiency. Schafferhans and Klebeg9have looked at the same issue from the perspective of the uncertainties that accompany structures built by homology modelling. They have developed a protocol that tries to make a link between the information present in homology models and the SAR inherent in binding data, both of which are generally more readily available than X-rays of drug-receptor complexes. The alignment for a 3D-QSAR model can derived from a homology model, as others have shown, but great care is required. In this work, the uncertainty in side-chain position is reflected by the use of Gaussian functions to describe the binding site. The effects of different protein conformations are handled by arithmetically averaging the Gaussians, and the ligands are aligned to the site using a SEAL-like algorithm; ligand flexibility is handled by using a database of precomputed conformers. The best alignment across all ligands is

284


decided by constructing a CoMSIA model and examining the @ statistic. The approach was validated using proteases at test cases, and seems to offer an better way of using homology models effectively in drug design. The design of selective ligands is a hard task, and there are not many modelling tools that can assist. However, Kastenholz et al.90have developed a new method for analysing the GRID fields around a set of protein active sites, and comparing them statistically. The field values form the variables for consensus PCA, which can determine the key components that will allow one to discriminate between the input protein families. These differences can also be displayed visually to highlight where interactions between a ligand and the binding site would be favourable for one protein, and disfavourable for another, thereby allowing the issue of selectivity to be addressed. The test proteins used were trypsin, thrombin and factor Xa, which are highly homologous, but for which there are highly selective ligands. The authors could show that the selectivity regions they determined matched the known SAR for the ligands. The method is scalable and could be used for much larger protein families. The data contained in the IsoStar database continue to be a rich source of information. Nissink et aZ.9’ have derived a new Gaussian-based descriptor of the propensity distributions for molecular interactions about a set of core groups. The results are very similar to those that can be obtained using GRID.92 The Gaussian formulation is slightly less predictive (75%) than Superstar (81%), but is 5-8 times faster to compute, making it more appropriate for use in docking protocols.

7 Cheminformatics

Cheminformatics is developing strongly as a subfield of modelling, as the storage, retrieval and mining of chemical information become part of the chemist’s toolbox. The LaSSI m e t h ~ d is ~ ~a ?novel ~ ~ method for performing similarity searches on a database. Rather than considering all descriptors as equal, LaSSI uses a singular value decomposition algorithm to project the compound database and the descriptors into a lower k-dimensional space. This has the effect of increasing the weight on some descriptors. The descriptors chosen are dependent on the composition of the database. The method is shown to perform as well as traditional approaches, and it also retrieves as similar classes of compounds that the traditional method misses. LaSSI seems to offer a very different way of approaching similarity searching. Clustering by 2D fingerprints is a very common procedure, and yet there are still many unanswered questions, principally around the choice of descriptor, and the selection of a statistically appropriate number of clusters. Wild and Blankleyg5have looked at these issues and have come to some interesting conclusions: MACCSlike keys are best for general diverse sets (e.g. corporate databases), whereas Daylight-like keys are the best for similar sets (e.g. combinatorial libraries); the best method for cluster-level selection is dataset-dependent. However, the Kelley method96 seems to have the best worst-case performance across the different datasets and descriptors. Xue et al.97have developed a method for using consensus


285

fingerprints, based on MACCS keys, to enhance screening performance. Consensus fingerprints are derived from a set of compounds by determining which bits are always set in the fingerprints of the component molecules. The consensus fingerprint is then used in combination with a query molecule in the calculation of the similarity coefficient. The effect of the consensus fingerprint can be adjusted by a scaling factor. In their hands, use of a consensus fingerprint gave better retrieval rates, with fewer false positives, than using MACCS keys alone. The use of BCUT descriptors as 2.5D descriptors has been in~estigated,~~ using the classification of kinase inhibitors as the test case. This is a tough example, as the homology between kinases in the binding region can be very high, and yet kmases can be inhibited by several distinct classes of compound. The comparative descriptors examined were Daylight keys and 3D pharmacophores. It was found that, in this case, 2D descriptors gave poor classification, and infomation contained in the BCUT and pharmacophore descriptors was complementary and effective at classification of a test set of novel kinase inhibitors. The BCUT descriptors associated with a particular classification could also be interpreted in terms of the kinase structure. They are also quick to compute, making this approach suitable for virtual screening. The enrichment of chemical collections is best achieved by the judicious use of both diversity and focus in the design of new chemical libraries. Diversity is needed to generate novelty, while focus is needed to maintain attributes that are usually associated with drug-like behaviour. The LASSOO program99 provides a means to getting to this end. Similarity between molecules is measured using MACCS keys. The score of a molecule depends on density of its neighbourhood environment with respect to a reference library of desirable compounds (known drugs) and the compounds already in the chemical collection. This biases towards picking molecules that have many close analogues in the reference library, on the assumption that this will ensure favourable characteristics, and against picking close analogues of what is already in the chemical collection. The method was validated by spiking the ACD database with drugs from the CMC database; LASSOO can provide five-fold enrichment over chance in selecting CMC compounds for addition to a screening set. Some brave computational chemists are still involved in trying to model chemical complexity as it relates to ease of synthesis. This area has been bedevilled by a lack of clean data, and by the many exceptions to the rules that organic synthetic chemists delight in. Barone and ChanonIo0 have devised a scoring system based on ring size and number of neighbours each heavy atom has. This measure is simple, and gives intuitively correct rankings in the cases examined. However, there are still many areas, such as chiral centres and heteroatoms, which are not dealt with.

8 Structure-based Drug design The success of structure-based design methods in assisting drug discovery efforts has prompted a large number of research efforts towards the issues of docking


286

and scoring, as described above. It has also driven research into de novo design methods, which has been coming back into fashion as a topic for research. Much of the early work"' was beset with issues of chemical feasibility and exponential explosion. Combinatorial chemistry and the development of new algorithms has offered a new way of addressing these issues. Douguet et a1.'02 have developed a genetic algorithm to modify a molecule according to a conservative set of rules to optimise a QSAR developed for retinoids. A sanity check is provided by restraints on physicochemical properties. While the GA parameters are described and tested in some detail, the molecular modification rules are not, which makes proper evaluation difficult. In a related paper,lo3 the building method elaborates a set of scaffolds with known building blocks and a well-defined set of chemical reactions. The building blocks are themselves generated by applying the reactions retrosynthetically to drug-like compounds from the World Drug Index database. The program was tested by trying to design analogues of a thrombin inhibitor. The fitness function was based on Tanimoto similarity and a 2D fingerprint comprising of atom and pharmacophoric path patterns. Initial results appear encouraging, and this will be an area to watch. A new program for de novo design, LigBuilder, has been published.lW The basic strategy is 'inside-out' growth, that is, defining the key interactions first then trying to construct a molecular skeleton that will connect these interaction sites. The new twist employed here is the use of a GA to circumvent the combinatorial explosions that occur when considering how many different ways there are of making the connections. This implies the use of a palette of fragments that can be encoded into a chromosome, rather than a completely general structure builder. Mutations are driven by the interaction fields in the protein, e.g. a CH in a hydrogen-bond acceptor region will be mutated to an 0. This operation greatly enriches the chemical diversity of the solutions generated. In the two validation cases discussed, it seems that the program can regenerate close approximations to known ligands from scratch. Other innovations in de novo design are provided by Leach et aE.,'05who try to

Table 3 Some successful applications of structure-based drug design ~~

Target

Structure

Method

Aldose reductaselo6

X-ray (1.8

Androgen receptor"' CDKll'*

Homology model Homology model

Matriptasel"

Homology model

Src SH2"' Renin'

X-ray (1.9 X-ray

A)

A)

Outcome

3D-database search + Lead discovered docking SAR rationalised MD Lead optimised in MD very few iterations Selective lead 3D-database search discovered docking In vivo lead FL097 In vivo lead Manual

+

287


introduce measures to enhance synthetic accessibility of the generated structures by decomposing the structure down into generalised synthons followed by 2D substructure searches for suitable reagents that have the same pharmacophoric and geometric properties as the initial structure from which they were derived, In terms of 'outside-in' design, the fragments chosen to link the key interaction centres are drawn from a pool of substructures that can be made easily (amides, benzodiazepines, imidazoles). Once a structure has been generated, the central fragment is disconnected (for an amide, this would be to an amine and an acid), and the resulting synthons are converted into generic queries for searching. All hits found are then reconnected and the complete molecule is redocked and scored. This method provides a nice way of exploiting both de novo design and combinatorial chemistry.

9 Reviews

No review would be complete without a self-referential summary of some of the other reviews published this year. I have chosen some which give a very good overview of some key research fields in modelling.

Table 4 Some key reviews published this year Topic

Authors

Methods for computing log P

Mannhold and van de Waterbeemd' I 2 Livingstone' l 3 Mullerl I 4 van de Waterbeemd, Smith, Beaumont and Walker''' Norinder and Osterberg116 MueggeI I ' Lipinski, Lombardo, Dominy and Feeney * Garcia-Echeverria, Traxler and Evans"' Lewis, Pickett and Clarklzo

Chemical descriptors Modelling GPCRs Drug absorption and pharmacokinetics Knowledge-based scoring functions Experimental and computational approaches to estimate solubility and permeability Kinase inhibitors Computer-aided molecular diversity analysis and combinatorial library design Compound classification, molecular descriptor analysis and virtual screening Similarity vs. docking for virtual screening Small molecule alignment methods Predicting drug toxicity Structure-based drug design

BajorathI2 Mestres and Knegtel 122 Lemmen and Lengauer'23 Cr~nin'~~ KlebeI2'

Number of References 108

249 80 223 34 42 50

220 247

93 90 142 45 54

288


10 Conclusions

The year starting May 2000 has seen many advances in modelling: the development of better scoring functions for docking, the rebirth of de now design and a growing interest in modelling phenomena related to ADME/Tox. This snapshot shows that the discipline of modelling is a healthy state, and that the field will continue to grow and advance, consolidating its position as an invaluable aid to drug discovery.

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24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

34. 35. 36. 37. 38. 39.

40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50

51

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6 Electric Multipoles, Polarizabilities, Hyperpolarizabilities and Analogous Magnetic Properties BY DAVID PUGH

1 Introduction

Chapter 1 of the first volume of this series of Specialist Periodical Reports was entitled ‘Electric Multipoles, Polarizabilities and Hyperpolarizabilities’ . A brief overview of the period up to 1998 was given, followed by the review of 19981999. Magnetic effects were not included. In the present article the aim is to provide the same kind of brief historical introduction to magnetic effects and then to review the literature on both electric field and magnetic field response functions for the period June 1999-May 2001. Magnetizability and hypermagnetizability are the magnetic analogues of the electric field polarizability and hyperpolarizability, but there are special features in the theory of the magnetic response functions which are not paralleled in the electric field theory. Also, the applications are very different in the two cases. In the last few decades the study of the electric field hyperpolarizabilities has been a major ingredient of the exploitation of molecular phenomena for optoelectronics. The emphasis has been on the reliable quantification of the frequency-dependent first and second hyperpolarizabilities. The study of the response of molecules to magnetic fields, on the other hand, is very often pursued with the interpretation of nuclear magnetic resonance phenomena as the ultimate aim. In particular, the bulk of the activity reported in recent years is directed at the nuclear shielding problem, which usually involves only the linear response to a zero-frequency applied magnetic induction field. The subject has been extensively reported in the NMR literature and here the intention is to review only the underlying theoretical approaches to the calculation of the nuclear screening tensor in the context of a more general review of magnetic properties. The papers referred to often go more deeply into the NMR application, in particular in connection with rovibrational averaging for the signal, but these aspects will not be covered. The range of magnetic effects reviewed here is mainly confined to the response of closed shell molecular structures to external fields. Inclusion of paramagnetic molecules,

’

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294

Table 1 Acronyms for gauge transformation procedures GIAO, LAO

Gauge Invariant Atomic Qrbitals, also known as London Atomic

LORG IGLO CTOCD IGAIM

-Localized Qrbitals/Localized Origins Individual Gauge for Localized Qrbitals continuous Iransformation of Qrigin of Go-ordinates individual Gauge for Atoms in Molecules

-Orbitals

spin-spin interactions etc. would increase the amount of material to be reviewed beyond reasonable limits. Notation. The symbols a , p, y are used throughout to denote the electric field polarizability, first and second hyperpolarizabilities respectively, suitably qualified by frequency factors where necessary. The magnetizability is denoted by and the nuclear screening tensor by 0. The numerous but well-known acronyms specifying the computational procedures are used without definition. The possibly rather less well-known acronyms for the principal gauge invariant procedures are given in Table 1.

x

2 Response of Closed Shell Molecules to Magnetic Fields

The electronic hamiltonian of a molecule in the presence of an external electrostatic potential, and magnetic vector potential, A, is

where pn = -ihV, is the momentum operator for the nth electron, pn + eA, is the modified operator in the presence of velocity-dependent magnetic forces, and @,, is the electrostatic energy of electron, n, in the nuclear framework. The sum is over all electrons and the subscripts specify that the fields are to be evaluated at the positions of the electrons. In many cases the response to a uniform magnetic induction field, B, is required, when the vector potential can be written as A = iB X r, which automatically satisfies the Coulomb gauge condition, V.A = 0. The hamiltonian can then be reduced to the form

6: Electric Multipoles, Polarizabilities, Hyperpolarizabilities

295

and, if the orbital magnetic moment operator is defined as, &, = ifie/2m(r X V) and a spin magnetic moment operator, &, is added, the Schrodinger-Pauli equation obtained from the hamiltonian is

The spin moment operator has been introduced arbitrarily here, but appears naturally if the equation is derived by reducing the relativistic Dirac equation. With the spin operator included the wavefunction must be treated as a two component spin function and the operators as 2 X 2 matrices. The second, third and fourth terms inside the first summation in equation (3) are the perturbations introduced into the hamiltonian by the effects of the external fields. The fourth term, describing the electric field perturbation, is linear in the external potential or electric field. The second and third terms give rise to linear and quadratic responses ro a constant, uniform magnetic field. Smaller terms, arising from the Dirac equation, which represent spin-orbit coupling etc. have been omitted. An account of earlier work and the general formalism is to be found in the book by Davies.2 2.1 Magnetic Susceptibility. - The definitions of the linear molecular magnetic susceptibility, xI/ are analogous to those of the electric dipole polarizability, a,,, and, as in the case of a, it can be defined in two ways, directly or through the expansion of the energy, (fF in powers of the magnetic field, when

xi/=-

(4)

~

The second order terms in A (or B) in the hamiltonian, acting on the unperturbed ground state wavefunction, lead (for a constant magnetic field in the z-direction) to a contribution,

xQ =

e2 2m

- -(0lx2

+ y210)

which is diamagnetic in the sense that the induced moment is opposed to the field. The superscript relates to the usual terminology referring to this contribution as the Langevin term (although it does not involve statistical averaging of a random distribution of permanent dipoles as in the Langevin-Debye equation). A paramagnetic term, alternatively referred to as the high frequency contribution,

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Chemical Modelling: Applications and Theory, Y,lume 2

analogous to the expression for the electric dipole polarizability, arises from the linear perturbation term in the hamiltonian,

where L = Cn(r,, X V,,), with the summation over all electrons, is the angular momentum operator. The two kinds of contribution to the susceptibilty can be separated by a carefbl study of the rotational magnetic moment of the molecule or by molecular beam, Zeeman effect or microwave spectroscopy.

2.2 Nuclear Shielding. - A comprehensive review of the theory has recently been provided by Helgaker, Jaszunski and R ~ u d There .~ are also a number of other reviews, within the last decade, in the NMR The most recent NMR Specialist Periodical Reports contain relevant articles by Jamieson and de Dios' and Yamaguchi.8 The nuclear shielding tensor, oij,of primary importance in nuclear magnetic resonance spectroscopy, is closely related to the magnetic susceptibility. It is defined through the equation giving the local field at a nucleus in terms of the perturbing external field, B Y = B,

- a,pBp

(7)

The procedure [due to Ramsey' (see also McWeeny")] for calculating a is to introduce the complete vector potential including that due to the nuclear magnetic moments, MN,of each nucleus,

N

where

AN=az M N X r ~

r3

(9)

where a = e 2 / 4 ; 2 ~ o= h ~1/137, is the fine-structure constant. In equation (9) the origin for the calculation of A N is taken at the Nth nucleus. This modified vector potential has to be inserted into the Hamiltonian of equation (3) and additional terms giving the energy of the interaction of the total magnetic induction field with the nuclear moments and also the nuclear spinnuclear spin interaction included. The total magnetic field is derived from equation (8) and (9) as


297

If the solution of the modified Schrodinger equation is expressed as a sum over states, the second order energy is given by

where H ’ includes all the additional terms in the hamiltonian that arise when the external magnetic field is applied and the summation is over all excited states. This energy can also be written as an expansion in terms of the external magnetic field and the nuclear moments, schematically,

where N,M spec@ the atomic nuclei. The partial derivatives, evaluated at zero field and nuclear moments, are the magnetizability, the nuclear shielding tensor and the nuclear spin-nuclear spin coupling matrix. In using equation (12) with values obtained from approximate wavefunctions the same considerations apply regarding variational and non-variational parameters as in the application of the Hellman-Feynman theorem and its extensions in the case of the electric field (see, for example, ref. 1). It should also be noted that the contribution from the nuclear potentials (9) to the magnetic field derived from (10) includes a singularity at each nucleus which leads to the Fermi contact terms. In computations of magnetizability and nuclear shielding the evaluation of the paramagnetic (high frequency) contributions from the summations in equations (6) and (1 1) is a challenging problem. In ab initio work very high level correlated calculations are needed.’ An alternative approach through an empirically adjusted DFT functional has recently been advocated.12Estimates of the magnetizability to a reasonable percentage accuracy are not too difficult to achieve but the calculation of the nuclear shielding tensors to an accuracy sufficient to be useful in NMR spectroscopy is not yet fully attainable.

’

2.3 Interaction of Molecules with Electromagnetic Fields: Higher Order Terms. - In the foregoing paragraphs the simpler magnetic effects have been introduced in a manner closely analogous to the usual treatment of the electric field response functions as discussed, for example, in the previous volume.’ It has been assumed that the molecule has been perturbed by a constant magnetic field, the effects of which are clearly separated from those of the electric field. If, on the other hand, the perturbation is produced by an electromagnetic wave, or by a combination of radiation and static fields, a number of new effects arise which have to be treated by more general methods. In a radiation field the vector potential, A , produces both magnetic and electric fields, which may or may not be supplemented


298

by an external coulomb field, I? As in the case of the electric field theory the molecule is represented by a multipole expansion based on a fixed origin of coordinates in the molecule. The flexibility of the gauge transformations allows the vector and scalar potentials to be expanded as Taylor series about the same origin and enables an unambiguous identification of perturbations in each order in terms of the field strengths to be made. The development of these methods is due to Bloch,13 Raab,14 Langhoff, Epstein and Karplu~,'~ Buckingham16 and others. A recent account, including systematic extension of the method and specializing towards the theory of the nuclear shielding tensors, has been given by Lazzeretti." A comprehensive set of formulae for all second and third order response hnctions is provided. The quantum chemical methods required to calculate the perturbed molecular wavefunctions needed to evaluate these functions will range over the same approximations that are familiar in the simpler cases, although the degree of success achieved at a given level of approximation will vary from case to case. A list of some of the effects included in this general formalism might serve to put reports of more recent calculations in context: Linear Response (2nd rank tensor): Electric dipole polarizability, a , magnetic dipole susceptibility, optical rotatory power, IC, nuclear shielding tensor at nucleus, I, 0'.

x",

Quadratic Response (3rd rank tensors): (1) electric field effects: Linear electrooptic (Pockels) effect, three-wave mixing, SHG. (2) radiatiodmagnetic field: Faraday Effect. n i r d Order (4th rank tensors): Kerr effect, Cotton-Mouton effect. 2.4 Gauge Invariance. - The electric and magnetic fields are measurable physical quantities and can be regarded as directly responsible for effects on the molecules. Given well defined fields there will be well defined molecular responses. The hamiltonians, on the other hand, are written in terms of the potentials, Vand A and these are not uniquely defined. The fields are given by,

B=VXA and are unchanged if the potentials are subjected to the following gauge transformation:

v + v---datf where f i s any twice differentiable scalar field.

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299

Since the transformation leaves the fields unchanged it should not lead to any changes in physical observables and the solutions of equation (3) and its timedependent version should be essentially unaltered. In particular the energy eigenfunctions of the time-independent Schrodinger equation should be the same. It can be shown by straightforward substitution that, if q!~is the eigenfunction of the untransformed equation for a particular eigenvalue, E, then J!+I exp(-(ie/h)f) is a solution of the equation derived from the potentials subjected to the gauge transformation (14) for the same eigenvalue. It is easy to establish that the presence of the unimodular phase factor exp(-(ie/h)f) in the transformed wavefunction will have no effect on observable quantities. For example, McWeeny” gives a proof in relation to magnetic properties based on the invariance of the current density. Nonetheless, although it is certain that the exact wavefunctions will transform in the above manner, it is by no means always the case that approximate solutions will do so. The question of the correct treatment of gauge invariance in the presence of magnetic fields is therefore not trivial.

2.4.I Change of Origin. - The vector potential giving rise to a uniform field can be written as

I A=-BXr 2

If the origin of co-ordinates is displaced through a constant vector, ro, then A

1

-+- B

2

1 2

X r - - B X ro

which is a gauge transformation represented by a function f, such that

Vf

=

1 --BXr, 2

=&

Here & is a constant quantity which does not vary with the electron co-ordinates in the new co-ordinate system. It is therefore possible to write the quantity, f, as f = &.r and the transformed solution of Schrodinger’s equation as

2.4.2 Gauge Invariant Atomic Orbitals (London Atomic Orbitals) (GIAOs or LAOS).- In molecular science the most well known example of the importance of preserving the correct transformation under gauge invariance is the analysis of the anisotropy of the magnetic susceptibility of conjugated molecules in terms of the ‘ring current’. In view of the importance of gauge invariance in magnetic problems this topic will be briefly examined here. The problem, as it first presented itself, was to modify the usual Hiickel treatment of benzene to allow for the effect of a

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uniform magnetic field normal to the plane of the ring. The molecular n-orbitals are formed as linear combinations of the atomic pz orbitals, each of which should provide a reasonable description of the state of a n-electron near a particular atom. Suppose the AOs have been chosen and parameterized for a particular atom in a co-ordinate system centred on that atom. If subsequent calculations for the molecule refer to a common origin for the vector potential, then the extent to which the AOs approximate the solutions of the differential equation in the neighbourhood of each atom will depend on its position relative to the common origin. This unsatisfactory situation is easily remedied by defining each A 0 with a phase factor,

v -+ vexp(-i*,.r), where A, is now the vector potential at the (fixed) nucleus of the atom a evaluated in the common coordinate system. When the AOs are defined in this way they will have identical status as approximate solutions on all atoms and the molecular orbitals constructed from them will be such that they lead to the same eigenvalues and physical properties for all choices of origin. In the original LondonI8 treatment and other earlier work, further approximations were made to extract the exponential factor from the integrals over the AOs and to restrict the effect of the perturbing magnetic terms in the hamiltonian to a shift in the a Huckel parameter. With these approximations the Huckel matrix is modified only by the inclusion of the above exponential factors. Expanding these gives the energy as a power series in the magnetic field strength and hence the magnetic susceptibility. This formalism provides an elegant qualitative explanation of the special effects that occur when conjugated rings are present. The approximations in the earlier work have been analysed and the theory extended by Pople and co-w~rkers,’~ Davies and others. Accounts can be found in the books by Davies‘ and a detailed exposition of the analysis of the gauge invariant Huckel determinant is provided by Salem.” Gauge invariant orbitals are routinely used in more recent work. More recent reviews of the ring current concept have been provided by Haigh and Mallion2’)22 and Gomes and M a l l i ~ n . ~ ~ Early calculations of the magnetizability of small molecules, often using methods adapted to the particular case, are due to Pople, Ransil, Karplus and Kolker, Stevens, Pitzer and Lipscomb, Hameka. References can be found in Davies2 and in the earlier Specialist Periodical Report by Hinchliffe and Bounds.24 Earlier work using LAOs has been extensively reviewed by D i t ~ h f i e l d . ~ ~ While the LAOs are now a well established ingredient of most computational procedures, the original interpretation of the properties of aromatic molecules in terms of ring currents has often been challenged (see for example Bilde and Hansen26and other references later in the review). The implementation of the LAO method within general HF programs was made much more recently by Wolinski et aZ.” and Haser et aZ.,28but within the last decade LAO based methods have been extended to all levels of calculation up to CCSD (see references in later sections). The application of the method in standard quantum chemistry packages involves a considerable amount of algebraic


30 1

rearrangement since the expansion of the gauge factors is necessary to identi@ perturbation series terms in powers of the magnetic field, B. Numerical evaluation of the integrals between LAOS can also be very time consuming. The gaussian atomic bases (apart from the inclusion of the gauge factors) range over sets which are similar to those used in non-magnetic calculations. It has been established that correlation effects are usually small for magnetizability calculations but are important in the calculation of nuclear shielding tensors.” 2.4.3 Other Approaches to Gauge Invariance. - The London method is clearly well adapted to LCAO methods and immediately removes major inconsistencies even when small basis sets are used. Since almost all modern quantum chemistry employs basis functions localized on atoms it is not surprising that the LAO procedure has remained the most popular method for achieving some measure of gauge invariance with limited basis sets. The complex phase factors are attached to the molecular orbitals, rather than to the basis functions, in the localized origin (LORG) method of Hansen and B ~ u m a nand ~ ~ in the individual gauge for localized orbitals (IGLO) method of Kutzelnigg and S~hindler.~’+~’ Kutzelnigg3’ has reviewed these methods. These methods are computationally less demanding than the LAO procedure. In calculations on small molecules the basis sets have in recent years approached the completeness condition and under these circumstances gauge invariance will be achieved automatically without introducing explicit transformations of the localized orbitals. Such procedures impose very heavy demands on computation time. Invariance under a displacement of the origin does not imply invariance under all gauge transformations, a condition that must be satisfied by the exact wavefunction. It has been established that for most purposes gauge invariance can be guaranteed if the one electron basis is complete. For smaller molecules studies of gauge invariance under a variety of transformations are sometimes conducted to assess the quality of the computed wavefunction. Ferraro et aZ.33have explored the Landau gauge which reduces the vector potential to a single transverse component. They apply it to the magnetic properties of some small molecules and compare the stringency of the conditions imposed on the wavefunction in Landau and Coulomb gauges. G e e r t ~ e n ~ ~ and , ~ ’Keith and BadeIj6 have introduced procedures that amount to a continuous transformation of the origin of coordinates as the point in space changes (CTOCD). The partition of the current density into diamagnetic and paramagnetic contributions is origin dependent and the advantage of the CTOCD method is that the transformation can be chosen to make the transverse part of the paramagnetic current vanish. Lazzeretti and coworker~~~*’* have applied and extended this method. Lazzeretti” has also employed the Bloch gauge, which expands the vector potential in terms of multipoles, to develop a systematic theory of the molecular response to electromagnetic fields.

2.5 Ab initio Calculations of Magnetic Response to 1999. - A number of groups have contributed to the implementation of magnetic property calculations

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within computational packages. A considerable amount of development of the formulae is required in order to cope with LAOs. Through the phase factors the LAOs depend explicitly on the perturbing magnetic field, which leads to additional terms in perturbation theory expansions in terms of the field strength. In the early 1990s Helgaker, Ruud and co-workers instituted a program of ab initio studies using LAOs. The calculations of magnetizability and nuclear shielding factors were at MCSCF level for small and at HF level for h y d . r o ~ a r b o n sincluding ~ ~ ~ ~ aromatic structures. In order to give a reasonable account of experimental results a scaling procedure had to be applied to the computed HF values. The work was extended to include solvation effects.4548An investigation of the spin-rotation and rotational g - t e n s o r ~ , ~which ~ - ~ ~are closely connected to the paramagnetic contribution to the magnetizability, was facilitated by the introduction of modified rotational London orbitals53that depend explicitly on the angular momentum as well as on the magnetic potential. They have also investigated the Cotton-Mouton e f f e ~ t , ’ ~birefringence .~~ induced in an isotropic medium by an applied magnetic field, which is the magnetic analogue of the Kerr effect and depends on the second hypermagnetizability. An initial attempt to calculate the Cotton-Mouton constant for liquid water using gas phase MCSCF values modified by a dielectric cavity effect did not succeed in reproducing the experimental sign or magnitude of the constant.56A subsequent attempt in which the first solvation shell was explicitly included led to a result in reasonable agreement with e~periment.’~ Later work on the water molecule has included the effect of vibrational and rotational contributions and led to a re-interpretation of the origins of the magnetic effects in the water molecule. Rovibrational contributions and vibrational averaging have also been included in many of the later

calculation^.^^^^^ Bishop and CybulskPM4 have produced a series of studies of magnetizabilities, shielding tensors and Cotton-Mouton hypermagnetizabilities for diatomic and other small molecules. Their approach has been through Hartree-Fock calculations corrected for correlation at MP2 and MP3 levels or through coupled cluster theory. Finite field calculations in the earlier work have been replaced by analytical methods. They compare results for nuclear shielding and hypermagnetizability at MP3, MCSCF L-CCD and CCSD(T) levels and conclude that accurate values can only be achieved when correlation is included at a very high level of c a l c u l a t i ~ n . ~ ~ ~ ~ ~ Gauss has performed correlated calculations at MP2 leve167*68 and Gauss and Stanton at CCSD

2.6 Current Density Functional Theory (CDFT). - The Kohn-Sham density functional expression has to be modified when magnetic fields are present so that the density functional depends on the paramagnetic current density. A local exchange-correlation term then has the form


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where

The theory, known as current density functional theory (CDFT), has been developed by Vignale et aZ.70-72 and applied by Handy and co-~orkers.’~

3 Review of Literature on Response of Molecules to Magnetic Fields: June 1999-May 2001 3.1 Ab initio Calculations. - Rudd and c o - ~ o r k e r shave ~ ~ made a study of the basis set dependence of multinuclear NMR parameters calculated at the HF level with LAOS. A number of high level families of basis sets have been investigated for the case of formaldehyde and the results have also been compared with those obtained by B3PW9 1-DFT calculations. They have attempted to extend the calculations of nuclear shielding factors to solutions with different solvent polarities, using their dielectric continuum model, and find good agreement with experimental results for formaldehyde in cyclohexane and acetonitrile. In further work75 the method has been applied to H2S and HCN. They have also in~estigated~~ the effect of spin-orbit corrections on to the nuclear shieldings for molecules H2X ( X = 0, S, Se and Te), HX (X = F , C1, Br and I) and CH3X (X = F, C1, Br and I). The uncorrected values are taken from previous MCSCF work. Rudd et al.” have also developed perturbation theory expressions for the pure vibrational contributions to the hypermagnetizability. They apply their results to calculations of the pure vibrational and zero-point vibrational contributions to the polarizability, magnetizability and hypermagnetizability of water and discuss the effects of vibrational motion on the Cotton-Mouton constant for water in the gas phase. More general applications of vibrational effects are discussed78 and estimates made (inter aEia) of magnetizabilities, nuclear shieldings and the rotational g-tensor for hydrogen fluoride, water, ammonia and methane. The parallelization of the DALTON code, including application to the calculation of magnetic properties, has been Jonsson et aL8’ have continued their MCSCF studies and calculated the hypermagnetizability and its anisotropy to interpret the Cotton-Mouton effect in C02, N20, OCS and CS2. They also assess the importance of the vibrational effects and show that they may sometimes be more important than the electronic contributions. Cernusak et a1.*’ have used the Coupled HF method to study ring currents in six cyclic isomers of (CH)2B2N2.Correlated calculations show that all six have planar structures and n-electron diamagnetic ring currents. The nuclear shielding produced by the ring current is compared with that of benzene. Ligabue et a1.82have calculated n-electron current density maps from Coupled

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Hartree-Fock theory using the CTOCD method (see Section 2.4) for anthracene, phenanthrene and triphenylene. The magnetizabilities and nuclear shieldings are also calculated in the same approximation and are found to agree well with experimental values. Ferraro and Caputog3have used the RPA (TDHF) method in conjunction with the CTOCD gauge transformation and a polarization propagator formalism to calculate magnetic properties of HF, H20, NH3 and CH4. have used LAOs to calculate nuclear shielding surfaces Wigglesworth et dg4 for water and its isotopes at the MCSCF level. The correlated surface for the proton is found to be similar to that obtained in an earlier uncorrelated calculation, but the oxygen surface is considerably changed in the correlated MCSCF method. An accurate force field has been used to represent the nuclear motion and good agreement with experimental isotope shifts has been found. Similar calculations have been performed for a ~ e t y l e n e . ~ ~ In the course of an investigation of the excited states of O2 Minaev and Minaevg6have calculated the magnetizability and rotational g-factor. These results are analysed in detail for the three lowest singlet states. Kiribayashi et ~ 1 have . ~developed ~ a version of the HF-LAO method for calculating magnetizabilities and have applied it to H2. Webb et aLg8have investigated the 19F shieldings in fluorobenzenes. They have used the CASSCF method for non-dynamic correlation effects and MP2 and MP4 corrections for the dynamic part. In both cases LAOs were employed. A comparison with reported experimental data has been made. The exact relativistic theory of magnetic shielding in one-electron atoms has been extended by Pyperg9 and Pyper and Zhang” to deal with a point charge dipole model of the nucleus. Relativistic calculations of the rotational g-factor of the hydrogen halides and noble gas hydride cations have been carried out by Enevoldsen et ~ 1 . ~ ’ 3.2 Density Functional Calculations. - Handy and c o - ~ o r k e r s ’ ~have - ~ ~produced several formalisms for the calculation of magnetizabilities and nuclear shielding factors within the framework of density functional theory. Some of the groundwork was laid in the earlier paper73which was discussed in a previous section. Wilson et u E . ~ review ~ results obtained with a variety of functionals and point out that any particular functional does best when used to calculate the property for which it was optimized. They conclude that the best pre-existing functionals produce nuclear shielding values which are intermediate in accuracy between the uncorrelated HF methods and the MP2 corrected calculations. The same authors have introduced a hybrid density functional tailored to the computation of accurate nuclear ~ h i e l d i n g s .Referred ~~ to as B 3 L Y P g A it includes 5% of the HF exchange term, supplemented by the B3LYP density functional. The shielding parameters obtained in this way for first and second row nuclei are in very much better agreement with experiment and approach the accuracy achieved with coupled cluster calculations. . ~also~ applied the method to make predictions of nuclear Wilson et ~ 1 have shielding constants in diamagnetic first-row transition metal complexes. The preceding papers have worked with the LORG algorithm for fixing the gauge. In subsequent work Helgaker et ~ 1 have . ~implemented ~ the method within the


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DALTON code which incorporates procedures for working with LAOs and found that there are significant improvements in the basis set convergence of the method when used in this way. This paper includes a useful comparison between the shielding factor results obtained with various levels of approximation from uncorrelated SCF through a range of DFT approximations up to high level correlated ab initio calculations. The MCSCF, MBPT(2), CASSCF and CCSD(T) calculations included are in good agreement with experimental results on small molecules, as are the results of the hybrid-DFT method. A comparison of attempts to reproduce the experimental pattern of chemical shifts in the large cation 1cyclopropylcyclopropylidenemethylshows that the very expensive LAO-CCSD(T) approach is indeed superior to the others, but that the hybrid method is rather better than other DFT or uncorrelated calculations. Menconi et aLq7 have further examined the theory exchange-correlation functional and discussed applications to the nuclear shielding in CO and H 2 0 . Ferraro9* has used the B3LW density functional with LAOs to study the nuclear shielding of ' H and I3C in some unsymmetric N,N'-dipyridyl ureas. He has related the chemical shifts to the conformations of the molecules and the averaging due to rapid internal rotation. Ormsby et aZ.99 have calculated "B chemical shifts in boron clusters using DFT/GIAO methods and interpreted the results in terms of bond rotamers. Bagno and BonchioIo0have calculate the nuclear shielding of the tungsten '83W nucleus in mononuclear tungsten complexes using DFT with effective core potentials. De Proft and Geerlings'" have reviewed DFT in relation to the study of aromaticity.

4 Review of Literature on Response of Molecules to Electric Fields: June 1999-May 2001 Work on smaller systems increasingly includes treatment of both electronic and vibrational responses. Some papers which include work on the development of the

vibrational theory, or are primarily concerned with large vibrational effects, are grouped together in a separate section, but many of the papers discussed in the other sections report results for both types of response for particular molecules. There is a substantial amount of work on linear conjugated molecules where papers, or series of papers, begin with calculations on small oligomers and eventually extrapolate results to make predictions about infinite polymeric systems. Readers should bear in mind that the decision as to whether to include such papers in one of the small molecule sections or in the Polymer section (4.9) is often rather arbitrary.

4.1 New Schemes for Calculation and Analysis of Properties. - Jonsson, Ruud and Taylor79 have described aspects of the parallelization of the DALTON quantum chemistry program in relation to its application to the calculation of the second and higher order properties of large molecules.

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Zhou and DykstraIo2have carried out large basis set ab initio calculations on a large set of organic molecules and attempted to analyse the dipole polarizability and second hyperpolarizability in terms of additive and transferable atomic contributions. Reasonable estimates of the appropriate tensor average properties can be obtained, although the agreement for 7 is worse than for a, but individual tensor components are poorly reproduced. Bishop and B o ~ f e r g u e n e 'have ~ ~ also attempted to partition molecular polarizabilities and hyperpolarizabilities into localized group contributions over the molecular frame. The theory has been applied to a series of push-pull polyenes. An interesting conclusion is that there is little similarity between the distributions for the vibrational and electronic contributions. McCoy and Sykes'04 have introduced a method based on Fourier transforming the ab initio wavefunction to generate a momentum space representation. Low momentum components of this function are then fitted to known values of the property under consideration for a series of molecules enabling the corresponding property of other molecules to be obtained by interpolation. Holm'o5 has reviewed some of the aspects of polarizability in the light of new experimental and theoretical methods. K ~ z y k ' ~ has ~ 3 'attempted ~~ to find upper limits for molecular hyperpolarizabilities. He argues that application of sum rules to the three level model enables one to prove that the two-level limit yields the absolute maximum susceptibility, which depends only on the first excited state transition energy and the number of electrons. This conclusion has been tested by showing that a large set of experimental B and y values never exceed these limits. The conclusions are very far reaching and if confirmed will have important consequences for the future direction of research in this field. Arrighini et al. have calculated exact results for linear and non-linear electric reponse functions for a non-interacting ensemble of charged particles confined within an impenetrable box and subjected to a uniform static electric field. Aiga et a1.*O9have developed a time-dependent density functional theory for systems in periodic external potentials. The new formulation has a number of advantages over the previous theory, Zyss et al.'" have been concerned with n-conjugated molecules with a saturated barrier between the donor and acceptor regions. In experimental and theoretical work on a substituted paracyclophane they have established that the p-hyperpolarizability is much greater than the sum of the contributions from the separate conjugated regions. The experimental results have been confirmed by calculations using the Collective Electron Oscillator approach which indicates an increased importance for electron- hole pair delocalization in higher order responses. The interaction through the saturated region is described as 'throughspace charge transfer'. Blanchard-Desce et al."' have described the results of work on new octupolar molecules. Ostroverkhov et al. have discussed the optimisation of the P-hyperpolarizability for SHG in chiral materials. Results for lambda shaped molecules obtained by hyper-Rayleigh scattering are analysed.


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Zhang' l 3 has investigated the y-hyperpolarizability in polyenes using a Lanczos-based density matrix renormalization-group scheme. Magnoni et aZ.114review experimental data in an attempt to define criteria for producing molecules with large negative y-values.

4.2 Ab initio Calculations on Atoms. - Marouli~"~ has investigated the static response functions for the Na atom. He has calculated values for the dipole polarizability, a, the quadrupole polarizability, a2,the dipole-dipole-quadrupole hyperpolarizability, B, and the dipole hyperpolarizability, p. Although large systematically optimized gaussian basis sets were employed the calculated dipole polarizability is appreciably larger thm all three available experimental estimates. There are no experimental values for thz hyperpolarizabilities, but the difficulties of obtaining convergence indicate that the calculation of the dipole hyperpolarizability is a considerable problem. The implications are that these techniques would not, at their present level of sophistication, lead to reliable results for the corresponding properties for larger systems, such as sodium clusters. Kobayashi, Sasagane and Yamaguchi'16 have developed the theory of the timedependent spin-restricted Hartree-Fock method for application to open shell systems (TDROHF). The expression for the cubic hyperpolarizability is obtained from the quasi-energy derivative (QED) method. The theory is applied to the investigation of the frequency-dependent y susceptibility of the Li, Na, K and N atoms. 4.3 Ab initio and DFT Calculations on Diatomic Molecules. - Jaszunski, Klopper and Noga'17 describe an analysis of the static dipole polarizability of the Hez dimer using high precision CCSD(T) calculations. The results for the polarizability and its anisotropy are not significantly different from CCSD results in the literature and serve to confirm their accuracy. Maroulis has continued his extensive program of studies of molecular multipoles and static response functions using the finite field method in conjunction with high level gaussian basis sets. Using the Coupled Cluster CCSD(T) method' l 9 he has determined the quadrupole, octupole and hexadecapole moments for CO and finds reasonable agreement with recent gas phase measurements of a high density CO-Ar mixture, although not with data on similar mixtures in the liquid phase. In a study of the polarizability of HI''9 a set of 287 contracted gaussian functions, which are expected to lead to wave functions near the nonrelativistic Hartree-Fock limit, was used Again the results are not in satisfactory agreement with experiment, even after relativistic corrections have been estimated. A study of the CS, SiO and SiS using MP2 and MP4 perturbative extensions to the Hartree-Fock solutions and coupled cluster calculations, leads to good agreement between calculated and measured dipole moments but is less satisfactory for the electric quadrupole moments. The dipole polarizability and its anisotropy have also been calculated. It is found that CS is less anisotropic than SiO. Vavali2' has put forward a scheme for finding the cluster and derived cluster amplitudes from the extended coupled-cluster method, in which cluster amplitudes

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are solved in a de-coupled manner. Using this procedure the dipole moment and fist hyperpolarizability tensor for HF and H 2 0 have been calculated with Sadlej's basis set. Reimers and Hush'22 have used a range of high level ab initio and DFT methods to calculate the dipole moment, polarizability and hyperpolarizabilities of CO and CN-. The work has been directed toward the interpretation of data obtained in a high precision experimental Stark-type experimental technique known as electroabsorption spectroscopy. The calculated results are said to show excellent agreement with the experimental data. Basis sets suitable for use in DFT calculations are given. Kedziora and S ~ h a t z have ' ~ ~ used ab initio methods with the inclusion of electron correlation through a second order polarization propagator (SOPPA) method to calculate properties of CO that are relevant to the interpretation of surface enhanced Raman spectroscopy. They have calculated the fi-equencydependent dipole-dipole, quadrupole-dipole, quadrupole-quadrupole and the electric dipole-magnetic dipole polarizabilities and their normal co-ordinate derivatives. Basis set convergence to about 20% for most of the quantities computed is claimed for the largest basis set SOPPA result, although there are more severe problems with some of the higher order off-diagonal tensor elements. Costa et al. 124 present results of many-body perturbation theory, coupled cluster and quadratic CI methods applied to the calculation of the polarizability and fist hyperpolarizability of NaH. It is shown that the nuclear relaxation contribution is substantial for this molecule and that it is appreciably affected by electron correlation effects. Serrano et al.'25 have used similar methods to calculate the dipole moment, polarizability and first hyperpolarizability of CaC and have used the results to predict the Rayleigh and Raman scattering activity. 4.4 Ab initio and DFT Calculations on Small and Medium-sized Molecules. 4.4.1 Water. - It is desirable that the electric moments and response functions of the water dimer should be calculated accurately Jn order to establish the reliability of approximations that can be extended to large water clusters and so to a better understanding of the aqueous environment. In a CCSD(T) calculation employing 370 gaussians126Maroulis found that the interaction contributions to a and 7 were rather small so that the mean values were close to twice those obtained for a single water molecule. This result still held when the geometry of the water molecules in the dimer was allowed to relax. Basis set size effects have been investigated with a view to proposing strategies for the investigation of larger clusters. Batista, Xantheas and J o n ~ s o n 'have ~ ~ calculated the molecular multipole moments for water molecular clusters and in ice (Ih) using ab initio methods. Taking the computed charge distribution for the whole cluster they have attempted to partition it in the manner of Bader and hence obtain multipole moments for the individual water molecules. Different partitioning schemes lead to widely different values. In all schemes the magnitude of the dipole increases with size of cluster, monotonically, from its value in the gas phase to its value in ice.

6: Electric Multipoles, Polarizabilities, Hvperpolarizabilities

309

Morita and Hynes128 have attempted an analysis of the sum frequency generation spectrum at the surface of water from ah initio calculations of normal mode shifts and hyperpolarizability in the frequency range of the OH stretch. Molecular dynamics has been used to generate a sample of structures. Agreement between the simulated and experimental spectra is good. The spectrum appears to depend on signals from molecules in a few standard orientations in the top two layers at the surface. 4.4.2 03,SOz, SeOz and Te02. - Maroulis et a1.129,130 have employed their finite field approach to investigate the static response functions of these triatomic molecules. In the case of ozone both open (ClV) and cyclic (D3h) forms of the molecule have been investigated. The results reported employ the CCSD(T) method and are expected to be more accurate than those obtained in an earlier publication. In both symmetries the hyperpolarizabilities of ozone are found to be drastically affected by the inclusion of more electron correlation in the more recent calculations. 4.4.3 Other Molecules. - The hexadecapole moment and the dipole and quadrupole polarizability of SF6 have also been calculated by Mar0u1is.l~~The results are lower than the best experimental estimates. Pluta and SadlejI3*have calculated the dipole moment and static a , #? and y and y tensors of urea and thiourea using three high level basis sets of increasing flexibility. Excellent agreement is found with experimental determinations of the dipole moment and linear polarizability. Frequency-dependent polarizabilities and hyperpolarizabilities are calculated in the TDHF approximation and the results are then scaled to allow for electron correlation and the effect of basis set extension. Estimates of the response functions for non-linear optical processes are obtained. The introduction of the sulfur atom is found to produce a large increase in the predicted efficiency for third order effects. Soscun et al.'34 report ah initio and DFT studies of the static and frequencydependent a and y tensors for ethyne. The coupled perturbed Hartree-Fock (CPHF) method has been used for the static calculations and TDHF for the frequency-dependent properties, which have been calculated at 633 nm. New basis sets have been proposed. Jacquemin, Champagne and Hattig134have investigated the longitudinal component of the frequency-dependent #?-polarizability for small polydiacetylene chains containing terminal amino and nitro donor and acceptor groups. The computations have been carried out at the Hartree-Fock, MP2, MP4 and Coupled Cluster levels and frequency-dependent CC results have been used to test the efficiency of additive and multiplicative corrections to approximate the frequency-dependent correlated values, The multiplicative corrections are found to reproduce the CC values to within a few percent while the additive correction underestimates the values by about 20%. continue their critical assessment of DFT schemes as Champagne et methods of calculating dipole moments and the a, /3, y response functions. In this paper they conclude that DFT, at least with the existing functionals, is extremely

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unreliable as a method of calculating B-values for push-pull n-conjugated molecules. Van Gisbergen et a1.136have also noted an apparent error in the DFT formulae which lead to unreasonable results for computed optical properties. K a ~ m a 'has ~ ~ used the unrestricted TDHF method to carry out ab initio calculations of the static and dynamic a and /3 for the SiH3 radical and on the SiH4 molecule. He has also reviewed the role of theory and modelling in the design of electronic and non-linear optical material^.'^^ Comparison of the two sets of results reveals, perhaps unsurprisingly, that there is little difference in the a values, while non-zero ,M and p contributions occur only when the SiH4 symmetry is broken (although /3 should in principle have a small non-zero value for a tetrahedral structure). Of more significance is the finding that the a and /3 spins in the unrestricted Hartree-Fock wave functions make substantially different contributions to the hyperpolarizability. Raptis, Papadopoulos and Sadlej139 have calculated the electronic structure of fully lithiated benzene and find very large increases in the dipole polarizability and y-hyperpolarizability, which they attribute to the effect on the molecular orbital structure of the molecule arising from the electronic valence structure of the lithium atom. It is suggested that several known stable organometallic systems involving alkali metals should be investigated. Reis et a1.l4O have carried out DFT calculations on the static polarizabilities and hyperpolarizabilities of bare boron clusters incuding up to 10 boron atoms. They find that the y-hyperpolarizability saturates when the cluster size reaches approximately five atoms. A maximum in the hyperpolarizabilty per atom occurs for the cluster containing six atoms. Yamaguchi and co-workers have continued their studies of radicals that have unusual resonance structures leading to the reversal of the electric dipole on Ah excitation and large negative contributions to the y-hyperp~larizability.'~'-'~ initio and DFT methods have been used to investigate 1,6,6-a-trithiapentaleneand a relationship between the unusual properties of y and the possibility of combining conductivity with third order non-linear optical response in n-conjugated systems involving sulfur is In studies of furan and its sulfur, selenium and tellurium homologues, Ohta et aZ.'46 have computed the static values of the a, p and y functions using HartreeFock ab initio methods with correlation corrections. For the heavier atoms an effective core potential was employed and justified by comparison, in the case of furan, with all electron calculations, when the response functions were found to be similar. As might be expected, the inclusion of the more polarizable heavy atoms increases the components of the a and y tensors. The p tensor, which in general is more critically dependent on a few non-centrosymmetric excitations of low energy, behaves more erratically in the different homologues. The results for y have been compared with optical Kerr effect measurements made by the same group and some discrepancies found for the higher homologues. Kamada et ~ 1 . l ~ ~ have therefore investigated the frequency dependence of y using the TDHF method when agreement is improved. Wang, Yang and Sun148have calculated the static values of p, a, p and y for


311

pyrrole, furan and thiophene at HF/MP2, MP3 and MP4 levels. They find that the correlation corrections have a substantial effect on the results, although if the results are arranged for each molecule in order of size the same sequence is found for all levels of calculation. Milledori and A l p a r ~ n e 'have ~ ~ investigated a group of 16 chalcogenides, C,&X, where X = 0, S, Se, Te. The static and frequency-dependent a and p tensors have been calculated by ab initio methods at the MP2 level using a number of high level basis sets. The vibrational contributions were found to be small in comparison with their electronic counterparts and do not show any clear dependence on the heavy atom, whereas the electronic p increases steadily along the series. Bartkowiak and MisiaszekIso used the coupled perturbed Hartree-Fock method and the sum-over-modes formalism to calculate the electronic and vibrational &tensors for 4-nitroaniline, 4-nitro-4'-aminostilbene, 4-amino-4'-nitrobiphenyl and 4-amino-4'-nitrophenylacetylene, all typical push-pull conjugated molecules of the kind that have been associated with second order optical non-linearities derived from their large p values. Their calculations refer to the gas phase and to chloroform and aqueous solutions, the solvent effects being included through a continuum self-consistent reaction field model. They demonstrate that the solvent effects are much greater for the vibrational hyperpolarizability than for the electronic contribution. Bhanuprakash and Ra0I5' have calculated the p-hyperpolarizability in the ab initio TDHF approximation for some zwitterionic organic compounds containing n-electron donor and acceptor groups separated by saturated C-C bridges. The geometries have been optimized at the HF/6-31G basis set level. A large hyperpolarizability, associated with charge transfer, is observed even when the bridge contains more than one single bond, indicating that these materials may have potential applications in non-linear optics. Szymusiak et .I.''* report results on model polyenes with modified indanone groups, studied by DFT, HF/3-2 1 G* ab initio and semi-empirical methods. Champagne, Fischer and B ~ c k i n g h a m 'have ~ ~ calculated the sum frequency hyperpolarizability, p( -30; 20, m), of two small chiral molecules, R-monofluoro-oxirane and R-(+)-propylene oxide. They employed a sum-over-states method based on CI involving singly excited configurations calculated at the 6-3 11++G** level. The excitation energies were scaled to fit the experimental UV absorption spectrum and checked against values from time-dependent density functional theory. The calculated values were much smaller than those reported on sum frequency generation experiments on aqueous solutions of arabinose. Janssen et al. Is4 have used calculated ab initio static hyperpolarizabilities as input data in a study that attempts to simulate hyper-Rayleigh scattering in liquid NOz. Using molecular dynamics they take account of positional and orientational inhomogeneities and quantifL the incoherent and coherent parts of the second harmonic generation. Local fields are found to influence the signal. The coherent part of the signal is significant, so that, on this score alone, it is necessary to work in dilute solutions when attempting HRS determinations of molecular polarizabilities. Simulations of this type should become increasingly important if really

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quantitative agreement between calculated hyperpolarizabilities and measurements in condensed phases is to be established. Molecules with multi-directional charge transfer and octupolar molecules have aroused interest in recent years, mainly in connection with avoiding the 'tradeoff' between high hyperpolarizability and undesirable properties also linked to the characteristic strong uni-directional charge transfer behaviour. The undesirable properties include the indisputable tendency of such compounds to absorb strongly in regions where optical transparency is required and their increased likelihood of forming macroscopic structures that are centrosymmetric. Greve et aE. have investigated molecules with multi-directional transitions and shown that, when compared with uni-directional molecules of the same size, they have higher values of the y-hyperpolarizability but show no redshifts in the absorption ' ~ studied ~ two series of octupolar molecules spectrum. Lee, An and C ~ Ohave using ab initio HF/6-3 1G calculations. They find that a, p and y increase with the donor strength and also with the polarity of the solvent, represented by the selfconsistent reaction field approximation. These results are in agreement with a valence bond charge-transfer model. l ~ ~studied the possible structures of squaric acid and the Spassova et ~ 1 . have monohydrogenosquarate anion at the MP2 level with 6-3 1lG** basis sets supplemented by a set of diffuse p and d functions. They use the finite field method to calculate the p- and y-hyperpolarizabilities and find that, compared with the acids, the former is reduced by 40% in the ion and the latter increased by a factor of more than two. Domagalska et aE.lsshave used HF/3-21G* ab initio and AM1 Finite field and SOS calculations to investigate new conjugated polyenes of 1,3-dialkyl-2thiobarbituric acid. Correlations between polarizability and integral IR intensities were found. The variation of the polarizability and hyperpolarizabilities with chain length is discussed. Borisov et ~ 1 . " have ~ used the ab initio RHF method with MP2 and the 6-3 lG* basis set to investigate CF3 substituted acrylates and their non-fluorinated analogues. The dipole moments and a and p have been calculated. Lee et ~ 1 . l have ~ ' investigated the two photon absorption cross-section and the y-hyperpolarizability of a series of quadrupolar molecules using ab initio finite field and SOS methods. 4.5 Semi-empirical Calculations on Molecules. - The number of semi-empirical packages that can be employed routinely to estimate the a, p and y tensors is proliferating, but they can still be classified into a few broad groups: (i)

Sum over States methods, which produce frequency-dependent functions from time-dependent perturbation theory summations that usually calculate the excited states and their matrix elements at SCI or SDCI level. The underlying quantum chemistry system is usually one that has been specifically parameterized for excited states, often CNDO(S) or ZINDO or other INDO implementations. The SOS method is also used in conjunction with the NDDO (neglect of diatomic overlap) theories, using the AMl,


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PM3 or MNDO hamiltonians for calculations on organic molecules. The MOPAC and AMPAC packages are the usual sources for these programs. The excited state spectrum calculated in the NDDO based methods is usually inferior to that obtained in CNDO(S) or ZINDO/INDO, but scaling is sometimes used to overcome this difficulty. It is essential to use SDCI or better if any reasonable treatment of the y-hyperpolarizability is to be achieved. (ii) Finite field methods for static properties, which are very often based on MOPAC/AMl . Usually these use the numerical approach with either an added uniform field or perturbation by point charges which can be added in the form of 'sparkles' in the MOPAC package. (iii) More sophisticated CPHF or polarization propagator methods are also sometimes used for frequency-dependent calculations with MOPAC type theory. In the following review of publications we attempt to identify the type of calculation that has been employed in each case. Goller and Grummt161have made a detailed study of the NDDO/CI/SOS approach. They have applied the method to a set of 236 donor-acceptor-nconjugated organic compounds and compared their results for the THG y-hyperpolarizability with finite field results obtained from MOPACIFF and AMPACIFF, both for the gas phase and with solvent corrections. They find, after making scaling corrections for correlation effects, that a useful correlation with experimental results is obtained allowing qualitative prediction of trends for small as well as large hyperpolarizabilities. Agreement is better when y is large, possibly because the experimental results are then more reliable. The difficulties found for the solvation effects of elongated molecules have been previously reported and are probably related to the inability of the solvation procedures to represent the local fields in needle shaped cavities. A computational package, VAMP/PECI/SOS has been produced. Grummt et all6* have applied this method to the study of the y-hyperpolarizability of hydroxystilbazolium salts and their betaines. Wang et u Z . ' ~ ~ have used the AMl/FF approach for static properties and ZINDO/SCI to study the P-hyperpolarizability of some azulene derivatives, High values are obtained and it is concluded that the azulene ring is a more efficient conjugation bridge between donor and acceptor than either benzene or thiophene. They use the same methods in a study of push-pull polyenes containing nonaromatic cyclic olefineslMand for tetra-substituted benzenes and pyrazines. Zhao et al. 165 have investigated the frequency-dependent p-hyperpolarizability of several series of substituted aromatic compounds using a CNDO/SCI SOS method. They have also'66 made experimental and theoretical studies of phenylurea and m-hydroxybenzoic acid. Zhu and Jiang'67have systematically employed the AM 1 hamiltonian with point charge fields represented by sparkles to calculate static p-tensors. They have investigated push-pull polyenes and correlate the polarizabilities and hyperpolarizabilities with the net charge on the polyene bridge between donor and acceptor.

3 14


The same method has been applied to the donor-acceptor substituted cumulenes'68and has re-established the conclusion that these molecules will have higher @-values than the polyenes, other things being equal. Donor-acceptor quinoid molecules have been analysed in the same way.'69 Comparisons are made between the non-linear responses of these molecules and the polyenes. Sheng and Jiang170 have extended the investigation of the quinone structures by carrying out INDO/S calculations on their frequency-dependent properties. Lipinski and Bartkowiak171 have investigated the effect of solvation on the p and y tensors for a number of charge transfer organic chromophores. Finite field and SOS semi-empirical methods were employed and the solvent was represented in by discrete Langevin dipoles distributed by a Monte Carlo method and by the continuum virtual charge method. Substantial effects related to solvation and conformational changes were found. B a r t k ~ w i a k 'has ~ ~ applied the same methods to the calculation of the @- and y-hyperpolarizabilities of aminobenzodifuranone . Lacroix et a1.173 have discussed the effect of proton transfer on the P-hyperpolarizability of hydrogen bonded merocyanine dyes. They use a PM3 calculation to calculate the energy barrier for the transfer and INDO/SOS to calculate the hyperpolarizabilities. The effect on P is strong and has a possible application in a switching device. Lacroix et a1.174have used the same method to calculate pVec for (4-nitrophenylazo)azulene, which has a very large experimental (EFISH) value. The monoclinic crystals of this material exhibit SHG signals 420 times larger than urea. The origins of these large effects are discussed. Averseng et al.,I7' using INDO/SOS, have investigated the effect of a spin transition on the P-hyperpolarizability of a bis(salicylaldiminato)Fe(II) Schiff base complex. They find that the effect of the transition is to increase the hyperpolarizability by about 25%, mainly due to a change in the molecular geometry. Zerner et a1.'76have used the ZINDO/SOS method in conjunction with AM1 structural optimizations to investigate dicyanomethylene-derived heteroaromatic dyes, The effect of E/Z conformational isomerism and the introduction of various combinations of end groups on Pvec(the part of the P tensor that is detected in EFISH experiments) is elucidated. Solvent effects are included through the selfconsistent reaction field approximation. The ZINDO/CI/SOS method has also been used by Liu et al.'77in a study of p for 4-(dicyanomethylene)-2,6-bis-(2'-thiophene-vinyl)pyran and its derivatives; by Fu et al. 1 7 8 ~ 1 7 9to study unsymmetric bis(phenylethyny1)benzene derivatives and barbituric acid and its derivatives; and by Breitung et al.lg0 to investigate the thiazole and thiophene analogues of donor-acceptor stilbenes. Zhang and Yan"' have used the AM1 method to study the phthalocyanines and their derivatives. The TDHF approximation has been employed in conjunction with the PM3 semi-empirical formalism in work by Kim et al. l g 2 on the @-hyperpolarizabilities of photoconductive chromophores. It was noted that the optical non-linearities of dipolar photoconductive molecules with carbazole, indole or indoline as donor units are large enough to be useful in electro-optic and photorefractive applications.


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Jalali-Heravi et al. 1 8 3 ~ 1 8 4 have investigated some tridentate salicylaldiminato Schiff base ligands and calculated the static P using the finite field method with AM1. The effect of various groups as dentates on the hyperpolarizability is discussed. Andraud et al. have made a theoretical analysis, using a CNDO/S method, of polyenic molecules of octupolar C3hsymmetry. The results have been analysed in a tensorial formalism to elucidate the optimum arrangement of groups for high P in octupolar molecules. Sato et a1.186have found large values of the P-hyperpolarizability as calculated by a semi-empirical method in molecules intended to be prototypes of a highly amphoteric and polar molecule. It is thought that such structures will lead to novel molecular properties. Cundari et a1.I8’ have computed a and y values for solutions of inorganic transition metal thiometalates and metalates. De and Ganguly188have computed the P-hyperpolarizability of a benzothiophene-acetophenone bichromophore and find a large value. Yamaguchi et al. 1 8 9 ~ 1 9 0have studied azobenzene dendnmers theoretically and experimentally. They have employed a CNDO/S-CI method to investigate their P-hyperpolarizabilities. Illien et a1.I9I discuss the large electron donor effect associated with the R3N+ N- donor. They calculate large values of the P-hyperpolarizability for R3N+N .(CH=CH) .NO*. ~

4.6 Vibrational Effects.

- This section includes papers that are primarily concerned with vibrational effects, including those that extend the theoretical framework and those that deal with particular molecules. Over recent years there has been a steady growth of interest in vibrational effects in the context of ab initio calculations o f linear and non-linear molecular response functions. It has been realized that in some cases vibrational may rival electronic contributions to the parameters controlling non-linear optical responses. This is particularly likely where the molecule is of higher symmetry (quadrupolar or octupolar rather than dipolar), and for lower frequency effects where there is little pre-resonant enhancement of the electronic contribution. The main features of the theoretical methodology for the calculation of vibrational response functions were established several years ago and the fundamental papers were reviewed in the previous volume. Recent developments have been the introduction of field induced ~o-ordinates,’’~ improved integration and the first relativistic studies. 194 Luis et a1.192,194 have shown that, in evaluating the contribution to the static vibrational hyperpolarizability that comes from the normal co-ordinate derivatives of the zero point energy, it is possible to replace the sum over 3N - 6 normal coordinates by transforming to a set of field induced co-ordinates, when only one term remains. The method has been applied to a typical push-pull polyene, NH2 -(C=C) -N02. Ingamells, Papadopoulos and Sadlej 193 have introduced a new semi-numerical method based on Numerov-Cooley integration to calculate the vibrational

’

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corrections to the static electric properties of diatomic molecules. This method of integration of the field-dependent vibrational equation allows a unified study of the separate contributions to the complete response fbnction to be made. They apply the technique to compute the vibrational contributions to the calculation of a and p of the KLi dimer. In a study of the coinage metal hydrides, CuH, AgH and AuH, the same group195have utilized the Numerov-Cooley technique in a relativistic calculation of the static a and p. They find that relativistic effects produce a large reduction, as compared with the non-relativistic values, in the vibrational contributions, especially in the p-values. The effect increases with the atomic number of the heavy atom. The same have introduced a method of calculating vibrational contributions with respect to arbitrary reference geometries. The order of magnitude of the correction required is estimated for the cases where the reference geometry is computed at the SCF, MP2, CCSD and CCSD(T) levels. Perpete, Quinet and ChampagneI9' have shown that vibrational contributions to the y-hyperpolarizability of various symbiotically substituted quadrupolar n-conjugated molecules are large. Using ab initio calculations at the HF level they calculate the response functions for the dc-Kerr effect, degenerate four wave mixing, hyper-Raman effect and electric field induced SHG. Quinet and Champagne198have made an ab initio study of the vibrational contributions to y in acetylene, ethylene and ethane, using RHF theory with MP2 corrections and including first order anharmonicity contributions. There is a general increase in the effect in going from sp3 to sp hybridized carbons. Vibrational effects account for 10-20% of the intensity-dependent refractive index (calculated at zero frequency) and about 50% of the Kerr constant for acetylene. Eckart et have investigated the vibrational contributions to a and /3 for cyclopropenone and cyclopropenethione and for their saturated and non-cyclic counterparts. While in most cases the vibrational contributions are about 10% of the total value of a tensor component, it is suggested that the vibrational contribution to the Pockels effect tensor, p ( - m ; cr), 0), might be as much as 50%. Eckart and Sadlej200have calculated the electronic and vibrational contributions to the dipole moment and a and p for the HF and H 2 0 dimers. They find relatively small vibrational contributions to y and a but the vibrational contribution to the first hyperpolarizability is predicted to be of the same magnitude as the electronic contribution. Okuno et aL2" have used ab initio methods in an attempt to find the origin of the large p-hyperpolarizability found in functionalized azobenzene dendrimers. They find that the interaction between monomers partly accounts for the formation of non-centrosymmetric structures and that, in these structures, the hyperpolarizability is further enhanced by the interaction of monomer units arranged in series. Tommasini et have developed a model for calculating the Transverse Acoustic Modes of all- trans-polyenes and discuss their relevance to the static vibrational polarizability.


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4.7 Calculations on Complexes, Dimers, Clusters, Excited States. - MaroulisZo3has used his finite-field technique to explore the hyperpolarizabilities of the rare gas diatoms He2, Ne2, Ar2 and Kr2 and the interaction hyperpolarizability of hydrogen fluoride with a neon atom. Champagne et a1.’O4 have stressed the importance of the inclusion of three-body effects when interpreting intermolecular interactions in terms of the usual equations involving the polarizability and hyperpolarizabilities. Hattig et a1.’O5 have made a high level ab initio study of the two-body interaction-induced polarizabilities and y-hyperpolarizabilities in helium and argon. The calculations were at the CCSD level and also with FCI in the case of He. These results are applied206to the calculation of the macroscopic dielectric constant, refractive index, Kerr effect and hyperpolarizability virial coefficient. Hohm et al.207have calculated the static dipole polarizability of P4 clusters using ab initio finite-field MP and coupled-cluster methods. The results have been compared with frequency-dependent measurements obtained from the gas phase refractive index. Korambath and Karna208have used ab initio TDHF to investigate the a and p functions for GaN, GaP and GaAs clusters. hcciardi et al. 209 have investigated the heteroleptic tetrapyrrole sandwich complex formed with porphyrinato-porphyrazinato-zirconium(1V) using timedependent density functional theory. They find that an accurate description of the UV/visible spectrum is obtained and the nature of the excitations is discussed and compared with previous work on bisporphyrins. A calculation of the P-hyperpolarizability fails to reproduce the experimental result, which appears to predict a negative value for the SHG p at all frequencies up to the experimental one. A pole is found in the calculated P fbnction close to the laser frequency, which would lead to the possiblity of negative values at higher frequencies. Reis and Papadopoulos’” have studied the B4 cluster using a range of high level theoretical methods. Cundari et aE.’“ have used an effective core potential model to study the nonlinear optical properties of a transition metal complex, osmium tetroxide. Yartsev212has considered the effects of electron correlation on a and y using a molecular dimer with two radical electrons as a model. Yartsev and Marcano213 explore the effects of electron correlation on the static y of dimers through the Hubbard model. Imase et al.’I4 have carried out an ab initio study on the 4-hydroxybenzoic acid6-hydroxy-2-naphthoicacid copolyester which is a rod-like molecule forming a nematic liquid crystal. The P-hyperpolarizability of the dimer is comparable to that of 4-nitroaniline and increases with degree of polymerization. Bance~icz’’~ has derived irreducible spherical tensor formulae for long range interaction-induced P and y pair hyperpolarizabilities for arbitrarily shaped molecules. For isotropic systems the expressions reduce to well established Cartesian tensor formulae. These formulae have applications, particularly in the interpretation of Rayleigh and hyper-Rayleigh light scattering. Larsson et al. 2 1 6 have employed a multiconfigurational method to investigate

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the hyperpolarizabilities of a solute in an electronically excited state. Calculations were carried out on 4-nitroaniline. 4.8 Fullerenes. - Perpete, Champagne and Kirtman2I7 have computed the vibrational contributions to the main non-linear optical response functions for the fullerene c60. They use normal co-ordinates obtained in a DFT approximation combined with Hartree-Fock ab initio calculations of the electrical properties. Vibrational contributions to the electro-optic Kerr effect and degenerate four wave mixing are found to be comparable with the electronic effects. Xie and Rao2’* have given a brief review of the third order optical nonlinearities (related to y ) in the higher fullerenes (c7O-c96) and compared their response to carbon nanotube and polyenic systems. They find that, in tubular fullerenes, doping can greatly increase y . Ruiz et aL2I9 have investigated the linear polarizability, a, of icosahedral fullerenes in the range C6O-c720 using the PPP (x-electron) method. They find that the calculated polarizability is given by the formula, a = 0.75R: in terms of the effective molecular radius. It is argued that the fullerenes should exhibit the maximum possible polarizability for carbon shells. Nomura et aZ.220have investigated the y-hyperpolarizability of c60 using CNDO/SOS methods. They have subjected the third order term in the SOS perturbation expansion to a group theoretical analysis and show that only excited states of ‘A1, and HI, symmetry can occur as the second intermediate state in the They then find sequence of matrix elements: (glrln>(nlr”’)(rz’Irln’’>(n’’lrlg). that the only significant positive contributions to y arise when the second intermediate is a doubly excited state. For centrosymmetric molecules the second intermediate must always be a doubly excited state, so that this conclusion can be attributed to the highly symmetric nature of the fullerene. Lin and Lee2*’examine the static y-hyperpolarizability for c60 using an AM1 method. The AM1 method, as opposed to PPP, includes all valence electrons and gives a better account of the way in which the o and x interactions change when a planar sheet is replaced by the icosahedral structure. They find low values for y , in agreement with recent experimental results. It is also found that this result is in accordance with a procedure for scaling the y-values of conjugated organic molecules according to their n-conjugation lengths. Lin et aZ.222have made a comparative study of the y-hyperpolarizabilities of carbon bowl and cage structures with 72 or 78 carbon atoms and correlated the results with various geometric parameters. Mishra et aZ.223have further elucidated the relationship between y and the pyramidalized angle in these compounds. Xie224has extended his work on zig-zag carbon nanotubes to investigate the effect of doping with boron or nitrogen on the y-hyperpolarizability, and Xie and Ra0225have calculated the y-hyperpolarizability for a number of chiral graphene tubules. Margulis226 has shown how general arguments based on the K-electron photoemission spectrum near the conduction and valence band energies can lead to an estimation of y for carbon nanotubes, provided it is assumed that virtual x-electron transitions between band states are responsible for the effect.


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4.9 Polymers. - A review of calculations of the polarizabilities and hyperpolarizabilites of stereoregular polymer systems has been provided by Champagne et ~ 1 . ~ ~ ’ Perpete et and Champagne et al.229have continued their investigations of long chain conjugated systems, with the inclusion of some correlation effects. The bond length alternation (BLA) in conjugated polymer chains and its effect on the a and y response functions have been studied. The work on BLA has employed HF ab initio calculations at the MP2 level with 6-3 11G” basis sets and the static response functions have been calculated at this level and with several DFT schemes. Champagne et aL2” have carried out RHF/6-3 1G calculations of the effect of electrical and mechanical anhannonicity on the longitudinal y-hyperpolarizability of eight homologous series: polyacetylene, polyyne, polydiacetylene, polybutatriene, polycumulene, polysilane, polymethineimine and polypyrrole. Chains containing up to 12 heavy atoms along the conjugated backbone have been investigated in order to approach the limiting infinite polymer behaviour. The importance of overall vibrational contributions and the particular contribution due to anharmonicity have been assessed for third order non-linear optical effects. have investigated solvent effects on the polarizability and Norman et hyperpolarizabilities of conjugated polymers by means of a semiclassical solvation model using results from ab initio calculations on the molecules. The solvent effects are found to attain maxima at fairly short oligimer lengths with the main axial contribution decreasing to zero at the polymer limit. The saturation behaviours in the gas and solution phases are different. Kulakowska and K ~ c h a r s k i ~have ~ * used ab initio procedures from the GAMESS and GAUSSIAN packages and the semi-empirical INDOE method to investigate the P-hyperpolarizability of maleimides and acrylates of the chromophoric derivatives containing azo and sulfonyl or nitro groups. Some acrylate monomers were found to have promising p-values and the agreement between the ab initio and semi-empirical results was good except where there were nitro groups involved. Schmidt and S p r i n g b ~ r ghave ~ ~ ~developed a method for the calculation of static hyperpolarizabilities of infinite conjugated polymers in which an external electric field is included directly in the DFT formalism. In the simpler case where the field does not break the translational symmetry they have applied the theory to trans-polyacetylene and polycarbonitrile and demonstrated the effects of heteroatoms in the backbone. They find that the properties change only by a small amount in the presence of a strong field. Silva et al.234have made a comparative study of the non-linear polarizabilities of solitonic polyenic chains using AM 1 semi-empirical and ab initio 6-3 1G with MP2 methods. Chernyak et al. 235 have calculated the y-polarizabilities of donor-acceptor substituted polymers. They find that in the frequency regime usually employed in optoelectronic applications, in the range intermediate between vibrational and electronic transitions, the response is dominated by the electronic effect. Bishop et dispute this conclusion and discuss related issues.

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Chemical Meddling: Applications and Theory, Volume 2

Bishop et al.237have used C P H F adapted for periodic systems to calculate the dipole moment and the static u, p and y for infinite polymer chains. The results are found to be in good agreement with large oligomer calculations. Gubler et aZ.238 have studied rod-like molecules of poly(triacety1ene) where the electrons are delocalized over a one dimensional path. A power law dependence of y on the number of monomer units is found which is in good agreement with quantum chemical calculations. The critical conjugation length for saturation is found to be about 60 carbon-carbon bonds. Robinson et aZ.239 have proposed a new type of electro-optic chromophore. The aim is to load the chromophores into a polymer matrix to produce electro-optic effect devices. The molecular properties of the chromophores have been obtained by quantum chemical calculations. Liang et al.240 have used a linear scaling localized-density-matrix (LDM) approach to calculate the non-linear optical responses of large polydiacetylene oligomers. Martin et aZ.,241as part of a combined experimental and theoretical study of a series of monodisperse capped poly(triacety1ene) oligomers, have calculated their y-hyperpolarizabilites by a method which combines the use of a Valence Effective Hamiltonian with a SOS procedure. Perpete et aZ.242 have evaluated the static electronic and vibrational P-hyperpolarizabilities of non-planar trans-cisoid polymethineimine oligomers at the HF/6-3 1G level. Vibrational contributions still increase linearly up to oligomers with 15 unit cells whereas the electronic contributions tend to saturate. An and Wong243have used ab initio finite field methods to study the long chain-length behaviour of the u and y functions of polyenic chains with donoracceptor substitutions and charged topological defects. Linear variation of a and y with chain length is found in the long chain length limit for all cases. Zhang and George244predict a huge enhancement of y in conjugated polyenes since they argue that the scaling dependence undergoes a dramatic change with the electron correlation strength. Zhang245considers also considers the effects of electron correlation and charge transfer in charge-transfer insulators and predicts the existence of giant non-linearities under optimum conditions. 4.10 Crystals. - Voigt-Martin et al.246 have developed a method producing complete crystal structure data from very small crystals by using a combination of electron diffraction and simulation methods. They point out that given this facility the non-linear optical properties of the crystal should be deducible from accurate quantum chemical calculations of the molecular hyperpolarizabilities. Munn et al.247have attempted to relate the macroscopic refractive indices and ~ ( susceptibility ~ 1 tensor for the iodoform and sulhr (S,) crystals and the crystalline complex CH31.3S, to the molecular polarizabilities and hyperpolarizabilities. The attempt is only partially successful and illustrates the difficulties that will be encountered in projects such as that described in ref. 246. Penhuis and M ~ n n ~have ~ ' performed calculations of the linear and non-linear optical properties of layers of Langmuir-Blodgett films and find that the internal field varies little after the first layer. They have investigated249Langmuir-Blodgett


32 1

films of stearic acid and the mesogen, 5-CT (4”-n-pentyl-4-cyano-p-terphenyl). Malagoli and have calculated the surface-induced quadratic non-linear optical response at 4-nitroaniline surfaces. The general question of how to allow for the modified electronic structure that pertains to molecules in crystals is with the methyl iodide-sulfur system, urea, Langmuir-Blodgett films and surface-induced SHG used as examples. The rigorous local field procedures described in the references in the preceding paragraph are applied by Reis et aZ.252in conjunction with molecular DFT calculations using specially designed basis sets and MP2 ab initio calculations with 6-31ffG** basis sets. They calculate first, second and third order susceptibilities for crystals of naphthalene, anthracene and rn-nitroaniline, using the molecular calculations as input data. They find good agreement for the first order susceptibility in all cases, provided that the calculated molecular a is distributed over all the heavy atoms in the molecule; and for x(2)in mNA provided the same distribution is used and the permanent local field is properly accounted for. The anisotropic Lorentz field factor approximation is in reasonable agreement with the more rigorous calculations for first order susceptibilities, but fails for the second order in mNA due to its inability to represent the effect of the permanent local field. has further investigated the hyperpolarizabilities of molecular crystals in connection with hyper-Rayleigh scattering. Bishop and Gu254have calculated the static polarizability and y-hyperpolarizability of finite T,-symmetry clusters from C€& to C281H172 in which fragments of the diamond structure are capped by hydrogens. Semi-empirical, MNDO, AM1 and PM3 hamiltonians and ab initio methods using STO-3g, 3-21G and 6-31G** basis sets have been used. The results are claimed to be consistent with the little experimental evidence available on diamond. Zhu et aZ.255have applied extended Hiickel theory to calculate the x(2)nonlinear susceptibility in a number of organic crystals. The method enables the effect of interactions between neighbouring cells to be included. Castet and Champagne256have developed a simple multiplicative scheme for the evaluation of the x(’) susceptibility of molecular crystals from the P-hyperpolarizability of small molecular clusters. The scheme is illustrated by an AM1 calculation for the mNA crystal.

x“)

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228. E.A. Perpete and B. Champagne, Theochem-J Mol. Struct., 1999,487, 39. 229. B. Champagne and E.A. Perpete, Int. J Quantum Chem., 1999, 75,441. 230. B. Champagne, J.M. Luis, M. Duran, J.L. Andres and B. Kirtman, J Chem. Phys., 2000,112, 1011. 231. Y. Luo, I? Norman, I? Macak and H. Agren, J Chem. Phys., 1999, 111, 9853. 232. J. Kulakowska and S. Kucharski, Eur. Polym. J , 2000,36, 1805. 233. K. Schmidt and M. Springborg, Phys. Chem. Chem. Phys., 1999, 1, 1743. 234. D.A. Silva, C.P. de Melo, B. Kirtman, T.T. Toto and J. Toto, Synth. Metals, 1999, 102, 1584. 235. V: Chernyak, S. Tretiak and S. Mukamel, Chem. Phys. Lett., 2000, 319, 261. 236. D.M. Bishop, B. Champagne and B. Kirtman, Chem. Phys. Lett., 2000,329, 329. 237. D.M. Bishop, EL. Gu and B. Kirtman, J Chem. Phys., 2001, 114, 7633. 238. U. Gubler, C. Bosshard, P. Gunter, M.Y. Balakina, J. Cornil, J.L. Bredas and F. Diederich, Opt. Lett., 1999, 24, 1599. 239. B.H. Robinson, L.R. Dalton, A.W. Harper, A. Ren, F. Wang, C. Zhang, G. Todorova, M. Lee, R. Aniszfield, S. Gamer, A. Chen, W.H. Steier, S. Houbrecht, A. Persoons, I. Ledoux, J. Zyss and A.K.Y. Jen, Chem. Phys., 1999, 245, 35. 240. W.Z. Liang, S. Yokojima and G.H. Chen, J Chem. Phys., 2000, 113, 1403. 241. R.E. Martin, U. Gubler, J. Cornil, M. Balakina, C. Boudon, C. Bosshard, J.P. Gisselbrecht, F. Diederich, I? Gunter, M. Gross and J.L. Bredas, Chem-Ezq J., 2000, 6, 3622. 242. E.A. Perpete, B. Champagne and D. Jacquemin, Theochem-J Mol. Struct., 2000, 529, 65. 243. Z. An and K.Y. Wong, J Chem. Phys., 2001, 114, 1010. 244. G.P. Zhang and T.F. George, Phys. Rev. B, 2001, 6311, 3 107. 245. G.P. Zhang, Phys. Rev. Lett., 2001, 86, 2086. 246. I.G. Voigt-Martin, G. Li, U. Kolb, H. Kothe, A.V. Yakimanski and C. Gilmore, Phys. Rev. B, 1999, 59, 6722. 247. R.W. Munn, J.F. Kelly and EM. Aicken, Chem. Phys., 1999, 245, 227. 248. M.I.H. Panhuis and R.W. Munn, J Chem. Phys., 2000,112, 6763. 249. M.I.H. Panhuis and R.W. Munn, J Chem. Phys., 2000, 113, 10691. 250. M. Malagoli and R.W. Munn, 3: Chem. Phys., 2000, 112, 6757. 251. R.W. Munn, M. Malagoli and M.I.H. Penhuis, Synth. Metals, 2000, 109, 29. 252. H. Reis, M.G. Papadopoulos, I? Calaminici, K. Jug and A.M. Koster, Chem. Phys., 2000,261,359. 253. R.W. Munn, J Chem. Phys., 2001,114, 5607. 254. D.M. Bishop and EL. Gu, Chem. Phys. Lett., 2000,317, 322. 255. X.L. Zhu, X.Z. You and Y. Zhang, Chem. Phys., 2000, 254,287. 256. F. Castet and B. Champagne, 3: Phys. Chem. A, 2001, 105, 1366.

7 Many-body Perturbation Theory and its Application to the Molecular Electronic Structure Problem BY S . WILSON

1 Introduction

In Volume 1 of this series, I provided a detailed account' of some of the more important aspects of the many-body perturbation theory and its application to the molecular electronic structure problem. I described the significant progress that had been made since my first review of this subject2 which appeared in Volume 4 of the Specialist Periodical Reports series Theoretical Chemistry published in 1981. I described not only developments in the non-relativistic theory, which moved, for example, from the study of the triple excitation component of the correlation energy3d in my 1981 review to the fully diagrammatic analysis of the fifth-order terms for closed-shell systems,' but also the development and practical realization of the fully relativistic many-body perturbation theory,*-'* which did not exist in 1981 and which is now recognized as an essential ingredient of any ab initio treatment of molecules containing heavy elements. I described the systematic development of the underlying approximation which is ubiquitous in molecular studies - the use of finite basis sets - the algebraic appro~imation,'~ which is continuing to drive down the basis set truncation error in molecular electronic structure calculations. I described the development of ccMPBT (concurrent computation Many-Body Perturbation Theory)'420 and of algorithms and computer programs capable of exploiting the power of contemporary computing machines. All practical applications of the methods of molecular quantum mechanics involve approximations. They involve the construction of what Pople and his coworkers2' have termed a theoretical model chemistry. I concluded my last review with a list of properties that one might wish a theoretical model chemistry to have. It read as follows: 1. provide well defined results for the energies of arbitrary electronic states for any arrangement of fixed nuclei (including dissociative processes), leading to a set of continuous potential energy surfaces; Chemical Modelling: Applications and Theory, Volume 2 0 The Royal Society of Chemistry, 2002

330


2. be amenable to systematic refinement; 3. be amenable to efficient parallel computation so that the elapsed time required for calculations does not increase too rapidly with the size of the system; 4. support energy and other expectation values which scale linearly with the number of electrons in the system; 5. provide a rigorous account of relativistic and quantum electrodynamic effects within a unified theoretical framework. No practical theoretical approach to the molecular electronic structure problem possesses all of these attributes, but the many-body perturbation theory provides a unique combination of properties which make it both the most widely used and also the most promising approach to the correlation problem in molecules today. It is accurate provided the reference hnction with respect to which the perturbation series is developed is appropriate since it facilitates the use of the large and flexible basis sets which form an essential ingredient of any high precision application. It is computationally efficient. Algorithms for low order perturbation theory calculations are well suited to a range of modern computer architectures including vector processors and parallel processors. Efficient algorithms facilitate the use of large basis sets and the application to extended systems. It can be systematically refined. Most particularly, relativistic and quantum electrodynamic effects can readily be incorporated by recognizing that Q.E.D. provides a covering theory for MBPT.” This chapter covers developments in the theory and the applications of manybody perturbation theory to the molecular electronic structure problem during the period June 1999 through to May 2001. It thus provides a snapshot of both theoretical developments and application areas at the turn of the century. The emphasis in this review is on applications, of which there are an ever growing number, particularly using finite- and low-order theory. In my last review,’ the increasing use of many-body perturbation theory in molecular electronic structure calculations was measured by interrogating the Institute for Scient$c Information (ISI) database” and the determination of the number of incidences of the string ‘MP2’ in titles and in keywords. This acronym is associated with the simplest form of many-body perturbation theory. It is the Rayleigh- Schrodinger perturbation expansion taken through second order in the energy with respect to the so-called Marller-Plesset reference Hamiltonian. In my previous review, we noted that the abbreviation ‘MP2’ was used in the title or keywords of just three publications in 1989. By 1998, the last year covered by that review, the number had grown to 854. In 1999, the number of occurrences dropped slightly to 821, but it increased again in 2000 to 883. At the time of writing (June 2001) there had been 260 incidences of the term ‘MP2’ in titles and keywords. For this review, we performed an analysis of the journals in which in 883 publications which appeared in 2000 the term ‘MP2’ occurred in the title andor keywords. The Journal of Physical Chemistry A contained the largest number of incidences, with 162, which corresponds to about 18% of the total. This was

7: Many-body Perturbation Theory

33 1

followed by Theochem-Journal of Molecular Structure for which there were 89 hits in the IS1 database or about 10% of the total, Journal of Chemical Physics with 69 or about 8% of the total, Chemical Physics Letters with 62 or 7% of the total, Journal of the American Chemical Society with 41 or 4.6%, Phys. Chem. Chem. Phys. with 34 or 3.9%, International Journal of Quantum Chemistry with 30 or 3.4%, Journal of Organic Chemistry with 26 or 2.9% and Journal of Computational Chemistry with 23 or 2.6% of the total. The wide range of journals in which these publications appear is indicative of the broad spectrum of application areas in which perturbative correlation treatments are being exploited. Furthermore, we can expect there to be many more publications reporting work which exploited many-body perturbation theory methods which are not included in the above analysis simply because, quite rightly, other details of a particular study were considered more important when assigning keywords than the standard theoretical method used to approximate the electronic structure of the targeted system(s). The plan of this report is as follows: in Section 2 an overview of the most important theoretical aspects of the second-order many-body perturbation theory is given. Section 3 contains a review of the numerous applications of the secondorder theory in its ‘MP2’ form which were published during the period June 1999 to May 200 1. The final section then contains a brief summary and a discussion of prospects.

2 Many-body Perturbation Theory through Second Order

The many-body perturbation theory, in both its diagrammatic and algebraic formulations, is described in a number of books23-29and review article^.^^^' It is usually based on the Rayleigh- Schrodinger perturbation expansion and assumes its simplest and most computationally efficient form when taken through second order in the energy for a system described in zero order by a single determinantal reference function. This is the lowest order in which a description of electron correlation effects is achieved. When employed together with the Hartree-Fock model hamiltonian as a zero-order hamiltonian one obtains what has come to be called the Mdler-Plesset expansion.38 In the section, we review the theoretical foundations of the Mdler-Plesset second-order theory. We begin in Section 2.1 with a straightforward derivation of the second-order Rayleigh- Schrodinger perturbation theory whose purpose is mainly pedagogical but which also serves to establish nomenclature. We then, in Section 2.2, turn our attention to the Msller-Plesset perturbation theory and, in particular, the second-order theory, MP2, that was presented in their original 1934 paper by Marller and P l e ~ s e t . In ~ ~ Section 2.3, the partitioning technique, developed extensively by Lo~din,~’~‘ is employed and the MP2 energy expression is obtained together with a remainder term in closed form rather than the infinite-order summation that results from the usual perturbation development. Msller-Plesset perturbation theory results from a specific choice of zero-order hamiltonian in the Rayleigh- Schrodinger perturbation expansion. The choice of

332


zero-order hamiltonian is considered from a more general perspective in Section 2.4. This discussion leads naturally to the scaling of the zero-order hamiltonian, which is discussed in Section 2.5. In Section 2.6, we turn our attention to multireference second-order many-body perturbation theory and then, after discussing the problem of the intruder states, which almost invariably plague the multireference Rayleigh- Schrodinger perturbation theory, we discuss the Brillouin-Wiper perturbation theory through second order, which is seen to hold some promise in alleviating these problems.

2.1 Rayleigh-Schrodinger Perturbation Theory though Second Order. - Let the quantum mechanical eigenvalue problem whose solutions are required be

HY; = €jYj, i = 0, 1, 2, . . .

(1)

where H is a self-adjoint hamiltonian operator. Suppose that the eigenvalue problem corresponding to some model description of the system

HoQj = I?;@;,

i = 0, 1, 2, . . .

(2)

where Ho is some self-adjoint model hamiltonian, has been solved. Equation (2) may be referred to as the zero-order or reference problem. Typically, in studies of atoms and molecules, equation (2) may correspond to the Hartree-Fock model or some other independent electron model. In order to develop a perturbation series for the systematic correction of the solutions of the zero-order problem, put

where

HI is termed the perturbation operator and A the perturbation parameter. As d is increased from zero to unity, equation ( 3 ) interpolates between the zero-order hamiltonian appearing in equation (2) and the perturbed hamiltonian appearing in equation (1). Substituting equation (3) into equation (1) yields (Ho +AH,)Y,(d) = €j(d)Yj(d),

i = 0, 1, 2, . . .

(5)

where the dependence of the eigenvalues and eigenhnctions on the perturbation parameter has been explicitly indicated. It will be assumed that €;(a) and Yj(A) are continuous functions of 1.We note that this assumption may be invalidated if the perturbation operator, HI, introduces new singularities or if it modifies the nature of existing singularities. Clearly E j = €,(O) and Q j = Y,(O). Consider the case of a single perturbation, H1, and a non-degenerate reference

333


function, (Details of the multiple perturbation formalism and of the handling of degenerate reference functions can be found e l ~ e w h e r e . ~ ~ ) Since the eigenvalue, €,(A), in equation ( 5 ) is assumed to be a continuous function of A, a power series expansion can be made

€,(A) = E l +

xAp~lp X

p =I

E , is the p t h order energy, or energy coefficient, in the perturbation series for the exact energy for the i f h state. The energy coefficients are given by

Similarly, for the wavefunction, Yl(A), an expansion is also made

in which xlr,is the p f horder wavefunction

Substituting the expansion for the eigenvalue and the eigenfunction, equations (6) and (8), respectively, into the Schrodinger equation, equation ( 5 ) , gives

Now the coefficients of powers of A in equation (10) can be equated to give in zero, first and second-order

Generally, for p > 1 these equations have the form

Equations (-1 1- 13) and (14) constitute the basic equations of Rayleigh-Schrodinger perturbation theory. From equations ( 1 1- 13) and (14), expressions for the perturbed wavefunctions,


334

xip,can be obtained. Without loss of generality, the perturbed wavefunctions can be taken to be orthogonal to the reference function, that is

(@ilxip)= 0,

p = 1, 2, . . .

(15)

If Ho is self-adjoint then the eigenfunctions of the zero-order problem are orthonormal

The reference function and the exact wavefunction satisfy the intermediate normalization condition

The first-order perturbation equation, (12), may be written

(Ei- &)xi1

= (HI - E i l ) @ i

Now, defining the projection operator onto the model function pi

I@i)(@il

and its orthogonal complement Ql = I

-

PI

and noting that PiIXiP)= 0,

p

=

1, 2, . . .

Qi may be applied to equation (18) from the left to give

Furthermore, defining the resolvant Ri by Ri(Ei - Ho)Qi = Qi

multiplying equation (22) from the left by Ri yields an expression for the firstorder wavefunction

The second-order perturbation equation, (13), may be written

which, when multiplied from the left by R,, yields

or, using equation (24) for the first-order wavefunction

In general, the p t h order wavefunction can be written in terms of lower-order wavefunctions

9=2

From equation (29), it appears that the p t h order wavefunction depends on the perturbed energy coefficients up to and including order p - 1. The energy coefficients in the perturbation expansion (6) are obtained from the perturbation equations (1 1- 13) and (14) by multiplying from the left by Q i and integrating over all the electronic coordinates. Recalling the orthogonality conditions (15) and the orthonormality condition (16) leads immediately to the expressions

or, generally,

The p t horder energy coefficient thus depends on the p - l t horder wavefunction. Substituting the expressions for the perturbed wavefunctions (24), (28) and (29) into the above expressions for the energy coefficients gives, for the second-order energy

Generally, the terms in the perturbation expansion for the energy are given by


336

u- I

q=2

The expressions given above for wavefunctions and energies depend upon the resolvant, Ri, which is defined in equation (23). In equation (23), Ql is the projector onto the orthogonal complement of the model function at.In terms of the eigenfunctions of the model eigenproblem, equation (2), this projection operator may be written

kfi

Hence equation (23) becomes

k#

I

kfi

so that

(37) kfi

kfi

and thus

The resolvant can be expressed in terms of the eigenfunctions and eigenvalues of the zero-order problem. Substituting equation (38) into the expressions given above for the perturbed wavefunctions and energies gives the sum-over-states formulae most often used in practical applications of perturbation theory. The first-order wavefunction in Rayleigh- Schrodinger perturbation theory for a single perturbation is given by equation (24). Using equation (38), this may be written as the sum-over-states expression

Similarly, the second-order wavefunction may be written

337


For the second-order energy coefficient in Rayleigh- Schrodinger perturbation theory, the sum-over-states formula is

This is the leading term in perturbative treatments of the electron correlation energy based on the Rayleigh- Schrodinger expansion.

2.2 Msller-Plesset Perturbation Theory. - Many-body perturbation theory with a Marller-Plesset reference hamiltonian is the most widely used approach to the correlation problem in atomic and molecular systems. Second-order theory, which is often designated 'MP2' and which was the order of theory originally presented by Msller and Ples~et,~* is computationally efficient and facilitates the use of very large basis sets which allows basis set truncation errors to be reduced to a level where other effects, such as relativity, are often more significant.1*53*54 In 1934, Marller and Plesset applied the Rayleigh-Schrodinger perturbation theory taken through second-order in the energy to the electronic structure problem in which the Hartree-Fock model is employed as a zero-order approximation. The Hartree-Fock wavefunction is a single determinant of the form

where the variable x, represents both space and spin coordinates. The single particle state functions or spin-orbitals cp,, i = 1, 2, . . . , N are occupied in the reference determinant. Associated with each of these spin-orbitals is a spin-orbital energy, E ~ . In addition to the occupied spin-orbitals q Z ,we have a set of unoccupied spin-orbitals cpa, a = N 1, N + 2, . . .. Formally, this set is infinite, but for calculations carried out within the algebraic approximation, that is by using a finite basis set, the number of unoccupied spin-orbitals is determined by the size of the basis set. The Hartree-Fock energy is the expectation value of the total electronic hamiltonian evaluated over the determinantal wavefunction (42). This is just the sum of the zero order and first-order energy coefficients. Explicitly, they take the form

+

and

338


where

where r12 is the interelectronic distance and the permutation operator P,2 interchanges the labels 1 and 2. The leading correction term to the Hartree-Fock energy arises in second-order of the Rayleigh- Schrodinger perturbation expansion. The sum-over-states formula (4 1) for the second-order energy coefficient in the Rayleigh- Schrodinger perturbation theory developed with respect to the reference state Qo given by (42) is

The functions Q k ( k # 0) are obtained by replacing one or more of the spinorbitals q i by qoin (42). The energy E0 is given by equation (43). The energy Ek is given by

Explicit evaluation of E~~ for a closed-shell system yields the expression

Here we have used the fact that for a closed shell Hartree-Fock system described by the canonical orbitals only the double replacements (i, j ) -, (a, b) give a nonzero contribution for the matrix elements (a0J HI I @ k ) . Equation (48) is the basic equation of 'MP2' theory. If we define

then (48) can be written in the more compact form

Each spin-orbital

(pP

can be written as a product of a (spatial) orbital,

GP, and a


339

one-electron spin function, a p so that the spin-orbital index p can be replaced by the combined indices P a p . Performing the spin integration explicitly, we have

S )second-order , energy then takes the form Putting GpeRs= ( P Q [ ~ / Y , ~ I Rthe

For real orbitals this expression can be simplified further to give

It is the form ( 5 5 ) that is employed in most practical calculations. The seminal paper by Mdler and Plesset entitled a Note on an Approximation Treatment for Many-Electron Systems appeared in Physical Review in October, 1934. The title and abstract are reproduced in Figure 1. In his book Quantum

OCTOBER 1. 1933

PHYSICAL REVIEW

VOLVME 46

Note on an Approximation Treatment for Many-Electron Systems CHR. M ~ L L EARN D M .S. PLESSET.* Institut for teoretisk Fysik, Copenhagen (Received July 14, 1934)

A perturbation theory is developed for treating systems of n electrons in which the HartreeFock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy arid the charge density of the system is zero. The expression for the second-order correction for the energy greatly simplifies because of the special pr0pert.v of the zero-order solution. It IS pointed out that the development of the higher approximation involves only calculations based on a definite one-body problem.

Figure 1

Title and abstract of’ the seminal paper of Mdler and Plesset published in Physical Review in 1934

340


Chemistry. The development of ab initio methods in molecular electronic structure theory,s5Schaefer writes: This ‘note’ is in fact jive pages long and is one of the key papers in the development of molecular quantum mechanics. Schaefer describes how The development of eficient [computational] methods . . . led to a resurgence of interest in techniques based on the ideas of this classic paper. For many years configuration interaction was regarded as the method of choice in describing electron correlation effects in atoms and molecules. The method is robust and systematic being firmly based on the Rayleigh-Ritz variational principle. The total electronic wavefunction, Y, is written as a linear combination of N-electron determinantal functions, Q i , Y =CQiCi. I

The expansion coefficients, C,, are determined by invoking the variation principle which leads to a secular problem of the form HC = €C

(57)

where H is the hamiltonian matrix

and C is a vector of expansion coefficients, C,. Now the non-relativistic hamiltonian is a semi bounded, self-adjoint operator in Hilbert space. The algebraic approximation results in the restriction of the domain of this operator to a finite-dimensional subspace S of Hilbert space. For an N electron system, the algebraic approximation may be implemented by defining a suitable orthonormal basis set of M ( > N ) one-electron spin-orbitals (usually solutions of the matrix Hartree-Fock equations) and then constructing all unique N-electron determinants. The number of unique determinants that can be formed is

v=(:)

(59)

and q is the dimension of the subspace S. The algebraic approximation restricts the domain of the hamiltonian to this q-dimensional subspace. In practice, difficulties arise in setting up and solving secular equations of high order. Only a small subset of the determinantal functions Q i can be used in the expansion. The

7: Many-body Perturhution Theory

34 1

expansion (56) may be limited to all configurations that can be generated by making double replacements with respect to some chosen single determinantal reference function. (A perturbative analysis of the configuration interaction problem reveals that such configurations will be the most important.) The problem with the limited (double excitation) configuration interaction expansion can most easily be seen by considering an array of n well-separated He atoms each of which is in its ground state. By ‘well-separated’ we mean non-interacting. Let us consider the application of limited (double excitation) configuration interaction to the calculation of the correlation energy for this system. For a single helium atom the double excitation configuration interaction method provides an exact solution of the problem within the chosen basis set. For the case n = 2 the method can describe the correlation of the electrons associated with one of the atoms but not correlation effects in both atoms simultaneously. The description of these effects requires the addition of quadruply excited configurations, but these are excluded from our double-excitation configuration interaction model. The problem is compounded when we consider the case n = 6 where sextuply excited configurations are required to recover the ‘exact’ solution. We see that the limited configuration interaction method becomes more and more problematic as the number of electrons in the system increases even for this simple model system in which electrons are only correlated in well-separated pairs. Now if we look at the perturbation theory description of this system using wavefunctions which are localized on the individual atoms we find that the second-order correlation energy for two helium atoms is just twice that of a single helium atom and in general

where & ( n ) is the second-order energy for an array of n well separated helium atoms. The proof that in higher orders of the perturbation expansion

is not straightforward and rests on the cancellation of certain ‘unphysical’ terms in each order p. For example, in third-order the Rayleigh- Schrodinger perturbation energy component has the form

which can be written as

342


The matrix elements arising in these summations can be evaluated by using the rules dues to Slater. If Q K r is a determinant differing from Q K in one spin-orbital (q:# q,) and QKrt is a determinant with two spin-orbital differences (ql # c p r , q: # q s ) then

where we have assumed that the total hamiltonian H can be written as a sum of one-electron and two-electron components h and g, respectively. Certain diagonal terms in the first term on the right-hand-side of equation (63) are exactly cancelled by the second term. The surviving terms scale linearly with the number of electrons and &(n) = nE3(l). In third order of the Rayleigh-Schrodinger expansion the only intermediate states which arise are doubly excited with respect to the reference function Q0 and so the cancellation occurs if we carry out a limited double excitation configuration interaction calculation. In fourth-order there is a cancellation between terms involving doubly excited intermediate states and terms involving quadruply excited states. In limited double excitation configuration interaction therefore the cancellation is incomplete. This incomplete cancellation of ‘unphysical’ non-linear terms explains the difficulties encountered in applying limited configuration interaction to the array of well-separated helium atoms. It was Brueckne?6 who first recognized that the Rayleigh- Schrodinger expansion, when suitably modified, could provide a valid many-body theory. Brueckner demonstrated the cancellation of ‘unphysical’ terms in low orders. The seminal paper by Brueckner entitled Many-Body Problem for Strongly Interacting Particles. II. Linked Cluster Expansion appeared in Physical Review in October, 1955. The title and abstract are reproduced in Figure 2. It is well known that a general proof was then obtained by G~ldstone,~’ H ~ b b a r dand ~ ~ H~genholtz.~~ Goldstone used the methods of Tomonaga,60s6’S ~ h w i n g e r ~and ~ , ~Feynmarf4 ~ together with the results of Gell-Mann and Low66and and introduced the use of diagrams both as a representation of the physical processes described by a particular algebraic expression and as a very convenient method of ‘bookkeeping’. The many-body perturbation theory was applied to atoms and some simple molecular systems by K e l l ~ ~ using ~ - ’ ~ a reference function obtained from finite difference solutions of the Hartree-Fock equations. The introduction of the algebraic approximation using finite basis sets during the 1970s’3>75-77 opened up applications to arbitrary polyatomic molecular systems. The second-order many-body perturbation theory Goldstone energy diagrams are shown in Figure 3. The first of these is the direct term and the second the

7: Many-body Perturbation Theory PHYSICAL REVIEW

VOLUhlE 100. NIThiBER 1

343 OCTOBER 1, 1955

Many-Body Problem for Strongly Interacting Particles. 11. Linked Cluster Expansion K.A. RRITECKNER Indaana Unaversaty, Bloornangton, Indaana (Received April 28, 1955) An approximate method developed previously t,o deal with many particles in strong interaction is examined in further detail. It is shown that the series giving the interaction energy is a devlopment in a sequence of linked or irreducible cluster terms each of which gives a contribution to the energy proportional t o the total number of particles. Consequently the convergence of the expansion is independent of the total number of particles. The origin of this simple feature is illustrated by showing that, a similar situation exists in the expansion of standard perturbation theory. The numerical convergence of the expansion is quantitatively discussed for the nuclear problem wherc it is shown that the correction arising from the first cluster term involving three particles is less thali the leading term by a factor of about lo-*. The smallness of the correction is largely a resiilt of the action of the exclusion principle.

Figure 2

Title and abstract Review in I955

of

the seminal paper of Brueckner published in Physical

( a ) ‘Direct’second-order Goldstone diagram

(h) ‘Exchange’second-order Goldstone diagram Figure 3

Second-order correlation energy diagrams for a closed-shell system described in zero-order by a single determinantal Hartree-Fock wavefunction constructed from canonical orbitals. The ‘direct’ second-order Goldstone diagram and only Brandow diagram in second-order is shown in (a). The ‘exchange’ Goldstone diagram is shown in (b)

344


corresponding exchange term. The diagrams are interpreted in the particle-hole picture in which excitation of a electron from an occupied single particle state simultaneously creates a hole below the Fermi level and a particle above it. The lower electron-electron interaction in each of the diagrams shown in Figure 3 creates two particle-hole pairs. The upper electron-electron interaction in each diagram sees the destruction of both of these particle-hole pairs. In the process represented by the direct diagram each particle-hole pair is first created and then destroyed, whereas in the process represented by the second diagram exchange occurs between the creation and destruction processes. In one of the alternative diagrammatic conventions due to bran do^,^^ exchange is incorporated in the electron-electron interaction. In terms of Brandow diagrams the second-order energy is represented by just the first diagram shown in Figure 3. Using the rules given in my previous review’ an algebraic expression correponding to the first diagram shown in Figure 3 can be written down. This expression is identical to that arising in Marller-Plesset perturbation theory, i.e. expression (1 56). 2.3 Partitioning and the Remainder Term. - The use of second-order MarllerPlesset perturbation theory in describing electron correlation in atomic and molecular systems has recently been critically re-examined by Wilson and H ~ b a ; . Some ~ ~ recently published have examined the higher-order terms in the Msller-Plesset perturbation expansion and the convergence behaviour of the perturbation series. From these investigations conclusions about the applicability of the low order theory have been drawn which, in view of the widespread use of MP2, demand more thorough analysis. By combining the Rayleigh-Schrodinger and the Brillouin- Wigner perturbation expansions Wilson and HubaE avoid an infinite-order expansion and obtain a closed expression which consists of the MP2 energy components together with a remainder term. The applicability of MP2 theory then rests upon the magnitude of this remainder terms rather than the behaviour of the higher-order terms on the perturbation series. The most recent of the publications to question the applicability of the low order theory studies is authored by Schaefer and his co-workersg7 who ask the question ‘Is Mdler-Plesset perturbation theory a convergent ab initio method? ’ These authors write

The tacit assumption behind chemical applications of MPn theory is that the perturbation series is convergent . . . They quote the conclusion of earlier work by Dunning and his co-workersg3

The current study, combined with other recent studies, raises serious doubts about the use of Mdler-Plesset perturbation theory to describe electron correlation effects in atomic and molecular calculations. The work of Schaefer et al.,87along with earlier work by Dunning et al.,83-86by

345


Olsen et d8cr82 and others rests on the assumption that the utility of lower-order Msller-Plesset perturbation theory can be inferred from the behaviour of the higher-order terms in the perturbation series. It is widely appreciated that MsllerPlesset perturbation theory and the equivalent many-body perturbation theory for a single determinantal reference function are not as robust as, for example, configuration interaction. Some care must therefore be exercised in applications to ensure that an appropriate reference function is employed. However, in a recent paper Wilson and H ~ b a t ’ employ ~ the partitioning technique due to L o ~ d i n ~to ~ -develop ~’ an expansion for the correlation energy which is a hybrid of the Rayleigh-Schrodinger perturbation theory and the (generalized) Brillouin- W i p e r expansion. A closed expression is obtained which consists of the ‘MP2’ energy together with a remainder term. Let the time-independent Schrodinger equation of which we aim to approximate the solutions be written

with

and the usual normalization condition

We wish to develop approximations to the eigenvalues of (65) with respect to the solutions of a model eigenproblem

with

(@J@,)

= 6,,

Let us define the projector on to the ground-state model eigenfunction as

p = I@o)(@ol and its orthogonal complement

Q=I-P which can be written in sum-over-states form as

346


If we assume the intermediate normalization condition (@ol~o)= 1

(73)

PIYO)

(74)

then I@o) =

Using (65) and (71) we can follow the partitioning approach pioneered by LGwdin39-51 and write

H ( P + Q)lYo) = EIWO)

(75)

and then, operating from the left by P and independently by Q, we obtain

PHPIYO)

+ PHQlYo) = EPJYO)

QHpIyo)

+ QHQIVo) = EQIYo)

from which QlY)can be eliminated to give

[PHP

+ PHQ(€

-

QHQ)-'QHP]/Yo) = €PIYO)

and then using (74) we have [PHP

+ PHQ(E

-

QHQ)-'QHP]1@0) = €(@o)

which can be written as an effective eigenproblem Heff(@o)= El@o)

where the effective hamiltonian has the form He, = PHP

+ PHQ(E - QHQ)-'QHP.

He, depends on the exact energy eigenvalue E. The exact energy, E, is then given by

Various types of perturbation theory may be obtained by using the operator identity

to expand the inverse in the effective hamiltonian (80). The Rayleigh-Schrodinger perturbation theory, which underpins the Mdler-Plesset theory, is obtained by putting

347


A = Eo

-

QHoQ

(83)

and BzQHlQ-AEo

where the level shift, A E o , is defined by & = Eo

+ AEo

and is usually expanded n

AEO =

C &?'Ak k=l

where E:) is the kChorder energy coefficient. This is, of course, an infinite expansion. Many studies of the utility of perturbation theory have concentrated on the behaviour of the higher-order terms in this expansion. Now the expansion (82) can be put in the alternative form ( A - B)-' = A-'

+ A - - 'B ( A

-

B)-'

(87)

For a given A and B iteration of (87) leads directly to (82). Equation (87) can, in

turn,be written in the more general form

where, in general, A # A' and B # B' but

and in applications to (78) A

-

B = A'

-

B'

I - QHQ

1

The choice

and

whilst putting leads to a Brillouin -Wigner perturbation e~pansion*~-~' A'=€

(93)

348

Chemical Modelling: Applications and l%eory, Volume 2

and

B' = QHQ

(94)

yields the generazized Brillouin- Wigner expansion, which was introduced by - ~ re' Lowdin in Parts I1 and XI1 of his studies of perturbation t h e ~ r y ~ ~and examined recently by HubaC and Wilson.87a Using (88) together with the Rayleigh-Schrodinger choice for A and B, that is (83) and (84), and the generalized Brillouin-Wigner choice for A' and B', that is (93) and (94), gives

Combining (80) and (81) gives

which upon substituting (95) gives

where the first term on the right-hand-side is the sum of the zero-order and firstorder energies:

In Msller-Plesset theory this sum is just the Hartree-Fock energy. The second term on the right-hand-side of equation (97) is the second-order Msller-Plesset energy, fiequently designated MP2,

Unlike the usual perturbation expansions, (97) is not an infinite order series, but a closed expression containing a remainder term which we designate 3R and which has the form

7: Man-y-body Perturbation Theory

349

or

The approximation to the total energy of an atomic or molecular system given by MP2 theory can be written in the form

The utility of the method rests on the requirement that the remainder term satisfies

for some arbitrarily chosen small z. Specifically, the use of the MP2 method does not depend on the behaviour of the individual higher order terms in the MarllerPlesset expansion. Let us remark that estimates of 3R can be obtained from other approaches to the correlation problem in atoms and molecules, such as, for example, limited configuration interaction or coupled cluster expansion^.^'^ By employing a hybrid partitioning scheme based on both the RayleighSchrodinger and the generalized Brillouin- Wigner approaches, it has been shown that the electron correlation energy expression arising in the MP2 theory can be written as part of a closed expression. The utility of the MP2 theory therefore rests not on the behaviour of the individual terms in the perturbation expansion but on the magnitude of the remainder term. We conclude this section by recalling Dirac’s observation on the applicability of perturbation theory in Chapter VII of his monograph on The Principles of Quantum Mechanicsg8 Even when the series does not converge, theJirst approximation obtained by means of it is usually fairly accurate.

2.4 The Choice of Zero-order Hamiltonian. - The utility of low-order perturbation theory is determined not only by the choice of reference function but also by the choice of zero-order hamiltonian. In the Marller-Plesset perturbation theory the N-electron Hartree-Fock hamiltonian, Ho, is used as a zero-order operator. It should be clear that any operator, A , obeying the commutation relation

can be used to develop the sum-over-states perturbation series in terms of the eigenfbnctions of H,, mk.The condition (104) implies that

350


where X is an arbitrary operator. The operator

represents a special case in which the operator X is taken to be the total electronic hamiltonian. This choice leads to the shifted denominator perturbation series which was first considered by E p ~ t e i nand ~ ~ N e ~ b e t .It~ gives ~ rise to a second-order energy expression of the form

where sijab is a denominator shift and has the form

The use of shifted denominators may be interpreted as the summation of certain diagonal terms in the higher-order Mnrller-Plesset expansion though infinite order. The choice of zero-order hamiltonian is, to some extent, arbitrary. By using a zero-order hamiltonian of the form93a

one can interpolate between the Marller-Plesset perturbation expansion and the shifted or Estein-Nesbet expansion by varying the parameter p. For example, in applications to the ground states of the Be and Ne atoms it was found93athat the shifted expansion was favoured for the former but the Marller-Plesset expansion for the latter. In general, any linear combination of operators each of which satisfies the commutation relation (104) can be employed as the zero-order hamiltonian and so

represents the most general choice.

2.5 Scaling of the Zero-order Hamiltonian. - A particularly simple modification of the zero-order hamiltonian is obtained by scaling. Let us recall that if, following Feenberg,94we modify the zero-order hamiltonian operator by multiplying it by an arbitrary scalar, p, say, the zero-order hamiltonian is then


and the perturbation is

so that the full hamiltonian is recovered when these two modified operators are added.95-99It can be easily demonstrated that the second-order energy is then given by 1

- E2

P

where E2 is the second-order energy corresponding to the unmodified zero-order hamiltonian. Now, if we know the exact correlation energy, E~ZZlarion, we can always put

so that the second-order energy given by the modified zero-order hamiltonian is equal to the exact correlation energy. Of course, in general E~Zu~lation is unknown. However, this simple modification shows that there is some choice of zero-order hamiltonian which yields the exact correlation in second order. In such a case the remainder term is zero, a. Values of p can be obtained from perturbation theory calculations taken to higher-order. The second-order energy provides a useful approximation to the total correlation energy

then the remainder term

Bz= E3+

E4

+ ... + E, + ...

can be assumed to be small. The energy coefficients in the scaled perturbation expansion, I$, Ey, E;, . . . , E;, . . . may be written in terms of the original (unscaled) energy coefficients as follows:

352


Explicitly, the first few orders take the form

Now, the total correlation energy may be written in the form

where p is chosen so that the magnitude of remainder term is reduced

In the past, a number of prescriptions for the determination of ,M have beep investigated. The most popular is to set the modified third-order energy coefficient to zero

Er

=0

(126)

an equation which can be solved for p to give

which, in turn, gives a modified second-order energy coefficient that is entirely equivalent to the [ 2 / 1 ] Pade approximant to the original perturbation expansion. It does not seem to be widely recognized that the setting of modified energy coefficients to zero in fourth and higher order does not lead to a unique value of p. Thus, putting the modified fourth-order coefficient to zero

gives two values of ,u


353

The alternative approach of putting the sum of the third- and fourth-order terms in the modified series to zero

also leads to two solutions 1 4E2

= -(-3E3

+ 3E2 f

(9E: - 2E3E2

+ Ei

-

8E2E4))

(131)

Putting higher-order coefficients, or combinations of such coefficients, to zero always leads to a multivalued solution. A procedure for the determination of a unique value of p from higher-order expansions was given by the present The modified second-order energy coefficient may be set equal to the sum of the coefficients through third order

that is =

E2 + E3

(133)

which can be solved for p to give

In fourth order, the modified second-order energy coefficient may be set equal to the sum of the second-, third- and fourth-order terms in the original series

This equation has a unique solution

In general, the modified second-order energy coefficient may be set equal to some higher-order approximation to the total correlation energy

Er

=&

This provides a unique prescription for the determination of p

(137)

354


I may, for example, correspond to the correlation energy estimate obtained from a full configuration interaction calculation or a cluster expansion. Scaling of the second-order correlation energy has been used to estimate the effects of basis set extension.’* Consider a calculation performed using a basis set designated SA. The relation between the modified second-order correlation component and the higher-order approximation to the total correlation energy then takes the form

giving

Now, consider a second calculation performed by using an extended basis set S B . For this basis set, the modified second-order energy coefficient and the higherorder approximation are related as follows:

The higher-order approximation can be written

If we make the assumption that p [ S B ]may be replaced by p[SA] then we obtain the estimate (143)

2.6 Multireference Seco d-order Many-body Perturbatio Theory, Intruder States and Brillouin-Wigner Perturbation Theory through Second Order. Multireference second-order Rayleigh-Schrodinger perturbation theory might be thought to provide the theoretical apparatus for the description of the electronic structure of molecular systems for which a single determinantal functions does not form an appropriate zero-order approximation. Multireference hnctions are required, for example, in the description of many dissociative molecular processes. In practice, the multireference Rayleigh- Schrodinger perturbation theory is not a robust technique. It cannot, therefore, be regarded as a general purpose ab initio electronic structure method. The problems arising in the multireference theory can be traced to the occurence of ‘intruder states’.


355

Eigenstates may be reordered when the perturbation parameter il is varied. In particular, eigenstates with energies lying above those in the reference space at il= 0 may have energies below the higher levels in the reference space for some [ill > 0. For 1 3 A > 0 these states are termed ‘intruder states’. For -1 d il< 0 these states are termed ‘back door intruder states’. Brillouin- Wigner perturbation t h e ~ r y ~ ~ is ~ ~seen ’ ~ as ’ ~a~solution ~ ’ ~ ’ to the demand for a robust multireference formalism. In this section we briefly survey the basic formalism and provide the necessary background. We wish to approximate the solutions of the time-independent Schrodinger equation

with

where 3-to is the zero-order hamiltonian and ‘FI1 is the perturbation. We assume that the solutions of the zero-order eigenproblem

7 - l o q = E,@,,

p = 0, 1,2, . . . .

(146)

are known. Let

{ a Pp;= 0, 1, 2, * . . p

-

1)

(147)

be a set of linearly independent functions which constitute the reference space, which we label P. Let P be the projection operator onto this reference space

and let Q be its orthogonal complement

so that the projectors P and Q satisfy the relations

356


P=P

e'=Q PQ=O

P+Q=I We can write the subspace S as

Let us consider the projection of the exact wavefunction, Y a ,onto the reference space P , i.e.

Y?,p=PY,,

a = 0 , 1 , 2 ,..., p-1

(152)

YL is sometimes called the modeZ function. Obviously, Yap can be written as a linear combination of the set (147)

The Y: are, in general, non-orthogonal but are assumed to be linearly independent. The exact wavefunction, Y u ,is expanded as follows

where 0,is the Brillouin-Wigner type propagator

which depends on the unknown exact energy eigenvalue E,. The exact wavefunction, Y u ,and the model fwnction, Y,: intermediate normalization conditions

satis@ the following

The wave operator, Q a , is defined by

wa= QaYL

(157)

357


so that application of the wave operator to the model function yields the exact wavefunction. Comparing (157) with (154), we have

or, re-writing (158) as a recursion,

Ba = 1 +B,'FI,B,,

(159)

which may be seen to be the Bloch equationlo2in Brillouin-Wigner form. We now introduce the 'effective' hamiltonian which acts in the reference space

The effective hamiltonian, fi,, operates only in the reference space P but has the exact energy, € a , as an eigenvalue, i.e.

Using (145) we can write the effective hamiltonian operator in the form

which can then be re-written in the form

a = 0 , 1 , 2, . . . , p - 1

Fl,=rnoP+PV,P,

(163)

where we have introduced the reaction operator, V,, which is defined as follows

Combining (159) and (164) gives a Lippmann-Schwinger-like equationlo3 in Brillouin- Wigner form

v, = X,+ X,B,V, It should be emphasized that V, is a state spec@ reaction operator corresponding to only one eigenenergy I,. Note that the wave operator arising in the Rayleigh-Schrodinger formalism, Q can be related to the wave operators B, in the Brillouin-Wigner method through the relation D-

1

B = CS2,Pa a=O


358

The exact energies, I,, are eigenvalues of different effective hamiltonian operators

i i , ~ , =~E,Y:, :

a

= 0,

1,2, ...,p

-

1

Let us consider a p-state system and obtain an explicit formulation of the multireference Brillouin- Wigner perturbation theory for this case. In the p-state case, we have a reference space spanned by p hnctions, Q0, O I , . . . , @ p - l . The projector onto this space is

fl=O

The corresponding wave operators are

and the reaction operators are

The effective Schrodinger equation has the form

=I~~\IIU~,

a

= 0, 1 , .

The effective hamiltonian operator takes the form

so that

..,p

-

I


359

For the state a = 0, we have a model function given by

The secular equation for the lowest state takes the form

where the matrix elements of the effective hamiltonian are

and

with p, Y = 0, 1, . . . , p - 1. Since 7-11is a two-particle operator, the configuration I@/) is at most a double replacement with respect to lap)in (176). The matrix elements of the reaction operator are obtained fiom the Lippmann-Schwingerlike equation, equation (165). Specifically, we have

Equation (175) has p roots of which we take only one. The exact energy, Eo, occurs in the denominator factors in equations (176) and (178) and the equation (175) must, therefore, be solved iteratively until self-consistency is achieved. If

360


we restrict the order of perturbation admitted in (178) then we realize a finite order multireference Brillouin- Wigner perturbation theory. A posteriori corrections to second order multireference Brillouin- Wigner perturbation theory based on the identity'm~'O1

where AEo = lo- Eo, can be applied to the matrix elements in (175). Rearranging (179) we have

The first term on the left hand side of (1SO) is a Brillouin-Wigner denominator and the term on the right hand side is a Rayleigh-Schrodinger denominator. The second term on the left hand side can be regarded as a correction term. For the state a = 0 the corrected matrix elements for the p-state case may be written

The matrix elements of the reaction operator Uo are determined recursively using the above equations. Recursion is continued until convergence to some tolerance is achieved for each matrix element. Only in the final iteration are the correction terms added to the matrix elements. The ground-state secular equation is then solved to obtain an estimate of the exact ground-state energy, lo, which can then be used to repeatedly construct and solve the secular equation until selfconsistency is achieved.

3 Some Applications of Second-order Many-body Perturbation Theory with a Msller-Plesset Reference Hamiltonian Marller-Plesset many-body perturbation theory taken through second order in the energy is the most commonly used ab initio molecular electronic structure method in contemporary quantum chemistry. For this report on many-body perturbation theory and its application to the molecular electronic structure problem we restricted our survey of applications to second-order Marller-Plesset perturbation theory. Even with this restriction, the number of publications appearing in the period covered by our review - namely, June 1999 to May 2001 - is sizeable. We recorded in the introduction that 883 publications containing the string ' M . 2 ' in the title or keywords appeared in the year 2000 alone. However, rather than review just a small subset of these publications we decided to try to convey the


36 1

extent and breadth of application areas of the 'MP2' method. We divided our literature search into two main parts. Firstly, we searched for publications with the string 'MP2' in their title in the period June 1999 to May 2001. This search, of course, will emphasize publications in which the 'MP2' method is an important aspect of the particular study and will overlook the majority of publications relating to the 'MP2' method in which it is employed as a routine tool for the study of the molecular electronic structure problem. Secondly, we searched for publications with the string 'MP2' in their title and/or list of keywords for the year 2000 in four primary research journals.

3.1 Publications with the String 'MP2' in Their Title. - Interrogation of the Institute for Scientijic Information (ISI) database to determine the number of incidences of the string 'MP2' in titles during the period June 1999 to May 2001, resulted in a list of 46 publications.1w149The list of titles of these publications, given in chronological order beginning with the most recent, serves to convey some idea of the variety of applications being reported. 1. Organic thermochemistry at high ab initio levels. 1. A G2(MP2) and G2 study of cyclic saturated and unsaturated hydrocarbons (includmg aromatics)'" 2. Binding energies of nitrile-containing proton-bound clusters: the performance of HF, MP2 and B3-LYP vs. G21°5 3. G2(MP2) molecular orbital study of [H3AlXH3]-(X = C, Si, and Ge) and H3A1YH3 (Y = N, P, and As) complexes'o6 4. Various local minimum structures of an aniline complex with carbon monoxide; DFT and MP2 calc~lations'~~ 5 . Low-order scaling local electron correlation methods. I. Linear scaling local MP2'" 6 . Determination of the bond length alternation of polyacetylene, polydiacetylene and polybutatriene from MP2 oligomeric investigationsIw 7. A theoretical investigation of the torsional potential in 3,3'-dimethyl-2,2 '-bithiophene and 3,4'-dimethyl-2,2'-bithiophene:A comparison between HF, MP2, and DFT theory"' 8. A comparative study of MP2, B3LYP, RHF and SCC-DFTB force fields in predicting the vibrational spectra of N-acetyl-L-alanine-"-methyl amide: VA and VCD spectral I I 9. Adenine-hydrogen peroxide system: DFT and MP2 investigation]I' 10. Effects of basis set and electron correlation on the calculated interaction energies of hydrogen bonding complexes: MP2/cc-pVSZ calculations of H20-MeOH, H20Me20, H20-H2C0,MeOH-MeOH, and HCOOH-HCOOH complexes''3 11. Molecular structure and vibrational Raman and infrared spectra of bromochlorofluoromethane and its silicon and germanium analogs: quantum-mechanical DFT and MP2 12. Protonated and methylated dimethyl sulfoxide cations and dications. DFT/GAOMP2 MMR studies and comparison with experimental data"' 13. Ab initio molecular dynamics studies of the photodissociation of formaldehyde, H2CO-+H2+CO:Direct classical trajectory calculations by Mp2 and density functional theory'I6 14. G3(MP2) calculations of enthalpies of hydrogenation, isomerization, and formation

362

15. 16. 17. 18. 19.

20.

21.

22. 23. 24. 25. 26.

27.

28. 29. 30.

31. 32. 33.


of bi- and tricyclic C8 and Clo hydrocarbons. The bicyclo[3.3.0]octenes and triquinacenesl l7 MP2, Tamm-Dancoff, and RPA methods based on the generalized HF solution"' Implementation and refinement of the modified-conductorlike screening quantum mechanical solvation model at the MP2 level"' A comparison of the B3LYP and MP2 methods in the calculation of phosphate complexes'20 Conformational properties of dimethylaminobenzonitrile in gas phase and polar solvents: ab initio HF/6-3 1G(d,p) and MP2/6-3 lG(d,p) investigationsI2' Heats of formation of small bicyclic hydrocarbons, spiropentadiene (C5H4), spiropentane (C5H8) and bicyclo[ 1.1.O]but-l(3)-ene (C4H4): a theoretical study by the G2M(RCC,MP2) method'22 Complete basis set, the MP2 ab initio, and hybrid density functional theory evaluation of ionization potential and electron affinity for PH, PH2, PHF, PF, and PF2123 Basis set effect on hydrogen bond stabilization energy estimation of the WatsonCrick type nucleic acid base pairs using medium-size basis sets: single point MP2 evaluations at the HF optimized Scalability of correlated electronic structure calculations on parallel computers: A case study of the RI-MP2 method12' Theoretical study of the thermal decomposition of N,N'-diacyl-N,N'-dialkoxyhydrazines: A comparison of HF, MP2, and DFT'26 G2(MP2) molecular orbital study of the substituent effect in the H3BPH3-,Fn (n=0-3) donor-acceptor complexe~'~' DFT and MP2 molecular orbital determination of OH-toluene-0-2 isomeric structures in the atmospheric oxidation of tolueneI2* Peptide models XXIII. Conformational model for polar side-chain containing amino acid residues: A comprehensive analysis of RHF, DFT, and MP2 properties of HCOL-SER-NH2I2' Conformations of silicon-containing rings. I1 a conformational study on silacyclohexane. Comparison of ab initio (HF, MP2), DFT, and molecular mechanics calculations. Conformational energy surface of silacycl~hexane'~~ A comparative study of the DFT and MP2 methods on molecular structure of diphosphadithiatetrazocineI I Molecular structure and IR spectra of bromomethanes by DFT and post-HartreeFock MP2 and CCSD(T) calculation^'^^ Effects of the higher electron correlation correction on the calculated intermolecular interaction energies of benzene and naphthalene dimers: comparison between MP2 and CCSD(T) calculation^'^^ Density functional theory study of the hydrogen-bonded pyridine-H20 complex: A comparison with RHF and MP2 methods and with experimental data134 HF and MP2 calculations on the transition states of S-"0- c y c l i z a t i ~ n s ' ~ ~ Metal-metal closed-shell interaction in M2X2 (M = Ag, Cu; X = C1, Br, I) and related compounds [Ag2Br2](PH3)3and [ C U ~ C ~ ~ ] ( Pan H ~RHF, ) ~ : MP2 and DFT

34. Calorimetric, computational (G2(MP2) and G3) and conceptual study of the energetics of the isomeric 1,3-and 1,4-dithiane~l~~ 35. Density functional and MP2 studies of germylene insertion into C-H, Si-H, N-H, PH, O-H, S-H, F-H, and C1-H bondsi3' 36. An MP2 study on pre-reactive complexes (CH2)20 XY (X, Y = H, F, C1, Br, I)139


363

37. Nh2’ and N4’+ dications and their N-12 and N-10 azido derivatives: DFT/GIAOMP2 theoretical studied4’ 38. Molecular structure of 2-butanimine, an unstable imine, as studied by gas electron diffraction combined with MP2 and DFT calculation^'^^ 39. MP2 correlation effects upon the electronic and vibrational properties of p ~ l y y n e ’ ~ ~ 40. The thermochemistry of TNAZ ( 1,3,3-trinitroazetidine) and related species: G3(MP2)//B3LW heats of formation’43 41. Thermochemistry of cyclopentadienylidene (c-C5H4, C,,), cyclopentadienyl radical C2”)and 1,3-cyclopentadiene (c-C5H6, C2u):a theoretical study by the (c-C, H5., G2M(RCC,MP2) method’* 42. Hyperfine structure of some hydrocarbon radical cations: a B3LYP and MP2 study’45 43. Proton affinities of fluoro derivatives of benzene, toluene, and rn-xylene from ab initio MP2 calculation^'^^ 44. Energetics and structure of glycine and alanine based model peptides: Approximate SCC-DFTB, AM1 and PM3 methods in comparison with DFT, HF and MP2

45. G3(MP2) calculation of the enthalpies of formation, isomerization and hydrogenation of cubane and cyclo~ctatetraenel~~ 46. Structures of carborane cations derived from the reaction of 2-propyl cation with diborane: DFT/IGLO/GIAO-MP2/NMR investigations of alkylborane and alkylcarborane patterns’49

These 46 publications include studies of the structure and bonding of molecules, themochemistry, ioniation potentials and electron affinities, vibrational spectroscopy, internal rotation, hydrogen bonding in both small molecules and in biomolecules, intermolecular interactions, reactive intermediates, polymers and solvation effects. Over 30% of these publications - 14 out of the 46 - present the results of applications of both ‘MP2’ and some form of density functional theory.I05,l07,110,112,1 I~ll6,120,l23,l26,l28-l32,l34,l36,l38,l40,l4l,l43,145,l47, I49 The comparison of the most widely used ab initio method with density h c t i o n a l theory, the most promising of semi-empirical methods, can be seen as one of the main thrusts of research activity during the period covered by this report. This interest is undoubtedly driven by the need for more accurate methods for handling larger molecular systems. On the one hand, density functional theory can be applied to larger molecular systems than can a6 initio methods but on the other hand only ab initio methods, o f which ‘MP2’ is the most efficient computationally, can be systematically refined. Three publications concerned with the application of ‘MP2’ methodology to larger systems appeared during the period covered by this report. The work of Schutz, Hetzer and WernerIo8begins a series of papers exploring the development of local electron correlation methods with low-order scaling by presenting a linear scaling local ‘MP2’. They describe a novel multipole approximation based on a splitting of the Coulomb operator into two terms

364


Putting

f(r ) = erf(or),

(183)

where w is a tunable decay parameter, they cast the Coulomb operator as a sum of a rapidly decaying, but singular, short-range part S(r) =

erfc(o r ) r

~

and a slowly decaying, but non-singular, long range part L(r) =

erf(o r ) ~

r

Electron repulsion integrals over molecular orbitals can then be separated into two integrals one of which is obtained by a transformation of the short-range integrals over the chosen basis set. This transformation can be carried out very efficiently by employing pre-screening methods. A multipole expansion is made for the long-range integrals. The paper by B e r n h ~ l d t also ’ ~ ~ examined the scalability of ‘MP2’ calculations but in this case the emphasis was on the exploitation of parallel computing methods in ‘MP2’ calculations using the so-called ‘RI-MP2’ in which approximations based on the resolution of the identity are used in performing the sum over states. In the paper of Li, Millam and Schlegel’16 ab initio molecular dynamics calculations for the photodissociation of formaldehyde,

are reported using direct classical trajectory calculations using ‘MP2’. Using a basis set designated 6-31I G(d,p) these authors demonstrated that by exploiting the speed and economics of contemporary computing machines the ‘primary bottleneck’ of coventional trajectory calculations can be avoided by computing energies, gradients of the energy and Hessian matrices as they are required. Direct classical trajectory calculations using density functional theory are also reported by Li, Millam and Schlegel.’16 Several studies employ the so-called G2 and G3 proced u r e ~ . ~ ~ ’The ~ effects ~ ~ of ’ ~the ~basis ~ ~set ~continued ~ ~ ~to ~demand ~ ~ ~ ~ ~ attention’13~123~’24 in electron correlation studies using ‘MP2’, particularly in studies of hydrogen bonding.’137’24

3.2 Publications with the String ‘MP2’ in the Title and/or Keywords. - The literature search for publications with the string ‘MP2’ in their title in the period June 1999 to May 2001, described in the previous subsection, will, of course, emphasize publications in which the ‘MP2’ method is an important aspect of

7: Manv-body Perturbation Theory

365

the particular study and will overlook the majority of publications relating to the ‘MP2’ method in which it is employed as a routine tool for the study of the molecular electronic structure problem. Here we describe the results of literature searches for publications with the string ‘ M . 2 ’ in their title and/or list of keywords for the year 2000 in four primary research journals. The Institute for Scientijc Information database was interrogated to determine the number of incidences of the string ‘MP2’ in titles and keywords in four specific primary research journals during 2000. The journals chosen were The Journal of Chemical Physics, Chemical Physics Letters, The Journal of Physical Chemistry A and The Journal of Physical Chemistry B. We describe the results obtained for each of these journals in turn.

3.2.1 Journal of Chemical Physics. - The declared purpose of The Journal of Chemical Physics, which is published by the American Institute of Physics, is ‘to bridge a gap between journals of physics and journals of chemistry by publishing quantitative research based on physical principles and techniques, as applied to chemical systems’. Subject areas covered by the journal include ‘polymers, materials, surfaceshterfaces, and biological macromolecules, along with the traditional small molecule and condensed phase systems’. Interrogation of the ISI database to determine the number of incidences of the string ‘MP2’ in titles and keywords of papers appearing in The Journal of Chemical Physics during the year 2000 resulted in a list of 69 publication^.'^^^^' The list of titles of these publications serves to convey some idea of the variety of applications being reported in The Journal of Chemical Physics. (Only two of these publication^"^^^'^ also occur in the list given in Section 3.1 .) 1. A mobile charge densities in harmonic oscillators (MCDHO) molecular model for numerical simulations: The water-water interactionIS0 2. An assessment of theoretical methods for the study of transition metal carbonyl complexes: [Cl,Rh(CO),]- and [Cl,Rh(CO)]- as case studies’” 3 . Low-order scaling local correlation methods 11: Splitting the Coulomb operator in linear scaling local second-order Msller-Plesset perturbation theoryI5? 4. An ab initio study of the interaction in dimethylamine dimer and trimerts3 5. Chiral discrimination in hydrogen-bonded complexes’54 6. Ab initio molecular dynamics studies of the photodissociation of formaldehyde, H2CO+H2 t C O : Direct classical trajectory calculations by MP2 and density functional theory’I6 7. Theoretical study of the N(’D)+O2(X3Zg)+O+NO reactionls5 8. Implementation and refinement of the modified-conductorlike screening quantum mechanical solvation model at the MP2 level”’ 9. Ground state gas and solution phase conformational dynamics of polar processes: Furfural 10. Approximating the basis set dependence of coupled cluster calculations: Evaluation of perturbation theory approximations for stable molecules’58 1 1. Coupled-cluster calculations on ferrocene and its protonated derivatives: Towards the final word on the mechanism of protonation of f e r r ~ c e n e ? ’ ~ ~

366


12. Relative energies of the C2H2S2isomers 1,2-dithiete and dithioglyoxal: Peculiar basis set dependencies of density functional theory and ab initio methodsIbO 13. Static electric properties of conjugated cyclic ketones and thioketonesI6' 14. C-H...O H-bonded complexes: How does basis set superposition error change their potential-energy surfaces?'62 15. Structure and stability of the N-hydroxyurea dimer: Post-Hartree-Fock quantum mechanical 16. CC2 excitation energy calculations on large molecules using the resolution of the identity appr~ximation'~~ 17. Ab initio study of OH addition reaction to isoprene'65 18. Theoretical study of kinetic isotope effects on rate constants for the H2+C2H+H+C2H2 reaction and its isotopic variants166 19. Improved quantum mechanical study of the potential energy surface for the bithiophene molecule'67 20. Isomeric structures and energies of H: clusters (n = 13, 15, and 17)16* 21. The reaction of benzene with a ground state carbon atom, C(3fJ)169 22. Anharmonic treatment of the lowest-energy conformers of glycine: A theoretical study'70 23. Potential-energy surfaces related to the thermal decomposition of ethyl azide: The role of intersystem crossing^'^' 24. Exploring the ab initiolclassical free energy perturbation method: The hydration free energy of water'72 25. Ab initio studies of anionic clusters of water ent tamer'^^ 26. Vibrational spectra and intramolecular vibrational redistribution in highly excited deuterobromochlorofluoromethane CDBrClF: Experiment and theory'74 27. Ab initio calculation and spectroscopic analysis of the intramolecular vibrational redistribution in 1,l ,1,2-tetrafluoroiodoethaneCF3CHFI 28. Accuracy of the energy partitioning data obtained by classical trajectory calculations on potential energy surfaces constructed by interpolation: H2CO+H2 +CO as an example'76 29. CCNN: The last kinetically stable isomer of cyanogen"' 30. Ionization energies of hyperlithiated and electronically segregated isomers of Li.(OH).pI (n = 2-5) clu~ters''~ 31. Spectroscopic and ab initio investigation of the v(0H) overtone excitation in transformic acid'79 32. Direct vibrational self-consistent field method: Applications to H 2 0 and H2C0"0 33. Structural and energetic properties of the Br-C2H2 anion complex from rotationally resolved mid-infrared spectra and ab initio calculations'*' 34. Rotationally resolved spectroscopy of a librational fundamental band of hydrogen fluoride tetramerlX2 35. Basis set convergence of correlated calculations on He, H2, and He2IX3 36. Potential energy surface and infrared spectrum of the Ar-H2C1+ ionic 37. Chlorine atom addition reaction to isoprene: A theoretical study'*5 38. Theoretical investigation of Ca.RG, Ca'.RG, and Ca2+.RG (RG = Ar and Ne) complexes' '6 39. A comparative ab initio and DFT study of neutral aniline 01igomers''~ 40. Accuracy of atomization energies and reaction enthalpies in standard and extrapolated electronic wave functionhasis set calculations'*' 4 1. Noble gas-metal chemical bonding? The microwave spectra, structures, and hyperfine constants of Ar-CuX (X = F, C1, Br)'"


367

42. Hydrolysis of sulfur trioxide to form sulfuric acid in small water clusters”’ 43. Ground and excited Hubbard states for buckminsterfullerene with uniform and alternating bond strengths’” 44. Comprehensive ab initio studies of nuclear magnetic resonance shielding and coupling constants in XH. ..O hydrogen-bonded complexes of simple organic molecules’92 45. Assessment of Gaussian-3 and density functional theories for a larger experimental test set”‘ 46. The performance of density-functional theory in challenging cases: Halogen oxidesIv4 47. A first principles study of the acetylene-water interacti~n’’~ 48. Rotational spectra of four of the five conformers of 1-pentenel” 49. A complete basis set model chemistry. VII. Use of the minimum population localization methodI9’ 50. Modeling proton mobility in acidic zeolite clusters. I. Convergence of transition state parameters from quantum chemistry”’ 5 1. Convergent summation of Msller-Plesset perturbation theory’99 52. Negative ion photoelectron spectroscopy of OH-(NH3)200 53. A theoretical study of polyimide flexibilityZ0’ 54. Closely approximating second-order Merller-Plesset perturbation theory with a local triatomics in molecules model2”’ 55. Electric fields in ice and near water clusters’”‘ 56. Microsolvation of the methyl cation in neon: Infrared spectra and ab initio calculations of CHT-Ne and CH; -Ne2’(’‘ 57. The structure and stability of Sb4H clusters: The importance of nonclassical structures205 58. Direct calculation of anharmonic vibrational states of polyatomic molecules using potential energy surfaces calculated from density functional theoryZo6 59. Electronic structure and dynamics of 0(3 P)+CO(’E+)collisions207 60. Interaction of the water dimer with n-systems: A theoretical investigation of structures, energies, and vibrational frequencies2”* 6 1 . Fluorobenzene and p-difluorobenzene microsolvated by methanol: An infrared spectroscopic and ab initio theoretical investigationZo9 62. Ab initio calculations o f [COY, .X,,]’+ complexes21o 63. Ab initio study of (NH3)2:Accurate structure and energetics’” 64. Ab initio investigations on the HOS02+O,-+SO, +H02 reaction2” 65. Gaussian-3 theory using scaled energies” ’ 66. van der Waals isomers and ionic reactivity of the cluster system para-chlorofluorobenzene/methano12l 4 67. Ab initio study o f the gas-phase structure and electronic properties of M-CH3 (M = Li, Na) and M-CCH (M = Li, Na, K): A combined post-Hartree-Fock and density functional theory study2” 68. Theoretical characterization of the excited-state structures and properties of phenol and its one-water 69. The microwave spectra and structures of Ar-AgX (X = F, C1, Br)”’

3.2.2 Chemical Physics Letters. - Chemical Physics Letters is devoted ‘to the analysis of phenomena in the domain of chemical physics, with an emphasis on theoretical interpretation. Experimental contributions are included if their results

368


are of direct importance for a theoretical analysis’. Interrogation of the IS1 database to determine the number of incidences of the string ‘MP2’ in titles and keywords of papers appearing in Chemical Physics Letters during the year 2000 resulted in a list of 62 publi~ations.~~*-~’~ The list of titles of these publications serves to convey some idea of the variety of applications being reported in Chemical Physics Letters. (Only three o f these also occur in the list given in Section 3.1.) 1. Ab initio calculations on indole-water, 1-methylindole-water and ind~le-(water)~~’* 2. Combined ab initio and anharmonic vibrational spectroscopy calculations for rare gas containing fluorohydrides, HRgF2” 3. Conformational flexibility of pyrimidine ring in adenine and related compoundsZZ0 4. An isomeric study of N:, N5, and N;: a Gaussian-3 investigationZ2’ 5. Photoinduced charge separation in indole-water clustersZZZ 6. A theoretical study of BZLihZZ3 7. The electronic properties of water molecules in water clusters and liquid ate?^^ 8. Could uranium(XII)hexoxide, U 0 6 (0,)exist?225 9. A theoretical study of pyrimidine photohydrates and a proposed mechanism for the mutagenic effect of ultraviolet lightZZ6 10. Computational studies on the ring openings of cyclopropyl radical and cyclopropyl cation227 11. An ab initio potential energy surface of Ne-LiH2” 12. Theoretical evidence for a new lund of intramolecular dihydrogen bond229 13. Squaramide as a binding unit in molecular recognition230 14. Application of the overlap model to calculating correlated exchange energiesZ3’ 15. Hydration of zinc ions: theoretical study of [Zn(H20)4](H20)~+and [Zn(H20)61(H20):+ 232 16. Heats of formation of small bicyclic hydrocarbons, spiropentadiene (C5H4), spiropentane (C,H,) and bicyclo[ 1.1.O]but-l(3)-ene (C4H4):a theoretical study by the G2M(RCC,MP2) methodiZ2 17. A theoretical study on the novel molecule OSiCO and its isomers233 18. Computational study of medium-sized cumulenethiones H2C,S (n = 3-9)234 19. Theoretical study on reforming of COZcatalyzed with BeZ3, 20. Evaluating the formation of salt-bridges: a molecular orbital study236 2 1. The pyrolysis mechanism of furan revisited237 22. Electronic structure investigation of the A1404rn01ecule~~~ 23. Hybrid density functionals and ab initio studies of 2-pyridone-H20 and 2-pyridone(H20)2239 24. Potential energy landscape for proton transfer in (H20)3H+:comparison of density functional theory and wavefunction-based methodsZ4’ 25. G2(MP2) molecular orbital study of the substituent effect in the H3BPH3-,,F, ( n = 0-3) donor-acceptor c~mplexes’~’ 26. The molecular structure and conformation of tetrabromoformaldazine: ab initio and DFT calculations241 27. The singlet and triplet states of HF2+:a theoretical study242 28. Electronic spectra and photophysics of the two stable conformers of anthracene dimer: evaluation of an ab initio structure prediction243 29. Infrared spectrum of HXeI revisited: anharmonic vibrational calculations and matrix isolation experiments244


369

30. Torsional barriers in biphenyl, 2,2’-bipyridine and 2-phen~lpyridine~~’ 3 1. A theoretical investigation of the N203- anion isomers246 32. Interaction of alkali metal cations and short chain alcohols: effect of core size on theoretical affinities247 33. The barrier to linearity of hydrogen s ~ l p h i d e ~ ~ * 34. Transition states for inversion and retention of configuration channels in the reactions of alkyl and silyl fluorides with a water molecule249 35. Theoretical study of geometrical effect on the deoxygenation of epoxide by singlet carbenes2’0 36. The tetrahydropyran .HC1 dimer: a theoretical studyz5’ 37. Theoretical studies on the nonlinear optical susceptibilities of 3-methoxy-4-hydroxybenzaldehyde 38. Theoretical study on the reversible storage of H2 by B e e s 3 39. On the origin of I3cand I4N hyperfine interactions in [co(CN)6l4- and [Rh(CN)6l4complexes in KC 1 host 40. Inversion of stationary point levels in a flat transition structure region on account of vibrational energy255 prismic, hsed cubic and dodecahe41. Ab initio studies of (H20)20H+ and (H20)21H+ dral clusters: can H30+ ion remain in cage cavity?256 42. Infrared depletion spectroscopy suggests fast vibrational relaxation in the hydrogenbonded aniline-tetrahydrofuran(C6HSNHz..0C4H8)complex257 43. Effects of the higher electron correlation correction on the calculated intermolecular interaction energies of benzene and naphthalene dimers: comparison between MP2 and CCSD(T) calculations133 44. The accuracy of molecular dipole moments in standard electronic structure calculations258 45. Theoretical study of microscopic solvation of LiCl in water clusters: LiC1(H20), ( n = 1 -4)2’9 46. Correlated frequency-dependent electronic first hyperpolarizability of small pushpull conjugated chains260 47. Ab initio molecular orbital study of the mechanism of photodissociation of formamide26I 48. The C-H ...& bonds: strength, identification, and hydrogen-bonded nature: a theoretical study262 49. Alternative pathways for the C2-C3 bond cleavage and Cz configuration inversion processes for the Rubisco-catalyzed carboxylation sequence263 50. Importance of secondary electrostatic interactions in hydrogen-bonding complexes: an investigation using the self-consistent charge and configuration method for subsystems2@ 51. A theoretical study on the molecular mechanism for the normal Reimer-Tiemann reaction26s 52. ONIOM as an efficient tool for calculating NMR chemical shielding constants in large 53. Ab initio and density functional predictions of the structure, gas-phase acidity and aromaticity of 1,2-dithi0-3,4-diselenosquaric 54. Relativistic core-valence correlation effects on molecular properties of the hydrogen halide molecules268 55. Ab initio study of the reorganization barrier for bent triatomic species269 56. A theoretical study on cytosine tautomers in aqueous media by using continuum models270 *

*

370


57. The intermolecular interaction in the charge-transfer complex between NH3 and F2. A subtle case27' 58. An ab initio and density functional study on the ring-chain tautomerism of (2)-3formyl-acrylic 59. An ab initio study on the insertion reaction of silylenoid H2SiLiF with H2273 60. A quantum chemical study of three isomers of N20274 61. c ~ s - [ P ~ ( N H ~ )coordination ~]~+ to the N7 and 0 6 sites of a guanine-cytosine pair: disruption of the Watson-Crick H-bonding 62. A spectroscopically effective molecular mechanics model for the intermolecular interactions of the hydrogen-bonded N-methylacetamide dimef?76

3.2.3 Journal of Physical Chemistry A . - The Journal of Physical Chemistry is devoted to reporting new and original experimental and theoretical basic research of interest to physical chemists and chemical physicists. The Journal of Physical Chemistry A publishes studies on molecules (dynamics, spectroscopy, gaseous clusters, molecular beams, kinetics, atmospheric and environmental physical chemistry, molecular structure, bonding, quantum chemistry, and general theory). Interrogation of the ISI database to determine the number of incidences of the string 'MP2' in titles and keywords of papers appearing in The Journal of Physical Chemistry A during the year 2000 resulted in a list of 162 publ i c a t i o n ~ The . ~ ~list ~ ~of~titles ~ of these publications serves to convey some idea of the variety of applications being reported in The Journal of Physical Chemistry A . (Only eight of these publications106~'10~1'2~117~126~'34~138~141 also occur in the list given in the previous subsection.) 1. Photoelectron spectroscopy of SO,- at 355 and 266 nm277 2. Molecular complexes between sodium and carbonyl compounds: Photoionization and ab initio molecular orbital studies2'* 3. Potential energy surface for the chlorine atom reaction with ethylene: A theoretical study279 4. Origin and nature of lithium and hydrogen bonds to oxygen, sulfur, and selenium28" 5. Studies on the trapping and detrapping transition states of atomic hydrogen in octasilsesquioxane using the density functional theory B3LYP method"' 6. Does a stacked DNA base pair hydrate better than a hydrogen-bonded one?: An ab initio studyzs2 7. Reaction coordinate and rate constants for nitrous acid cis-trans i s o m e r i ~ a t i o n ~ ~ ~ 8. 0 Bond activation by cooperative interaction with rzsz atoms: Be+nH2, n = 1-3284 9. Polarizabilities of carbon dioxide and carbodiimide. Assessment of theoretical model dependencies on dipole polarizabilities and dipole polarizability anis~tropies~~~ 10. Thermochemistry of hydrochlorofluorosilanes: A Gaussian-3 studyzs6 1 1. Absolute binding energies of alkali-metal cation complexes with benzene determined by threshold collision-induced dissociation experiments and ab initio theory287 12. Water-mediated base pairs in RNA: A quantum-chemical study288 13. An ab initio post-Hartree-Fock comparative study of 5-azacytosine and cytosine and their dimers with guaninezs9 14. G3(MP2) calculations of enthalpies of hydrogenation, isomerization, and formation

37 1

7: Many-bodv Perturbation Theory

of bi- and tricyclic C8 and Clo hydrocarbons. The bicyclo[3.3.0]octenes and triquinacenes I I' 15. Ab initio study and NBO interpretation of the anomeric effect in CH2(XH& (X = N, P, As) compounds290 16. Gas-phase photoemission study of 2-mer~aptobenzoxazole~~' 17. Thermochemical property, pathway and kinetic analysis on the reactions of allylic isobutenyl radical with O2: an elementary reaction mechanism for isobutene 18. Theoretical study of the reaction of C1. with C3H2293 19. Theoretical calculations and matrix-isolation FT-IR studies of hydrogen-bonded complexes of molecules modeling cytosine or isocytosine tautomers. 7. 2hydroxypyridine/2-oxopyridineComplexes with H20294 20. Theoretical study of the CF2=CHI HF+CF CH reaction2y5 21. G2 molecular orbital study of [H3A1XH]- (X = NH,PH, ASH, 0, S, and Se) and H3AlYH (Y =OH, SH, SeH, F, C1, and Br) donor-acceptor complexes296 22. Calorimetric and theoretical determination of standard enthalpies of formation of dimethoxy- and trimethoxybenzene 23. Six structures of the hydrazine 24. Combined mass spectrometric and ab initio study of the point contacts between 9methyladenine and the amide 25. Gas phase reactions of HONO with NO2, 03,and HCI: Ab initio and TST

-

26. An ab initio study of the kinetics of the reactions of halomethanes with the hydroxyl radical. 2. A comparison between theoretical and experimental values of the kinetic parameters for 12 partially halogenated methanes3'' 27. Matrix infrared spectra and density functional calculations for GaNO, InNO, and ~ 1 ~ 0 3 0 2

28. Methylenecyclopropane-boron trifluoride van der Waals complexes; an infiared and DFT study303 29. Unimolecular reactions of proton-bound cluster ions: Competition between dissociation and isomerization in the ethanol-acetonitrile dime9°4 30. Kinetics of hydrogen abstraction from chloromethanes by the hydroxyl radical: A computational study305 3 1. Theoretical study of intramolecular hydrogen transfer in thioformohydroxamic acid and its aceto and fluoro-substituted derivatives306 32. Anharmonic vibrational spectroscopy calculations for novel rare-gas-containing compounds: HXeH, HXeC1, HXeBr, and HXeOH307 33. Structures of NO;(H,O),, and (HN03)(H30+)(H20),-2( n = 2-4) c l u s t e r ~ ~ " ~ 34. The shielding constants and scalar couplings in N-H..-O=C and N-H.-.N=C hydrogen bonded systems: An ab initio MO study309 3 5 . Ab initio molecular orbital and density functional studies on the stable structures and vibrational properties of trans- and cis-az~benzenes~'~ 36. Polarizable model potential function for ion-methanol systems3'' 37. Atomic mean dipole moment derivatives and GAPT charges3I2 38. Heats of formation of hydrofluorocarbons obtained by Gaussian-3 and related quantum chemical computations313 39. A conformational study of the alpha-L-aspartate-containingdipeptide"' 40. The electron localization function signature of the amide bond exhibits nitrogen lone pair character3I5 4 1 . Theoretical study on the reaction mechanism of CO, with Mg3I6

372


42. Energetics of Br-H-Br- formation from HBr dimer anion: An ab initio study3” 43. Molecular geometries at sixth order Msller-Plesset perturbation theory. At what order does MP theory give exact geometries?’” 44. New channels in the reaction mechanism of the atmospheric oxidation of toluene3I9 45. Rearrangements of 2-nitrobenzyl compounds. 1. Potential energy surface of 2nitrotoluene and its isomers explored with ab initio and density functional theory methods320 46. An ab initio and density functional theory investigation of the structures and energetics of halide ion-alcohol complexes in the gas phase321 47. Gas-phase reaction pathways of aluminum organometallic compounds with dimethylaluminum hydride and alane as model 48. Rotational spectra of the less common isotopomers, electric dipole moment and the double minimum inversion potential of H 2 0 . .HC1323 49. Experimental and computational study of hydration reactions of aluminum oxide anion ~ I u s t e r s ~ ~ ~ 50. The structures of difluorodiisocyanatomethane, CF2(NC0)2:X-ray crystallography, gas electron diffraction, and quantum chemical calculations325 5 1 . Infrared depletion spectroscopy suggests mode-specific vibrational dynamics in the hydrogen-bonded aniline-diethyl ether (C6H5-NH2.. -OC4H10) complex326 52. Protonation of nucleic acid bases. A comprehensive post-Hartree-Fock study of the energetics and proton affinities327 53. Toward a low-barrier transition-metal-free catalysis of hydrogenation reactions: A theoretical mechanistic study of HAIS-catalyzed hydrogenations of ethene (X = F, C1, and Br)328 54. Molecular vibrations of pteridine and two symmetric tetraazanaphthalene~~~~ 55. Theoretical study of the tautomeric/conformational equilibrium of aspartic acid zwitterions in aqueous solution33” 56. Theoretical investigation of the neutral/zwitterionic equilibrium of gamma-aminobutyric acid (GABA) conformers in aqueous ~olution”~ 57. How do strong hydrogen bonds affect the acidities of carbon acids? An ab initio molecular orbital study’32 58. Strongly bonded bimolecular complexes between HCN and HNC333 59. Structure and conformation of bis(methylthio)methane, (MeS)2CH2,determined by gas-phase electron difiactiom and ab initio methods334 60. Ab initio calculation of nonbonded interactions: Are we there yet?335 61. The molecular structures and energetics of Cl2CO, ClCO, Br2C0, and BrC0336 62. The n-butonium cation (n-C,H:,): The potential energy surface of protonated n*

63. Theoretical study of the thermal decomposition of N,N’-diacyl- N,N’-dialkoxyhydrazines: A comparison of HF, MP2, and DFT’26 64. Theoretical studies of substituent and solvent effects on protonation equilibria of ben~aldehydes~~~ 65. Matrix-isolation and mass-spectrometric studies of the thermolysis of [Me2N(CH2),]GaMe2.Characterization of the monomeric organogallanes Me,GaH, MeGaHz, and MeGa339 66. Ab initio modeled matrix trapping sites, PES asymmetry, and automerization in the Adcyclobutadiene system340 67. Photochemistry of butatriene - Spectroscopic evidence for the existence of allenyl~arbene~~’

7: Many-bodv Perturbation Theory

373

68. Transoid, ortho, and gauche conformers of n-Si4Cllo:Raman and mid-IR matrixisolation spectra3'* 69. Theoretical and experimental studies of the reaction of OH with isoprene3" 70. A theoretical study of the different conformations of N,N,N',N'-tetramethyleth~lenediarnine~'~ 7 1. High-level ab initio calculations of dihydrogen-bonded c~rnplexes~'~ 72. Theoretical investigation of the interaction between 2-pyridone/2-hydroxypyridine and ammonia3" 73. An ab initio study of potential energy surfaces for N8 isomers347 74. Detection of a higher energy conformer of 2-phenylethanol by millimeter-wave spectros~opy'~~ 75. Assessment of Gaussian-3 and density functional theories for enthalpies of formation of C I -C16alkanesJ4' 76. Kinetic analysis for H 0 2 addition to ethylene, propene, and isobutene, and thennochemical parameters of alkyl hydroperoxides and hydroperoxide alkyl radicals35o 77. Tautomerization of nucleobase model compounds: The 4-pyridinol and 4( 1H)pyridinone monomers and their dimers3" 78. Ab initio studies for geometrical structures of ammonia cluster cations"' 79. Conformational analysis of N,N,N',N'-tetramethylsuccinamide:The role of CH. . -0hydrogen bonds353 80. Are the thiouracils sulfur bases in the g a s - p h a ~ e ? ~ ~ ~ 8 1. Raman and infrared spectra, conformational stability, normal coordinate analysis, vibrational assignment, and ab initio calculations of 3,3-difl~orobutene~~' 82. Microwave spectrum, conformational equilibrium, intramolecular hydrogen bonding, tunneling, and quantum chemical calculations for 1-ethenylcyclopropan- 1 - 0 1 ~ ~ ~ 83. Characterization of aromatic-amide( side-chain) interactions in proteins through systematic ab initio calculations and data mining analyses357 84. Free energetics of NaI contact and solvent-separated ion pairs in water clusters358 85. A h initio molecular orbital calculations on NO+.(H,O), cluster ions. Part I: Minimum-energy structures and possible routes to nitrous acid formation3" 86. Interaction energies of hydrogen-bonded formamide dimer, formamidine dimer, and selected DNA base pairs obtained with large basis sets of atomic orbitals360 87. Diatomic halogen anions and related three-electron-bonded anion radicals: Very contrasted performances of Mdler-Plesset methods in symmetric vs dissymmetric cases'"' 88. Proton affinity and protonation sites of aniline. Energetic behavior and density functional reactivity indices362 89. Photodissociation of HN3 at 248 nm and longer wavelength: A CASSCF study363 90. Ab initio study of the X- +RCOY displacement reactions with R = H, CH3 and X, Y = C1, B?'" 91. Importance of charge transfer and polarization effects for the modeling of uranylcation complexes365 92. A solid-state NMR and theoretical study of the I7O electric field gradient and chemical shielding tensors of the oxonium ion in p-toluenesulfonic acid monohydrate '6h 93. A theoretical study of the potential energy surface and rate constant for an O(3P) + HOz reaction3h7 94. Gas-phase nuclear magnetic resonance study of Berry pseudorotation of SF,.

374


Comparison of experimental and calculated kinetic parameter's and falloff kinetic s368 95. Conformers of n-Si6Me14:Ab initio, molecular mechanics, and additive increment methods369 96. Nitrosodifluoroamine, F2N20370 97. Microwave spectrum, conformation, dipole moment, and quantum chemical calculations of 1-amino- 1-ethenylcy~lopropane~~' 98. An ab initio study of the kinetics of the reactions of halomethanes with the hydroxyl radical. 1. CH2Br2372 99. Quantum chemistry study of the van der Waals dimers of benzene, naphthalene, and anthracene: Crossed (D.(,) and parallel-displaced ( CZh)dimers of very similar energies in the linear p o l y a c e n e ~ ~ ' ~ 100. Theoretical study of proton transfer in hypoxanthine tautomers: Effects of hydration374 101. Infrared and ab initio study of the chloride-ammonia anion ~ o m p l e x " ~ 102. Experimental and theoretical analysis of thevibrational spectra and theoretical study of the structures of 3,6-dichloropyridazine and 3,4,5-tri~hloropyridazine~~~ 103. Anharmonic vibrational spectroscopy of hydrogen-bonded systems directly computed from ab initio potential surfaces: (H,O),, n = 2, 3; Cl-(H,O),, n = 1, 2; H+(H20),, n = 1,2; HzO-CH30H377 104. 0-Substituted vinyl cations: Stabilities and electronic properties37x 105. Revised and expanded scale of gas-phase lithium cation basicities. An experimental and theoretical 106. Bond additivity corrections for quantum chemistry methods3" 107. A journey from generalized valence bond theory to the full CI complete basis set limit38' 108. Ab initio study of XH; (X = B, A], and Ga) isomers3x2 109. Solvation of the hydroxide anion: A combined DFT and molecular dynamics study383 110. Activation of small alkanes in Ga-exchanged zeolites: A quantum chemical study of ethane d e h y d r ~ g e n a t i o n ~ ~ ~ 1 1 1. Matrix isolation and ab initio study of the hydrogen-bonded complex between H 2 0 2and (CH3)203Xs 112. The structure of the 2-norbornyl cation: The Jt-complex and beyond3" 113. Density functional theory study of the hydrogen-bonded pyridine-H20 complex: A comparison with RHF and MP2 methods and with experimental data' 114. Hydrogen bonding in methyl-substituted pyridine-water complexes: A theoretical study'34 1 15. Pyrolysis of furan: ab initio quantum chemical and kinetic modeling 116. Electronic structure calculations on the reaction of vinyl radical with nitric 117. Theoretical study of the excited state properties and transitions of 2-aminopurine in the gas phase and in solution3") 1 18. Conformational geometries and conformation-dependent photophysics of jet-cooled 1,3-dipheny1propane3" 119. Decomposition and isomerization of the CH3CHC10 radical: ab initio and RRKM 120. Acetic anhydride in the gas phase, studied by electron diffraction and infrared spectroscopy, supplemented with ab initio calculations of geometries and force fields393

7: Manv-body Perturbation Theory

375

12 1. Halogen bonding in fluoroalkyl halides: A quantum chemical study of increasing fluorine s~bstitution~'~ 122. Raman spectroscopic measurements of scandium(II1) hydration in aqueous perchlorate solution and ab initio molecular orbital studies of scandium(II1) water clusters: Does Sc(II1) occur as a hexaaqua complex395 123. Ab initio calculations of cooperativity effects on clusters of methanol, ethanol, 1propanol and methanethi01~~~ 124. Rotational spectra of seven conformational isomers of 1-hexene3" 125. Tropospheric oxidation mechanism of dimethyl ether and methyl formate"' 126. Initial reactions in chemical vapor deposition of Ta205from TaCl, and HzO. An ab initio study3" 127. The reaction of C,H, with CO: Kinetic measurement and theoretical correlation with the reverse proce~s'"~ 128. Intramolecular proton transfer in glycine radical cation4" 129. Polyiodine and polyiodide species in an aqueous solution of iodine plus KI: Theoretical and experimental studies'"' 130. Structural and conformational studies of N,N-dichloroethanamine and N,N-dichloro-2-propanamine by gas electron diffraction combined with ab initio calculation^^^)'

13 1 . High-level ab initio calculations of torsional potential of phenol, anisole, and ohydroxyanisole: Effects of intramolecular hydrogen b ~ n d ~ ' ~ 132. Calculated vibrational spectra for CH,,OH,, species405 133. Ab initio study of the Ne('S)-CN('Z' ) van der Waals complex406 134. Electronic and vibrational structures of corannulene anions4" 135. Molecular transition metal oxides: ab initio and density functional electronic structure study of tungsten oxide clusters408 136. Theoretical analysis of concerted and stepwise mechanisms of Diels-Alder reaction between butadiene and ethylene'"' 137. Interactions of molecular hydrogen with alkali metal halides in argon matrices: A computational model"" 138. The acetylene-ammonia dimer as a prototypical C-H.. .N hydrogen-bonded system: An assessment of theoretical procedures4'' 139. Geometry, vibrational frequencies, and ionization energies of BeX, (X = F, C1, Br, and 1)": 140. Structural studies of higher energy conformers by millimeter-wave spectroscopy: Oxalic acid4' 141. Ab initio mechanism and multichannel RRKM-TST rate constant for the reaction of C1( PJ with CH2C0 (ketene)'I4 142. Vibrational analysis of the ground states of trifluoroacetyl fluoride and trifluoroacetyl chloride4" 143. Computationally efficient methodology to calculate C-H and C-X (X = F, C1, and Br) bond dissociation energies in haloalkanes4Ih 144. Kinetic energy release distribution in the dissociation of toluene molecular ion. The tropylium vs benzylium story continues4" 145. Far-infrared spectrum, ab initio, and DFT calculations and two-dimensional torsional potential fimction of dimethylallene (3-methyl- 1,2-butadiene)'l8 146. Structure and spectra of HOCl(H20), clusters, n = 1-4: A theoretical calc~lation~'~ 147. Experimental and theoretical studies of metal cation-pyridine complexes containing Cu and Ag""

376


148. Characterization of reaction pathways on the potential energy surfaces for H+S02 and HS+02421 149. Dual-level direct dynamics of the hydroxyl radical reaction with ethane and haloethanes: Toward a general reaction parameter method422 150. Theoretical thermochemistry of the 1-buten-3-yn-1-yl radical and its chloro derivatives423 151. Ab initio MO and density functional theory study of substituent effects on electron attachment to benzyl chloride^'^' 152. An ab initio investigation of the reactions of 1,l- and 1,2-dichloroethane with hydroxyl radical425 153. Bonding of NOz to the Au atom and Au(ll1) surface: A quantum chemical study426 154. Density functional and MP2 studies of germylene insertion into C-H, Si-H, N-H, P-H, 0-H, S-H, F-H, and Cl-H bonds427 155. Ab initio MO study of diverse Si3H,' isomers'" 156. Ab initio study of unimolecular decomposition of n i t r ~ e t h y l e n e ~ ~ ~ 157. Proposed mechanism of 1,l-diamino-dinitroethylene decomposition: A density functional theory 158. p-Quinone dimers: H-bonding vs stacked interaction. Matrix-isolation infrared and ab initio study43' 159. Ab initio study of energetics of protonation and hydrogen bonding of pyridine Noxide and its derivative^'^^ 160. Thermochemistry of N302433 161. Fluorinated organosilicon cations: A comparison of potential energy surfaces for SiC2Xi where X is H or F and n = 1, 3, and 5434 162. Diaminocarbenes; Calculation of barriers to rotation about CcarbeneN bonds, barriers to dimerization, proton affinities, and "C NMR shifts435

3.2.4 Journal of Physical Chemistv B. - The Journal of Physical Chemistry: B publishes studies on materials (e.g. nanostructures, micelles, macromolecules, statistical mechanics and thermodynamics of condensed matter, biophysical chemistry, and general physical chemistry), as well as studies on the structure and properties of surfaces and interfaces. Interrogation of the ISI database to determine the number of incidences of the string 'MP2' in titles and keywords of papers appearing in The Journal of Physical Chemistry B during the year 2000 resulted in a list of 18 publ i c a t i o n ~ The . ~ ~list ~ ~of~titles of these publications serves to convey some idea of the variety of applications being reported in The Journal of Physical Chemistry B. 1. A QWMM study of the conformational equilibria in the chorismate mutase active site. The role of the enzymatic deformation energy contribution436 2. Theoretical study of the alkaline hydrolysis of a bicyclic a~a-beta-lactam~~' 3. Many-body effects in systems of peptide hydrogen-bonded networks and their contributions to ligand binding: A comparison of the performances of DFT and polarizable molecular mechanics438 4. An ab initio molecular orbital study of the hydrogen sorbed site in Co/MoS2

5. Which functional form is appropriate for hydrogen bond of amide~?~"


377

6. Zinc’s effect on proton transfer between imidazole and acetate predicted by ab initio calculations4’ 7. C-He - -0 contacts in the adenine. .uracil Watson-Crick and uracil.. -uracil nucleic acid base pairs: Nonempirical ab initio study with inclusion of electron correlation effectsu2 8. A deuterium labeling, FTIR, and ab initio investigation of the solution-phase thermal reactions of alcohols and alkenes with hydrogen-terminated silicon surfacesu3 9. Lattice resistance to hydrolysis of Si-0-Si bonds of silicate minerals: ab initio calculations of a single water attack onto the (001) and (111) beta-cristobalite surfaces10. Sorption behavior of 1-butene in perfluorocarbon type ion- exchange membranes doped with various silver saltsu5 1 I . Ground-state properties of nucleic acid constituents studied by density functional calculations. 3. Role of sugar puckering and base orientation on the energetics and geometry of 2 ’-deoxyribonucleosides and ribonucleosidesM6 12. Molecular properties from combined QWMM methods. 2. Chemical shifts in large m01ecules~~ 13. A b initio and density functional study of the activation barrier for ethane cracking in cluster models of zeolite H-ZSM-548 14. Isomerization of fluorophors on a treated silicon surface449 15. Theoretical study of urea and thiourea. 2. Chains and ribbons450 16. Nature of intercalator amiloride-nucelobase stacking. An empirical potential and ab initio electron correlation study45’ 17. Hydroxyl radical reactions with phenol as a model for generation of biologically reactive tyrosyl radicals452 18. Complexes of pentahydrated Zn2+ with guanine, adenine, and the guanine-cytosine and adenine-thymine base pairs. Structures and energies characterized by polarizable molecular mechanics and ab initio calculations453 *

4 Summary and Prospects

It is difficult not to sympathize with the views expressed by Professor B.T. Sutcliffe in a lecture delivered in April 1997 at a workshop on ‘Quantum Systems in Chemistry and Physics’ held in Jesus College,

‘It is hard to avoid feeling that electron correlation problems are just like the bibical poor: they are always with us. Each new administration in Quantum Chemistry, just as each new administration in politics, hopes to have a solution to the problem.’ The term ‘electron correlation’ was first employed by Wigner and S e i t in ~ ~1933 ~ ~ in a study of the electronic structure and cohesive energy of metals published in The Physical Review. Just one year later and in the same journal, Marller and Plesset3*presented what is today the most widely used method for describing correlation effects in molecules,’ the method which is nowadays most often referred to as ‘MP2’, Marller-Plesset second-order theory. In this review, we have

318


demonstrated the wide range of applications to which this method has been applied over the past two years, providing a snapshot of research activity in this area at the turn of the century. Modern application of the Msller-Plesset theory rest upon both theoretical and computational developments mainly during the second half of the twentieth century. On the theoretical side, the proof of the linked diagram theory of many-body perturbation theory must rank as the most important of these developments. However, the introduction of a fully relativistic (four-component) formulation is also very significant since it extends the range of applications to molecules containing heavy and even superheavy atoms. The introduction of the algebraic approximation thereby facilitating applications to arbitrary molecular systems is perhaps the most significant computational advance. When implemented systematically the algebraic approximation can support calculations of increasing accuracy. In spite of this progress, problems remain and the description of electron correlation in molecules will remain an active field of research in the years ahead. The most outstanding problem is the development of robust theoretical apparatus for handling multi-reference treatments. Methods based on Rayleigh- Schrodinger perturbation theory suffer ftom the so-called ‘intruder state’ problem. In recent years, it has been recognized”’ that Brillouin- Wigner perturbation theory shows promise as a robust technique for the multi-reference problem which avoids the ‘intruder state’ problem. Wigner’s first contribution to Brillouin- Wigner perturbation theory appeared in 1935, the year following the publication of Marller and Plesset’s seminal work. Wigner’s paper’’ was published in Mathematischer und Natunvissenschaftlicher Anzeiger der Ungarischen Akademie der Wissenschaften and was entitled On a modijication of the Rayleigh-Schrodinger perturbation theory. Wigner notes that the ‘. . . series converges much more rapidly than the power series of Schrodin-

ger.’

Under certain conditions the Brillouin- Wigner perturbation theory forms the basis for a many-body theory. Whether it can provide a robust multireference many-body theory of electron correlation effects is the subject of current research.

Acknowledgments The support of the Engineering & Physical Sciences Research Council under Grant GR/M74627 is acknowledged.

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8 Quantum Topological Atoms BY P.L.A. POPELIER AND P.J.SMITH

1 Introduction This report follows on from the first report’ in the RSC Specialist Periodical Report series, this time with a slightly more adventurous title that specifies more precisely the character of the theory of ‘Atoms in Molecules’ (AIM)2 pioneered by Bader and co-workers. In the first report we reviewed AIM from its very origin up to June 1999. The reader is referred to the first report for information on what AIM is and on software that implements it. Very recently two new book^^,^ have appeared that may help the uninitiated to learn AIM or to deepen one’s level of understanding. A recent popular account’ may also be of interest. In this report we briefly discuss papers referring to Bader’s ground-breaking 1990 monograph published in the period of July 1999 to June 2001 (see Disclaimer, Section 13). Twice as much material is now included in more or less the same space as in the first report. In order to provide enough room for comments and discussion on each paper we decided not to include any Figures and introduced a few acronyms (Table 1). It should be made clear that it was not our purpose to summarize the full content of a paper. Instead we have briefly mentioned its area of interest or the main question driving the work it reports on, focusing on just the AIM part. As a result we have presented each article as ‘seen through AIM glasses’. For example, if a paper explores in depth the PES of a compound, we have not extensively reported on the levels of theory used or even the details of energies, geometries or extremum characterization. Rather, we might have mentioned how for example the presence of RCPs helped in deciding upon the structure of say a transition state. Given the large number of unrelated areas in which AIM is applied, this way of reporting makes it often hard or impossible to understand from our text the work published in a paper. For example, because of space restrictions we could not show details of an intricate organometallic species that are highly relevant to the main message of a paper. In other words, even if an AIM analysis were peripheral to the main body of the paper we would still put the spotlight on it. Although for most papers we attempted to quote the authors’ opinions and statements with almost journalistic precision we were occasionally tempted to express our concern about the quality of these opinions. Chemical Modelling: Applications and Theory, Volume 2 0 The Royal Society of Chemistry, 2002


392

Table 1 List of acronyms‘ used in the text Acronym

Full name

APT NIC S BCP RCP HB AIM ELF IAS PES MO MEP

Atomic Polar Tensor Nucleus-Independent Chemical Shift Bond Critical Point Ring Critical Point Hydrogen Bond Atoms in Molecules Electron Localization Function Interatomic Surface Potential Energy Surface Molecular Orbital Molecular Electrostatic Potential

“We confine ourselves to specialized acronyms directly related to AIM and to unconventional ones. Well-known acronyms such as MP2 or HF (Hartree-Fock) are not included.

Figure 1 shows the number of references to Bader’s 1990 monograph. The growth conveyed in the first report is clearly still sustained. If this trend continues it is safe to say that about two hundred papers a year appear making use of or commenting on AIM. Since some AIM-related papers do not quote the 1990 monograph this is an underlimit. A word on the categorization of the papers is in place here. We faced the same dilemma as for the first report: does one classify the papers in terms of chemical classes or in terms of methods and concepts? As before we have opted to give 250

200

f 0

f

e

150

Y-

lo0

z 50

0

199019911992199319941995199619971998199920002001 Year

Figure 1 A histogram of the number of literature citations per year to Baderk book ‘Atoms in Molecules. A Quantum Theory’, 1990

8: Quantum Topological Atoms

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priority to method and concepts. To give a hypothetical example, if a paper discusses a reaction path involving peptides, one should first look under the heading ‘reactions’ and then ‘biological compounds’. We have tried to maintain the same style of categories as in the first report but were obliged to abandon quite a few very special categories and introduce brand new ones, filled with just a few papers. In any event we apologize in advance for possible classification inadequacies - it is often difficult to ensure that a category does justice to the main message of a paper. In order to help in surveying how we classified the papers we inserted Table 2, which lists the paper content of each category. Some workers regard AIM and ELF as two different methods to analyse quantum systems. Although we have maintained this distinction throughout the text for ease of reference and discussion we feel uncomfortable perpetuating it.

Table 2 Survey of papers discussed in this report. Each paper is marked by the prime author(s) and the publication year 2.1

2 Theoretical Alternative partitioning

2.2

Electron correlation

Cioslowski, Liu et al. 1999; Lopez-Boada, Pino et al. 1999

2.3

Algorithms-software

Kenwright, Heme et al. 1999; Biegler-Konig 2000; Flensburg and Madsen 2000; Volkov, Gatti et al. 2000; Biegler-Konig, Schonbohm et al. 200 1; MacDougall and Heme 200 1; Popelier 200 1

2.4

Transferability

Aicken and Popelier 2000; Bader and Bayles 2000; Fradera, Duran et al. 2000; Matta and Bader 2000

2.5

Pseudopotential

Sierraalta and Herize 2000

2.6

Intermolecular interaction

Gough, Dwyer et al. 2000; Kosov and Popelier 2000; Kosov and Popelier 2000; Dehez, Chipot et al. 2001; Popelier and Kosov 2001

2.7

Transition probability

Bader, Bayles et al. 2000

2.8

Entropy

Ho, Clark et al. 2000

2.9

General extensions

Bader 2000; Bader 2001

2.10

Quantum Monte Carlo

Mella, Morosi et al. 2000

2.1 1

Magnetic coupling

Chevreau, Moreira et al. 2001

3.1

3 Chemical Bonding Theory

Lazzeretti, Caputo et al. 1999; Uberuaga, Batista et al. 1999; Zitto, Caputo et al. 2000; Cerofolini, Meda et al. 2001; Haiduke, Hase et al. 2001; Oliferenko, Palyulin et al. 2001; Zitto, Caputo et al. 200 1

Pendas, Blanco et al. 1999; Alkorta, Barrios et al. 2000; Hernandez-Trujillo and Bader 2000; Ma and Wong 2000 continued overleaf


394

Table 2 (continued) 3.2

Heavy main group elements Power 1999; Abboud, Alkorta et al. 2000; Allen, Fink et al. 2000; Barrientos, Redondo et al. 2000; Costales, Kandalam et al. 2000; El-Bergmi, Dobado et al. 2000; Gonzalez, Mo et al. 2000; Pacios, Galvez et al. 2000; Scalmani, Bredas et al. 2000; Schoeller 2000; Yufit, Howard et al. 2000; Zanchini 2000; Alkorta, Rozas et al. 2001; Gal, Decouzon et al. 2001; Luana, Pendas et al. 2001; Molina, Dobado et al. 2001

3.3

Surface science

Marquez, Lopez et al. 1999; Ge, Jenkins et al. 2000; Jenkins and King 2000; Bollani, Piagge et al. 2001; Yamagishi, Jenkins et al. 2001

3.4

Fluorides

Kaltsoyannis and Price 1999; Alkorta, Rozas et al. 2000; Arulmozhiraja and Fuji 2000; Mitzel, Losehand et al. 2000; Tsirel’son 2000; Francisco, Costales et al. 2001

3.5

Transition metals

Bernad, Esteruelas et al. 1999; Buljan, Alemany et al. 1999; Abarca, Gomez-Sal et al. 2000; Braden and Tyler 2000; Chen, Petz et al. 2000; Diaz, Suarez et al. 2000; Dobado, Molina et al. 2000; Fortunelli and German0 2000; Frenking and Frohlich 2000; Gade 2000; Garcia-Yebra, LopezMardomingo et al. 2000; Guzyr, Prust et al. 2000; Isea, Rivas et al. 2000; Kovacs 2000; Luna, Gevrey et al. 2000; Negishi, Yasuike et al. 2000; Wang, Tang et al. 2000; Weiss, Winter et al. 2000; Calatayud, Silvi et al. 2001; Wiberg and Zilm 2001

3.6

van der Waals

Dopfer, Roth et al. 1999; Lopez, Alonso et al. 1999; Daza, Dobado et al. 2000; Esseffar, Bouab et al. 2000; Laidig, Speers et al. 2000; Valdes, Rayon et al. 2000; Lamsabhi, Bouab et al. 2001; Rayon and Sordo 200 1

3.7

Agostic

McKean, McGrady et al. 2001

3.8

Radicals

Alkorta, Rozas et al. 1999; Arnaud, Vetere et al. 2000; Uc, Garcia-Cruz et al. 2000; Joshipura, Vinodkumar et al. 2001

3.9

Alkali and alkaline earth oxides and halides

Lievens, Thoen et al. 1999; Luana, Costales et al. 1999; Schulz, Smith et al. 1999; Groen, Oskam et al. 2000

3.10

Organic

Janczak and Kubiak 1999; Kidemet, Mihalic et al. 1999; Muchall and Werstiuk 1999; Okulik, Peruchena et al. 1999; del Rio, Menendez et al. 2000; Dobado, Grigoleit et al. 2000; Glasovac, Eckert-Maksic et al. 2000; Janczak and Kubiak 2000; Okulik, Peruchena et al. 2000; Werstiuk and Muchall 2000; Ho, Szarek et al. 200 1; Jaszewski and Jezierska 200 1; Ruggiero and Williams 2001

395


Table 2 (continued) 3.1 1

Aromaticity

Alkorta and Elguero 1999; Cossio, Morao et al. 1999; Ferrer and Molina 1999; Cyranski, Stepien et al. 2000; de Lera, Alvarez et al. 2001; De Profi and Geerlings 2001; Krygowski and Cyranski 2001; Minkin and Minyaev 2001; Morao and Hillier 2001; Williams 200 1

3.12

Minerals

Rakovan, Becker et al. 1999; Kirfel and Gibbs 2000

3.13

Populations

Nolan and Linck 2000

3.14

Bond and valence indices

Kar, Angyan et al. 2000

3.15

Solid state

Aray, Rodriguez et al. 2000; Janczak and Kubiak 2000; Scherer, Spiegler et al. 2000; Mori-Sanchez, Pendas et al. 2001; Wozniak and Mallinson 2001

3.16

Organometallic s

Leoni, Pasquali et a1. 1999; Lupinetti, Jonas et al. 1999; Uhl and Breher 2000

3.17

Chemical shift

Kuivalainen, El-Bahraoui et al. 2000

3.18

Biological

Diaz, Suarez et al. 2000; Mercero, Fowler et al. 2000; Topol, Nemukhin et al. 2000; Vank, Sosa et al. 2000; Munoz, Sponer et al. 2001

3.19

Noble gases

Wong 2000; McDowell2001

3.20

Zeolites

Valerio, Plevert et al. 2000; Berthomieu, Krishnamurty et al. 2001

3.21

Hypervalency

Dobado, Martinez-Garcia et al. 2000; Molina and Dobado 2001

3.22

Polymers

Hernandez, Losada et al. 2000

4 X-Ray Diffraction

4.1

Organic compounds

4.2

Minerals

Koritsanszky, Buschmann et al. 1999; Buschmann, Koritsanszky et al. 2000; Mallinson, Barr et al. 2000 Iversen, Latturner et al. 1999; Tsirelson, Ivanov al. 2000; Tsirelson, Avilov et al. 2001

rt

4.3

Metals

Krivovichev and Filatov 1999

4.4

Hydrogen Bonding

Herbstein, Iversen et al. 1999; Abramov, Volkov et al. 2000; Gopalan, Kulkarni et al. 2000; Macchi, Iversen et al. 2000

4.5

Comparison between theory Dahaoui, Pichon-Pesme et al. 1999; Peres, Boukhns and experiment et al. 1999; Arnold, Sanders et al. 2000; Borbulevych, Antipin et al. 2000; Koritsanszky, Zobel et al. 2000; Kulkarni, Gopalan et al. 2000; Koritsanszky and Coppens 200 1

4.6

Transition metals

Bianchi, Gervasio et al. 2000; Lippmann and Schneider 2000; Zhurova, Ivanov et al. 2000; Bianchi, Gervasio et al, 2001; Blatova, Blatov et al. 200 1 continued overleaf

396


Table 2 (continued) 4.7

Biological

5.1

5 Laplacian Surface science

5.2

Electron pair localization

Bader and Heard 1999; Popelier 2000

5.3

Transition metals

Aray, Rodriguez et al. 2000; Choi and Lin 2000; Rohmer, Benard et al. 2000; Sarasa, Poblet et al. 2000

5.4

Heavy main group elements Blattner, Nieger et al. 2000; Deubel 2001

5.5

Non-linear optics

Gopalan, Kulkarni et al. 2000

6.1

6 Hydrogen Bonding Reviews

Calhorda 2000

6.2

Dihydrogen bond

Kulkarni 1999; Calhorda and Lopes 2000; Grabowski 2000; Grabowski 2000; Grabowski 2000

6.3

Groups 13/15

Rozas, Alkorta et al. 1999; Tarakeshwar and Kim 1999

6.4

C-H.. .X

Masunov, Dannenberg et al. 200 1; Vila, Mosquera et al. 2001

6.5

Organic

Calvo-Losada, Suarez et al. 1999; Marsman, van Walree et al. 2000; Grabowski and Pogorzelska 2001; Quinonero, Frontera et al. 2001

6.6

Cooperative effect

Gonzalez, Mo et al. 1999; Rincon, Almeida et al. 200 1

6.7

Blue-shifted

Cubero, Orozco et al. 1999; Muchall 200 I

6.8

Biochemical

Dobado and Molina 1999; Fidanza, Valiensi et al. 2000; Vank, Henry-Riyad et al. 2000

6.9

With ions

Daza, Dobado et al. 1999

Abramov, Volkov et al. 2000; Benabicha, PichonPesme et al. 2000; Dittrich, Flaig et al. 2000; PichonPesme, Lachekar et al. 2000 Aray, Rodriguez et al. 1999; Ricart, Habas et al. 2000

6.10

Isotope effects

Gawlita, Lantz et al. 2000

6.1 1

Low barriers

Pantano, Alber et al. 2000; Rozas, Alkorta et al. 2000; Vishveshwara, Madhusudhan et al. 200 1

6.12

Intramolecular

Gilli, Bertolasi et al. 2000; Jensen, Vank et al. 2000; Dubis and Grabowski 2001; Ferreiro and RodriguezOtero 2001; Grabowski 2001; Pacios, Galvez et al. 2001; Pacios and Gomez 2001

6.13

n-Systems

Tarakeshwar, Kim et al. 2000; Grabowski 2001

6.14

Kinetic energy density

Galvez, Gomez et al. 2001

6.15

Organometallic

Alkorta, Rozas et al. 2001; Clot, Eisenstein et al. 200 1


397

Table 2 (continued)

7.1

7 Topology of other functions ELF

7.2

Electrostatic potential

Leboeuf, Koster et al. 1999; Martinez, Calaminici et al. 2001

7.3

Intracule/extracule

Cioslowski and Pernal 2000; Fradera, Duran et al. 2000; Fradera, Duran et al. 2000; Fradera, Duran et al. 2000

8.1

8 Reactions Organic

8.2

Inorganic

Aray, Rodriguez et al. 2000; Niu and Hall 2000; Delabie, Vinckier et al. 2001; Fortunelli, Leoni et al. 2001; Ijjaali, El-Mouhtadi et al. 2001

8.3

Transition metals

Teruel and Sierraalta 1999; Luna, Amekraz et al. 2000

8.4

Mass Spectrometry

Oliveira, Ferreira et al. 2000

8.5

Rotation barrier

Platts and Coogan 2000

8.6

Biological

Topol, Nemukhin et al. 2000

9.1

9 Ionic materials Thermodynamics

Pendas, Costales et al. 2000

9.2

Phase Change

Blanco, Costales et al. 2000; Blanco, Recio et al. 2000

9.3

Impurity-doping

Baruah, Kanhere et al. 2001

10 Spectroscopy

Venuti and Modelli 2000

Berski, Silvi et al. 1999; Fourre, Silvi et al. 1999; Joubert, Picard et al. 1999; Llusar, Beltran et al. 1999; Lundell, B e r s l et al. 1999; Noury, Krokidis et al. 1999; Chesnut 2000; Fuentealba and Savin 2000; Fuster, Sevin et al. 2000; Fuster, Sevin et al. 2000; Fuster and Silvi 2000; Fuster and Silvi 2000; Joubert, Silvi et al. 2000; Silvi and Gatti 2000; Berski, Jaszewski et al. 200 1; Calatayud Andres et al. 200 1; Chevreau, Fuster et al. 2001 ; Molina, Dobado et al. 2001

Fang, Yalcin et al. 1999; Fu, Wang et al. 1999; Gonzalez, Luna et al. 1999; Meng, Zheng et al. 1999; Okulik, Peruchena et al. 1999; Takahashi and Kira 1999; Thamattor, Snoonian et al. 1999; Thorsteinsson, Famulari et al. 1999; Wang, Fang et al. 1999; Berski, 1999; Wiberg and Shobe 1999; Arnaud, Adamo et al. 2000; Calvo-Losada and Suarez 2000; Chesnut 2000; Fang and Li 2000; Lamsabhi, Alcami et al. 2000; Molina, El-Bergmi et al. 2000; Nguyen, Mayer et al. 2000; Reid, Hernandez-Trujillo et al. 2000; Sakata 2000; Vivanco, Lecea et al. 2000; Wang, Fu et al. 2000; Wiberg and Freeman 2000; Poater, Sola et al. 2001; Poater, kSola et al. 2001; Valdes and Sordo 2001

398


This is because in our view AIM and ELF are both two unequal branches of the same topological theory, with divisions that are largely historical. Bader and coworkers have invested plenty of research time in deepening AIM and formulating it as a theory of quantum subsystems, whereas ‘elf-ology’ refers to the systematic application of the idea of topological partitioning to a particular field, namely the electron localization function.6 The ELF theory has introduced a few useful concepts such as synapticity, integrated properties over ELF basins and appealing pictures but does not have profound quantum mechanical roots in the way AIM does. From an algorithmic point of view AIM and ELF are essentially identical, and we believe that their combined future lies in an extended quantum topological theory that fully embraces the topological idea for any scalar field. The topological study of the Laplacian of the electron density is very similar to ELF. In a similar vein the former does not have such a firm quantum mechanical basis as the topology of the electron density. At present the Laplacian lags behind ELF in terms of computational implementation. This gap is currently being narrowed by work done in our laboratory.’~~ In summary, although the chemical community at large has gradually welcomed the topological approach as a mature and reliable means to obtain chemical insight from wavefunctions the mainstream option is still based on an analysis involving MOs and the Hilbert space of basis functions. We look forward to an extended and unified topological theory that can take on the continuous challenge of extracting chemical information from reduced density matrices.

2 Theoretical

2.1 Alternative Partitioning. - Inspired by the appealing character of the topological approach to identify atoms in molecules or condensed phases Uberuaga et aL9 proposed a new method to find the zero-flux surface. An ‘elastic sheet’ is created represented by a swarm of particles. Two kinds of forces act on the particles: the component of the gradient of the density normal to the elastic sheet, and an interparticle force which only acts in the local tangent plane of the sheet. Although this method can converge where other methods fail” it tends to round off sharp cusps or sharp points. The hypothesis that geometrical derivatives of the electric dipole moment can be used as ‘effective charges’ of the atoms in a neutral diatomic molecule was initially proposed by Van Vleck. After decades of development the APT became customarily applied. In their spunky paper Lazzeretti et a1.l’ described a version of the APT approach that they claimed is probably superior to other physically based methods, and certainly more reliable on theoretical grounds than some loosely defined orbital-based approach. In their critical review of population methods they presented AIM favourably. Zitto et a l l 2 formulated atomic populations different from those of AIM. They employed quantum-mechanical relationships involving the Atomic Polar Tensor and the related nuclear electric shielding, which provide clearcut recipes to evaluate atomic populations in molecules within the dipole acceleration formalism


399

of quantum mechanics, by rewriting the Thomas-Reiche-Khun sum rule. Numerical evidence was found for the additivity and transferability of atomic populations, within a halosubstituted alkane series. Haiduke et a l l 3 rejoiced in the fact that experimental mean dipole moment derivatives, obtained from the polar tensor formalism (and calculated from molecular wavefunctions) are more accurately related to the C core ionization energies than Mulliken, CHELPG or AIM charges. As such the potential model, which describes the core ionization process, permits the interpretation of the dipole moment derivatives as atomic charges. Sadly the authors fail to realize that according to AIM a molecular dipole moment is broken down into two separate types of contributions: a charge transfer term and an atomic polarization term. The polar tensor formalism ignores the latter term, and would make it very hard to assign atomic charges to C and 0 in carbon monoxide, compatible with their electronegativity difference and still reproducing the nearly vanishing molecular dipole moment of CO. Oliferenko et a1.14 proposed a novel method to compute atomic charges. The starting data for the computation are the classical topology of a molecule and electronegativities of either constituent atoms or atomic orbitals. A comparison of the obtained charges with those produced by means of alternative computational schemes proves that the new model provides fast, convenient and reliable methods to calculate theoretically justifiable atomic charges, according to the authors. In their paper on the experimental determination of net atomic charge Cerofolini et al. I 5 proposed to use X-ray photoemission spectroscopy. They assert that the definition of a net charge of an atom is conceptually difficult and that as of yet no experiment has been developed to determine such a net charge. However, they are comforted by the AIM theory, which they see as giving physical meaning to an atomic charge, hence making the difficulties associated with it technical in character rather then conceptual. Zitto el ~ 1 . applied ’ ~ an additive scheme for resolving average molecular electric dipole polarizabilities into atomic contributions, based on the acceleration gauge for the electric dipole, to a series of methyl and ethyl derivatives. The authors believe AIM to be a sophisticated and aesthetically appealing formulation but wish to follow an alternative path, which supposes that an atom within a given molecule is a region of space that defines the actual domain of an atomic operator. Such a domain is not necessarily closed and is not uniquely defined, depending on the form of the operator itself. The reliability of partitioning methods based on this definition can only be established via empirical criteria and numerical tests.

2.2 Electron Correlation. - Lopez-Boada et al.” reported on a new derivation of correlation functionals. They used AIM to compute the correlation energies for atomic fragments and simply added the values for the whole system. The topological analysis demonstrated its utility in the detection and characterization of various types of electron correlation, illustrated’* via the electron intracule density of two ‘Eistates of H2.

400


2.3 Algorithms and Software. - In the area of computational fluid dynamics identifying regions of separated airflow is very important in the design of aircraft because it causes control problems and reduced lift. Kenwright et al.I9 presented two algorithms that extract separation and attachment lines and discuss their strengths and weaknesses. The authors mentioned the mathematical similarities between their application area and AIM. According to Biegler-Konig2’ the calculation of average properties of atoms in molecules and interatomic surfaces is a difficult problem that requires the evaluation of 2D and 3D integrals over regions with non-trivial borders. He presents an updated algorithm based on work he and others did in the early eighties, which involves a Jacobian arising in the volume integral due to a transformation between Cartesian and natural atomic coordinates. This algorithm is incorporated in the AIM2000 program. Volkov et a1.*’ evaluated net atomic charges and atomic and molecular electrostatic moments via a new program called TOPXD. Large differences are found between charges derived from monopole parameters and those from AIM analyis of the experimental model density. Flensburg and Madsen22 implemented their algorithm to obtain AIM charges from experimental densities, which is analogous to the PROMEGA algorithm, and tested it on the accurate densities of methylammonium hydrogen succinate monohydrate and methylammonium hydrogen maleate. MacDougall and H e n ~ epresented ~~ a new visualization tool called EvolV, which enables the interactive exploration of topological features based on the molecule’s electron density. Using novel volume rendering techniques, the entire molecule can be probed without obstruction by opaque surfaces or preselection of specific orbitals or contours. The authors investigated V 2 p calculated for several biomolecules. P0pelie9~ proposed a novel algorithm to compute AIM charges. Via the divergence theorem the 3D volume integral yielding the atomic charge can be expressed purely in terms of 2D surface integrals. Hence, it can be proven that an atomic charge is equal to the flux of the electric field of the whole molecule through the atom’s complete boundary. The proposed algorithm opens an avenue to invalidate the oft-quoted drawback that AIM charges are computationally expensive, especially because it does not suffer from the infamous geometrical cusp problem. In their poorly referenced article Biegler-Konig et al.25 described the objectives and features of AIM2000, the rebirth of AIMPAC. Amongst other attributes twoand three-dimensional visualization of density functions has been implemented. 2.4 Transferability. - Fradera et a1.26 defined an atomic hole density by combining the topological atom with either the exchange or the exchangecorrelation density. Its contour map revealed the partial delocalization of p of an atom into the rest of the molecule. The new function was illustrated by N2-, CN-, NO+, CO, H2C0 and HCOOH. The transferability of N and 0 within the two series was studied by means of atomic similarity measures. Bader and Bayles2’ showed that the transferability of group properties is in


40 1

general only apparent as a result of compensatory transferability wherein the changes in the properties of one group are compensated for by equal but opposite changes in the properties of the adjoining group. These compensating changes are in some cases vanishingly small, but even when the energy changes are in excess of 80 kJ mol-' the experimental heat of formation is still predicted to be additive to within 0.4 kJ mol-' . The operation of compensatory transferability is illustrated for the linear homologous series of hydrocarbons and polysilanes and for the formation of pyridine from fragments of benzene and pyrazine. Matta and Bade98 studied the effects of conformation and of tautomerization on geometric, atomic and bond properties of leucine, which exhibit transferability. The effects of HB are determined and a set of geometrical conditions for the occurrence of such bonding is identified. The effects of transforming neutral leucine into its zwitterionic form on its atomic and bond properties are shown to be localized primarily at the sites of ionization. Aicken and P0pelie9~focused on the inherent error present in atomic properties obtained from volume integration. They proved that it is impossible to predict the size of an angular Gauss-Legendre grid (outside the beta sphere) that guarantees a preset error L( a).The erratic relationship between the integration error and the grid size prompts a statistical interpretation of atomic integration, at a purely practical level. This work led to an estimate of safe error bars of atomic properties for atoms occurring in biological molecules with reasonably sized integration grids. The most stable properties were found to be the energy and the population. Finally, they have observed that the influence of the grid orientation is less if L(Q) is small, and that population and energy are the least affected. 2.5 Pseudopotential. - Sierraalta and Herize3' proposed a new scheme for the development of ab initio core potentials, which includes only valence electrons and one auxiliary s-type atomic function. They showed that the failure of largecore effective-core-potential (ECP) approaches in reproducing bond topological properties can be remedied. 2.6 Intermolecular Interaction. - The convergence of the multipole expansion for electrostatic potentials of topological atoms was in~estigated.~' This issue was addressed in order to explain why, in spite of their potentially highly nonspherical shape, the electrostatic moments generated by topological atoms show good convergence behaviour. Calculations demonstrate that this fast convergence is due to the decay of the electron density. Gough et al.32calculated the C-C and C-H stretch polarizability derivatives for straight chain, branched and cyclic hydrocarbons. Their interpretation of the CH-stretching intensities has been based on AIM, which they use to compute atomic populations in the presence and absence of an applied field. Interestingly strain is associated with a decrease in dc?/arcc and an increase in ac?/ar,. The exact atomic electrostatic potential was calculated and compared to a multipole expansion using spherical The authors prove that the convergence of this expansion is faster than previously assumed, even for com-

402


plicated atomic shapes, drawn from a set of molecules including molecular nitrogen, water, ammonia, imidazole, alanine and valine. Dehez et ~ 1 probed . ~ the~ accuracy of a novel approach for evaluating induction energies. Fully distributed polarizability models obtained with AIM are used to generate large grids of induction energies much more rapidly than with a finite perturbation approach. Test calculations show that this method is able to provide compact, yet optimal distributed polarizability models. Popelier and K O S O Vproposed ~~ an atom-atom partitioning of the (super)molecular Coulomb energy on the basis of the topological partitioning of p. Atom-atom contributions to the molecular intra- and intermolecular Coulomb energy are computed exactly, i.e. via a double integration over atomic basins, and by means of the spherical tensor multipole expansion, up to rank L = .& & 1 = 5 . The convergence of the multipole expansion is able to reproduce the exact interaction energy with an accuracy of 0.1-2.3 kJ mol-' at L = 5 for atom pairs, each atom belonging to a different molecule constituting a van der Waals complex, and for non-bonded atom-atom interactions in single molecules. We provide computational details of this method and apply it to (C2H2)2,(HF),, (H20)*,butane, 1,3,5-hexatriene, acrolein and urocanic acid.

+ +

2.7 Transition Probability. - Bader et ~ 1 investigated . ~ ~ for the first time transition probability densities for electric dipole transitions. The definition of the atomic contributions to the oscillator strength enables one to determine the extent to which a given electronic or vibrational transition is spatially localized to a given atom or functional group. Their newly introduced concepts are applied to the Rydberg-type transitions observed in the electronic excitation of a nonbonding electron in methanal and ammonia. 2.8 Entropy. - Ho et ~ 1 . ~ calculated ' the Shannon information entropies in position and momentum space of thirteen simple molecules at RHF/6-3 1++G** level in vacuo and in the Onsager aqueous field. The AIM population shows that upon solvation there is an increase in electron localization for ammonia, water, hydrogen cyanide and hydrogen peroxide. For the other molecules there exist electronic charge withdrawals from atoms with negative atomic charges and electronic charge transfers to positively charged atoms, which at local level, one may consider as a delocalization.

2.9 General Extensions. - Bade$' applied ideas of AIM to the atomic force microscope (AFM). In a quantum system, the force exerted on the tip is the Ehrenfest force, a force that is balanced by the pressure exerted on every element of its surface, as determined by the quantum stress tensor. The surface separating the tip from the sample is an IAS. Thus the force measured in the AFM is exerted on a surface determined by the boundaries separating the atoms in the tip from those in the sample, and its response is a consequence of the atomic form of matter. This approach is contrasted with literature results that equate it to the Hellmann-Feynman forces exerted on the nuclei of the atoms in the tip. Bader39 refited a variety of objections40 raised against AIM. In particular he


403

shows how the topology of p of an excited state of methanal is homeomorphic to the ground state's topology. Furthermore he states that the use of the cusp condition in the definition of an atom is completely acceptable up to energies that would result in the formation of a plasma and for a Hamiltonian that includes relativistic effects. Thirdly, ring and global attractors can be found in excited states, but they are physically (brief and finite life time) and structurally unstable. The author states that whatever new topological features the electron density may be found to exhibit, as a consequence of its averaging over nuclear motions for example, they may be incorporated into an expanded theory to provide a still deeper understanding of the behaviour of matter. Finally he reaffirms the validity of the variational procedure in the context of Schwinger's principle, by emphasizing the use of a special class of trial functions, i.e. those whose variation corresponds to continuous changes in the coordinates of the physical system.

2.10 Quantum Monte Carlo. - Mella et aL4' analysed the electron densities of both LiH and [LiH,e+] sampled during Quantum Monte Carlo simulations and saw an almost complete charge transfer from the Li atom basin, which bears a net positive charge, to the H atomic basin. 2.11 Magnetic Coupling. - Chevreau et aL4' investigated the charge density of triplet and broken symmetry states relevant to magnetic coupling in systems with localized spin moments. A comparison of p for triplet and broken symmetry solutions obtained from different computational methods together with an AIM and ELF topological analysis supports the description of these systems through the Heisenberg Hamiltonian.

3 Chemical Bonding

3.1 Theory. - The occurrence of "As in the experimentally determined electron densities of Si and Be has been surrounded by controversy. Invoking simple arguments Pendas et aZ.43examine their quantum mechanical densities and conclude that promolecular atomic shell structure is the basic organizing principle behind the occurrence of "As. The authors state they are a normal step in the chemical bonding of homonuclear groups, if analysed in the appropriate range of internuclear distances, which occurs far away from the equilibrium geometry for most elements. Ma et aE.44studied the properties and reactivities of ketene, thioketene, and selenoketene at G2(MP2) level. NBO analysis led to a strong negative charge at the C, atom, consistent with the experimental NMR data, according to the authors, whereas AIM gives a virtually neutral atom. They believed that the NBO approach provides a more realistic charge distribution of the ketenes. Their reasoning was based on combining a charge-separated resonance canonical with the fact that the 13C NMR chemical shift for Cp has shifted downfield indicating a decrease of negative charge on this carbon. According to Ma et aE. the strongest evidence came fiom the decrease in dipole moment upon substitution. A smaller

404


dipole moment along the series indicated a smaller degree of charge separation, despite the charge centres (C, and X) being fwrther apart when 0 is substituted by S or Se. Hernandez-Trujillo and Bader45 studied the evolution of the electron densities of two separated atoms into an equilibrium molecular distribution, and considered a range of interactions fi-om closed-shell with and without charge transfer, through polar-shared, to equally shared interactions. The 'harpoon mechanism' operative in the formation of LiF was found to exert dramatic effects on the electron density and on the atomic and molecular properties. The virial, the Hellmann-Feynman and the Ehrenfest force theorems provided an understanding of the similarities and differences in the bonding. Alkorta et ~ 1 compared . ~ ~models to correlate pBCpand bond distance. They proposed a logarithmic relationship covering van der Waals and HB interactions as well as traditional covalent bonds. A unique equation was devised to correlate all the H-X or C-X bonds. 3.2 Heavy Main Group Elements. - In his authoritative review47 Power introduced the issue of multiple bonds between heavier main group elements (period three and beyond) as a central theme in organometallic chemistry. Extrapolation from the bonding picture of second period elements dominating organic chemistry to elements such as Ga, Si, Ge, Sn etc. is wrong. For example alleged formal triple bonds between two Ge atoms cannot be viewed as alkyne analogues. Power mentions the topological approach as a possibility to resolve the quandary. Pacios et ~ 1 investigated . ~ ~ the structures of SiX3M (M = Li, Na; X = H, F, C1, Br) compounds by uncorrelated and correlated methods with 6-3 11++G(3df,pd). There was little sharing of electron density between atoms in all of the structures, and a region of charge shared by atoms has only been found for Si-H bonds in the hydrogenated silanes. The global bonding picture is clearly supported by the atomic charges and dipole moments that reveal a strong ionic character in all structures, with the possible exception of inverted SiC13Li and SiBr3Li. Abboud et ~ 1 focused . ~ on ~ the P4.. .Li+ ion in the gas phase, which they colourfully described as a planetary system. Protonation of gaseous tetraphosphorus P4(g) lead to a new chemical bond, the essentially covalent 2e-3c bond P.. .H.. .P. From AIM they learnt about the fundamental differences between the nature of the interactions of P4with H+ and Li+. In the former the P-H bonds formed are covalent, while the interactions with Li+ are essentially electrostatic. Costales et aL5' studied the chemical bonding of small (monomer, triatomic, and dimer) neutral clusters of AlN, GaN, and InN. The large charge transfer in the Al-N bond and the significant deformation of the A1 electron shells marked this bond as a highly polar shared interaction. On the other hand, Ga-N and InN bonds are non-shared interactions, with smaller charge transfers and polarizations. In all cases, the existence of a N-N bond weakened the metal-nitrogen bond. The emerging bonding picture depended only on the reliability of the electron densities. Partially triggered by the atmospheric interest of metal carbides Barrientos et


405

ul.” embarked on a computational study of AlC3. They pointed out that a topological analysis of p at the MP2 level showed that AlC3 is a truly fourmembered ring, since no transannular C-C bonding was observed. ZanchiniS2 performed a computational study on the kinetic behaviour of the cyano-isocyano isomerization and the nature of the transition states of relevant Si and C containing compounds. In the silylcyanides a type of AIM bond order was computed to be half of the bond order values of the cyanides. This unexpected result could contribute to rationalization of the calculated values of the isomerization energy, indicating a larger ionic character in the Si compounds. In view of their interest in the stabilization in neutral bicyclic sulfoxide compounds El-Bergmi et ~ 1 performed . ~ ~ B3LYP/6-3 1 +G* calculations on 5thiabicyclo[2.l.l]hex-2-ene S-oxide derivatives. The GIAO method and AIM revealed a stabilization of the S atom with the double bond for the ex0 configuration, in agreement with the experimental results. Yufit et aLS4performed an AIM analysis of the experimental charge density distribution in triphenylphosphonium benzylide. Based on charges and BCP properties they assigned multiple character and high ionicity to the ylide bond and a different role of the phenyl groups in charge delocalization in the PPh3 fiagment. The analysis of intra- and inter-molecular contacts located the interactions that are responsible for the conformation of the molecule and determine the packing of the molecules in the crystal. Allen et a1.” joined a lively debate on the bond order of certain recently synthesized GaGa and GeGe bonds. They argued that while any analysis of the calculated electron density is arbitrary, one of the most self-defining was AIM. The calculation of an AIM bond order for trans-MeGeGeMe yielded a GeGe covalent bond order of 2.097, which is reasonably close to 2. They computed bond orders for other model systems and compared results with MO interpretations. Gonzalez et ~ 1 examined . ~ ~the structure and relative stabilities of the different Sb4H+ clusters by means of high level ab initio calculations. From a Laplacian map they learnt that the Sb-Sb bond of the neutral compound had been replaced by two Sb-H linkages in the side-protonated species. In other words, according to the authors, the H atom was covalently bound to the two neighbour Sb atoms through a three-centred bonding orbital. Schoeller5’ reported on the electronic properties of substituted phosphanylcarbenes and allegedly used AIM to analyse in more detail the nature of bonding, without making clear exactly how. . ~ various ~ density functionals to look at the structure of Scalmani et ~ 1applied isolated, monomeric M-CH3 (M = Li, Na) and M-CCH (M = Li, Na, K). The CH3 group was considered as one entity while reporting the Mulliken and NPA atomic charges. According to the authors AIM did not suffer from the unevenly balanced description of the C and H atoms drawn in by a particular basis set. Curiously the fact that the Mulliken population analysis showed that the metalcarbon bond in Li compounds, though being strongly polarized preserves a covalent character, remained unchallenged by the authors. Gal et aLS9 looked at the acidity trends in a,P-unsaturated alkanes, silanes,

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germanes and stannanes with FT-ICR techniques and G2 ab initio and DFT calculations. Their AIM analysis was restricted to a cursory monitoring how Pb changed upon changes in bond length. Luana et aL6’ focused on chemical bonding in cyclophosphazenes based on the electron and pair density of HF/6-3 1G** wavefinctions of a collection of (NPX2), derivatives. The PN BCP point properties lay between those of XNPX3, formally a double NP bond, and those of X2NP&, formally a single NP bond. The charge on the ring N remained almost invariant, -2.3e, in all cyclotriphosphazenes, whereas the charge of the ring P varied from +2.9 to +4.0e, depending on the electronegativity of the X group. The partitioning of the pair densities indicated that about 0.63 electron pairs were shared between each P and its two N neighbours in the ring, this value being typical of a polar but largely ionic bonding situation. These topological properties did explain the chemistry of cyclophosphazenes and agreed well with the available experimental densities. The AIM analysis supported the main conclusions from the traditional Dewar’s model of phosphazenes. Molina et aL6’ joined a lively debate on the nature of multiple bonding in ‘gallynes’. They combined a topological analysis of the electron pair density and p and decided that the Ga-Ga bond in bent [HGaGaHI2- was a result of the sharing of two electron pairs at the HF level. Alkorta et aZ.62 looked at molecular complexes between Si derivatives and electron-rich groups with B3LYP and MP2. The electron density was characterized by means of AIM and the interaction nature was studied with the NBO method. 3.3 Surface Science. - The need for an AIM analysis was recognized in work63 on the interaction of CO and NH3 on Cu( 100) and Pt( 1 1 1) surfaces. In a computational study Jenkins and Kinga probed the role for induced molecular polarization in catalytic promotion focusing on CO coadsorbed with K on Co( lOiO}, which had profound implications for the industrially important Fischer-Tropsch reaction. The supposition that adsorbed H atoms might become reactive to polarized CO due to their charge transfer with the surface may be strengthened by examining the topology of V2p . The charge accumulation zone in the vicinity of the 0 atom highlighted a reactive surface for electrophilic attack, while the exposed C atom was susceptible to nucleophilic attack. Ge et aZ.65used AIM to quantify the surface charge transfer and spin structure at an atom-resolved level in CO-adsorbed Ni{ 1lo}. Bollani et aZ.66developed a novel way to self-assemble aromatic molecules as monolayers on a Si(100) surface through the formation of a direct Si-C bond. Such systems served as chemical gas sensors. The charge on C directly bonded to Si changed from -0.74 to -0.81 going from the (SiH3)3Si-C&5 molecule to the (SiH3)3Si-C6H5...CO system according an RHF optimized geometry. The increase in the number of electrons on the aromatic ring lead to the observed blue shift in the ring C-H stretching. Yamagishi et aZ.67 carried out large-scale DFT calculations for benzene adsorption on the ferromagnetic substrate Ni{ 111}. AIM is used to obtain local


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information from their plane wave calculations. Some bonds were characterized via BCP properties such as V 2 p and E . 3.4 Fluorides. - A computational study by Kaltsoyannis and Price@' on BF3+, AlF,', CFJ2+and SiF32+predicts that the planar CZvgeometry is much more stable than the D3h one, with the exception of the B3LYP result for BF,+. An AIM analysis supports the experimental conclusion that the unique element-F bond is weaker than the shorter element-F bonds. A DFT study was performed for the first time to obtain the Li+ ion affinities of various perfluorocarbons that cause gl~bal-warming.~~ Complex geometries were obtained and the ion-molecular bonding nature was discussed on the basis of atomic charge, p and its Laplacian. The change in the V 2 p values at the BCP of the interacting C-F bonds upon complexation is directly proportional to the bond weakening, which according to the authors demonstrates the superiority of these parameters in bond analysis. Tsirel~on'~ considered the character of chemical bonding in F2 within the AIM context. He showed that the parameters of the distributions of the electronic density and kinetic energy density are qualitatively the same as for other molecules with covalent bonding. In the case where molecules are easily deformed, for example by electrostatic interactions with the environment, molecular structures can differ spectacularly between the gas phase, solution, or solid state. In this context Mitzel et aL7' looked at ( N ,N-dimethy1aminoxy)trifluorosilane.An AIM examination shows that the main aspects of bonding in this compound are not changed in the solid state and that the Si . . . N attraction is not of covalent nature, but rather due to strong electrostatic and dipole interactions. Alkorta et aZ.72 studied the effect of fluorine substitution on a series of HB systems with the aid o f ab initio methods. The atomic properties of a series of charge transfer and HB complexes have been compared. Francisco et al.73were interested in the structure and bonding in MgF2 clusters. An AIM analysis reveals that MgF2 is a highly ionic molecule, the net charge of Mg being about +1.8e, and that most basis set effects are due to the poor convergence properties of the atomic electron dipole moments. This suggests a polarizable ions model that is shown to account for the trends found in most of the properties studied. The origin of the bending problem in these compounds is traced back to the polarizability of the cation. 3.5 Transition Metals. - Carbene complexes of the chromium triad have proven to be attractive reagents in modern organic chemistry. Since the advent of the catalyst R u C ~ ~ ( C H C H = C P ~ ~ ) ( Pmuch R , ) ~ interest has been dedicated to the synthesis of the alkenylcarbenes of the Fe triad. Bernad et aZ.74 are the first to describe the alkenylaminocarbenes of Ru. The ellipticities of the Ru-C, and C,N bonds are reported to be low. In their ab initio study on CuM02 (M=Al, Ga, Y) delafossite-type oxides Buljan et aL7' analysed the bonding in the Cu layers. The topological analysis revealed the existence of weak attractive d'O -d'O interactions that were suggested

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Chemical Modelling: Applications and Theory, k l u m e 2

to be important in the determination of the electrooptical properties of doped delafossites. (Cp” = A new homoleptic diplatinum complex [Pt2(GaCp*)2(p2-GaCp*)3] pentamethylcyclopentadienyl)exhibiting a central unit of two platinum atoms was An analysis of the Pt-Pt bond with the help of NBO and AIM ~ynthesized.~~ suggested weak attractive interactions only. The intra- and intermolecular bonds in AuCl(C0) and its dimer were studied77 at HF, DFT and MP level. For the dimer, the good agreement between the solidstate structure and the simple head-tail model suggested that the crystal geometry is due primarily to electrostatic interactions though a more elaborate analysis of the charge density revealed also weak covalent Au-Au and C-Cl bonds. Besides using the Mulliken, NBO and ESP methods Fortunelli and German0 used their own s o h a r e to locate critical points based on a density grid generated by GAUSSIAN. They believed that ‘the subtle differences between the various population analyses become more apparent when a more refined theoretical tool, such as the Bader analysis, is employed since the Bader approach gives a more complete and interesting landscape’. A combined experimental and theoretical study on the bond characterization of Cr-L (L = 0, N, C) multiple bonds was applied78 to a series of Cr-complexes. Bond characterizations were also shown in terms of V2p where the inner VSCC is embedded. The topological properties associated with the BCP of Cr-L multiple bonds in these compounds indicated a strong covalent bond character. In order to gain insight into the origin of the preference for six-coordination of hydrated Zn2+ ions Diaz et al.79 performed ab initio calculations on [Zn(H20)4](H20)82+ and [Zn(H20)6](H20)62+. Using AIM it was found that all water Zn2+ interactions in both clusters are closed-shell in view of the BCP properties. In his reviews0 on highly polar metal-metal bonds in ‘early-late’ heterodimetallic complexes Gade expected the topological approach to find more widespread use in future studies of metal-metal bond polarity. A consistent picture of the nature of the Ti-Co bonds in dinuclear complexes was obtained by a combination of AIM and an ELF analysis. The covalent metal-metal bond order was less than 0.5, and high partial charges of > 0.5e emphasized the highly polar character of the Ti-Co bonds. Isea et a1.81 set out to determine the lowest energy conformation of the [Zn2,(CN)55]- ion using ab initio techniques. They used AIM to characterize both the normal bonds and the intermolecular interactions responsible for the helical structure of the compound. The Zn-N and Zn-C turned out to be ionic. Negishi et al.82 researched the electronic properties of copper cyanide cluster anions [Cu,(CN),-; n = 1-6, rn = 1-61 using photoelectron spectroscopy supplemented with B3LYP/p-VDZ calculations and an AIM analysis. Although the bond between Cu and the CN group was essentially ionic, similarly to the alkali halide clusters, some covalent character was also included in the Cu-CN bond. According to AIM the net charge on the Cu atom was around + O h . The partially covalent character in Cu-CN was attributed to the linear structures of the copper-cyanide clusters and generates the structural difference between the Cu-


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CN clusters and the alkali halide clusters, which formed compact cubic or planar structures. Chen et aLg3 performed B3LYP and CCSD(T) calculations on the carbon complex (CO), FeC. The bonding situation was analysed with NBO and a detailed AIM analysis. It was noted that the small positive charge of C in (C0)4FeC did not contradict the classification of the carbon ligand as a nucleophile. Carbon monoxide also reacted as a C-nucleophilic agent, although the C of CO carried a positive charge. The shape of the charge distribution had a stronger influence on the chemical reactivity than the atomic partial charge. The data supported the suggestion that the Fe-C bonds of (CO),FeC and (C0)4FeC-BC13 have a significant covalent character. Dobado et al.84 show multiple bonding in four-coordinated Ti(1V) compounds. The main goal of their work paper was to characterize Ti-0 bonding in a series of Ti(1V) methoxides with reference to analogous Ge(1V) complexes, which were proven to have only Ge-0 single bonds. In summary the linear Ti-0-C angles in the titanium alkoxides were rationalized via polarized triple Ti-0 bonds, on the basis of NBO, ELF, AIM, and electron delocalization indexes. On the other hand, the 18-e Ge compounds displayed bent Ge-0-C angles compatible with a standard single Ge-0 bond, with two additional electron lone pairs on 0. Garcia-Yebra et ~ l . reported *~ on the synthesis of alkynyl- and vinylideneniobocene complexes and an unexpected q 1-vinylidene-v2-alkyne isomerization. They used AIM to locate CPs. For example, the existence of the metal-CH interaction was confirmed via the presence of two BCPs, one between Nb and H and the other between Nb and C. A RCP between Nb, C and H was also found. Guzyr et d g 6 carried out an extensive study on the formation of Mo and W containing organometallic oxides serving as model compounds for catalysis on metal oxide surfaces. In a rather curious and cursory statement the authors said that no Mo-Mo BCP was detected, but that a RCP was found in (q5c 5MeS)3M03(p-o)2 (p-CH2)(p3-CH)Braden and Tylerg7carried out DFT calculations on three 19e Fe-containing sandwich complexes. According to AIM the partial charge on the iron atom was 0.75. Interestingly they concluded that consideration of a single molecular orbital did not allow a reliable estimate of the net charge or spin density around an atom in a molecule. Kovacs8*used DFT and relativistic effective core potentials to study the Ln2X6 (Ln = La, Dy; X = F, C1, Br, I) dimers. The ionic bonding in the dimers was characterized by NBO charges as well as by BCP properties. In view of the tremendous interest in the chemical processing routes to ceramics Abarcag9 studied the ammonolysis of Ti(1V) derivatives. DFT calculaand the topology tions have been performed on [ { Ti2(q5-C5MeS)2C13(NH3))(p-N)] confirmed the existence of an intramolecular N-H.. .C1 HB. Luna et al.90 investigated the gas-phase reactivity of orthophosphoric acid (H0)3P=O with Cu+ via mass spectrometry experiments and DFT calculations. For some of the minima of the [Cu,H3P04]+hypersurface, the nature of the different Cu-0 bonds as well as the character of the orbitals involved were analysed by means of AIM and NBO.

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In their review on the nature of the bonding in transition metal compounds Frenking and Frohlich” commented extensively on AIM. They mentioned that it has been shown that the topological analysis of p and its Laplacian revealed helphl information about the electronic structure of a molecule. Another very attractive feature of the AIM method was that the results of the topological analysis directly gave the atomic subspaces of a molecule and the bonding connectivity between the atoms. The authors claimed that although AIM was the most straightforward partitioning scheme it had not yet been fully accepted by the chemical community. They believed that one reason is that it is sometimes difficult to connect traditional bonding concepts with the results of an AIM analysis. They claimed that the bonding connectivity given by BPs does not always agre-ith the notion of a chemical bond. Also, the authors thought that AIM charges suggested that molecules are much more ionic than generally assumed and predicted by orbital based methods such as the NBO model. Wiberg and Zilm92 used the ‘Atoms in Molecules’ based IGAIM method to calculate the 19F NMR shielding for the alkyl fluorides from methyl fluoride to t-butyl fluoride and to separate it into the contribution from each of the MOs. In their study on the structure, energetics and bonding of VO,+ and VO, ( n = 1-4) systems Calatayud et aZ.93found that AIM and Mulliken charges show parallel trends though the former’s absolute values are slightly larger. As a general rule the electron transfer towards 0 was less than l e whereas the V electron loss was less than 2. The values of the net charges were consistent with the pictures that emerge from the geometries. 3.6 van der Waals. - Lopez et uE.’~ developed a methodology to express the wavefunction of a complex A.. .B in terms of the contributions made by different electronic configurations, which they combined with a BCP analysis of the tetrahydrofuran. . .HC1 complex. They concluded from their microwave spectroscopy and ab initio study that the complex could be regarded as a combination of electrostatic and charge transfer contributions. Dopfer et aL9’ recorded the infrared photodissociation spectrum of the T-shaped complex CH3CNH+-H2 and supplemented their study with MP2 calculations. The AIM charges showed that the centre of charge was close to the nitrogenbound carbon atom. Daza et aZ.96studied complexes of H202withNO’, CN-, HCN, HNC and CO and characterized intermolecular interactions, ranging from electrostatic for the H202.. .NO+ complex to different types of hydrogen bonding. Valdes et aZ.” explored the PES of the tetrahydropyran.. .HC1 system at the MP2/6-31G(d,p) level of theory. According to the authors the energy density computed at the BCP is negative (shared interaction) for both conformers, which they believe means that the type of interaction in these systems results from an accumulation of p at the BCP (covalent interaction). A NEDA analysis of Glendening and Streitwieser corroborates this latter point emphasizing the importance of the charge transfer contribution, in agreement with early findings by Klemperer and co-workers. Laidig et aL9* examined the complexation of Li+, Na+, and K+ by water and

8: Quuntum Topological Atoms

41 1

ammonia. The driving force for complexation is the stabilization of the alkali cations by increased nuclear-electron attraction between the heavy nucleus and the electron density on the solvent heavy atom, a density that is increased significantly on complexation as a result of the transfer of electronic charge from the hydrogen atoms. This stabilization is at the expense of increased repulsion between the charge distributions and nuclei within the adducts. Esseffar et al.99 give an account of UV-visible spectra and intermolecular charge-transfer spectra of several complexes between thiocarbonyl compounds and I2 supplemented with ab initio calculations. A BCP between I and the amino H is localized demonstrating an intramolecular HB. They note that it appears between the iodine that points to the S atom, and not the terminal one as hypothesized before for imidazolidine-iodine complexes. Rayon and Sordo"'o located a set of BCPs in extrema of the potential energy surface of the CzH2...SO2 complex explored at MP2 and QCISD levels with Pople and Dunning basis sets. Three classical properties evaluated at the S . . .O, S. . .nand 0. . .H BCPs are reminiscent of van der Waals molecules. Lamsabhi et al. studied intermolecular charge-transfer complexes between a wide range of carbonyl compounds and ICl via UV-visible spectroscopy. Ab initio calculations at HF/LANL2DZ* and MP2(full)/LANL2DZ* were carried out in order to clarify the structures of these CT complexes. The electron densities as well as the energy densities found at 0-1 BCPs are typical of electrostatic interacti ons. 3.7 Agostic. - In their combined IR and DFT study McKean et aZ.Io2zoomed in on the vibrational properties of a P-agostic ethyl ligand. The AIM part of this study repeats earlier workIo3they seem to be unaware of.

3.8 Radicals. - Methane, which is a very weak acid in the gas phase, completely changes its behaviour upon removal of one electron. In a studyIo4 on the interaction between CH,+- radical and noble gas atoms five to seven stationary points were located depending on the noble gas element. Significant charge transfer is observed only in a Kr complex in accordance with its high stabilization energy. Very small electron densities and positive values of V 2 p ( r )at BCPs are observed as an indication of van der Waals or weak HB interactions. One complex with Kr is an exception since V 2 p ( r )indicates a shared interaction. Uc et a1.'O5determined the CPs on the PES describing the addition of OH. to toluene at post-HF level. The position of the two BCPs points between the hydrogen atom of the OH radical and the Cip,y,and Cparu,which are detected in the prereactive complex, could indicate that a three-centre orbital is formed, similar to those in diborane. In their computational study of radical additions to substituted olefins Arnaud et a1.1°6 used AIM to detect potential HBs formed at different stages of the reaction. In one case the authors were not able to locate any BCP between H atoms of the methoxy radical and F atoms of the fluoroethene substrates. On the other hand, in cases where AIM revealed the presence of weak HBs p at the BCPs

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seemed to indicate that these interactions do not play a significant role in determining the preferred conformations of the products.

3.9 Alkali and Alkaline Earth Oxides and Halides. - A paperlo7on the heats of formation of alkali and alkaline earth oxides and hydroxides computed AIM charges. In agreement with experimental results, all the alkali and alkaline earth hydroxides as well as the alkali oxides are predicted to be exothermic molecules. However, the alkaline earth oxides represent endothermic species. A striking feature is that the second-row oxides and hydroxides are less exothermic than their first-row or third-row analogues. Intriguingly, the AIM charges also show non-monotonic behaviour in going from the first- to the second- to the third-row systems; the positive charges on the metal in the second-row hydroxides are greater than in the first- or third-row analogues, while the positive charges on the metal in the second-row oxides are smaller than in the first- or third-row analogues. In their AIM discussion on 120 alkali halide perovskites Luaiia et LIZ."* discussed the shape of the ions and showed how the topological description contained the classical picture in terms of slightly deformed spheres. They enthusiastically concluded from this and previous work that AIM supplied a rigorous foundation for important historical concepts like ionicity, index of coordination, coordination polyhedra or atomichonic volume in a solid. Lievens et aZ. Io9 reported on experimental and theoretical investigations of the ionization potentials and structures of lithium monoxide clusters. In agreement with NPA the AIM analysis predicted an oxygen charge of -2.03 in Li30. In Li40 the most striking fact was the presence of a NNA in the centre of the threemembered lithium ring with a charge of -1.39. The authors stated that since valence electrons of lithiums are highly delocalized within this ring, charge assignments based on the partitioning of the Hilbert space of the atomic orbitals may not have physical meaning. Groen et U Z . " ~ studied the structure and vibrational properties of the LiCeX4 (X = F, Cl, Br, I) complexes. The AIM analysis is in agreement with the strong ionic interactions in the complexes. 3.10 Organic. - Okulik et u1.l" looked at proponium cations at ub initio level and typified six different stable structures: four proponium cations and two van der Waals complexes. A topological analysis of p in the C-proponium cation shows significant differences with respect to structures of the H-proponium cations. The detection of an unstable CP and the larger ellipticity found for the C-H* and H*-H* bonds made clear why these species undergo a structural change. Dobado et al. ' I 2 performed B3LYP/6-3 11+G** calculations on 5-0x0, 7-0x0 and 5-oxo-7-amino[ 1,2,4]triazolo[1,5-~]pyrimidinesand compared them with Xray data. The electronic properties of the anionic forms have been evaluated via AIM, enabling the rationalization of the coordination mode preferences observed experimentally for the anions. The phenonium ion was initially a major subject of the classical-non-classical


413

carbocation debate, and has been considered non-classical in nature for years. Del RIOet al. ' I 3 investigated a dozen phenonium ions at B3LYP/6-3 lG* level. The analysis of the Kohn-Sham orbitals and an AIM analysis of [C6H5-C2H4]+clearly show that back-bonding from the phenyl cation moiety to the ethylene fragment determines the formation of the three-membered cycle. This makes the shielding of the ips0 C similar to that for an sp3 C, while an extension of the conjugation occurs as both n systems merge with each other. Curious about the impact of fusing a strained ring onto benzene Glasovac et al. * I 4 studied benzocyclobutene via Fourier transform mass spectrometry and ab initio calculations. The conjugation of the carbanions was probed from BCP ellipticities. Werstiuk and Muchall' l 5 performed post-HF calculations on the 2-norbornyl cation in order to elucidate its structure via AIM. The topology of the cation's electron density is that of a T-form, with C6 being tetracoordinate, and not that of a pentacoordinate, bridged species. The bridged cation, which is not a minimum on the potential energy surface, has been characterized for the first time and it has been shown that, upon bridging, the cation pushes out more positive charge onto the H atoms on C6 thereby weakening the corresponding C-H bonds. Janczak and Kubiak'16 described the crystal and molecular structure of 2,3dicyanonaphthalene, which is a naphthalocyanine precursor. A pattern of long and short bonds in the 2,3-DCN ring was observed in the ab initio optimized molecule. The differences in C-C bond lengths and angles in the naphthalene ring were interpreted via AIM. A substitution effect of both C-N groups on p and the location of the BCPs was discussed. Jaszewski and Jezier~ka"~used DFT to study the EPR parameters of heterosubstituted vinyl radicals and found that AIM showed that increasing electronegativity of the substituent lead to a strong enhancement of the ellipticity of the C,=CB bond and to an increase of the Cp atomic charge. Ruggiero and Williams''8 sought insight from p in an attempt to answer whether oxiranones are a-lactones or zwitterions. For that purpose they analysed electron density distributions for oxiranone and hydroxyoxiranone in vacuo [MP2/ 6-3 1+G(d,p)] and in water [SCI-PCM/MP2/6-3 1 +G(d,p)//HF/6-3 1 +G(d,p)] and compared them with those for cyclopropane, cyclopropanone and oxirane. They conclude that oxiranone possesses a RCP in vacuo, and may be considered as an a-lactone with considerable ionic character in the endocyclic C,-0, bond. In water, oxiranone has neither a RCP nor a BCP for C,-On, and may be considered as a zwitterion, whose carboxylate group has a net charge of -0.63. Geometrically, however, the molecule still possesses an acute-angled three-membered ring with a C,CO, angle of only 69". Electronically, hydroxyoxiranone is acyclic and zwitterionic even in vacuo,but geometrically it still looks like an a-lactone. ~ on unusually short bond lengths in oxirane and Ho et ~ 1 . ' 'concentrated derivatives. The n-complex-back-donation model gives similar conclusions to AIM. The shortening of the C-C bond in the oxirane ring and of the neighbouring C-C bond arise mainly fiom the substituent groups. However, neither the n-complex-back-donation model or AIM offers a satisfactory explanation for the elongation of one of the C - 0 bonds.

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An ab initio study by Okulik et d.‘” on isobutonium cations revealed five stable structures, two of which are van der Waals complexes. Properties at the BCPs were computed to characterize the bonds and to understand the delocalization that operates through the 0 bonds in saturated molecules. The H isobutonium cations are higher in energy and likely to undergo a structural change in view of some high ellepticities. Muchall and Werstiuk”’ performed an extensive AIM study on the structure of cation. The topology confirms a hyperthe 1,2,4,7-anti-tetramethy1-2-norbornyl conjugatively stabilized 2-norbornyl cation, which is not bridged and does not show a 3c-2e bond as had been reported. This study clearly shows that it is necessary to consider the full information available from the charge density if the goal is to obtain information about the structure of a molecule. A cursory AIM analysis’22was performed on a computed electron density of tris(2-cyanopheny1)-1,3,5-triazine, which was also determined by routine X-ray crystallography and spectroscopy. Some remarks related with the phase transition and the molecular architecture are given. The challenge of elucidating the structure of the 2-chloroallyl cation prompted &demet et to embark on a combined computational and FT-IR investigation. In going from the trans-1-chioroallyl cation to the 2-chloroallyl cation the charge on chlorine decreases from 0.333 to 0.156 (NPA) or from 0.127 to -0.001 (AIM), respectively. Also, the C-C1 bond order, calculated by the AIM method changes from 1.445 in the trans-I-chloroallyl cation to 1.183 in the 2-chloroallyl cation.

3.11 Aromaticity. - Alkorta and E l g ~ e r o ’wondered ~~ if aromaticity can be described with a single parameter. They investigate the problem of magnetic versus geometric criteria of aromaticity using NICS/GIAO/B3LYP calculations on benzene and distorted benzenes, and decided to use NICS at different distances including at the RCPs. The authors conclude that a relationship exists between these criteria but only for specific subsets. Four bowl-shaped polyaromatic hydrocarbons (C30H12)were in~estigated’~~ with DFT in the context of carbon nanotubes. Bond orders were determined via the electron density at the BCPs. The authors note a discrepancy between the deduced bond order and the ellipticity. Bonds at the inside of the bowl, where p is highest, show smaller ellipticity values than those at the periphery where p is lower. studied the aromaticity and regiochemistry of 1,3-dipolar Cossio et al. cycloadditions computationally. They investigated the aromaticity of both transition structures and reaction products of the reactions scrutinized via NICS values evaluated at RCPs, which they consider as characterizing rings unambiguously. While the NICS values computed in solution are lower than those obtained in the gas phase, the differences are very small. Therefore, aromaticity does not appear to be very sensitive to solvent effects in the compounds under study. Cyranski et a1.12’ mentioned the use of AIM in connection with the quantification of aromaticity but this not apply it in their study. In order to gauge aromaticity Morao and Hillier’28evaluated the NICS values at the RCPs for 22 para-substituted benzylic cations. The topology proved useful


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since for substituted benzenes the RCP is no longer the geometrical centre of the ring but is slightly displaced. In his review on homoaromaticity Williams'29 asserted that theory plays an even more important role in homoaromaticity than in aromaticity. The interruption of the direct aromatic delocalization in a homoaromatic results in an attenuation of the properties of aromaticity. He mentioned an AIM application in this area by Cremer and co-workers. In their review on cyclic aromatic systems with hypervalent centres Minkin and M i n y a e ~ ' ~mentioned " high values of the covalency ratio factors for specific fused-ring chalcogens in line with an AIM analyis, which points to the existence of a chemical bond between formally non-bonded chalcogen and oxygen atoms involved conjugated five-membered rings. These findings justified the attribution of a broad variety of such compounds to the pseudo-heterocycles containing nelectron sextets in the ring and, therefore, acquiring substantial aromatic character. In their review on structural aspects of aromaticity Krygowski and Cyranski13' mentioned NICS and HOMA indices being evaluated at the RCPs of benzene, naphthalene, anthracene, pyrene, tetracene, triphenylene, chrysene, perylene and coronene. Also they referred to AIM'S 'total energy decomposition scheme', which showed that the energy difference between trans-p-dinitrosobenzene and cis-p-dinitrosobenzene was 3.2 kJ mol-' , when only the ring is considered. In their review on conceptual and computational DFT in the study of aromaticity De Profi and G e e r l i n g ~ stated ' ~ ~ that the central function in DFT is p whose topology has been used to quantify the aromaticity of molecules. In their study on the aromatic character of electrocyclic and pseudopericyclic reactions de Lera et al.133 examined the thermal cyclization of (2 Z)-hexa-2,4-5trienals and their Schiff bases. They used the NICS at the RCP to detect aromaticity, since a large negative NICS value is associated with aromatic character. - In their study'34 on the surface microtopography, structure, chemistry, and reactivity of goethite Rakovan et al. displayed a contour map of V 2 p showing the sites for possible nucleophilic (near the Fe atoms) and electrophilic attack. The 0 - H bond has a significant covalent character whereas the Fe-0 bond is predominantly ionic. Kirfel and G i b b ~ 'concentrated ~~ on the electron density distributions and bonded interactions for the fibrous zeolites natrolite, mesolite and scolecite and related materials via accurate single-crystal X-ray diffraction. BCP properties of calculated densities supplement their study and relief maps of V 2 p are inspected. A study of the lumps and holes in the VSCC of microporous zeolites such as natrolite were promised. The authors anticipated to be able to delineate the sites of reactivity.

3.12 Minerals.

3.13 Populations.

- In their colourful article Nolan and L i n ~ k reported '~~ 631+G* calculations on over 75 substituted ethanes and higher alkanes, and determine the charges on the C and H atoms by topological- and orbital-based

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methods. They established that the relative charges on the hydrogen atoms calculated by the NPA and AIM methods are highly correlated. A model that involved through-space interactions between the charged group and the alkane framework accounted for the observed data. The charge calculations indicated that a substituted C has a charge that is more dependent on the nature of the atom to which it is attached than on that atom's charge or on other atoms in the substituent.

3.14 Bond and Valence Indices. - Atomic charges, bond indices (two-centre and three-centre) and valences have been ~ a l c u l a t e d 'for ~ ~ a number of closed-shell molecules using HF and KS orbitals. It was observed that, compared to the HF orbitals, the KS orbitals predicted a slightly enhanced covalency. Beside AIM the authors employed Mulliken, Lowdin and Natural Population Analysis and noted that the AIM scheme led to a more polar charge distribution than NPA. 3.15 Solid State. - The X-ray geometry of 1,3-dicyanobenzene has been c~mpared'~' with a HF/6-3 1+G(d) structure in the gas-phase. The distortion of the benzene ring in 1,3-dicyanobenzene has been compared and discussed in relation to the 0- and p-isomers of dicyanobenzene. The differences in the C-C bond lengths of the phenyl ring have been analysed in terms of p. The electron density and its Laplacian were analysed in terms of the topological properties at the BCPs. The topology of p for fcc transition metals (p-Co, Ni, Cu, Rh, Pd, Ag, Ir, Pt, and Au) was studied'39 and all BCPs in the unit cell systematically calculated. The network of bond paths describing the atomic connectivity confirms that the crystal graph of these metals is the result of packing octahedra and tetrahedra. A good correlation between the experimental cohesive energy and p at the BCP corroborates that this parameter provides a measure of the bond strength in transition metals. Scherer et al. 140 clarify features of intramolecular bonding via a topological analysis of experimental and theoretical charge densities in S4N4.Although crystal architecture is often controlled by directional interactions like HB this work appears to be the first experimental study to reveal a simple 3D directional interaction involving facing charge concentrations and charge depletions as a transferable architectural principle in a molecular crystal. The Laplacian was instrumental in showing this. Wozniak and Mallin~on'~'studied proton sponges, such as 1,g-bis(dimethylamino)naphthalene (DMAN), which are aromatic diamines characterized by exceptional basicity. For these compounds protonation causes substantial redistribution of p, which may be traced by observing the changes in the properties of p at BCPs. Mori-Sanchez et a1.142examined the polarity inversion in the electron density of the BP crystal, and found unusual properties using AIM. The standard polarity Bd+Pd-found at the zinc-blende equilibrium geometry suffers a reversal under the application of hydrostatic pressure. The inversion occurs through an intermediate


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situation in which the P valence shell is transferred to a non-nuclear maximum before being caught by the B atom. 3.16 Organometallics. - Leoni et a1.143performed a crude AIM analysis by locating BCPs from the total electron density in the P-H-Pt-CO plane of protonated [(But HP)Pt(p-PBu' 2)2 Pt(CO)(H)]CF3SO3 and [(PBu' 2H)(H)Pt(pPBu',),Pt(CO)(H)]CF, SO3,Their analysis comes to the same conclusions as those obtained from their structural data. An agostic Pt-H-Pt interaction occurring in a derived complex was characterized by NMR but not studied with AIM. A computational studyla on trends in molecular geometries and bond strengths of the homoleptic d'' metal carbonyl cations [M(CO),]"+ (MI+ = Cu+, Ag+, Au+, Zn2+,Cd2+,Hg2+,n = 1-6) probed the nature of Group 11 and 12 metal carbonyl cations, using MO, NBO and AIM arguments. Interestingly, the value of the energy density H evaluated at the BCP between Hg and C in [Hg(C0)2]2+shows an appreciable amount of covalency. Uhl and B r e h e ~ - used ' ~ ~ AIM to understand the bonding in newly synthesized carbaalanes. In particular, the B3LYP/6-3 1+G(d)) electron density of the model compounds A18H8(CH)5(R = H, CCH) shows that each C is connected to one H and four Als via BPs. Each A1 is is connected to one H and three C atoms. The bridging H is connected to four Als, but no BP is discovered between any two Als of the A18 acube, which is in line with the average Wiberg bond index. 3.17 Chemical Shift. - Kuivalainen et a1.'46showed that the 31Pchemical shift measured in 16 substituted O,O-dialkyl 0-aryl phosphorothionates correlate very well with the average value of the BCP-P distances for the P-S and the three P-0 bonds ( r 2 = 0.95). The shortening of the average BCP-P distance causes more effective electronic shielding of the phosphorus nucleus. 3.18 Biological. - In view of the recent medical interest in A13+ occurring in biological systems Mercer0 et a1.147used DFT to study A13+ and Mg2+ interactions with asparagines and glutamine and smaller model systems. Bonding interactions were investigated with NBO and AIM. It was observed that the charge transfer is larger in the aluminium complexes, Prompted by their interest in Zn metallo-B-lactamase Diaz et a1.14' performed quantum chemical calculations on model systems of the active site. They judge the topological analysis of p to be a usefbl complement to geometrical observations. In particular they found that all of the specific Zn-ligand interactions in these complexes may be characterized as closed-shell interactions. However, they note that the Zn-ligand bonds have a partial covalent character ('intermediate'). The Zn-ligand BCP properties depend notably on the hard-soft character of the ligand-+Zn*+ charge transfer, revealing more clearly how the Zn coordination environment is affected by ligand-exchange and protonation processes. performed a B3LYP conformation study on the 21st amino acid Vank et and N-acetyl-Lside chain via N-formyl-L-selenocysteinamide(For-~-sec-NH~) selenocysteine- N-methylamide (Ac-L-Sec-NHMe) in their y backbone conforma-

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tion. Three conformations exhibited intramolecular HBs as detected by the topology of p computed at the B3LYP/6-3 1G(d,p)//RHF/3-2 1G level. Munoz et al.lS0 investigated the influence of divalent metal cation binding on the nucleic base pairing. A topological analysis included AIM and ELF.

3.19 Noble Gases. - Wong asked the intriguing question whether a chargetransfer molecule containing He, Ne or Ar could exist, and reports’” the prediction of a metastable neutral helium compound. Geometries were fully optimized with the cc-pVTZ basis set at the B3LYP and CCSD levels, and relative energies obtained from CCSD(T). HHeF is predicted to exist with a significant potential energy well of 33 kJ mol-’ at Wong’s best level. The HNgF (Ng = He, Ne, Ar) compounds are best described as charge-transfer species. The bonding of HHeF, HArF, and HKrF consists of covalent Ng-H and ionic Ng-F contributions. M~Dowell’~* performed ab initio calculations at the B3LYP and MP4(SDQ) level on H-Ar-Cl. The NBO analysis places a larger positive charge on the Ar atom while the AIM value is about the same (0.32) for both Ar and H. The calculated positive Lapiacian of the electron density at the Ar-Cl BCP indicates that the Ar-C1 bond is dominated by electrostatic attraction, whereas the negative value for the Ar-H bond indicates a covalent interaction. 3.20 Zeolites. - Valerio et a1.’53studied the effect of substituting a Si centre with B in zeolites using ab initio calculations on model clusters with formula R3SiOBMR3 (M = H, Na; R = OSiH3). In contrast, in the presence of Na, ammonium or hydroxonium ions, the B sites are tetrahedral and their stability increases. This behaviour was ascribed to the nucleophilicity of the bridging 0 being lower in SiO-H than in the other cases. This picture is corroborated by the calculated net charge values obtained for 0 ( 5 ) , from a Bader-type analysis, which are - 1.60 in NaB and -1.34 in HB. Berthomieu et ~ 1 . lworked ~ ~ on model clusters of a Cu(I1)-Y zeolite using DFT in order to investigate the electronic properties of the metal site involved in the zeolite’s catalytic activity. The charge on Cu was around +0.8, which is far from a formal value of +2, according to the authors’ comments. Only for Cay, did the AIM charges differ significantly from the Mulliken ones.

3.21 Hypervalency. - Dobado et al.’55used the electron pair density in conjunction with AIM and calculated NMR chemical shifts to characterize the bonding properties for nine pnicogen and chalcogen ylide structures, at the B3LYP and MP2/6-3 11+G* level. No evidence was found to support the banana bonding scheme. Instead, different bonding schemes were found to be dependent on the electronegativity of the X atom in the C-X bond. Molina and do bad^'^^ revisited the nature of the 3c-4e bond via AIM and ELF. They performed B3LYP/6-311++G** calculations on a series of formally hypervalent compounds showing linear three-centre geometries. Their results supported the 3c-4e model for the linear structures but revealed only a small contribution fiom this model for the T-shaped structures. In addition there was no evidence to support the 3c-4e bond scheme for the bipyramidal compounds.


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3.22 Polymers. - Driven by the possibility of producing advanced materials that combine the electronic and optical properties of semi-conductors and metals with the mechanical behaviour and processing advantages of polymers Hernandez et al. embarked on a computational and spectroscopic study of dihexylbithienoquinonoid derivatives. An AIM analysis showed that two unexpected ring structures appeared as a consequence of the closed-shell interactions between the alkyl chain and (i) the S atoms upon substitution in the 3-position and (ii) the CN groups upon substitution in the 4-position. These ring structures could be the cause of similar trends in charge density properties of the central double bonds and its neighbours, on one hand, and the outermost double bonds and its neighbours, on the other hand.

4 X-Ray Diffraction 4.1 Organic Compounds. - Koritsanszky et aZ.’’* reported the structure and charge density of diisocyanomethane, derived from low-temperature X-ray diffraction data and obtained by ab initio calculations. Fairly strong indications were found for the effect of the crystal field on the topology of the density due to the electrostatic nature of intermolecular interactions, which polarize the density. Topological indices assign a higher bond order to the experimentally determined formal triple bonds than to the computational ones. An AIM analysis was performed’” on experimental charge densities of 1,ldifluoroallene and tetrafluoroallene, and they were compared to those obtained by MP2/6-3 11++G** calculations. It is noted that a small shift in the CP location in the C-F bond can lead to considerable changes in the value of V’p. 4.2 Minerals. - In their study on anhydrous sodalites Iversen et all6’ state that a proper understanding of zeolites needs to go beyond the determination of structure. In view of the challenge posed by computations on extended inorganic systems they advocate the use of AIM in a direct analysis of experimental electron distributions. Tsirelson et al. 16’ report on a topological analysis of an accurate experimental X-ray charge density of perovskite KNiF3. The topological coordination numbers of K and Ni are the same as the geometrical ones, whereas topological coordination for the F atom (6) differs from the geometrical value. The K-F interaction in KNiF3 can be considered ionic, while the Ni-F bond belongs to the polar covalent type. No correlation of the topological ionic radii with crystal or ionic radii was found. Tsirelson et a1.16’ measured very accurate electron structure factors and used them in a high-resolution quantitative study of the electrostatic potentials in LiF, NaF, and MgO crystals. A topological analysis of the electrostatic potential, defining the features of the electrostatic field and the Coulomb force field in a crystal, was developed. In addition to an AIM analysis this approach provides a more complete description of the atomic interactions according to the authors.

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Chemical Modelling: Applications and Theory, Elume 2

The application of this approach to the characterization of bonding in a crystal has been demonstrated. Lippmann and S ~ h n e i d e r 'performed ~~ a topological analysis of high-energy synchrotron-radiation data from cuprite, Cu20. The data were found to be well suited for the evaluation of hrther properties, e.g. the electrostatic potential and the Laplacian. The results clearly c o n h e d the ionic character of the Cu-0 bond.

4.3 Metals. - The AIM term of 'closed-shell interaction' is adopted in a study'@ on structural units based on anion-centred metal tetrahedra (m;X = 0, N; A = metal), which focuses on a rationale to understand the packing. 4.4 Hydrogen Bonding. - The crystal structure of benzoylacetone (1-phenyl- 1,3butanedione) has been determined'65 at four temperatures between 8 and 300 K. Application of multipolar analysis and topological methods to the charge density obtained from the combined lowest temperature X-ray and neutron data provides evidence for an intramolecular HB with partly electrostatic and partly covalent character, and large p-delocalization in the cis-enol ring. Anisotropic atom-atom potentials based on X-ray molecular charge densities were applied in the evaluation of intermolecular interactions and lattice energies of crystals of glycylglycine, DL-histidine and DL-proline, p-nitroaniline and p amino-p'-nitrobiphenyl.'66The intermolecular HBs in the crystals were identified by topological analysis of the experimental charge densities. Macchi et aZ.167pointed out that 0-H.. .O interactions in the solid state have been the subject of recent scientific discussion. Based on ab initio calculations on isolated anions it was suggested by other workers that the H.. .O contacts should be regarded as pseudo HBs, in spite of the very short 0...O and H.. .O distances. In the view of these other workers, the conformation adopted by the two anions in the solid state minimized repulsion but did not produce a chemical bond. Macchi et al. did not support this picture and found their evidence in a topological analysis of the HBs in a 15 K experimental charge density of potassium hydrogenoxalate. Interestingly they found that when a molecule undergoes a strong polarization, gas-phase calculations cannot predict its properties in the crystal, even if the solid-state conformation i s used. Gopalan et al. 168 investigated the experimental charge density of N-methyl-N(2-nitropheny1)cinnamanilide.The trends associated with the various HBs bonds are clarified by plotting V 2 p versus p. Two interaction regions can be clearly marked: one region where the HBs exhibit low values for both V 2 p versus p and the other, where both quantities are disproportionately higher. Interestingly all the bonds involving the nitro-oxygens belong to the former region and those from the amidic oxygen fill the latter. The amidic oxygen obviously forms stronger HBs compared to the more ionic nitro-oxygens, which appears to govern the molecular geometry and packing in the solid state.

4.5 Comparison Between Theory and Experiment. - Several multipole refinement^'^^ fitted the X-ray and neutron diffraction data of a single crystal of


42 1

dihydrogen phosphate equally well. Although, locally, p depends only slightly on the refinement strategy its derivatives and Laplacian may vary greatly. Errors in positioning the CP lead to a large error in V2p, an inaccuracy that is also observed in theoretical calculations due to basis set quality. However, this problem does not show up for intermolecular interactions. The electron density of L-cystine has been accurately The deformation density distribution and AIM analysis clearly reveal disulfide bridge characteristics and sulfur lone pair electron regions in accord with high-level ab initio calculations. In terms of p’s topology it is now known that SS bonds are weak single covalent bonds. The almost tetrahedral distribution of the VSCC of the sulfur atom is consistent with sp3 hybridization. Borbulevych et al. obtained an experimental charge density of the Meisenheimer complex of potassium 3-methyl-5 ’,7’-dinitro-5’,8’-dihydrospiro( 1,3-oxazolidine-2,8‘-quinolinide) and compared it to computed densities. The influence of the cation on the electronic structure of the anion in the crystalline phase is the most probable cause for the differences detected between the two charge densities. To better understand this effect, ab initio calculations of different anion-K+ systems were performed and an AIM analysis was carried out. Kulkarni et al. 172 extensively review methods of obtaining charge densities from X-ray diffraction and theory and discuss typical case studies, amply featuring AIM. In addition to dealing with the multipolar formalism for treating the experimental data, the program packages for orbital calculations in free molecules and crystals are mentioned. In particular, the authors discuss ring and cage systems, intermolecular HBs, polymorphism and non-linear optical crystals. point out the current considerable interest in using quantum Arnold et chemical methods to investigate bonding in molecules of ever increasing size and to help predict and refine the structures of molecules using spectroscopic observables. They report high-resolution single-crystal synchrotron X-ray difiaction data on L-asparagine H20. The authors show that p and its curvature (expressed via the Hessian eigenvalues) at BCPs (including the HB ones) are all in good accord with values computed either with HF or DFT. Koritsanszky et al. 174 collected high-resolution X-ray diffraction data at 15 K for potassium hydrogen (+)-tartrate. Three refinement models were tested to examine how the data resolution and restrictions on the anisotropic displacement parameters affect the topology of the static densities. The Laplacian distribution, especially for the polar bonds, was found to be highly sensitive to the resolution of the data included in the fit and to the treatment of the displacement amplitudes. In an authoritative review on modern high-resolution crystallography Koritsanszky and Coppen~’~’ spent about 40% of their printed space on AIM. The authors claim that AIM provides a powerful tool for the interpretation of X-ray determined charge densities. They give examples (e.g. transannular interaction in syn- 1,6:8,13-biscarbonyl[14lannulene and in tetrasulfur tetranitride) of information being obtained from an AIM analysis well beyond classical atomic connectivity. With few exceptions, a topological equivalence between experimental and theoretical densities (i.e. the same number and type of CPs) exists, and the

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agreement in terms of BCP properties is found to be generally satisfactory, outstandingly good for non-polar bonds but less satisfactory for polar bonds.

4.6 Transition Metals. - Zhurova et al. 176 studied the virtual ferroelectric KTa03 via accurate single-crystal X-ray diffraction. The geometrical and topological coordination numbers of Ta and K are the same: 6 and 12, respectively. At the same time, in spite of the fact that the nearest K - 0 and 0-0 distances are the same, oxygen is bonded to only two Ta atoms and four K atoms. Therefore, the topological coordination number of the 0 atom is only 6. The Ta-0 bond is classified as ‘intermediate’ (between ‘closed-shell’ and ‘shared’). Bianchi et al.177retrieved the experimental electron density of Mn2(CO)lofrom a multipole analysis of accurate X-ray diffraction data at 120 K. A BP connects the two Mn atoms, but no cross BP was found between one Mn and the equatorial carbonyls of the other. The distribution of V2p indicated ‘closed-shell’ interactions for the metallic Mn-Mn bond and the dative Mn-CO bonds. An extensive analysis of the electron density, its Laplacian, a kinetic energy density G, the potential energy density Vand the total energy density E, (= G + V ) , all evaluated at the BCP, leads to a classification. The Laplacian around the CO ligands shows that the carbon centroids of negative charge move toward the Mn atom indicating the polarization sense of the CO molecules. Blatova et a1.178investigated 135 n-complexes of rare-earth atoms (Ln) with Voronoi-Dirichlet Polyhedra (VDP), in anticipation of supposed technical problems with AIM integration. Ordinarily, an agostic interaction is determined by the geometrical analysis of crystal structure through the presence of sufficiently short Ln.. .H distances, taking into account the lengthening R-C bonds in the groups R-C-H.. .Ln. The use of VDP enables one to supplement and define these criteria more exactly. determined the orthorhombic crystal structure of CO~(CO)~(,UBianchi et CO)(p-C402H2)and found that two C-H.. .O bonds connected the molecules. The presence of a Co-Co BCP indicates for the first time the existence of a metal-metal bond in a system with bridged ligands. The BCP properties of the intramolecular bonds and of the intermolecular interactions show features similar to those found in Mn,COlo, confirming the authors’ previously established bonding classification for organometallic and coordination compounds.

4.7 Biological. - Pichon-Pesme et a1.’80 determined the electron density and electrostatic properties of Tyr-Gly-Gly and Gly-Asp from high-resolution X-ray diffraction data at 123 K. Topological properties of the charge distribution are discussed and compared with those derived from other experimental studies on peptide molecules, and the characteristics of the BCPs of the C=O, C-N and C-C bonds are analysed. Dittrich et al. extracted the charge density of glycyl-L-threonine dihydrate from a synchrotron data set collected at 100 K. The topology of the experimental density is analysed and compared with the topology obtained experimentally for the constituting amino acids and to that derived from HF calculations on the isolated molecule. All covalent and HB BCPs as well as the Laplacian CPs were


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located, thereby deriving quantitative topological data for the peptide and side chain bonds. Bond topological indices in the dipeptide compare well with those of the corresponding bonds in the building amino acids, thus suggesting transferability of electronic properties of atoms and functional groups. Discrepancies between theoretical and experimental results could be attributed to crystal field effects. Benabicha et al. 182 obtained the experimental electron density distribution in glycyl-L-threonine dihydrate at 110 K. According to the authors one of the most interesting results is the linear correlation between the A, curvature and the kinetic energy density G at the H.. .O BCP. Abramov et al. ' 83 used experimental X-ray charge densities from low-temperature data in the evaluation of the intermolecular interactions and lattice energies of crystals of glycylglycine, DL-histidine and DL-proline. Ab initio geometry optimizations of the three isolated molecules show that intramolecular HBs were formed in all three molecules, as indicated by the formation of topological BPs. The most striking geometry change occurs in the His molecule since the imidazole ring of this molecule rotates by about 180" to allow intramolecular HB.

5 Laplacian of the Electron Density 5.1 Surface Science. - Aray et af. a n a l y ~ e d the ' ~ ~interaction of a CO molecule with the (100) surface of the fcc d-block transition metals by the topology of V2p. This atomic graph for the top- and second-layer atoms is an octahedron with six vertices or local maxima linked by 12 edges and eight faces joining those vertices. The topology of V2p predicts the orientation of the CO, and although there are several difficulties in using V2p to predict the experimental heats of adsorption, its value at the appropriate CP is useful. Ricart et al. carried out a computational study of C 0 2 activation on Pt( 111) induced coadsorbed K atoms. On the clean surface, C 0 2 is undistorted and weakly bound. However, when coadsorbed K atoms are included in the model, a chemisorbed bent C 0 2 species on top of a surface Pt atom is found. The most salient feature is the appearance of BCPs between C and the Pt atom directly below and between K and 0 atoms of the C 0 2 molecule. The authors judge that the topology reinforces the hypothesis of the direct participation of K in the activation of CO. 5.2 Theory: Electron Pair Localization. - Bader and Heard'86 found that the Laplacian of the conditional pair density, L(e*,r) for same-spin electrons can be homeomorphically mapped onto L(r) = -V2p by placement of the reference electron e* at positions of maximum localization of the Fermi hole for an (a, p) electron pair. Their mapping appeared faithhl in that every maximum in L(r) can be associated with a corresponding maximum in L(e*, r). This mapping indicates that the charge concentrations of L(r) denote the spatial regions in which there is a partial condensation of the pair density towards individually localized electron pairs. The authors claim that, unlike L(r), which is the limiting form assumed by

424


L(e*, r) for complete localization of the Fermi hole of e*, ELF has no direct relationship to L(e*, r). Popelier's inve~tigation'~' into the full topology of - V 2 p detected and classified 43 CPs in water. He discovered the systematic structure of these CPs based on their connectivity and extended the concept of Valence Shell Charge Concentration (VSCC) with its 'mirror image' for which he proposed the name Valence Shell Charge Depletion (VSCD). The core graphs were called Core Shell Charge Concentration (CSCC) and the Core Shell Charge Depletion (CSCD).

5.3 Transition Metals. - In order to further elucidate the non-classical feature of dinuclear transition metal bis( p-q2-silane) complexes Choi and Lin analysed'88 V 2 p on the plane defined by the [Pd(p-)i72-HSi)]2~nit. The Laplacian shows higher electron density concentrations along the Si.. .H BP when compared to other nonclassical mononuclear y2-silane complexes previously studied. Aray et a1.l" analysed the topology V2p of bulk 3d transition metals and zoomed in on the atomic graphs. The authors were able to correlate the value of V 2 p at the local minima with the experimental heats of adsorption for the O2 and CO molecules on the 3d transition metals. Sarasa et a1.190carried out DFT calculations on the organometallic clusters [(q-C5H5)TiNI4and [C13MoN], with the aim of comparing the energies and the electronic structures of the cubane and planar forms for these two clusters. For [(y-C,H5)V(p3-P)],P2 they computed V 2 p at a P-P BCP and found a shared interaction, indicating covalency. In their review on the structure, reactivity, and growth pathways of metallocarbohedrenes M&, transition metal/carbon clusters and nanocrystals, Rohmer et al. 19' spent a whole section on Laplacian applications. 5.4 Heavy Main Group. - Blattner et a1.'92glance at plots of V 2 p for the cation [(NH3)2P(=NMe)2]+. The two donor ligands interact only weakly with the cationic heteroallene. The bonding features of the 'free cation' [P(=NMe)#-, namely the large N-P-N angle and the P-N double-bond character are preserved. D e ~ b e l 'feels ~ ~ that experiments concerning thianthrene 5-oxide (SSO) as a probe for the electronic character of oxygen-transfer reactions need to be reinterpreted. The SSO molecule has a sulfide group, which is attacked by electrophilic oxidants, and a sulfoxide moiety, which is oxidized by nucleophilic oxidants. An AIM analysis of thianthrene 5-oxide reveals that there is an area of charge depletion at the sulfoxide group. The location of this area indicates that the attack of nucleophilic oxidants on SSO is sterically hindered. Therefore, the SSO probe makes oxidants such as dioxiranes appear to be more electrophilic than they actually are.

5.5 Non-linear Optics. - Gopalan et al. 194 looked at experimental charge density to study the effect of the noncentric crystal field on the molecular properties of organic NLO materials. Although extensive lists of BCP properties and Laplacian


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contour maps and CPs are given the authors do not feel called upon to mention any conclusions drawn from them.

6 Hydrogen Bonding

6.1 Reviews. - In her well-written feature article C a l h ~ r d a 'reviewed ~~ HBs of varying strength and agostic bonds described in organic, inorganic, biological and organometallic chemistry. She believes that AIM is very promising way to study these weak bonds and to determine their origin, allowing one to distinguish between agostic and HBs in some ambiguous situations. The quality of the results depends, however, on the level of theory used in calculating p.

6.2 Dihydrogen Bond. - Kulkarni i n ~ e s t i g a t e dthe ' ~ ~existence of intramolecular dihydrogen bonding in the main group elements taking advantage of AIM. The AH,-XH, complexes (with A = Li, B, A1 and X = F, 0, N, C1, S, P) in general do not show intramolecular dihydrogen bonds in their equilibrium structures, but do in the transition state of the dehydrogenation reaction AH,XH,+AH,-I -XHmP1+H2. Interestingly he concludes that formation of dihydrogen bonds seems to be the driving force in dehydrogenation reactions. In his ab initio study on BeH2 as a proton-accepting molecule for dihydrogen bonded systems G a b r ~ w s k i applied '~~ AIM. The interactions of the Ir-H.. .H-X type (X=O, N) were studied19*computationally in models of a neutral organometallic complex and two cationic derivatives. From an AIM analysis it follows that a HB between the hydride and the protonic hydrogen is found only in the neutral complex. Furthermore the only HBs appeared to be formed between hydrogen atoms and fluorine atoms (of the anion19*).The authors pointed out that a short H.. .H distance in Ir-H.. .H-X arrangement is not in itself a diagnostic for a dihydrogen bond. Grabowski a new kind of intramolecular dihydrogen bond, which he calls n-electron delocalization-assisted and studied within the framework of AIM HB criteria. Grabowski2" carried out high-level ab initio calculations on dihydrogen-bonded complexes with HF as a proton-donating molecule and CH4, SiH4, BeH,, MgH2, LiH and NaH as proton-acceptors. The MP2/6-311++G** level of theory is sufficient for a description of dihydrogen-bonded complexes. An AIM examination shows that p at the H.. .H BCP may be a very useful parameter in describing the strength of H-bonding. 6.3 Group 13/15. - Rozas et used the values of p, V 2 p and H at the BCP to characterize bonds in the monohydride and monofluoride derivatives of B, Al, N and P. Surprisingly strong HBs are found in the B derivatives. When pBCpM a.u. the authors refer to a bond as a van der Waals interaction and when pBCp==: lop2a.u. they designate it as a HB. Benzene-A1X3 and ethene-A1X3 (X = H, F, Cl) interactions were investigated202using Frenking's charge decomposition analysis, NPA and the V 2 p contour

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maps in conjunction with MP2 wave functions. The benzene-BX3 complexes are predominantly bound by weak van der Waals forces whereas the benzene-A1X3 complexes are bound by weak chemical bonds. The most interesting facet of the AlX3 complex is the charge reorganization induced on the benzene atoms. 6.4 C-H. , .X.- Masunov et al.‘03 performed HF/D95** calculations on methane, acetylene and HCN in electric fields of various strengths to keep an eye on the shortening of the C-H bond upon HB formation. Electron density analyses using three different methods (Mulliken, NBO and AIM) all showed a shift of electron density from the H-bonding H toward the bulk of the molecule (although they disagreed with each other in several other ways). Vila et aL2@’analysed with AIM the minimum energy Hartree-Fock level structures of methoxymethane. The position and properties of the BCPs, and the variation of several integrated properties of the 0 and H atoms (N(S2), Ip(Q)1, E(s1) and vol(s1)) from the monomer to the dimer, clearly show the presence of C-H.. .O HBs in the five minima of the dimer. The binding energies for every minimum were also obtained and their order has been interpreted in terms of the AIM results. 6.5 Organic. - Tang et ~ 1 . ~performed ~ ’ an AIM analysis on the hexagonal

dimers of C2H5, N2H3and 02H. The electron density at the HB BCP correlated at least semiquantitatively with the HB energy. The Laplacian depletion and Laplacian concentration of the electron density also indicated the formation of the HB. A quantum chemical investigation206of the solvent effects on the competition between the Wolff transposition and 1,2-H-shift in /3-hydroxy-ketocarbenes in aqueous solution was carried out at the B3LYP/6-31G** level. The authors note that the subsequent changes of both the topological nature of p(r) and BCP properties for the intermolecular bonds in the transition states can be extremely useful in analysing the specific solvent effects between these related structures. Marsman et al.207 investigated infinite, undulating chains of intermolecularly hydrogen bonded dione dioximes in the solid state. The experimental density of (E,E)-2,2,5,5-tetramethylcyclohexane1,3-dione dioxime was scrutinized topologically. The description of the two distinct 0 - N bonds was not fully adequate: to obtain negative Laplacian values at their BCPs, hexadecapole parameters for C, N and 0 had to be used in the refinement. In the context of supramolecular chemistry Quinonero et a1.”* performed a topological analysis of the electron density in complexes between derivatives of squaric acid and NH4+.There are excellent relationships between either p or V 2 p at the RCP (arising upon complexation) and geometrical and energetic parameters. Grabowski and Pogorzelska’” performed MP2/6-3 1 1++G** calculations on modelled systems of water dimers with fixed short 0-H donating bonds to show that such dimers are energetically stable. The results suggest that short 0-H bonds may be stabilized by intermolecular forces in crystals. The 0-H.. .O bonds are analysed with AIM.


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6.6 Cooperative Effect. - Gonzalez et al.2’0studied the structure and the relative stability of the ethanol dimer and the cyclic ethanol trimer using DFT. Cooperative effects are reflected in the electron density of the BCPs. Rincon et al.’” investigated the energetic, structural, electronic and thermodynamical properties of (HF),, in the range n = 2-8, by ab initio methods and DFT. An AIM analysis reveals a linear correlation between the binding energy per HB and the density at the BCP and a covalent bond order. Based on these correlations HB cooperativity is associated with the electronic delocalization between monomers units. 6.7 Blue-shifted. - Cubero et a1.2’2used AIM to examine the nature of what they termed the ‘anti-hydrogen bond interaction’. Upon formation of the anti H-bond the X-H bond shortens and its vibrational frequency blue-shifts contrary to a Hbond. Amongst others the topological properties of p have been determined for a series of C-H.. .ncomplexes, which exhibit either anti H-bond or normal H-bond character. According to a set of topological criteria2I3 utilized to characterize conventional H-bonds no relevant difference was found in the two classes of CH . . .ncomplexes. Since the anti H-bonds are just a special type of H-bond the qualifier ‘anti’ is misleading and was abandoned in later work. Mucha1l2I4focused on the the anti-hydrogen bond in aromatic N-sulfinylamines with ortho H atoms. From AIM she learns that a C-H.. .O bonding interaction exists that meets all the characteristics of an anti-hydrogen bond.

6.8 Biochemical. - Different modified DNA bases have been identified by treating mammalian cells with hydrogen peroxide (HP) and it has been proven that the biological effect of HP is mediated by hydrogen bonded HP adducts, which could be due to the enhanced HP stability by adduct formation. To shed light on this issue Dobado and Molina’” investigated topologically the hydrogen bonding appearing in complexes between adenine and HP They discovered six cyclic complexes showing two HBs of different strength. Fidanza et al. * I h perfomed a conformational and topological analysis of the charge density in guanine-a-dicarbonyl adducts. It follows from the analysis of the structural results and AIM that the strength of the HB interaction between the iminic H of the guanine and the ketonic 0 follow the order: the adducts G-methylglyoxab G-glyoxab G-phenylglyoxal and in ketones: the adduct G-diacetyb G-phenylbenzoyl. With few exceptions this ordering is in agreement with the mutagenic activity reported before. Vank et al.2’7 investigated the effect of replacing 0 with S and Se in terms of successive protonation of Po43-, SP033- and SeP033-.No HBs were found in any of the neutral or anionic species according to an AIM analysis of the wavefunction at the B3LYP/6-3 11lG(d,p) level of theory.

6.9 With Ions. - Daza et aL2’*compared the geometrical parameters, hydrogenbonding properties, vibrational frequencies and relative energies for several hydrogen peroxide.. .X (F-, C1-, Br-, Li+, Na+) complexes. The molecular interactions have been characterized using AIM. The HB interaction showed the

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disposition of maxima in V2p(r) in C1- towards V2p(r) minima in the hydrogens. On the other hand, the positive V2p(r) region on Naf is directed towards the maxima in V2p(r) on the oxygen’s electron pair.

6.10 Isotope Effects. - Gawlita et ~ 1 . researched ~ ’ ~ the variation of the C-D vibrational stretching frequency in primary and secondary alcohols. They affirm that the bond order of C-D in the 2-propanol-methoxide complex decreases by 7.5%, erroneously assuming that V 2 pdetermines this quantity. 6.11 Low Barrier. - Rozas et examined the HB basicity of a series of ylides containing N, 0, or C as heavy atoms, as well as the influence of the formation of the complexes on their structure. They arrive at the following classification: weak HBs ( E < 12.0 kcal mol-’) yield positive values for both V 2 p and the energy density H at the HB BCP, medium HBs (12.0 < E < 24.0 kcal mol-’) have V 2 p > 0 and H < 0, while strong HBs (and hence low barrier HBs) ( E > 24.0 kcal mol-’) have V 2 p < 0 and H < 0. Pantano et ~ 1 . used ~ ~ Car-Parrinello ’ MD to compute the influence of thermal fluctuations on the proton hopping properties of a model substrate for enzymatic reactions, benzoylacetone. A plot showing the projection of ELF on the plane defined by the 0-H-0 moiety unambiguously demonstrates the presence of electron pair density between the hydrogen and the two H-bond donors, suggesting the presence of a three-centre covalent bond. The character of the bond is 0,as evidenced by analysing the Kohn-Sham one-electron wavefunction, in full agreement with AIM. Vishveshwara et aZ.222mention the growing evidence that some enzymes catalyse reactions through the formation of short-strong HBs. They investigated with MP2 and DFT the process of proton transfer between hydroxyl and imidazole groups, which serves as a model of the crucial step in the hydrolysis of RNA by the enzymes of the RNase A family. The nature of bonding between the catalytic unit of the enzyme and the substrate in the model is looked at with AIM. 6.12 Intramolecular. - Gilli et embarked on a combined crystal-structural, IR and NMR spectroscopic, and QM investigation in search of evidence for intramolecular N-H. . .O resonance-assisted hydrogen bonding in P-enaminones and related heterodienes. The most controversial assumption of their electrostaticcovalent H-bond model, namely the covalent nature of very strong 0-H.. .O bonds has been suggested by a number of different authors, but has only recently received authoritative support by very accurate X-N electron-density measurements that identified a ‘covalent’ BCP with a negative Laplacian along the H. . .O bond in both (-)CAHB and RAHB cases. used ab initio calculations to examine fluorinated cis o-cresols. Jensen et An AIM analysis confirms that there is indeed an HB between the H and F atoms in all three cases. This HB along with the adjacent atoms can be considered to be part of a hexagon, which is twisted above or below the plane of the phenyl ring. Pacios et al.225looked at 13 conformers of non-ionized glycine identified previously in ab initio correlated calculations at the MP2/6-3 11++G** level by


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means of AIM. They identified intramolecular HBs in some structures. The intramolecular effects related with the stability of every conformer were explored by analysing the potential energy contributions arising from the molecular fragments NH2, CH2, CO and OH computed within the AIM framework. This AIM study of intramolecular interactions complements previous studies on glycine based on traditional empirical considerations. Ferreiro and RodrigUez-Otero226conducted an HF study with simple basis sets on 1,4-, 1,5- and 1,8-dihydroxyanthraquinone,which exhibit two intramolecular HBs. Using AIM the HB strength was assessed to be quite strong and markedly additive in nature despite the substantial stiffness of the studied molecules. Pacios and G ~ m e z ~generated ~’ the MP2/6-3 11++G** wavefunctions of 13 stable conformers of gaseous glycine. They used AIM to examine the intramolecular interactions and explored different intramolecular effects in every conformer via atomic and BCP properties. The existence of intramolecular HBs on some conformers was demonstrated, while the presence of other stabilizing interactions arising from favourable conformations was shown to explain the stability of other structures in the PES of glycine. Dubis and Grabowski228examined conformational isomers of methyl pyrrole-2carboxylate by IR and RHF calculations. In one of the conformers they emphatically found no H . . .O BCP and could hence rule out an intramolecular HB. G r a b ~ w s k estimated i~~~ the strength of 0 - H . . .O intramolecular HBs via AIM operating on MP2/6-3 1 1++G** wavefunctions of malonaldehyde derivatives. Correlations between the HB strength and topological parameters are discussed.

6.13 x-Systems. - Tarakeshwar et al.230looked at the interaction of the water dimer with both the olefinic and aromatic n-systems using both the supermolecular MP2 and Symmetry Adapted Perturbation Theory (SAPT). They show how the water-n interaction influences the HB bond characteristics of the water dimer by evaluating the HB strengths, via BCP properties and Espinosa’s work. G r a b ~ w s k i ~applied ~’ AIM to characterize HBs in HF.. .H20, HF.. .NH3, HF. . .LiH, (H20)2,(HCOOH)2, C2H2.. .H20, H 2 0 . . .NH3 and (C2H2)2complexes. For such heterogeneous samples he introduced a new measure of HB strength. 6.14 Kinetic Energy Density. - Galvez et aZ.232studied approximate kinetic energy density for intermolecular regions in HB dimers. They followed Abramov’s approach, which computes G(r) fiom p ( r ) , to show that it can be reliably used in spatial intermolecular regions in HF and water dimers.

6.15 Organometallic. - Alkorta et al.233studied the ability of the (CO)4C0complex to act as an HB acceptor. In order to consider the importance of the formal negative charge of this compound in the interaction with other systems, the isoelectronic neutral (C0)4Ni complex has been considered as an HB acceptor. Electronic changes have been assessed with AIM and NBO. The lack of a formal charge in the Ni organometallic compound seems to be responsible for the absence or extreme weakness of the HB complexes. In other words, even though

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the Ni complex is isoelectronic to the Co one the charge in the metallic atom is responsible for the HB formation. Clot et aZ.234were preoccupied with the question of how HB affects ligand binding and fluxionality in transition metal complexes. Their DFT study on interligand HBs involving HF and H 2 0 benefitted from AIM via the presence or absence of crucial BCPs and their properties.

7 Topology of Other Functions 7.1 ELF. - In order to understand depleted homopolar bonds better Llusar et aZ.235 performed an ELF study of the N-N, 0-0 and F-F bonds. Deformation density maps are known to show a depletion of p between formally covalent bonds. However a topological analysis of the ELF function demonstrates that bond strength appears to be correlated with the disynaptic basin populations. From a qualitative point of view, the splitting of a disynaptic basin into two monosynaptic ones upon bond stretching is the signature of the covalent bond. Noury et aZ.236reported on the details of the algorithms behind the generation of the 3D ELF grids, the assignment of ELF basins and integrations of properties over these basins. The issue of speed versus accuracy is discussed. Since the discovery of extensively delocalized optically accessed chargetransfer states of solid Xe doped with halogen, noble gas clusters containing charged atoms have been of great interest. Ionic clusters trapped in condensed phases provide a fascinating interface between isolation and continuous matter. In this wider context a DFT in~estigation~~’ has been carried out to determine the equilibrium geometry, binding energy and vibrational properties of Xe2H3+.The bonding properties of this compound have been studied according to topological analysis of ELF, which indicates that the terminal hydrogens are bound by covalent interactions. The centre H is bound to the neighbouring xenons mainly by electrostatic forces even though a non-negligible fraction of covalent nature is found. The bonding in hypohalous acids has been investigated from the topological analysis ELF at the B3LYP and HF level by Berski et aZ.238They notice many interesting facts, for example that the very large values of the relative quantum fluctuation (lambda) is >0.8 for V(O,X), suggesting that the covalent electron density is almost entirely delocalized over other basins. The topological analysis of the ELF function supports the concept of probonded electronegativity and its usefulness as a tool for prediction of the nature of the oxygen-halogen linkage. Joubert et aZ.239used DFT to determine the structure and relative stability of Al-containing species involved in cryolitic melts. A concomitant topological analysis of ELF proved to be a powerful tool to obtain insight into the bonding properties. Fourre et aZ.240found that their ELF-based topological description of the chemical bond provides a deeper insight into the bonding evolution upon excitation, ionization, and electron attachment in carbonyl and imine compounds than the classical MO point of view. The changes occurring in the valence basin


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populations are, on the one hand, consistent with the findings of the classical descriptions in terms of delocalized MOs or resonance structures and, on the other hand, provide a chemical picture of the electron cloud reorganization. Fuentealba and S a ~ i n looked ~ ~ ' at the electronic structure of the ground state of monoxides MO and carbides MC2 with M an alkaline-earth metal atom. From their ELF study they learn that in the monoxides, and not in the carbides, a change in the type of bonding occurrs in going fiom Be to Ba. Be0 is the only compound presenting some covalent bond character while the other oxides are clearly ionic. All carbides are ionic and bent. C h e ~ n u tused ~ ~ ~ELF to study the bonding in simple four-coordinate species involving nitrogen and phosphorus and compared it to that in their conventionally singly and doubly bonded counterparts. Despite evidence suggesting the presence of a conventional multiple bond in certain cases of the four-coordinate species, the ELF study shows this not to be the case. Joubert et al.243presented a systematic study of the structural and bonding properties of selected lanthanide trihalide molecules, LnX3 (Ln = La, Gd, Lu; X = F, Cl). An ELF analysis revealed typical ionic bonding properties, emphasizing the increasing ionic character of the Ln-X bonds through the rare-earth series. Moreover the authors pointed out a strong distortion of the outer core shell of the metal. Armed with ELF Fuster and Silvi244looked at hydrogen bonding. In the weak HB case, the reduction of the localization domain yields two domains in the first step, which can be partitioned afterwards into valence and core domains. In contrast, for medium complexes the core-valence separation is the first event, which occurs during the reduction process, and therefore the complex should be considered as a single molecular species. The symmetrical strong HB has a protonated basin V(H) at the bond midpoint, a topology indicative of an incomplete proton transfer and a rather covalent bond. carried out an ELF analysis on five-membered (C4H4NH, Fuster et C4H4PH, C 4 H 4 0 and C4H4S) and six-membered (CSHSN,C5H5P) heterocycles. The bonding in these molecules is discussed on the basis of the valence basin populations. The authors show that the ELF values at the (3,-1) CPs between disynaptic basins related to a given centre provide a criterion to determine substitutional sites. In the five-membered rings, except pyrrole, the heteroatom lone pair(s) appear to have populations slightly larger than 2e-, which is not consistent with the standard resonance picture unless a strong a-back-donation compensates the n-donation. Using ELF'S topology Fuster and proposed a simple rule to determine protonation sites in bases, namely that the protonation occurs in the most populated valence basin of the base that yields the least topological change of the localization gradient field. Their description supports the Legon-Millen model, which generalizes the VSEPR concepts to HB complexes. Fuster et al.247 performed a topological ELF analysis to investigate electrophilic aromatic substitution. In the case of n-donor substituents the aromatic domain is first opened close to the substituted C and then in the vicinity of the metu C, whereas for attractor substituents it is first opened in the ortho and para

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positions. They suggested that the local Pauli repulsion plays a noticeable role in the orienting effects, which are complementary to the charge transfer effect involved in standard quantum chemical pictures. Silvi and Gatti248were interested in a direct space representation of the metallic bond and for that purpose performed periodic HF calculations on the bcc lattices of Li, Na, K, V, and on the fcc ones of Al, Ca, Sc, Cu. A topological analysis of p and of ELF reveals that, except for Al, the valence basins have a synaptic order larger than 2, and that the metallic bond appears to be a partial covalent bond, which is often multicentric and characterized by a low population of the valence basins (less than 1.Oe-) and by synaptic orders as large as 6. In their article on RHC=NO iminoxy radicals Berski et al.249used a topological analysis of ELF supplemented with AIM. It was shown that 2 to E isomerization resulted in a redistribution of electron density from the N and 0 lone electron pairs to the N-0 and C=N bonds. The AIM analysis of E isomeric forms showed quite the same trends for both charge and spin reorganization within the atomic basins. ~ ’ ~ the topological analysis of ELF as a mathematical Chevreau et ~ 1 . presented vision of Lewis theory. The authors mused on whether the chemical bond is a myth or reality and affirm that their method partitioned molecular space into regions with a clear chemical sense. Calatayud et al.251 generalized the molecular isodensity contour analysis originally proposed by Mezey to ELF. In their approach the basins are ordered with respect to ELF function values at the CPs that determine the reduction of the reducible localization domains. They show how tree diagrams can be used as a generator of mathematical definitions of chemical concepts and illustrate their new tool on VO, and VO,+ ( n = 1-4).

7.2 Electrostatic Potential. - An investigation252of the topology of the MEP led to a chemical interpretation of the CPS in terms of lone pairs, n bonds, hybrid orbitals and other electronic structure elements. A Poincare-Hopf relationship for the MEP connects electronic structure elements and electrostatic reactivity. Martinez et al.253investigated the topology of the electrostatic potential of Nb30, Nb3S and Nb,Se. The different relative stabilities of two- and threedimensional structures can be explained on the basis of BCPs in the molecular electrostatic potential. The change in the nature of the bonding between the three compounds cannot be explained by hypervalency or by a qualitative change in the molecular orbitals or orbital energies.

7.3 Intracule-Extracule. - The properties of the electron intracule and oneelectron densities of the ground state of harmonium were determined254 and expressed in terms of asymptotic expansions and Pade approximants. Topologies of both densities were investigated in detail. An attractor composed of a cage critical point and a (1,- 1) critical sphere. Fradera et ~ 1 . ~reported ’~ on the topological features of molecular intracule I and extracule densities E and their Laplacian distributions computed at the HF and CISD and interpret them in terms of VB structures. Within this framework

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+

they also studied the dissociation reactions H2 -, H' + H' and H3+ + H2 H+. The evolution of the values of I and E , in those points associated with VB structures, is found to be in qualitative agreement with the change of the weight of each VB structure to the total wavefunction along the reaction coordinate. The contributions of the correlated and uncorrelated components of the electronpair density to atomic and molecular intracule I and extracule E densities and their Laplacian functions were a n a l y ~ e dat~ ~the ~ HF and CI level. The correlated components of the I and E densities, and their associated Laplacian functions, reveal the short-ranged nature and high isotropy of Fermi and Coulomb correlation in atoms and molecules. In general, it has been found that the uncorrelated I and E have the same topological structure as the parent functions I and E functions. Fradera et aE.257 analysed electron-pair interactions fiom intracule I and extracule densities E (and their Laplacians) in the H transfers in hydride, proton and H radical reactions, modelled by CH3/CH4+, CH3/CH4-. The electronic nature of the H atom being transferred in the three systems can be differentiated by the topologies of the corresponding I and E. However, the analysis underlies also the ciifficulties in interpreting the topologies of contracted electron-pair densities, as different electron-electron interactions may contribute to the same point in the intracule or extracule spaces.

8 Reactions 8.1 Organic. - Meng et aZ.2s8traced the intrinsic reaction coordinate of the reaction of triplet 0 with CS2 and analysed its stationary points. The topological analysis of p was employed to investigate the changes of chemical bonds in the reaction process. Wiberg and ShobezS9examined the structures of the spiropentyl cation and its rearangement products at correlated level. Based on AIM charges they found that most of the charge in the cations is located in the hydrogens. Remarkable differences in the retention transition structures between the 1,3silyl migrations in formylmethylsilane and the 1,3-silyl migrations in allylsilane are revealed by detailed analysis26oof geometries, natural bond orbitals, and the Laplacian of the electron density. The presence of BCPs and RCPs was monitored and some typical properties were evaluated at the BCPs. Calculations at HF/6-31G* and B3LYP/6-31++G** have been used to study the mechanism of the cycloaddition reaction between thioketene and thioformaldehyde.261 In addition an AIM analyis was carried out to study the electronic structures of stationary points for the two reactions under investigation. With an eye on mass spectrometry the mechanism of the formation of cationic fragments from peptides was studied262at Hartree-Fock level. In addition to the energetics of the fragmentation, the electronic structure and bonding of the main stationary points have been analysed via AIM. Bond making and breaking was followed in detail, both numerically as well as visually by superimposing evolving molecular graphs on contour maps of the Laplacian of p. Thamattoor et al.263 investigated the ring expansion of 1-methylcyclopropylcar-

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bene to 1-methylcyclobutene experimentally and computationally. They showed that the barrier to ring-expansion is significantly smaller in 1-methylcyclopropylcarbene than in cyclopropylcarbene. The origin of the increased rate of ringexpansion is due to stabilization of the positive charge that occurs at the incipient tertiary carbon that is attached to the migrating carbon centre. Both the AIM and NPA charges are in agreement but curiously the NPA charges are larger in magnitude, which is unusual. The gas-phase reaction of hydrochloric acid and ethylene to give chloroethane was investigated2u in the framework of spin-coupled theory. The authors found little evidence supporting the occurrence of a four-membered transition state. AIM confirmed that the breaking of the H-Cl bond occurs before the C-Cl bond formation. A computational study265on the dimerization reactions of ketene imine and bis(trifluoromethy1)ketene imine monitored the topology of its four-membered ring transition states. Gas phase reactions between C+(*P)and formic acid were studied266at the G2 level. The insertion of C+ into the C-H bonds apparently led to the formation of a COC three-membered ring. However, an inspection of the topology of p of these structures shows that there is no BCP between the CH group and the oxygen atom, and therefore the CH is only bonded to the other carbon atom. Molina et al.267used AIM and ELF together with NMR chemical shifts to investigate three-membered heterocycle oxides of N, P, and As. An aromatic stabilization was found for the P and As rings. However, the N derivatives displayed a net negative hyperconjugation in the N - 0 bond formation, without ring aromaticity observed for their electronic properties. The reaction paths of their Meisenheimer rearrangements have been examined by AIM and ELF at the B3LYP/6-3 11 G* level. Sakata268does not believe that a method for a qualitative understanding of the wavefunction with high accuracy has yet been established, nor that it is known how to connect the change in a wavefunction along the reaction coordinate with the enhancement or depression of chemical reactivity. Although not using it Sakata called AIM an attractive attempt to clarifL these issues. In their article on intramolecular fluorine migration via four-member cyclic transition states Nguyen et al.269analysed p values at BCPs in transition states of fluorinated aldehydes and acyclic ketones reactants at the B3LYP/6-3 lG**// B3LYP/6-3 1G** level. Fang and Li270performed a computational study of cycloaddition reactions between ketenimine and olefin. For the model reaction (H2C=C=NH H2C=CH2), the stationary points are located at the HF/6-31G, MP2/6-31G* and B3LYP/6-31++G** level. The topological analysis shows that both transition states are open ring structures. Vivanco et al.271queried the origin of the loss of concertedness in pericyclic reactions. They studied several [3+2] thermal cycloadditions between azomethine ylides and nitroalkenes both computationally and experimentally. An analysis of the energy density H at relevant BCPs reveals that the lithium-heteroatom interactions are electrostatic.

+


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The reactivity of uracil thio derivatives has aroused great interest in chemical investigations because of their biological and pharmacological activity. Lamsabhi et al.272confirmed that thiouracils are bases in the gas phase. In all cases, the protonation takes place at the heteroatom attached to position 4. In the neutral and protonated species p at the C4-C5 BCP significantly increases upon protonation, while that of the C5-C6 bond sizably decreased. Wang et aZ.273examined the mechanism of the cycloaddition reaction between isothiocyanic acid and methylenimine and performed an AIM analysis on the encountered stationary points. Reid et al.274studied hydroxycarbene as a model for the homolysis of oxy- and dioxycarbenes. The atomic and molecular properties are examined using the theory of atoms in molecules. Curiously, the authors interpret the fact that p for the 0 - H BCP decreases to approximately zero at long distances as behaviour is consistent with the breaking of the 0 - H bond. Other unusual statements are made. Arnaud et al.275performed a detailed study of the addition of HCN to methanimine in the gas phase, which is the key step of one of the most realistic processes suggested for the prebiotic synthesis of amino acids. A wealth of BCP properties is monitored during the process of bond making and breaking. One of the most important aspects of the Diels-Alder reaction is the understanding of the factors influencing the endolexo stereoselectivity. To analyse the origin of the Diels-Alder stereoselectivity, the [4+2] cycloaddition of furan and maleic anhydride has been studied experimentally for many years as an interesting model system and revisited computationally by Calvo-Losada and S ~ a r e z . *At~ ~the MP2/6-31G* level the electron density of the endo has an additional RCP and a CCP. Wiberg and Freeman277re-examined the kinetics of the base-catalysed pennanganate oxidation of benzaldehyde. The stability of the dianion radical, PhC0,- , is interesting in itself. The AIM analyis of its MP2 wavefunctions agreed with the Mulliken population analysis that the C 0 2 group had a unit negative charge and the phenyl ring had the odd electron and a negative charge. Thus, the low dissociation energy derives at least in part from the possibility for transferring the odd electron of the radical as it is formed to the phenyl ring, thus separating the two negative charges. Poater et al.278analysed the effects of solvation on the pairing of electrons in molecules in a series of molecules and in the Menshutkin reaction between ammonia and methyl chloride. AIM atomic populations and localization and delocalization indices have been used for describing the electron-pair characteristics. This analysis shows that solute-solvent interactions modify the electronpair distribution of the solute increasing the polarization of the molecular bonds. Poater et al.279presented new insights in chemical reactivity reactivity drawn from an analysis of electron pairing. They judged that a physically accurate description of the electron pairing in atoms and molecules had to be based on the electron-pair density. Within AIM atomic localization and delocalization indices could be defined that describe the intra- and interatomic distribution of the electron pairs in a molecule. The authors considered these indices as a physically

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sound and numerically accurate extension of the Lewis model. They studied the electron-pair reorganization taking place in five different reactions: two intramolecular rearrangements, a nucleophilic substitution, an electrophilic addition and a Diels- Alder cycloaddition. For each reaction they performed a comparative analysis of the electron-pairing patterns in reactants, transition states, and products. Valdes and Sordo280studied the PES of pentamethylene sulfide.. .HCl at MP2 level. An AIM analysis shed some light on the nature of interactions in the chairlike conformers. The values for p, V 2 p and the energy density at the relevant BCP for both complexes are consistent with a shared interaction. Interestingly the energy density of the equatorial conformer (that being more stable according to experimental data) is clearly more negative suggesting a higher degree of covalency in the interaction. Other BCPs were located and used for interaction characterization.

8.2 Inorganic. - Aray et aZ.28’ correlated the topology of p of pyrite-type transition metal sulfides with their catalytic activity in hydrodesulfurization. The most active catalysts are characterized by intermediate values at the M-S BCP. This result supports the consistency of transition-metal-sulfide-catalysed hydrodesulfurization with the Sabatier principle. mentioned a Laplacian application in their review. Niu and Fortunelli et al.283 focused on the protonation mechanism of a compound containing a Pt trianguZo cluster and characterized a kinetic intermediate via NMR and DFT calculations. The authors used GAUSSIAN98’s unsupported AIM analysis to give a more quantitative idea of the strength of selected bonds by simply locating the BCPs. They find a substantial reduction in a CO bond order and a concomitant strengthening of the C-Pt interaction. Delabie et al.284used DFT and CCSD to examine reactions of 3d transition metal atoms with N20, producing the metal oxide and N2. It is shown that AIM charges show more ionic bonds compared to Mulliken Population Analysis, but the two methods show the same trend along the 3d series, namely that charge transfer from the metal to N 2 0 becomes more important when 2 increases. Ijjaali et al.285explored the singlet and triplet PES involved in the gas-phase reactions between N+ and ammonia with high-level ah initio techniques. The electron density at the BCP for the singlet is slightly higher and the energy density more negative than for the triplet. 8.3 Transition Metals. - Teruel and Sierraalta undertook286ab initio calculations on M o ( O S ~ H ~ ) L ( S ~ C Nmolecules, H~)~ for L = C1, Br, I, NCS and CN, to study the effect of ligands on the reactivity of siloxy compounds and performed an AIM analysis. For example, the analysis of - V 2 p ( r ) CPs at the Mo valence shell showed a local charge concentration region susceptible to electrophilic attacks but only in the case of L = CN. However, for L = C1, Br, I and NCS, local charge depletion sites were found in a bond-free region. Luna et aZ.287examined the gas-phase reactions between Cu+ and urea by means of mass spectrometry techniques and a computational study of relevant


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reaction paths. Several bonds were categorized via BCP properties AIM whereas charges were obtained from NBO. 8.4 Mass Spectrometry. - According to Oliveira et a1.288the most distinct characteristic of gas-phase ion chemistry is that the molecular ions so generated are stable species even though the corresponding neutral counterparts are not at all stable, which renders the gas-phase ion chemistry very rich and versatile. They use the presence of BCPs to understand interactions in the C2H4Br+system and to deny the existence of a ring structure in a transition state.

8.5 Rotation Barrier. - Platts and C ~ o g a nreported ~ ~ ~ DFT calculations on the various conformations of tetraformylhydrazine and rationalized the barrier height with AIM. It is found that the barrier to rotation about N-N is dominated by destabilization of N atoms and weakening of the N-N bond, and that the balance between steric and electronic factors determines the overall barrier. 8.6 Biological. - Topol et al.290performed HF and B3LW calculations to analyse the reaction profiles of threc. cyclic disulfide species with model zinc finger domains in the HIV-1 nucleocapsid protein (NCp7). They noticed that NPA atomic charges provide valuable information about p, and are mostly consistent with the AIM data. Both types of qualitative analysis help to understand binding properties in the molecules, to recognize attacking points, and to select a reaction coordinate for the reagents considered in this paper. According to the conclusions of the AIM and NBO analyses, an attack of the electrophilic agent should be directed on the weakest bond in the zinc-finger domain, namely, on one of the Zn-S bonds.

9 Ionic Materials 9.1 Thermodynamics. - Pendas et ~ 1 . applied ~ ~ ' AIM to partition static thermodynamic properties in condensed systems, with special attention for the definition and behaviour of atomic compressibilities. Transferability of basins together with their properties among compounds seems feasible, provided that certain volume requirements are met.

9.2 Phase Change. - Blanco et al.292applied Bader's topological analysis to the study of the B 1-B2 phase transition in the alkali halides. The results shed light upon the phase stability rules of the traditional ionic model: by using topological ionic radii, a connection between the topologies of p and the energy surface is found. The topological description of the transition that emerges puts an emphasis on the creation of two new BCPs in passing from coordination 6 to 8, and makes a unique definition of structural change. Topological isomerization, as a case of structural change, is mainly dominated by geometric relations involving topological ionic radii. Further relations between the structural diagram and the energy surface features are also investigated.

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Prompted by the need for a theoretical formulation of the mechanisms of solid-solid transitions in various areas of condensed-matter physics Blanco et al.293proposed a symmetry-based, non-displacive mechanism for the first-order B3 B 1 phase transition exhibited by many binary semiconductors. Potential energy surfaces and an AIM analysis were used to describe the atomic rearrangements, the energy profile along the transition coordinate, and the effects of the external pressure on this profile.

*

9.3 Impurity/Doping. - Baruah et aZ.294focused entirely on p’s topology of impurity doped small Li clusters (Li,Mg and Li,Be clusters for n = 1-6). The authors reported that Li-Li bonds were observed for clusters with n 4 in the Li,Mg series, while for n 3 5 the Li atoms form bonds only with the impurity atom. These statements are incorrect since they ignore the presence of nonnuclear attractors, which were nevertheless extensively mentioned.

10 Spectroscopy

Electron transmission spectroscopy was used by Venuti and M ~ d e l l ifor ~~~ determining the energies of vertical electron attachment to the empty n* orbitals of ethene, 1,4-~yclohexadiene, 1,4,5,2?-tetrahydronaphthalene and 1,4,5,8,9,10hexahydroanthracene, where the number of ethene double bonds, which interact through space and through the CH2 bridges, increases along the series. The authors claim that Mulliken atomic charges are not very reliable and basis set dependent, which is why they resorted to AIM. Their results are quite different from those previously reported, i.e. the negative charge on the ethenic carbon atoms is predicted to be one order of magnitude smaller, -0.04 and -0.05 in ethene and 1,4-~yclohexadienerespectively. Moreover the potential variation upon CH2 substitution goes in the opposite direction.

11 Opinions and Plans Some authors do not make use of AIM but air certain views about it in their papers or express an interest in applying it in the near future. For example, Abramov et al. report296on the evaluation of molecular dipole moments from multipole refinement of X-ray diffraction data. They promised that a further analysis would include integration of the basins obtained from the experimental electron density in order to obtain further information about the observed differences between theory and experiment. Wang et aZ.297 would like to pinpoint the overall charge transfer between the metal and the adparticle in their study on lateral interactions in coadsorbate layers. They believe that a rigorous implementation of these ideas is impossible because the charge and dipole moment on the adparticle are not observables in the quantum mechanical sense. They commented that several approximate schemes have been


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devised to define charges and dipole moments on adparticles, such as the Mulliken population analysis and AIM but concluded that none of these schemes is unique. Lamzin et u Z . ~ ~ * mention that the electron density multipole parameters obtained from small molecules have been proposed to be transferable to macromolecules, which they regard as a crystallographic adaptation of AIM. They claim that, although the transferability had been demonstrated to give good results for polypeptides, its applicability to proteins had not previously been shown. Cirujeda et al. 299 think that defining the space of the molecule that corresponds to a given atom is not an easy task and can only be done in a rigorous quantum mechanical form within AIM. Furthermore they state that because the computation of the AIM atomic spin population is very costly one normally resorts to simplified and approximate procedures, such as the Mulliken population analysis. Such a method is fast, but its results are dependent of the unitary transformations of the MOs. Romero et ~ 2 1 studied . ~ ~ ~the character of the HB in ice I-h. They did not find evidence for any covalent nature of the HB and quote the view of AIM on this matter, which classifies the H-bond interaction as between closed-shell molecules, similarly to van der Waals compounds, which is in line with their conclusions. Finally it appears that sometimes AIM is misunderstood. For example, Joshipura et ~ 1 . ~ "believe that the absence of overlap between atomic basins is a weakness of AIM. In their paper they misrepresent the electron density of a diatomic as a mere sum of atomic densities, between which there is supposedly no overlap. Of course, the atomic basins have been obtained from the diatomic's actual electron density, whithout recourse to a simple superposition of spherical atoms, so in summary their criticism is actually beside the point.

12 Conclusion

The chemical community at large generally considers AIM as a respectable theory with many appealing features. It has come to embrace AIM as a useful tool to extract chemical insight from ab initio calculations. The span of research areas that employ AIM concepts and discuss AIM properties is astonishingly wide, as can be seen from the Table of Contents. However, currently the emphasis remains on bond characterization via BCP localization and the evaluation of typical properties at them. As far as atomic charges are concerned, they appear more frequently than in the first report and are quite often given in conjunction with NBO or NPA. The use of AIM bond orders is also on the increase. In summary, we are confident that the topological approach (including ELF analysis) is steadily becoming a new paradigm in the quest for chemical insight from modern wavefbnctions.

13 Disclaimer The primary set of references for this manuscript was retrieved from the Web of Science (an IS1 MIMAS service). The completeness of the first automated selection of papers depends on the accuracy of the information present in this

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database, the determination of the non-trivial publication time window (July 1999 to June 2001) set by our publisher, and the accuracy of the reference to Bader’s 1990 monograph in the original paper. Papers containing AIM-related work without referring to this monograph were unfortunately not included. The vast majority of papers were screened electronically as pdf files, otherwise a hard copy was inspected, if readily available from our local libraries. Papers were included only if they actively make use of AIM or comment on it in sufficient depth. We apologize in advance to authors for the possibly erroneous omission of their work but we believe that we have provided a complete and fair reflection of the research activity in the AIM area.

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