Chemical Modelling Applications and Theory
Volume 1
A Specialist Periodical Report
Chemical Modelling 
Applications and Theory Volume 1 A Review of the Literature Published up to June I999 Senior Reporter A. Hinchliffe. Department of Chemistry, UMIST, Manchester, UK Reporters F.M. Aicken, UMIST, Manchester, UK J. Ladik, FriedrichAlexander University ErlangenNiirnberg, Erlangen, Germany R.I. Maurer, University of Essex, Colchester, UK S.E. O'Brien, UMIST, Manchester, UK P.L.A. Popelier, UMIST, Manchester, UK D. Pugh, University of Strathclyde, Glasgow, UK P. Pyykko, University of Helsinki, Finland C.A. Reynolds, University of Essex, Colchester, UK T. E. Sirnos, Democritus University of Thrace, Greece M. Springborg, University of the Saarland, Saarbriicken, Germany H. Stoll, University of Stuttgart, Germany S . Wilson, Rutherford Appleton Laboratory, Chilton, Oxfordshire, UK
.

ROYAL SOCI€IY OF CHEMISTRY
ISBN 0854042547 A catalogue record for this book is available from the British Library.
0The Royal Society of Chemistry 2000 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 U K 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 writing of The Royal Society of Chemistry, in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK,or in accordance with the terms of the licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to The Royal Society of Chemistry at the address printed on this page. Published by The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 OWF, UK For further information see our web site at www.rsc.org Typeset by Paston PrePress Ltd, Beccles, Suffolk Printed by Athenaeum Press Ltd, Gateshead, Tyne and Wear, UK
Preface
Richard Dixon and Colin Thomson were the Senior Reporters for the Specialist Periodical Reports ‘Theoretical Chemistry’, which ran for four Volumes. Richard summarized the state of the art, 1974, in his Foreword to Volume 1 when he wrote: ‘This is the first volume of the biennial series of Specialist Periodical Reports devoted to Theoretical Chemistry. Theoretical Chemistry is an extremely wide subject, since it provided the background for the interpretation of so many chemical phenomena, and it is therefore necessary to define the scope of these volumes. Quantum theory plays an important role in theoretical chemistry, both through the application of valence theory to the interpretation of molecular structure, and also in the development of spectroscopic models based on quantum mechanics, which are used in the determination of structural information from experimental spectroscopy. Indeed, to many chemists theoretical chemistry is synonymous with quantum chemistry. Quantum chemistry will thus constitute a major part of this series. There is, in addition, a second important aspect of theoretical chemistry, particularly concerning chemical reactions, where the dynamics of molecular motion and their statistical behaviour is more important than specific quantum effects. This aspect will be included in the general coverage of the series. The intended coverage of the series may thus be summarised as: the quantum theory of valence, with application to the calculation of the structure and properties of molecules, and to the calculation of potential energy surfaces for chemical reactions; theoretical aspects of spectroscopy; the dynamics of chemical reactions; intermolecular forces; and developments in fundamental theory and in computational methods.’ Many of the topics treated in Volumes 14 are still of interest to a small band of professional theoretical chemists. Twenty years ago, Molecular Mechanics was in its infancy. A number of calculations had been reported for hydrocarbons, but force fields were rather primitive, as was the graphical user interface. In my opinion, it was the GUI that was responsible for the exponential growth of Molecular Mechanics and Molecular Dynamics. Many
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Chemical Modelling: Applications and Theory, Volume I
people now associate the acronyms MM and M D with theoretical chemistry, despite the classical nature of the techniques. It is now fashionable to speak about Modelling, which is the generic term for anything to do with the structure, properties and reactions of atoms, molecules and materials. The term does not imply quantum mechanics. Again, we have seen the action of market forces and the rise in the number of consumers of Theoretical Chemistry. These are our colleagues who believe that Theoretical Chemistry has something to offer them, and the market has responded with sophisticated black boxes such as Gaussian98, Hyperchem 5.1 and so on. We should be pleased at the success of our marketing. There was always going to be an overlap between this new SPR and the original Theoretical Chemistry volumes, but things have moved on from the 1970s and so it seemed appropriate to start again with a new name. After much heartsearching, we decided on ‘Chemical Modelling: Applications and Theory’ to capture the spirit of the new Millennium, and to emphasize that Theoretical Chemistry is no longer dominated by quantum mechanics. The atom plays a key role in chemistry, and it seemed appropriate to ask Theodore Simos to start off our new SPR with a review of the status of atomic structure calculations. Stephen Wilson contributed a chapter to Volume 4 of Theoretical Chemistry entitled ‘Manybody Perturbation Theory of Molecules’, in which he described the beginnings of diagram techniques. Again, it seemed appropriate to ask Stephen to tell us how things now stand. He has done just this, with key references through May 1999. John Slater preached the message of Density Functional Theory (DFT) in his book ‘The Calculation of Molecular Orbitals’, published in 1979. In fact DFT had been widely used by solid state physicists for many years before that. Norman March’s chapter in Volume 4 of the Theoretical Chemistry SPR showed that chemists were beginning to come to grips with DFT by that time. The 1990s have seen the growth and growth of density functional theory. In comparison to the computerresourcebound MPn methodology, DFT gave us the first genuine tool for treating electron correlation in the electronic ground states of large molecules. Michael Springborg’s article will bring you completely up to the minute in this growth area. The idea of Molecular Atoms can be traced back to Richard Bader’s work in the 1960s. It didn’t get a mention in the original Theoretical Chemistry SPR, but is now a widely used technique. It is now an option in packages such as Gaussian98. Consumers will have to get used to the atomic charges, which seem to make much more sense than those of conventional Population Analysis. I therefore asked Paul Popelier, Fiona Aicken and Sean O’Brien to bring us up to speed. I didn’t think our new series would have credibility without a mention of Relativistic Pseudopotentials, and I have to thank Pekka Pyykko and Hermann Stoll for providing a concise summary of the state of the art. Janos Ladik contributed a chapter entitled ‘The Electronic Structure of Polymers’ to the 1981 Volume 4 of the Theoretical Chemistry SPR. This again
Preface
vii
is a growth area, and Janos has very kindly written a masterly article for our new SPR summarizing the latest trends. David Pugh remarked that there seemed to be very much more to write about electric and magnetic properties than when David Bounds and I wrote our own Theoretical Chemistry SPR contribution all those years ago. New techniques in nonlinear optics and nonlinear spectroscopy have given a new impetus to the accurate calculation of quantities such as the dipole hyperpolarizability. A visitor to the discovery labs of any pharmaceuticals company will note the importance attached to biology and biochemistry. Richard Maurer and Chris Reynolds show us how seriously modelling is taken in the world of biology. Starting a new SPR is never easy, and there was the problem of where the contributors should start their accounts; since time began? five years ago? An SPR should be the first port of call for an uptotheminute account of trends in a specialist subject rather than a dull collection of references. My solution was to ask contributors to include enough historical perspective to bring a nonspecialist up to speed, but to include all pertinent references through May 1999. Volume 2 will cover the literature from June 1999 to May 2001 and so on. In subsequent volumes, I shall ask those Contributors dealing with the topics from Volume 1 to start from there. New topics will be given the same generous historical perspective opportunity as Volume 1 but will have to cover the literature to May 2001 + n, where n = 0, 2, 4 ... This process will continue until equilibrium is reached. You may have noticed that there are holes in the coverage of the topics. Perhaps you are wondering about ONIOM, QSAR, theories of liquids, scattering and so on. One can only do so much in the first volume. Perhaps you would like to become a ‘Reporter’ yourself? I am always willing to listen to convincing ideas for topics and presentations, and you can reach me at
[email protected] Alan Hinchliffe Manchester, 1999
Contents
Chapter 1 Electric Multiples, Polarizabilities and Hyperpolarizabilities By David Pugh 1 Introduction 2 Perturbation of Molecules by Static Electric Fields: General Theory 2.1 Analytic Derivatives of the Energy 3 FrequencyDependent Polarizabilities: General Theory 3.1 TimeDependent Perturbation Theory: The Sum over States Method 3.1.1 Second Order Effects 3.1.2 Third Order Effects 3.2 Measurement of the Dynamic Hyperpolarizabilities 4 Methods of Calculation: Development from 1970 to 1998 4.1 Permanent Multipoles 4.2 Static Polarizabilities and Hyperpolarizabilities 4.3 Dynamic Response Functions 4.4 The First Hyperpolarizability of Organic Donor/ Acceptor Molecules 4.5 Calculations of the Second Hyperpolarizability 5 Review of Literature: 1998May 1999 5.1 Dipole and Quadrupole Moments 5.2 Polarizabilities and Hyperpolarizabilities of Small Molecules 5.2.1 Diatomic Molecules 5.2.2 Butadiene 5.2.3 Static Polarizabilities and Hyperpolarizabilities by ab initio Methods 5.2.4 Dynamic Polarizabilities and Hyperpolarizabilities by ab initio Methods 5.2.5 Density Functional Calculations 5.2.6 Clusters and Small Homologous Series 5.2.7 Excited State Polarizabilities
Chemical Modelling: Applications and Theory, Volume 1 0The Royal Society of Chemistry, 2000
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5 6 7 7 7 8 8 10 11 13 14 14 15 15 17 18 19 19 20 21
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Chemical Modelling: Applications and Theory, Volume 1
5.3 Polarizabilities and Hyperpolarizabilities of Larger Molecules 5.3.1 Ab initio Calculations 5.3.2 SemiEmpirical Methods 5.3.3 Linear Conjugated Chains 5.3.4 Vibrational Polarization 5.3.5 Fullerenes 5.3.6 Solvent Effects, Crystal Fields 5.3.7 New Theoretical Developments References Chapter 2 Atomic Structure Computations By T.E. Simos 1 Introduction 2 Methods with Coefficients Dependent on the Frequency of the Problem 2.1 Exponential Multistep Methods 2.1.1 The Derivation of ExponentiallyFitted Methods for General Problems 2.1.2 ExponentiallyFitted Methods 2.1.3 Linear Multistep Methods 2.1.4 PredictorCorrector Methods 2.1.5 New Insights in ExponentiallyFitted Methods 2.1.6 A New Tenth Algebraic Order ExponentiallyFitted Method 2.1.7 Open Problems in Exponentially Fitting 2.2 Bessel and Neumann Fitted Methods 2.3 Phase Fitted Methods 2.3.1 A New Phase Fitted Method 2.4 Numerical Illustrations for ExponentiallyFitted Methods and Phase Fitted Methods 2.4.1 The Resonance Problem: WoodsSaxon Potential 2.4.2 Modified WoodsSaxon Potential: Coulombian Potential 2.4.3 The BoundStates Problem 2.4.4 Remarks and Conclusion 3 Theory for Constructing Methods with Constant Coefficients for the Numerical Solution of Schrodinger Type Equations 3.1 Phaselag Analysis for Symmetric Twostep Methods 3.2 Phaselag Analysis of General Symmetric 2kStep, k E N Methods 3.3 Phaselag Analysis of Dissipative (NonSymmetric) Twostep Methods 3.4 Phaselag Analysis of the RungaKutta Methods
21 21 22 24 26 27 28 29 30 38
38 39 39 40 41 42 44 49 54 58 58 66 71 73 74 76 77 77 84 84 85 87 89
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3.5 Phaselag Analysis of the RungaKuttaNystrom Methods Methods with Constant Coefficients 4.1 Implicit Methods 4.1.1 PStable Methods 4.1.2 Methods with NonEmpty Interval of Periodicity 4.2 Explicit Methods 4.2.1 Fourth Algebraic Order Methods 4.2.2 Sixth Algebraic Order Methods 4.2.3 Eighth Algebraic Order Methods 5 VariableStep Methods 6 PStable Methods of High Exponential Order 7 Matrix Methods for the OneDimensional Eigenvalue Schrodinger Equation 7.1 Methods of Discretization 7.1.1 Methods Which Lead to a Tridiagonal Form of the Matrix A 7.1.2 Methods Which Lead to a Pentadiagonal Form of the Matrix A 7.1.3 Methods Which Lead to a Heptadiagonal Form of the Matrix A 7.1.4 Numerov Discretization 7.1.5 Extended Numerov Form 7.1.6 An Improved FourStep Method 7.1.7 An Improved ThreeStep Method 7.1.8 An Improved Hybrid FourStep Method 7.2 Discussion 8 RungaKutta and RungaKuttaNystrom Methods for Specific Schrodinger Equations 9 Two Dimensional Eigenvalue Schrodinger Equation 10 Numerical Illustrations for the Methods with Constant Coefficients and the VariableStep Methods 10.1 Methods with Constant Coefficients 10.1.1 Remarks and Conclusion 10.2 VariableStep Methods 10.2.1 Error Estimation 10.2.2 Coupled Differential Equations 10.3 Remarks and Conclusion Appendix References 4
Chapter 3 Atoms in Molecules By P.L.A. Popelier, F.M. Aicken and S.E. O’Brien 1 Introduction 1.1 What Is AIM?
91 93 93 93 104 110 110 110 111 114 117 119 119 120 120 120 120 120 121 121 122 123 123 124 125 125 126 127 127 128 132 133 140 143
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Chemical Modelling: Applications and Theory, Volume I
1.2 Scope 1.3 The Roots of AIM 1.4 The Development of AIM 1.5 Software 2 Theoretical 2.1 Open Systems 2.2 Molecular Similarity and QSAR 2.3 Electron Correlation 2.4 Transferability 2.5 Multipoles 2.6 Molecular Dynamics 2.7 Partitioning 3 The Laplacian 3.1 Alternative Wave Functions 3.2 Relation to Bohm Quantum Potential 3.3 Protonation 4 Electron Densities from Highresolution Xray Diffraction 4.1 State of the Art 4.2 Comparison between Experimental and Theoretical Densities 4.3 Hydrogen Bonding 4.4 Organic Compounds 4.5 Transition Metal Compounds 4.6 Minerals 5 Chemical Bonding 5.1 Theory 5.2 Ligand Close Packing (LCP) Model 5.3 Hypervalency 5.4 Organic Compounds 5.5 Transition Metal Compounds 5.6 Minerals 5.7 Solid State 5.8 Compounds of Atmospheric Interest 5.9 Van der Waals Complexes 6 Hydrogen Bonding 6.1 Review 6.2 Relationships 6.3 Cooperative Effect 6.4 Bifurcated Hydrogen Bonds 6.5 Lowbarrier Hydrogen Bonds 6.6 Dihydrogen Bonds 6.7 Very Strong Hydrogen Bonds 6.8 Organic Compounds 6.9 Biochemical Compounds 6.10 Compounds of Atmospheric Importance
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Reactions 7.1 Organic Compounds 7.2 Inorganic Compounds 8 Conclusion 9 Disclaimer References
Chapter 4 Modelling Biological Systems By R.I. Maurer and C.J. Reynolds 1 Introduction 2 GProtein Coupled Receptors 3 ProteinProtein Docking 3.1 Traditional Docking Approaches 3.2 Sequencebased Approaches to Docking 4 simulations on the Early Stages of Protein Folding 5 Simulations on DNA 5.1 Particle Mesh Ewald 6 Free Energy Calculations 6.1 Free Energy Calculations from a Single Reference Simulation 6.2 Multimolecule Free Energy Methods 6.3 Linear Response Method 6.4 Free Energy Perturbation Methods with Quantum Energies 6.5 Force Fields 7 Continuum Methods 7. I Parameter Dependence 7.2 pK, Calculations 7.3 Binding Studies 7.4 Protein Folding and Stability 7.5 Solvation and Conformational Energies 7.6 Redox Studies 7.7 Additional Studies 8 Hybrid QM/MM Calculations 8.1 Methodology Developments 8.2 The Models 8.3 The Link Atom Problem 8.4 Miscellaneous Improvements 8.5 The ‘Onion’ Approach 8.6 Applications 8.6.1 NickelIron Hydrogenase 8.6.2 /?Lactam Hydrolysis 8.6.3 Bacteriorhodopsin 8.6.4 The Bacterial Photosynthetic Reaction Centre 8.6.5 Other Studies
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CarParrinello Calculations Acknowledgement References
232 233 233
Chapter 5 Relativistic Pseudopotential Calculations, 1993June 1999 By Pekka Pyykko and Hermann Stoll 1 Methods 1.1 Introduction 1.2 Model Potentials 1.3 ShapeConsistent Pseudopotentials 1.4 DFTBased Pseudopotentials 1.5 SoftCore Pseudopotentials and Separability 1.6 EnergyConsistent Pseudopotentials 1.7 CorePolarization Pseudopotentials 1.8 Concluding Remarks 2 Applications by Element 3 Some Applications by Subject 3.1 New Species 3.2 MetalLigand Interactions 3.3 ClosedShell Interactions 3.4 Chemical Reactions and Homogeneous Catalysis 3.5 Chemisorption and Heterogeneous Catalysis 3.6 Other Acknowledgements References
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Chapter 6 DensityFunctional Theory By Michael Springborg 1 Introduction 2 Fundamentals 2.1 Wavefunctionbased Methods 2.2 Approximating the Schrodinger Equation 2.3 Densityfunctional Theory 2.4 Hybrid Methods 3 Structural Properties 3.1 Structure Optimization 3.2 Examples of Structure Optimizations 4 Vibrations 5 Relative Energies 5.1 Dissociation Energies 5.2 Comparing Isomers 6 Chemical Reactions 6.1 Transition States 6.2 Hardness, Softness and Other Descriptors
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Contents
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Weak Bonds 7.1 van der Waals Bonds 7.2 Hydrogen Bonds The Total Electron Density The Orbitals Excitations Spin Properties 1 1.1 NMR Chemical Shifts 11.2 Electron Spin 1 1.3 Electronic SpinSpin Couplings 1 1.4 Nuclear SpinSpin Couplings Electrostatic Fields Solvation 13.1 Dielectric Continuum 13.2 Point Charges Solids 14.1 Band Structures 14.2 Applications Liquids Surfaces as Catalysts Intermediatesized Systems Conclusions Acknowledgements References
Chapter 7 Manybody Perturbation Theory and Its Application to the Molecular Electronic Structure Problem By S. Wilson 1 Introduction 1.1 A Personal Note 2 Theoretical Apparatus and Practical Algorithms 2.1 Quantum Electrodynamics and Manybody Perturbation Theory 2.1.1 The NDependence of Perturbation Expansions 2.1.2 The Linked Diagram Theorem 2.2 Manybody Perturbation Theory 2.2.1 Closedshell Molecules 2.2.2 Openshell Molecules 2.3 Relativistic Manybody Perturbation Theory 2.3.1 The Dirac Spectrum in the Algebraic Expansion 2.3.2 Manyelectron Relativistic Hamiltonians 2.3.3 The ‘No Virtual Pair’ Approximation 2.3.4 Quantum Electrodynamics and Virtual Pair Creation Processes
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2.4 The Algebraic Approximation 2.4.1 Gaussian Basis Sets and Finite Nuclei 2.4.2 Eventempered Basis Sets 2.4.3 Symmetric Sequences of Basis Sets 2.4.4 Universal Basis Sets 2.5 Higher Order Correlation Energy Components 2.5.1 Fourth Order Energy Components 2.5.2 Fifth Order Energy Components 2.5.3 Higher Order Energy Components 2.6 The Use of Multireference Functions in Perturbation Theory 2.7 Concurrent Computation Manybody Perturbation Theory (ccMBPT) 2.7.1 Parallel Computing and Its Impact 2.7.2 Concurrent Computation and Performance Modelling: ccMBPT 2.8 Analysis of Different Approaches to the Electron Correlation Problem in Molecules 2.8.1 Configuration Mixing 2.8.2 Coupled Electron Pair and Cluster Expansions 3 Applications of Manybody Perturbation Theory 3.1 Graphical User Interfaces 3.2 Universal Basis Sets and Direct ccMBPT 3.3 Finite Element Methods Applied to Manybody Perturbation Theory 4 Future Directions Acknowledgements References
Chapter 8 New Developments on the Quantum Theory of Large Molecules and Polymers By Janos J . Ladik 1 Introduction 2 The Treatment of Large Molecules Using Solid State Physical Methods Developed for Aperiodic Chains 2.1 The Negative Factor Counting Methods with Correlation and Methods to Calculate Effective Total Energy per Unit Cell of Disordered Chains 2.1.1 The Matrix Block Negative Factor Counting Method 2.1.2 The Inclusion of Correlation in the Calculation of Density of States of Disordered Chains 2.1.3 The Calculation of Effective Total Energy per Unit Cell 2.2 Application to Proteins and Nucleotide Base Stacks
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2.3 Possible Application of the Negative Factor Counting Method to Large Molecules Correlation Corrected Energy Band Structures of Different Periodic Polymers 3.1 Methods 3.1.1 Inverse Dyson Equation with MP2 Self Energy 3.1.2 Formulation of the Coupled Cluster Method for Quasi 1D Polymers 3.1.3 Analytic Energy Gradients 3.2 Examples of Correlation Corrected Band Structures of Quasi ID Polymers Application of First Principles Density Functional Theory (DFT) to Polymers 4.1 Methods 4.2 Examples of LDA Calculations on Polymers Nonlinear Optical Properties of Polymers 5.1 Theory of Nonlinear Optical Properties of Quasi 1D Periodic Polymers 5.1.1 Solid State Physical Methods 5.1.2 Large Clusters and Extrapolated Oligomers 5.2 Results of Calculations of NLO Properties and Their Discussion 5.2.1 Solid State Physical Calculations 5.2.2 Extrapolated Oligomer Calculations Conformational Solitons in DNA and Their Possible Role in Cancer Inhibition Acknowledgement References
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1 Electric Multipoles, Polarizabilities and HyperpolarizabiIit ies BY DAVID PUGH
1 Introduction
The study of the response of molecules to external perturbations provides one of the principal sources of information on molecular behaviour. The theory of the response functions is therefore of fundamental importance in molecular science. Over the last two decades the development of optoelectronics has given an additional impetus to work on the nonlinear response of molecules to electromagnetic fields. The nonlinearities provide the means for amplification, modulation and changing the frequency of optical signals, in the same way that the nonlinear characteristics of valves and transistors facilitate these operations in conventional electronics. Current activity in theoretical modelling of the response functions to some extent reflects this dual motivation. At the more fundamental end of the range, ab initio calculations on small molecules using highly sophisticated theoretical methods are being applied with considerable success. On the other hand, semiempirical methods, with a good deal of reparametrisation for specific types of molecule and type of calculation, grounding the calculations on experimentally determined spectroscopic and dipole parameters, have been applied to a vast range of compounds with the aim of identifying those with large hyperpolarizabilities of the kind that might lead to applications of the material in optoelectronics. In this article only the response to the electric field will be treated. Magnetic effects were also included in an earlier SPR', but the great expansion of the field in recent years has necessitated a subdivision of the material. Specific applications in optoelectronics and reviews of molecules currently of direct interest in that field can be found in a number of books and edited Effects interpretable only through quantum electrodynamics are treated in the books by Loudon" and by Craig and Thirunamachandran. l 1 The plan of this review is as follows: Sections 2 and 3 attempt to give an overview of the general theory for static and dynamic effects respectively. Chemical Modelling: Applications and Theory, Volume 1 0The Royal Society of Chemistry, 2000
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Section 4 describes specific methods of calculation, with some emphasis on new developments in the period from about 1970 to the present. Section 5 is a literature review of the period 1998May 1999.
2 Perturbation of Molecules by Static Electric Fields: General Theory If the electrostatic potential created at the point, r, by the external field is V(r), the perturbed hamiltonian operator is,
where the sum is over all the constituent particles of the molecule, Ho is the unperturbed hamiltonian and qa is the charge on the particle at fa. Expanding the potential in a Taylor series in the displacement from an origin in the molecule (often conveniently taken as the electronic charge centroid) leads to the multipole expansion
for a neutral molecule, where the field and its derivatives, Fijk... = (a(”)Fi/ar$rk...), are now to be taken at the coordinate origin and the repeated index summation convention is being used. The expressions
a
are respectively the components of the dipole, quadrupole and octupole operators. Traceless forms of the higher multipole operators have been given by Buckingham.12 Much of the work reviewed will refer to the case of a spatially uniform applied field, when the perturbed hamiltonian reduces to
The foregoing formula refer to changes in the hamiltonian of the system. The polarizability and hyperpolarizability terms arise from the changes in the wavefunction induced by the perturbed hamiltonian.
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3
Denoting the unperturbed and perturbed normalised wavefunctions (usually for the ground state) by $o and $ respectively, we have, for the uniform field,
where the perturbed wavefunction has been expanded as a power series in the field. Formulae for the expectation values of the multipole components can be written directly in terms of the perturbed wavefunction as ($[pit$), ($IQijl$) etc. In particular the expression for the dipole expectation value can be used to define the permanent dipole moment and the polarizability and hyperpolarizabilities:
Here, p(O) is the permanent dipole moment and the tensors a, fl and y are respectively the linear polarizability and the first (quadratic or second order polarizability)) and second (third order or cubic) hyperpolarizabilities. 2.1 Analytic Derivatives of the Energy.  If the the nth order perturbation to the wavefunction has been calculated equation (6) gives a general prescription for calculating the nth order polarizability tensor. In cases where the perturbed wavefunction is calculated by a variational method, the calculation of the polarizabilities can be simplified by making use of relationships that have been established between the analytical derivatives of the energy. If equation (5) is differentiated with respect to the field the result (assuming for simplicity that the wavefunction is real) is
For variationally determined wavefunctions the second term can be shown to vanish, giving the HellmannFeynman theorem,
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Chemical Modelling: Applications and Theory, Volume 1
so that, on comparing equations (8) and (6), the dipole moment and polarizabilities can be identified with the derivatives of the energy at vanishing field strength. In many methods these derivatives can be evaluated analytically, in the sense that they can be obtained as functions of the integrals that have already been computed in the the determination of the ground state. In order to calculate any of the polarizabilities it is necessary to obtain some of the correction functions to the unperturbed wavefunction defined in equation (6), but extensions of the type of argument used in the HellmanFeynman theorem have shown that, again for variationally determined wavefunctions, a knowledge of the nth order correction to the wavefunction enables the polarizability of order (2n + 1) to be c a l ~ u l a t e d . ' ~ ~The ' ~ ~ 'introduction ~ of Lagrangian has also allowed some of the computational advantage obtained by exploitation of the (2n + 1) rule to be extended to nonvariational wavefunctions. The selection of either a variational or a nonvariational method still has an important bearing on the procedures subsequently employed for the computation of the polarizabilities. In a variational method the energy must be optimized, in the presence of the perturbing field, with respect to any of the parameters that are allowed to vary in the wavefunction. In most calculations of response functions the relevant parameters are the molecular orbital coefficients and the state vector coefficients in a configuration interaction (CI) procedure. In a few cases the orbital exponents may also be allowed to vary. It follows that the variational procedures are those where a full HartreeFock minimization is carried out in the presence of the perturbation (Coupled HartreeFock methods, CHF and CPHF) and, if CI is included, methods where both the CI coefficients and the orbital coefficients are optimized (Multiple Configuration HartreeFock, MCSCF). Perturbation methods (MoellerPlesset (MP2, MP4 etc.) are clearly nonvariational. The CI method [other than full CI (FCI)], without redetermination of the molecular orbital coefficients in the presence of the perturbation, is also nonvariational as is the coupledcluster method (CC), where blocks of configurations are added without optimization of the individual components within each block. In the nonvariational cases the dipole moment and polarizabilities calculated from equation (6) will differ from those obtained via equation (8). As approximate solutions are improved to approach the exact function these differences become less significant, even for the nonvariational methods, since the HellmanFeynman theorem holds for the exact wavefunction.
3 FrequencyDependent Polarizabilities: General Theory
The general theory of timedependent response functions has been described in many publication^.^^'^^'^ The response is nonlocal in time and the Fourier transforms of the general timedependent functions lead to the definitions of the frequencydependent response functions which are the quantities most easily related to experimental measurements and potential applications. The notation
I : Electric Multipoles, Polarizabilities and Hyperpolarizabilities
5
can be established by writing the formulae for the first three molecular response functions explicitly
The timedependent field components are here taken to be the complex forms
F(t) = Fexp(o,t),
p = 1,2,3.. .
which combine in the nonlinear interaction to give an induced dipole component at the sum frequency, Cop.The negative sign indicates an ‘output’ frequency. For real fields combinations of real and imaginary parts give additional numerical factors when two or more of the input frequencies are degenerate. [The case of second harmonic generation, below, illustrates this point.]
3.1 TimeDependent Perturbation Theory: The Sum over States Method.  The most direct approach to the calculation of the timedependent polarizabilities is to apply standard timedependent perturbation theory to the evolution of the wavefunction in the timedependent Schrodinger equation. The wavefunction is formally expanded in terms of the complete set of molecular eigenfunctions (ground and excited states) and the solutions are obtained in terms of matrix elements of the perturbation between these states and the corresponding eigenvalues. As an example we write the formula for the second harmonic generation (SHG) hyperpolarizability in schematic form:
where the notation is intended to indicate that a number of combinations of products involving permutations of the excited state indices, n, m with the terms containing the energy denominators are to be taken. The omission of the ground state, indicated by the primed summation, implies that the matrix elements are interpreted as rnm = ($nlrl$m)
 ($olrl$o>dnm
(12)
The subtraction of what is essentially the ground state dipole from the diagonal terms is the consequence of a canonical transformation that ensures that the ground state can be omitted from the perturbation sums, thereby removing a
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Chemical Modelling: Applications and Theory, Volume 1
term that becomes infinite at zero frequency. This modification of the matrix elements for polar molecules has important consequences for the qualitative interpretation of second order nonlinearities. Explicit formulae up to third order are given in the paper by Ward and An earlier paper by Ward2' was well known as a standard reference for perturbation formulae for nonlinear optical phenomena. Hameka and Svendsen22pointed out that Ward's formulae could lead to infinite terms of several different kinds. Their general criticism of the sum over states approach as a method of calculation is debatable. More specific points were resolved in the Orr and Ward paper by the introduction of an implicit rearrangement of the third order terms, which removed some apparently infinite contributions. That such a rearrangement can be made was also noted by Butcher and Cotter2 and has been discussed a little more explicitly by Morley et aZ.23It follows from the elementary perturbation theory. The earlier part of the Orr and Ward paper addresses the question of line widths using the method of averages of Bogoliubov and M i t r o p o l ~ k ybut ~ ~ this technique is not necessary to derive the above formulae. Another extensive set of timedependent perturbation theory formulae for many kinds of bulk and molecular response function can be found in the article by Flytzanis,'* which uses a density matrix approach. Line widths are discussed, among other aspects, by Butcher and M ~ L e a nand ~ ~ by Wherrett.26 The sum over states (SOS) method, which is simply the numerical implementation of the timedependent perturbation theory, will be discussed as a computational method in the next section. Here we use it to establish some general features of the timedependent polarizabilities. First, it is apparent from the nature of the formulae that, under the right circumstances, near resonance with one of the excitation energies may lead to one term dominating the response. Preresonant enhancement can be studied through SOS formulae provided all the photon energies are such that no combination produces a transition energy lying within the line width. 3.1.1 Second Order Eflects.  Second order effects occur only when the molecule lacks a centre of symmetry. The diagonal quantities, r,,, vanish for all states of centrosymmetric molecules and equation (12) indicates that excited states that have dipole moments very different from the ground state will be particularly effective in producing high quadratic susceptibility. The excitation of such a state from the ground state will involve charge transfer across the molecule and high hyperpolarizabilities in organic molecules have been associated with the presence of such charge transfer states. Early in the history of the nonlinear optics of organic materials a twostate model was i n t r ~ d u c e d in , ~ which ~ only the ground state and one excited state were retained. This model has been widely used, apparently successfully, to identify molecules with large /?hyperpolarizabilities. An excited state with large charge transfer from the ground state (large r,,) and high oscillator strength (essentially r&) is required. For preresonant enhancement the excitation energy of n should be a little above the doubled photon energy, thereby leaving both fundamental and doubled beams in a transparent region but substantially reducing one of the energy denomi
I : Electric Multipoles, Polarizabilities and Hyperpolarizabilities
7
nators. Given that the favoured wavelengths for fibre optic communications are in the 1 to 2 micron region, donor/acceptor substituted aromatic molecules with one or two rings are an obvious choice. The archetypical molecule is 4nitroaniline. Other second order effects, especially the linear electrooptic (Pockels) effect, are treated in much the same way although the absence of the preresonant effect at the doubled frequency would be expected to reduce the plausibility of the two state model. 3.1.2 Third Order Efects.  Third order response, like first order response, is present in all molecules and so the associated effects (Kerr effect, frequency tripling etc.) will occur in centrosymmetric and noncentrosymmetric materials. Third order effects are much weaker than second order effects, which is the reason why so much effort has been expended in the investigation of noncentrosymmetric materials likely to have large second order responses. In the third order the extra link in the chain of matrix elements means that excited states not directly accessible from the ground state via the dipole operator now come into play. Amongst these is the ground state itself and it is found that the states with ground state intermediate make a predominantly negative contribution to the susceptibility. The overall susceptibilities (calculated and measured) are usually found to be positive In noncentrosymmetric molecules diagonal terms involving charge transfer type states make a positive contribution and should therefore enhance the overall response. There is some evidence that noncentrosymmetric molecules have larger third order polarizabilities than centrosymmet ric molecules. 3.2 Measurement of the Dynamic Hyperpolarizabilities. The most commonly reported measurement of a second order molecular property is the electric field induced second harmonic generation (EFISH) measurement on a solution where the active molecule is present as In comparing experiment with theory it is pertinent to note that the measured quantity is the vector part of the second order SHG tensor projected onto the direction of the molecular dipole (x). Since the BsHG tensor has up to 18 components the experimental information for testing the details of a theory is usually by no means complete. Other methods of estimating PsHG are through hyperRayleigh scattering3' and through measurements of solvatochromic shifts in solution.31The latter method depends on the rather arbitrary choice of a cavity radius which does not drop out of the final answer. Frequency tripling and EFISH (from centrosymmetric molecules with no fl) are the usual sources of experimental data on the yhyperpolarizabilities. 4 Methods of Calculation: Development from 1970 to 1998
This section and the next section attempt to introduce the computational methods currently in use, with emphasis on their development in the period 19701998. Literature references are necessarily selective and are intended to
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Chemical Modelling: Applications and Theory, Volume I
identify examples of the application of the techniques described. In many cases, where a representative example of the work of a group occurs in the more complete survey of 1998/1999 (Section 5), earlier publications in the preceding period have been omitted, to save space. The first three subsections are mainly concerned with the methods for calculation of multipoles and static and dynamic response functions, respectively, and the subsequent sections with hyperpolarizabilities (static and dynamic) for particular types of molecule. A comprehensive review of the literature by Hasanein ( 1993)32is a convenient source, with a more complete listing of earlier material; and the rather earlier article by Dykstra et al. ( 1 9 ~provides ) ~ ~ a clear account of the logical structure of static polarization theories built around the mathematical structures obtained by analytical differentiation. Reviews of hyperpolarizability calculations will be found in recent specialist symposia.893495 4.1 Permanent Multipoles.  The techniques for extracting the multipoles from a calculated electronic wavefunction are straightforward and the accuracy of a particular ab initio result will be determined by the quality of the basis set and the correlation corrections, whether through CI or one of the MBPT or CC methods or through the particular functional used in DFT. Over the period 1970 to the present, the quality of the basis sets used has improved enormously, but to obtain accurate dipole moments it is still necessary to chose carefully balanced sets of polarizing functions. Recent estimates often indicate that correlation corrections may amount to as much as 10% in the value of the dipole moment. Some recent calculations are reported in Section 5. 4.2 Static Polarizabilities and Hyperpolarizabilities. In the case of the static polarizabilities the methods employed in semiempirical and ab initio work are essentially the same. Naturally most of the calculations on small molecules are now based on ab initio methods while semiempirical systems (often of the MOPAC/MNDO genre) come into play for larger molecules. There is a distinction between two groups of methods. The first is the finite field te~hnique.~’ In this case finite perturbations representing the external fields are added to the molecular hamiltonian and the calculation of the ground state wavefunction and energy is carried out as for the unperturbed molecule. The finite field method can be applied in conjunction with any quantum mechanical method that is available for molecular calculations. There are two principal subdivisions of the finite field method. In one of these terms of the form 1, qaryFi representing the effect of a constant finite electric field are added to the h a m i l t ~ n i a n , while ~ ~ . ~in~ the other a set of fixed charges is placed around the molecule to simulate the external field.37The second method produces nonuniform fields and with a suitable arrangement of the charges can be used to extract the response to field gradients and hence address higher multipole effects. In either case the most frequently reported procedure employs the HartreeFock method and computes the polarizabilities from the numerically determined power series for the energy against the field strength. Rather stringent limitations exist on the magnitude of the field necessary to achieve
I : Electric Multipoles, Polarizabilities and Hyperpolarizabilities
9
sufficient accuracy without obscuring individual terms. The method becomes progressively more difficult from the point of view of numerical analysis as the order of the effect increases.38 The second main group of methods analyse the response to successive powers of the perturbation separately using analytical rearrangements of the perturbed equations. The procedure is typified in the Coupled Perturbed HartreeF o ~ kmethod ~ ~ (CPHF). * ~ ~which ~ produces ~ variationally optimized solutions in each order. Since the results represent a solution of the variational HartreeFock equations to each order they satisfy the energy derivative equations for the polarisabilities and the (2n + 1) rule for the derivatives can be used to simplify the calculations. Corrections to the perturbed H F solutions can be made through MP2 or MP4 perturbation theory. This second group of variation/perturbation methods can be utilized most efficiently if systematic use is made of derivative techniques. Dykstra et aZ.33 have derived a series of equations for successive derivatives of the Schrodinger equation which provide the most effective way of organizing calculations of higher order polarizabilities, making use of recursion relations between different orders and fully exploiting the (2n + 1) rule method. The theory applies to any Schrodinger equation but in its implementation a choice of hamiltonian and basis must ultimately be made. When used with HartreeFock wavefunctions the technique is known as the Derivative Hartree Fock (DHF) method. An important development in the 1980s was the extension of the (2n + 1) rule to nonvariational functions through the introduction of Lagrangian multiplier techniques by Handy and SchaeferI6 and Helgaker and Jorgensen. l7 The mathematical basis of the method of analytical differentiation has been clarified by the work of King and Komornicki.'' The gradual introduction of better gaussian basis sets coupled to vastly increased computing power has led to great improvements in the accuracy of calculations. Particularly significant has been with later extensions and the introduction of the double bases of the specific addition of polarising functions, including some with high angular momentum ( I values). Examples of ab initio calculations using the finite field method have been Maroulis and coworkers have developed a systematic approach to finite field calculations of the static polarizability and hyperpolarizabilities of small molecules, which has been extensively reported.& In a series of publications commencing in the early 1990s, AndrC, Champagne and collaborators have investigated conjugated linear organic systems (polyacetylenes, polyynes, polyphenyls) initially using 63 1G basis sets and other methods. They have extended the calculations to include electron correlation effects via MP2 to MP4 perturbation theory and find significant changes in the behaviour of the linear polarizability and the yhyperpolarizability. They find that careful optimization of the structures at the MP2 level is essential to obtain the correct bond alternation which, in turn, has a critical effect on the values of a and y. They have also investigated the vibrational contributions to a and y (see Section 5 for references). Semiempirical methods, usually of the MNDO/MOPAC genre, have also
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Chemical Modelling: Applications and Theory, Volume I
been used extensively with the finite field technique. The method has been to investigate a number of molecules employed by Zyss and that have been of special interest in crystalline studies of nonlinear optical effects. The results obtained by the standard versions of the finite field technique give the static hyperpolarizability. In many publications an estimate of the frequency dependent values is then made by applying the empirical frequency factors in the two state model to the static value. 4.3 Dynamic Response Functions.  The perturbation series formula or spectral representation of the response functions can be used only in connection with theories that incorporate experimental information relating to the excited states. Semiempirical quantum chemical methods adapted for calculations of electronic excitation energies provide the basis for attempts at direct implementation of the sum over states (SOS) approach. There are numerous variants using the PPP,50y51 CNDO(S),5255INDO(S)56*57 and ZIND05*levels of approximation. Extensive lists of publications will be found, for example, in references 5 and 34. The method has been much used in surveying the first hyperpolarizabilities of prospective optoelectronically applicable molecules, but is not a realistic starting point for quantitative calculation in unparametrized calculations. For ab initio work and calculations based on more general semiempirical approximations, such as the MOPAC/ MNDO system, computational methods that avoid the necessity for the introduction of data referring to large numbers of excited states must be used. From the timedependent Schrodinger equation,
it follows that
and that the Frenkel variational p r i n ~ i p l e ~ ~(here * ~ ' ~in~ 'an abbreviated form applicable only to real functions),
holds. Solution of the variational equations will ensure only that (YIH  iti(aY/at)lY)is constant, but the wavefunction is indeterminate to a phase factor which can be chosen to satisfy equation (15). If
I : Electric Multipoles, Polarizabilities and Hyperpolarizabilities
11
then
In solving the equations the Fourier transformed versions are used. ‘Forced’ solutions at the appropriate frequencies give the response functions and ‘free’ solutions will lead to eigenvalue equations giving the transition frequencies of the unperturbed system.” The quantity W(t) is known as the quasienergy and is unambiguously defined in terms of CP if equation (15) is to be satisfied.19962*63The equations for determining cf, from the Frenkel principle are independent of the equation for W, but W can be expressed in terms of CP and the response functions as derivatives of W.6467 Alternatively they can be obtained from Q, using the dipole operator. The two procedures are analogous to the application of equation (8) or equation (6) in the static case. The methods based on differentiation of W, the Quasi or PseudoEnergy, are known by the appropriate acronyms, QED or PED. In either case, in ab initio work, the choice of basis functions follows the usual levels of approximation currently used in static calculations. The timedependent HartreeFock method (TDHF)6872 (also known as the random phase approximation (RPA) especially when used in solid state calculations) represents the timeevolving states in terms of a single determinant which develops through the mixing of the unperturbed virtual orbitals with the occupied orbitals. Higher levels of approximation corresponding to the inclusion of correlation effects at MBPT73974975 and MCSCF have been The procedures described in the previous paragraph are directly concerned with the calculation of the timeevolution of the wavefunction itself. The approximations that must be introduced are made by the explicit truncation of the full multiconfiguration manyelectron basis set. Alternative approaches calculate the response functions directly via the polarization propagator (PP)8098’or the equations of motion method (EOM).82983 In these approaches the polarisation response is first written exactly in terms of an inverse operator and approximation of the full multiconfiguration expansion of the evolving state is effected by truncation of the operator manifolds (in second quantization formulae), in terms of which the hamiltonian and perturbing operators are expanded. For any particular one electron basis, from which manyelectron determinantal states are constructed, the most complete solution is given by full (F) CI. The PP and EOM methods differ from the wavefunction methods in the way that they truncate the FCI expansion. Details of the theory underlying these advanced methods can be found in the literature.*978G83 The EOM method has recently been developed in conjunction with the coupled cluster formalism to provide a powerful, correlated, 4.4 The First Hyperpoladzabilityof Organic Donor/Acceptor Molecules. Much of the interest in organic materials for optoelectronics, that arose in the late
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1970s and 1980s revolved around the possible applications of molecules with exceptionally large first (/3) hyperpolarizabilities. Such molecules have the potential to produce second order optical effects much larger than those obtainable with inorganic materials. The exemplar of molecules of this type was 4nitroaniline (pNA). Work by Albrecht5* and Lalama and gar it^^^ established the utility of the SOS method for pNA and Garito and Teng28 found good agreement over a range of frequencies with the experimental values of BsHG determined by EFISH measurements, for both pNA and 2methyl4nitroaniline (MNA). A number of then developed extended SOS computational packages based on the CNDO/S and INDO methods. The parametrization of such methods has often been adjusted to reproduce experimental electronic excitation energies and dipole moments for the type of molecule under consideration so that the SOS procedure has a strong empirical bias. The spectroscopic information is usually taken from peak positions in solution spectra so that the transition energies in the SOS sum should be those actually operative under the conditions of the EFISH measurement. Promising organic structures have been studied in very large numbers. There has been special interest in the behaviour of donor/acceptor substituted linear conjugated systems, including polyenes, polyacetylenes and cumulenes, since these molecules show large nonlinear increases in the hyperpolarizability with chain length before saturation is reached. Optimum conditions in terms of the nature of the substituents and control of bond alternation have been e ~ p l o r e d . ~ ~ ~ ~ ~ * ~ ~ First hyperpolarizability calculations have often represented the excited states by an expansion in terms of singly excited configurations with a H F ground state. It can be argued that semiempirical parametrizations such as CNDO/S are designed to take account of the correlation effects associated with the inclusion of higher configurations through the parametrization. Certainly, if the parametrization is based on single excitations, it seems appropriate to use them. Other implementations of the SOS method use singly and doubly excited configurations and the latter are essential for third order (y) response functions. The usual interpretation of the relationship between the theoretical and experimental results has been questioned by Willets et al.89These authors draw attention to the various conventions for defining the dipole and energy expansions (equations 5,6,8). One kind of ambiguity arises from the difference between a Taylor series expansion where factors (l/n!) are explicit and the power series representation, generally used in perturbation theory, where they are included in the polarizabilities. A second source of confusion is due to the symmetry factors that may be included or omitted when there are degenerate input frequencies and a third arises from the possible factor o f f which occurs when defining a real field from positive and negative complex exponentials. The conclusion of Willets et al.89is that the reported experimental values of /3sHG from EFISH should be multiplied by a factor of 3 before being compared with the values obtained from the perturbation theory expansion. This means that the commonly quoted figure of 16 x esu for the effective EFISH value for pNA at 1064 nm should be 50 x which considerably worsens the quantitative agreement between the CNDO SOS results and experiment.

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An attempt at an ab initio calculation of the pNA hyperpolarizability by Sim e t al.38 using double zeta quality basis sets with diffuse polarizing functions within the finite field/HF/MP2 regime gave values which are even lower than those obtained by the semiempirical methods. These calculations are at zero frequency but have been corrected to 1064 nm using the usual two state model. Bartlett and S e k i n ~ ” ~have ~ ’ discussed the relationship between hyperpolarizabilities calculated by ab initio methods and by SOS formalisms. They conclude that only the most powerful recent methods for correlated calculations can be expected to produce accuracy of about 10% for gas phase hyperpolarizabilities, while ab initio calculations interpreted in terms of the SOS expansion give very poor results when the equivalent number of states is small. There are still many uncertainties about the true values of the first hyperpolarizability of pNA. Assuming that the interpretation given by Willets et al. is correct the source of the large discrepancy has to be explained. One relevant factor is that the ab initio calculations clearly refer to the isolated molecule whereas the experimental results are obtained in solution. All the solvents used have some polar character and there are wide variations in the results obtained in different solvents. Whether the enhancement of the polar characteristics with the associated implicit change in the excitation frequencies would be sufficient to bring the ab initio result into closer agreement with experiment is not currently known. One final source of uncertainty in the experimental values which would affect both solution and solid state results is that the standard quartz reference value for the strength of an SHG signal does not seem to be very securely established in absolute terms. 4.5 Calculations of the Second Hyperpolarizability.  It is essential to include
at least doubly excited configurations in the SOS series for the yhyperpolarizability. Even in the absence of any configuration interaction they will be coupled through the chains of four matrix elements that occur in the third order SOS expansion. In many cases they have a crucial effect in determining the sign of y since in SCI the negative terms with intermediate ground state are usually dominant. A qualitative theory of the response of certain small symmetric radicals has been developed by Yamaguchi and c o  w ~ r k e r s which ~ * ~ ~illustrates ~ this point both theoretically and experimentally. The theory has been partially confirmed by ab initio finite field calculations at HF/MP2/MP4 level. For the y and higher hyperpolarizabilities the question of how much CI should be included, even in the semiempirical theories, has to be ‘addressed. Soos and R a m a s e ~ h a have ~ ~ ~ developed ~’ a full CI (‘Model Exact’) approach at the PPP level using the correction vector method and generating a full set of configurations through the generalized valence bond (GVB) theory. Their application to the conjugated polyenes has shown that increasingly higher orders of CI are necessary as the chain length is increased and has produced reasonable quantitative results for the yhyperpolarizability. Albert e t al.55have generalized this technique to a SDCI study of the polyene hyperpolarizabilities at the
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CNDO/S level. A representative selection of work using current ab initio methods is included among the papers reviewed in Section 5.
5 Review of Literature: 1998May 1999
5.1 Dipole and Quadruple Moments.  Dipole moments are, of course, routinely calculated and reported as part of the output of many standard molecular modelling computational packages. In the majority of molecular structure studies of chemical interest the available experimental data will refer to solution or neat liquid measurements and reported values may vary by 10 or 20%. The computed values from most modern semiempirical and ab initio packages will fall within this range and are usually sufficiently accurate for use in studies that correlate chemical or biological activity with molecular properties (e.g. QSAR methods), In this context Gerber94has described a new method of obtaining reasonable estimates of dipole moments using electronegativities and charge shifts in Huckel theory and reports accuracy of within 5% over a range of several hundred compounds. An approach for large composite molecules introduced by Torrens et al.95*96 treats the dipole moments and polarizabilities of oligomers through the interaction of dipoles and induced dipoles of the separate units. The method has been applied to benzothiazoles, amino acids and peptides. Cohen and Tantir~ngrotechai~~ have investigated the performance of new exchangecorrelation functionals within the usual electron density schemes and compared calculated dipoles and multipoles for first and second row molecules with those obtained by established ab initio and electron density methods. The results obtained with the new functionals compare favourably with those of the previous methods and, in particular, give a value for the dipole moment of CO which is in good agreement with experiment. Full ab initio CI studies of BH and H F have been completed by Halkier et al.98 The dipole moments and traceless quadrupole moments have been calculated using correlationconsistent basis sets of the Dunning type augmented by diffuse polarizing functions. Full CI gives an exact solution in the configuration space defined by the basis set and provides a benchmark against which the results of truncated expansions can be compared. The authors have also carried out calculations using the MP2, MP3, MP4, CCSD, CCSD(T) and CCSDT approximations, with the basis sets used for FCI and also with larger basis sets that could not be employed with FCI. From the results obtained with the smaller basis sets, the authors conclude that the CCSD(T) method gives results within about 0.2% of the FCI values (the basis set limit). While the FCI dipole moment of H F differs by about 1% from the accurately determined experimental value of 1.803 & 0.002 Debye, the result of the CCSD(T) calculation with a larger basis set is found to fall within the experimental limits. It is also significant that there is a steady monotonic change in the calculated values for each type of approximation, as the basis set is enlarged. The dipole of BH is
1: Electric Multipoles, Polarizabilities and Hyperpolarizabilities
15
calculated in the same way to be 1.356 f 0.002 Debye, which is within the error bounds of its less precisely determined experimental value, 1.27 f 0.2 1 Debye. These results support the view that the CCSD(T) approximation is capable of reproducing molecular properties with a high degree of accuracy. In other studies using the CCSD(T) method Halkier and Coriani'' have calculated the quadrupole moment of C2H2as 21.88 (in units of C m2), to be compared with two recent experimental estimates of 20.9 and 20.5 and Halkier, Coriani and Jorgensen"' have found a value of 4.99 f 0.03 for N2 as compared to a recent experimental value of 4.65 & 0.08. The authors recalculate a correction term that has been applied to the experimental data and find that the correct value, for comparison with the theory, should have been 5.01 & 0.08. Using the same kind of augmented correlationconsistent basis sets employed in the above coupled cluster methods, but working with the complete active subspace (CASSCF and CASSCF( + 1, + 2)) approximations to the CI expansion, Lawson and Harrison'" have investigated the variation with interatomic distance and spatial distribution of the quadrupole moments of P2, S2 and C12 . The CT and 71 contributions to the quadrupole are resolved and the poor results obtained at the SCF level are attributed to the inadequate representation of the system in the SCF approximation for P2 and S2. Calculations of atomic quadrupole moments are still of interest, particularly for heavier atoms where relativistic accuracy is required. Sundholm lo2 has reported a finiteelement MCSCF method to calculate the quadrupole of Ar+ and finds a final value of 0.5271 au in comparison with a recent experimental determination of 0.5208 au. The HartreeFock value is 0.57213 au and the valence shell correlation correction, amounting to 0.04844 au, accounts for most of the additional contribution, a result which may be of some significance for molecular calculations involving atoms in the intermediate range of atomic numbers. At the most fundamental end of the spectrum of molecular theory, Bhatia'03 has recalculated the ground state properties of the hydrogen molecular ion without making use of the BornOppenheimer approximation. He employs a generalized Hylleraas form for the wavefunction but includes higher powers of the internuclear separation to simulate the localized motion of the nuclei. An accurate value of the permanent quadrupole moment is obtained and compared with the results of other calculations. A study of the variation of the quadrupole and polarizability of H z with internuclear distance has also been carried out by Hutson and Ernesti, lo4 primarily as part of an investigation of the HeH2 system.
5.2 Polarizabilities and Hyperpolarizabilities of Small Molecules.  5.2.I Diatomic Molecules.  The methods usually employed for diatomic response function calculations are often identical with those used for polyatomic molecules but some features are more typical of diatomic studies; the use of very large atomic basis sets, the more frequent use of complete CI as a benchmark, the inclusion of relativistic effects and calculations that do not assume the BornOppenheimer approximation. Work on diatomics is also
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found in the references discussed in later subsections where polyatomics are treated in the same papers. In this subsection we arrange the material roughly in order of the number of electrons in the molecule. Values for the static polarizabilities of H2f, obtained without use of the BornOppenheimer approximation, are reported by Bhatia and Drachman. The agreement with recent measurements is good. Shigeta et al.'06 have formulated a nonBornOppenheimer density functional theory and have used it to calculate the polarizability and hyperpolarizabilities of the H2 molecule and its isotopes. An extensive study of the BH molecule has been presented by Ingamells et al.'07 The static a, p and y tensors are computed by finite field and perturbation methods using HartreeFock, MP2, MP4 (including single, double, triple and quadruple substitutions) , the Brueckner variant of the CCSD(T) method and DFT with the B3LYP functional. Contributions from vibration at MP2 level and rotation at HartreeFock level are calculated. The frequency dependence of the vibrational contribution is analysed. Rozyczvo and Bartlett'08 have continued the discussion of their EOMCI method applied to the frequencydependent hyperpolarizability of H F in a reply to the criticisms made by Hattig et a1.1°9 who have responded with further comments. l o Kumar et al.' have developed an extended coupledcluster functional method for molecular properties and compared the results of a numerical and analytical approach to the calculation of the dipole moment and polarizability of H F and C H + as a function of bond length. MBPTR12 and coupled cluster CCSD(T)R 12 calculations with a 1ls8p6d5f/9~8p6d5fbasis set, which should be close to the basis set limit, have been employed by Tunega and Noga'12 to calculate the dipole and quadrupole and static a, p and y tensors of LiH. The frequency dependent polarizabilities, transition probabilities and some excited state properties of N2 are investigated with CCS, CC2 and CCSD approximations by Christiansen et al.' l 3 In a further publication''4 the CH+ ion and the N2 molecule are the subject of a study of the effect of the systematic inclusion of triple excitations in the CC3 model. A significant improvement over CCSD is claimed for the polarizability anisotropy and refractivity of N2. Merawa et al.' l6 have carried out timedependent gaugeinvariant (TDGI) calculations of the dynamic polarizabilites and hyperpolarizabilities of the 14 electron molecules, CO, N2 and BF. Their atomic basis sets include up to 100 gaussian functions. They find that the behaviour of N2 and CO is similar while that of BF is very different. M a r ~ u l i s " has ~ applied the finite field method to a study of HC1. In a systematic analysis with large basis sets, MBPT and CC techniques, the dipole, quadrupole, octupole and hexadecapole moments have been calculated at the experimental internuclear distance. The polarizability and several orders of hyperpolarizability have been calculated and the mean a and yvalues for the 18electron systems HC1, HOOH, HOF, A, F2 , H2S are compared. Fernandez et a1.'18 have calculated the frequency dependent a, and y tensors for HCl and HBr using the Multiple Configuration Self Consistent Field method (MCSCF), including the effect of molecular vibration. The results show good agreement with available experimental and theoretical data.
''
'
''
"J
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Density functional calculations on the dipole polarizabilities of N2, HZ, Fand H F (Ventura et al."') have been compared with experimental values and the results of correlated ab initio methods. In all the cases studied it was found that the results, in increasing order of accuracy, were arranged as, HF < MP2 < BLYP < B3LYP < (MP4, B3PW91, QCISD, CCSD), implying that the latest DFT functional method compares well for the static dipole polarizability with some of the most accurate correlated ab initio methods. A full discussion of the basis set variation is also given. In a study of the interhalogen compounds de Jong et al.12' find that relativistic contributions increase the electric dipoles moment while the quadrupole moment and the dipole polarizability vary irregularly. The relativistic contribution to the dipole and quadrupole moments of molecules containing I2 is between 10% and 20% of the whole and the contributions are not negligible in the other cases. Fernandez et a1.121 have applied CCS and CCSD to calculations of the frequencydependent interaction induced hyperpolarizability of the A2 dimer. The calculated virial coefficients are compared with available experimental data. Maroulis et al. 122 have applied their static polarizability, finite field technique to a study of the 22 electron diatomics CP, BC1, CCl+ and PO+. The vibrational contribution to the ground state polarizability has also been calculated. The dipole polarizability and other properties of YbF have been investigate in the unrestricted DiracFock approximation by Parpia' 23 and the static second hyperpolarizability of the Cu2 dimer has been calculated in a correlation corrected UHF study by Shigemoto et al. *24 5.2.2 Butadiene.  The ab initio calculation of the linear polarizability and second hyperpolarizability tensors for transbutadiene has attracted a great deal of attention. The molecule is the smallest stable structure showing nelectron delocalisation and is the first member of the fully conjugated polyene series. Extended conjugated structures are associated with low excitation energies and large response functions. Maroulis et ~ d . ,in' ~an~ experimental and theoretical study, describe calculations of the static atensor with a series of large basis sets. Correlation effects are included both through MP2 and MP4 perturbation expansions and through CCSD and CCSD(T) calculations. In all cases the finite field method is used to find all the tensor components, from which the average polarizability, CC and the depolarization ratio, Aa, are calculated. New light scattering measurements of the latter are reported in the paper. The values of a, and ayY are found to be relatively insensitive to correlation effects, but their inclusion leads to a change of about 20% in the values of aZzand ax=and consequently Aa. After applying corrections for vibrational effects and frequency dispersion, agreement with experiment is excellent. A comparison with earlier results is also given in the paper. Maroulis' 26 has also investigated the static hyperpolarizability tensor (y) by the finite field method. The molecular geometries and levels of correlated calculation are as in reference 125, although in this case some very large basis
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sets are included for which only the SCF approximation is practicable. The full tensor is computed and the consistency of results obtained with different correlation procedures and basis sets is systematically evaluated using cluster analysis and graphtheoretic methods. When correlation is included the large yvv component is drastically changed from the SCF value but a high level of consistency between correlated calculations is achieved provided adequate diffuse functions are included in the basis set. Using estimates of the correlation correction obtained with the smaller basis sets and added to the SCF value of the larger set (a standard device in ab initio studies) and applying a correction for frequency dispersion, good agreement with experimental EFISH measurements is attained. Variation of the hyperpolarizability with bond length is also investigated and, again, a comparison with the results of previous calculations is given. Rozyczko and Bartlett 127 have applied their Equations of MotionCoupled Cluster (EOMCCSD) technique in a study of the dynamic hyperpolarizability of butadiene. Their theoretical value of the average y (20.3 au) compares well with experimental determinations of 20.2 and 27.7 au. The zero frequency limit again gives the static values which compare well (for the same geometry) with those of reference 125. The paper addresses claims'28 that the TDHF or RPA method, which is essentially uncorrelated, can provide an adequate account of dynamic hyperpolarizability. While there is quite good agreement between the static values obtained with TDHF and those from highly correlated approaches,'26 plots of the frequency dependence, for a number of experimentally accessible response functions (dcSHG, THG etc.), show substantial differences. Absolute agreement with experiment is rather hard to assess in either case but the authors point to a much greater instability in the predictions of the TDHF approximation when used with different basis sets. Kirtman et al.129have also commented on the claims made for TDHF in reference 128 and have elicited a reply from the a ~ t h 0 r s . l ~ ' References 108, 109,110, 127, 129 and 130 highlight the difficulties that are still encountered in attempts at accurate ab initio calculations of higher order molecular properties. Perhaps more significant than the dispute about the methodology of the EOMCC method (concerned with renormalization terms when the reference state is omitted, time reversal symmetry etc.) are the rather arbitrary decisions as to whether orbital relaxation should be included or excluded in the high level correlated calculations. In reference 129, referring to TDHF, it is said to be well known that one can often obtain fortuitous agreement between theory and experiment by stopping at 'some low and/or incomplete level of treatment'. It may be that, in some cases, this comment is also applicable to the more sophisticated current treatments. Calculations on butadiene also appear inter alia in other papers discussed below dealing with sets of small molecules. The work described above appears to represent the most fully correlated level of calculation for this molecule. 5.2.3 Static Polarizabilities and Hyperpolarizabilities by ab initio Methods. Using methods similar to those described in reference 125, Maroulis and
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collaborators have computed the full yhyperpolarizability tensor of water, including a study of the basis set limit and an analysis of electron correlation effectd3' They have applied the same method to the polarizability, first and second hyperpolarizabilities of the linear molecules HCN and HCP'33 and to the linear polarizability of the cyclic form of ozone.'33 Holm et again using similar methods, have investigated the static polarizability of As4. A version of Coupled Cluster theory for use in the calculation of linear response functions (LRCCSD) has been developed by Piecuch et and applied to the case of ammonia where the dipole and parallel polarizablity has been calculated as a function of the symmetric stretch and inversion internal coordinates. Coriani et have also used CCSD response theory to calculate the electricfieldgradient induced birefringence in H2, N2, C2H2 and CH4. Yamada et al.'37 have investigated the effect of structural changes on the second hyperpolarizability of ethene. They find very large differences in the variation of the hyperpolarizability with rotation angle as calculated by HartreeFock and correlated methods. Shigeta et have discussed the polarizability of ethene using a path integral method. 5.2.4 Dynamic Polarizabilities and Hyperpolarizabilities by ab initio Methods. Hattig et al.139 have developed analytical expressions for the frequency dispersion coefficients of third order properties in coupled cluster quadratic response theory and have implemented this approach using the CCS, CC2 and CCSD coupled cluster models to calculate the first hyperpolarizability of the NH3 molecule. The quasi energy derivative method has been employed by Kobayashi et a1." in MP2 calculations of the electrooptic effect and SHG Bhyperpolarizabilities for HF, H20, CO and NH3. Analytical third derivatives are given and the response functions now have poles consistent with the MBPT wavefunctions on which the calculation is based. Lembarki et al.I4' calculate the first order frequency dependent correction to the wavefunction in terms of the set of singly excited HartreeFock configurations and from it obtain the dynamic polarizability and first hyperpolarizability for H20, CO and HF. A comparative study of the dynamic polarizability of a set of 13 small molecules, calculated by four standard ab initio procedures has been presented by Dalskov and Sauer.142 The methods (with increasing levels of electron correlation included) are the TDHF(RPA), two second order polarization propagator methods [SOPPA and SOPPA(CCSD)] and the CCSDLR approximation. Extensive comparison between these methods and with other theoretical and experimental results are provided. Multiconfiguration SCF response theory has been used by Rizzo and R a h m a r ~ to ' ~ obtain ~ the dispersion behaviour of the polarizability and second hyperpolarizability of acetylene. Most of the measurable third order response functions are calculated and a rough estimate of vibrational effects is included. 5.2.5 Density Functional Calculations.  Hinchliffe et al.144*145 give the results of static polarizability calculations for ethene, butadiene and hexatriene using 18
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different DFT formulations. They also report work on azulene, fulvalene and naphthalene and compare the results of DFT with those obtained with the AM 1, MNDO, PM3 semiempirical hamiltonians. Millefiori and A l p a r ~ n e ' ~ ~ have studied ethene, benzene and naphthalene using several DFT formulations. They find that the best results are obtained with the B3LYP version which is claimed to reproduce the average polarizability to within 2% of its experimental value. They have also i n ~ e s t i g a t e dthe ' ~ ~static o! and p for the chalcogenophenes (C4H4X7X = 0, S, Se, Te), where they find that both the linear and quadratic response increases steadily with atomic number. In the case of the chalcogenophenes both DFT and ab initio calculations have been carried out. An extensive study of the static and dynamic hyperpolarizabilities (/I and y ) using timedependent density functional theory has been provided by van Gisbergen et al.14' The authors compare results obtained using three approximations to the exchangecorrelation potential for N2, C02, CS2, C2H4, NH3, CO, HF, H20 and CH4. The importance of using an asymptotically correct form for the exchangecorrelation potential is established and the necessity for large basis sets is also emphasized. The paper provides a review of earlier results, both DFT and ab initio, for the set of molecules considered. Fuentalba and re ye^'^^ report DFT calculations on the electric dipole polarizabilities of Li,H, clusters . The variation of the polarizability with the cluster size and also with the isomeric structure is discussed and related to variations in the hardness, which is taken as proportional to the energy gap. 5.2.6 Clusters and Small Homologous Series.  Raptis et al.150 report studies on the polysulfanes, H2S,, using MNDO and ab initio methods. The static and dynamic polarizabilities and hyperpolarizabilities are calculated and the authors find that the forms of change exhibited by a and y are similar, so that the variation of the former may be used to predict that of the latter. Large changes in the hyperpolarizability with molecular structure are observed. Reiss and Papadop~ulos'~'have used a range of ab initiu methods, from restricted HartreeFock and unrestricted HartreeFock (UHF) to calculate the static polarizability and second hyperpolarizability of B4 clusters. UHF gives more reliable geometries at the expense of spin contamination and it is found that there is considerable oscillatory behaviour as the level of MBPT is increased (MPn, n = 2,3,4). Maroulis and Xenides,lS2using the methods described in reference 126, have calculated the linear and nonlinear polarizabilities of Li4 clusters and discuss the relationship between theory and experiment for the static dipole polarizability . The static dipole polarizability of mixed sodium/lithium has been investigated by Antoine et al.'53using DFT and finite field CISD methods and compared with the authors' molecular beam measurements. Fuentalba*'* has calculated the static polarizability of C, (n = 1,8) clusters by a DFT/finite field method with large basis sets. As an example of the significance of the results, it is found that the difference in polarizability between cyclic and linear C6 clusters is such as to lead to the prediction that these two components could be resolved when a jet is passed through an electric field. The static a, /I and y tensors of Be,, (n = 1, 4)
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clusters have been calculated by Begue et al.’” as a function of basis set size using the CPHF method. It is also found that the inclusion of electron correlation at the MRDCI, MP4SDQ(T) and CCSD(T) levels significantly reduces the values of the mean dipole polarizability. Dynamic polarizabilities are also calculated using the TDGI method. Intercluster dispersion coefficients, c 6 (in the term  C6/r6, are deduced from the polarizabilities. 5.2.7 Excited State Polarizabilities.  Jonsson et al.lS6 have attempted to simulate excited state polarizabilities by means of the optimization of a single determinant ground state. These excited state polarizabilities are given by the double residues of the cubic response functions. The method has been applied to H20, 0 3 , HCHO, C2H4,C4H6,cyclobutadiene, pyridine, pyrazine and stetrazine and the results compared with others obtained from multideterminant optimized excited states. 5.3 Polarizabilities and Hyperpolarizabilities of Larger Molecules.  Ab Initio Calculations. At the most highly correlated level Christiansen et al.lS7 have used the CCS, CC2 and CCSD models to calculate the static polarizability of furan. Dispersion effects are included to make an estimate of the frequencydependent polarizability. Static dipole polarizabilities are included amongst the properties of diazaborinines, triazaborinines and azaboroles calculated at the MP2 level by Doerksen and Thakker. 15* Additive atom and bond polarizability models, accurate to within a few percent, are constructed for a larger set of 104 planar molecules. The presence of boron in the isomers reduces the accuracy of the additive models compared with that attained in heterocycles containing only C, N and 0. Stout and Dyk~tra’’~ have also been concerned with transferable parameters in an additive model of the molecular polarizability. They have extracted the parameters from ab initio calculations on a large set of organic molecules. Dipole magnitudes and average polarizabilities are reproduced by the additive model to ‘very good accuracy’ while only ‘good accuracy’ is claimed for the dipolar orientation and the polarizability anisotropy. Additivity of group contributions in substituted benzenes to the polarizability and refractive index has also been investigated by Sylvester et al.,l6’ who find that, while the overall polarizability is not additive, procedures applied to separate tensor elements are successful. Maertens et al.16’ have interpreted results on the flhyperpolarizability of pushpull systems consisting of a dithienyl conjugated system containing a ketogroup ‘spacer’ within the backbone, the ends of the system being capped by donor and acceptor groups. The inclusion of an electric field in the ab initio calculations allowed the effects of varying donor/acceptor strength to be varied (cf. references 162, 163). Ruud et al.lM have attempted to introduce an integral screening procedure into direct SCF calculations of the second hyperpolarizabilities of large molecules. The screening simulates the correlation effects that are implicit macroscopically in the introduction of a dielectric constant. The introduction
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Chemical Modelling: Applications and Theory, Volume I
of the screening makes it possible to calculate second hyperpolarizabilities for atoms with more than 140 atoms. Kamada et a1.'659'66have studied the effect of heavy atoms on the second hyperpolarizability of furan, thiophene, selenophene and tetrahydrofuran by a H F calculation with augmented basis sets. The heavier atoms are found to enhance the hyperpolarizability but nconjugation no longer contributes to its value in the presence of the heavier heteroatoms. Kucharski et al.167 have calculated the static Phyperpolarizability of new sulphonamide amphiphiles using finite field SCF and INDO/S methods. In the latter case a solvent correction (SCRF option) was also included. The ab initio and INDO/S results for the isolated molecule were similar while the inclusion of the solvent correction increased the values by about 5565%. Kassimi and Lin lci8 have calculated the dipole moment and static polarizability of azasubstituted thiophene derivatives within the HartreeFock approximation. For a representative subset, correlation up to the MP4(SDQ4) level has been included. The results are expected to be accurate to within a few percent. Swart et al.'69 have reported RHF, TDDFT and Direct Reaction Field (DRF) calculations on the polarizabilities of a set of 15 organic molecules. They find that the RH F results are inferior to those of the other two methods and that the DFT method with the LB94 functional gives the best results for the polarizability anisotropy in molecules wid1 nbonds. Howard et al. 170 have calculated the static polarizabilities of alkylsiloxanate and methoxysiloxanate anions using D F T with the BLYP functional. Nakano, Yamada and Yamaguchi17' have developed a theory of the yhyperpolarizability of molecules with equivalent resonant structures corresponding to reversal of the sign of the dipole (symmetric resonance with inversible polarization, SRIP). Such structures give rise to a situation where the negative terms (involving the ground state intermediate) that can be identified in the SOS expansion (see Section 3) make the major contribution to y , which becomes negative. (In nonlinear optics negative y leads to selfdefocusing rather than selffocusing.) The nitronyl nitroxide radical is such a case and the authors have verified their qualitative conclusions by correlated CASSCF and DFT c a l ~ u l a t i o n sof ' ~ the ~ static second hyperpolarizability. Still pursuing their studies of molecules with unusual ground states the group has also used correlated ab initio and DFT methods to investigate the static second hyperpolarizabilities of trithiapentalene and dioxathiapentalene. 1733174~175
5.3.2 SemiEmpirical Methods.  Continuing attempts to rationalize the phyperpolarizability of donor/acceptor (pushpull) organic molecules in terms of comparatively simple models are exemplified by the work of Sheng and Jiang162 where the twolevel model is applied to quinone derivatives. A comparison with other pushpull conjugated systems is made and then it is shown'63 that the transition between quinoid and benzenoid behaviour and its relation to the strength of the donor/acceptor pairs can be simulated by introducing static electric fields (cf. reference 161) in a finite field AM 1 approach. Blanchard, Esce and B a r z o ~ k a shave ' ~ ~ associated a two level model approach with a mixture of
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limiting chemical resonant structures in work which enables them to interpret the variation of the quadratic polarizability with chain length in a series of pushpull polyenes. The effect of 0r coupling in a series of donor/acceptor organic molecules including zwitterions and betaines has been investigated within an INDO/S sum over state approximation by Rao and B h a n ~ p r a k a s h . 'Yuan ~ ~ et al.'78 have investigated structural effects on p using the CNDO/S CI sum over states approach. An INDO/S CI method has been employed by Albert et af.'79to study the effect of substitutional modification of interplanar dihedral angles in a series of quinopyrans. By adjusting the length of conjugation pathway and the strength of primary and auxiliary donor/acceptor groups, the (electronic) infrared absorption properties and the quadratic response can be drastically altered. In some cases the p B product is found to be exceptionally large. Theoretical calculations of a CFC substitute, CHC12CF3, have been carried out by Cabral18' using DFT. The calculated polarizability agrees well with an experimental value. Lambert et af. 8 1 have made semiempiricalPM3TDHF calculations on three dimensional phosphonium ion chromophores. They find that, while the linear optical properties of the 3D chromophores behave additatively, the first hyperpolarizability is distinctly enhanced in the 3D structures as compared to the 1D derivative. have used a perturbed HartreeFock method with the PM3 Choi et Hamiltonian to analyse the dynamic a, p and y response functions of thiophene, furan, pyrrole, 1,2,4triazole, 1,3,40xadiazole and 1,2,4thiadiazoIe monomers and oligomers. The PM3 method is also the basis of a study of the static a and y response functions of tetrakis(phenylethyny1)ethene. 183 Barlow et a1.'82have developed a simple orbital model for the interpretation of the quadratic response of metallocene based chromophores of the form metallocene(rbridge)acceptor. The orbitals have been obtained by DFT. Vance et al.'" have used a simple two state model to analyse the first hyperpolarizability of cyanide bridged structures containing metallic ions. While the initial impetus for the study of the nonlinear response of organic molecules was linked to the single donor/acceptor pair response along a well defined charge transfer axis, interest in more general classes of compound has been growing. In terms of planar conjugated systems the generalization can be described as an extension from a one dimensional to a two dimensional response. Within this category are found dipolar molecules with C2" symmetry and octupolar molecules, which have no dipole but lack a centre of symmetry. Wolff and Wortmann'86 have reviewed the 2D approach to organic materials for nonlinear optics. The 'missing states' method of Dirk and K ~ z y khas ' ~ been ~ revived by Tomonari and Ookubo'88 to analyse sumoverstates results for the anisotropy of the a and B tensors of a series of molecules of CZv symmetry. Spreiter et al. 89 have investigated donoracceptor substituted molecules based on the 2D conjugated tetraethynylethene framework and calculated /?hyperpolarizabilities using a semiempirical method; Lee et al. have calculated the molecular polarizability and Bhyperpolarizabilities of octupolar donor
''
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Chemical Modelling: Applications and Theory, Volume 1
substituted triphenylmethane dyes using a four state model, which is the natural generalization of the two state model commonly used for linear donoracceptor molecules. Wang et a1.19' have carried out AMl/finite field calculations of the static first hyperpolarizability of substituted ketene N,Sacetals and compared the results with experimental values obtained from solvatochromic shifts and from hyperRayleigh scattering. They attribute differences in the determined values to the 2D character and octupolar contribution of the molecules. Dekhtyar and R ~ z e n b a u m describe '~~ and analysis of the frontier orbitals in polymethine dye molecules of CZv symmetry. A Green function approach exemplifies the analysis by calculations on squarylium and thiosquarylium with a variety of end groups. The importance of helical structures for enhancing the first hyperpolarizability has been discussed by Panda and Chandrasekhar 193 and SOS theory has been used by Moreau et al.'94 and Monshi et to investigate have developed excited state polarizations with solvent effects. Torrens et a scheme to predict molecular polarizabilities from the effect of interacting dipoles. 5.3.3 Linear Conjugated Chains.  It is well established that linear conjugated structures have particularly high linear and nonlinear polarizabilities and there is continued interest in computing their response functions. Many simple unsubstituted chains are close to being centrosymmetric so that they have only cc and y polarizabilities, although chains constructed from polar monomers may have systematic contributions to p from each repeat unit. Finite chains, with donor and acceptor groups at opposite ends, may have very high /3 values. In all cases the problem of obtaining the correct behaviour as the chain length goes to infinity is important. 'Saturation' is usually taken to mean that the response function must become proportional to N , the number of monomers, for long enough chains, although the variation with N for short chains may be much more rapid. In quantum chemical terms the problem of the variation with N is ultimately one of 'sizeconsistency'. Most of the work currently reported is on finite chains, with a possible extrapolation to the limiting case. In a few cases a band structure for the infinite chain is calculated and the response function of the infinite polymer found from the spectrum of band to band transitions. There has been much discussion in recent years of the effect of bond alternation, particularly in combination with variations in donor/acceptor strength, on the response functions. The papers reviewed here mainly represent further development of the above themes. Del Zoppo et aZ.197have made an ab initio study of a model pushpull polyene and calculated the dependence of the polarizability and first hyperpolarizability on the bond alternation. Vibrational and solvent effects are simulated in the calculation. Jacquemin et aZ.'989'999200 have continued a series of studies of polymethineamine. They seek to establish, through studies on small oligomers, that a 63 1G basis set with a suitable form of MP4 is consistent for the investigation of the small chains and plausible for longer ones. They carry out band structure
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calculations at this level on several stereoregular forms of the infinite chain and estimate the longitudinal polarizability per unit cell for each structure. They revert to finite oligomers to establish that the /lhyperpolarizability evolves monotonically with chain length to its polymeric limit  contrary to earlier results at the HartreeFock level . It is also notable that the asymptotic value of p per monomer is found to be 7.7 times larger than that predicted at the HartreeFock level. Geskin and Bredas20'*202 attempt to relate the yhyperpolarizability of zconjugated chains to local contributions. The local contributions change as the chain length varies and it is found that the end parts of the molecule dominate the response at moderate chain lengths, while the contributions from the inner sections are responsible for the slow chainlength saturation. The Hubbard model, which has played a significant part in the earlier development of nonlinear optical response theory of molecules, has been revived by Shehadi et aL203to explain the properties of small bridged metallic polymers. The HubbardPeierls hamiltonian has also been used by Shuai et al.,204in conjunction with a symmetry adapted density matrix renormalization group formulation, to calculate a number of properties, including third harmonic generation in transoc t a tetraene. Fanti and Zebretto205 describe a scheme for predicting the yhyperpolarizabilities of long conjugated polyenes by scaling procedures based ab initio calculations. The computed HOMOLUMO gap is essentially replaced by the observed optical gap, which would relate to an excitonic transition rather than to a band to band transition. Good agreement with polymers containing up to 70 double bonds is obtained. Champagne et aL206 have studied solvent effects, through a continuum model, on the a and y response functions of polyacetylene chains in the TDHF approximation. They find large increases in the values which they relate to the solvatochromic shifts in the lowest optically allowed transition. Density functional theory has also been assessed207in connection with the calculation of the same response functions, but has been found to be inadequate due to the inability of the exchange/correlation potentials to satisfactorily represent the effects of the ends of the polymer. Schmidt and Springborg208have calculated the static hyperpolarizability of polyacetylene and polycarbonitrile in DFT in the presence of external fields. Zhang and Lu209have used finite field semiempirical calculations to analyse the onset of saturation in the a and y functions for the polyphenyls and polypyridines with up to 17 units. They suggest that a charge transfer effect due to end groups will become a small perturbation as the chain length increases and they attempt to establish additivity relations for the contributions of the interior groups. Kress et aL2'0 and Saxena et aL211 discuss breather (multiphonon bound) states in polyenes and their effect on the hyperpolarizability within the TDHF approximation. The static and dynamic polarizability of the polyyne (CZnH2) series is treated in the TDHF and correlated second order polarization propagator methods by Dalskov et aL212The calculated polarizabilities are extrapolated to the infinitely
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Chemical Modelling: Applications and Theory, Volume I
long chain limit by a number of different techniques. Werncke et al.213have analysed their results on the static hyperpolarizabilities of the thiacyanines in terms of semiempirical calculations. Harnada2l4 has investigated the static molecular and crystalline susceptibilities (y, x ( ~ ) )for polysilanes. The method involves a combination of the CPHF method and the orientedgas model for the crystal internal fields. The results were used to propose a molecular design method for polysilane based materials. Soliton effects in polyenic chains have been studied by Silva et al.2'5 Methods for analysing the polarizability and nonlinear optical response of polymers that can be broken into statistically oriented segments or into active chromophores distributed on a backbone have been described by Pretre et Nanavati et and Dworczak et aL218 5.3.4 Vibrational Polarization.  The earlier work on the nonlinear optical properties of organic molecules identified the exceptionally large response functions as arising from specific electronic effects. While this conclusion remains largely true in the more clear cut examples of the 'intramolecular charge transfer complex', especially when preresonant enhancement is present, there has been an increase in awareness of quite large vibrational contributions to the hyperpolarizabilities that may be present in many cases. In terms of measurable quantities, such effects would be important where one of the frequencies is low, as in the electrooptic effect. A difficulty encountered in the calculation of vibrational polarizabilities in the finite field method is that reoptimization of the molecular structure at different field strengths, which is necessary to include the effect of the nuclear relaxation, usually produces spurious rotations of the whole molecule. To remove them, constraints in the form of the Eckart conditions2199220 must be imposed on the relaxation. Luis et have developed a procedure for incorporating the Eckart conditions into the finite field method and have used it to the investigation of isotope effects on a, /?and y in H20, HDO and D20. Vibrational effects arise in two ways: from the change in the equilibrium molecular geometry (pure vibrational effect) when a field is applied; and from the change in the second derivative of the molecular potential surface which affects the zero point energy (zeropoint vibrational averaging, ZPVA contribution). These contributions differ in their variation with frequency. Kirtman et a1.222have analysed the two effects. Bishop et have derived formulae which give the dynamic vibrational a, fl, and y tensors to second order in the electrical or mechanical anharmonicity. Ingamells et al.224have investigated the vibrational contributions to the static and dynamic polarizability of ethene using ab initio methods at HartreeFock and MP2 level. They find that static pure vibrational contributions are comparable with the electronic contribution while the ZPVA contribution is estimated at the HartreeFock level to be about 10%. For purely optical and ir effects (SHG and THG) the ZPVA is the leading vibrational correction. Champagne225 has addressed the question of whether the disagreement between the calculated electronic yhyperpolarizability of carbon disulfide and experiment can be accounted for in terms of a vibrational effect. From
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calculations with a 631G* basis augmented by three diffuse functions, including MBPT corrections, he finds that the pure vibrational contributions to the static, dcKerr and electric field induced SHG (ESHG) second hyperpolarizabilities are respectively 26.5%, 6.8% and 0.8%, leaving a substantial gap between theory and experiment. Champagne et a1.226have continued their studies of merocyanines and have shown that CHF/63 1 1G* calculations predict a substantial fl(vibrational)/ p(e1ectronic) ratio in the quinoid form of these molecules. In the aromatic form the electronic component predominates and changes in direction of the major hyperpolarizability component between the two forms is thought to relate to the antiparallel aggregation recently found between quinoid and aromatic merocyanines. The vibrational and electronic Phyperpolarizabilities of the POM (4methylpyridine Noxide) and NPP (N(4nitrophenyl)~prolinol) molecules have been investigated by the same using CHF and double harmonic oscillator schemes with 63 1lG** basis. The vibrational/electronic ratio is considered as a function of the chemical nature of the nconjugated system, the NLO process and the response time. Bishop et a1.228 have discussed the relationship between static vibrational and electronic hyperpolarizabilities of pushpull molecules within a twostate, valencebond, charge transfer model. Ab initio calculations do not agree with deductions from the two state model. Several cases where the vibrational hyperpolarizability appears to be much larger than the electronic have been encountered. Quinet and Champagne229have investigated the vibrational second hyperpolarizability of CH4,F, molecules and find that vibrational contributions should be of quantitative significance for most quantities. Perpete et al.230have studied the vibrational second hyperpolarizabilities of polyyne chains. Convergence to an asymptotic limit about 7% less than the electronic contribution is predicted, but the vibrational/electronic ratio is reduced by a factor of 2 or 3 when electron correlation is included through MP2/63 1G terms. Lefebvre and C a r r i n g t ~ n have ~ ~ ' used DFT to study the vibrational polarizabilities of Na3. They attempt to explain a previously noted discrepancy between measured and calculated electronic polarizability through a vibrational correction, but find that the vibrational effect is too small. Cho232has applied the four state model to the analysis of the vibrational contributions to the first hyperpolarizability of octupolar molecules. He finds that the vibrational contribution increases with the bond length alternation. Rumi et al.233have investigated the vibrational contributions to the yhyperpolarizability of perylene and terrylene, using ab initio methods at the 321G level.
5.3.5 Fullerenes.  Jonsson et al.234have carried out analytical HartreeFock calculations, expected to be near the basis set limit, of a, y and the magnetizability for the C70 and c84 fullerenes. The results are compared with earlier calculations on Cb0 and the electronic structures of the molecules discussed. Moore et al.235have made semiempirical AM1 finite field calculations of the static yhyperpolarizability of Cm, C70, five isomers of c78 and two isomers of Cg4. The results are interpreted in terms of bonding and structural features.
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Chemical Modelling: Applications and Theory, Volume I
Wan et have used the SuSchriefferHeeger model to calculate a for five different c78 isomers and predict a great variation of optical properties with symmetry and shape. Lu0237 has attempted to establish a power law for scaling the static yhyperpolarizabilites of the fullerenes as a function of the number of carbon atoms. c60 does not fit into the relationship, a result attributed to its exceptional electron localization. An intermolecular potential model of C60with distributed dipole interactions has been used by Gamba238to obtain the polarizability and multipole moments. Measurements of the third order response of fullerenes in and correlated with chemical CS2 have been reported by Huang et structure. 5.3.6 Solvent Eflects, Crystal Fields.  This report is concerned with molecular properties and full coverage of intermolecular effects and solid state susceptibilities is not attempted. The papers reviewed in this section have been selected because they contain material closely related to the calculated properties of individual molecules. For example, calculations based on the electronic band structures of semiconductors etc. are excluded, but a few papers relating molecular crystal susceptibilities to the molecular hyperpolarizabilities are included. Cammi et al.240 have suggested a reformulation of quantum mechanical programs to include solvent effects through a realistic reaction field based on an appropriately shaped cavity. The method has been applied24' to calculate SCF static and dynamic solvent adjusted a, fl and y tensors for acetonitrile in water have extended a semiclassical ellipsoidal cavity and benzene. Norman et model for liquids and solutions to encompass vibrational polarizabilities. Champagne et have investigated solvent effects on the a and y functions of polyacetylene chains within a polarizable continuum model. They find substantial increases in the response functions which are partially accounted for by the solvatochromic shifts in the lowest electronic excitations. The field factors that relate the microscopic and macroscopic response along the longitudinal axis tend to unity with increasing length, as would be expected for a needle shaped cavity. Cho244has used a two state model to analyze the response of a pushpull polyene in solution.. A computer simulation of liquid benzene by Janssen et using molecular dynamics based on input from gas phase ab initio static hyperpolarizabilities has successfully reproduced measured values of the refractive index and yhyperpolariza bili ty . Nakano et al.246 have calculated the static a and y tensors for linear H 2 N 0 dimers using DFT as part of a study of molecular clusters. Papadopoulos and Sadlej247find, through CCSD calculations, that the polarizability and second hyperpolarizability of a Be atom are significantly reduced when it is embedded in a cluster of He atoms. Kirtman et al.248use ab initio methods to benchmark model calculations in which a bundle of hexatriene molecules is intended to simulate polyacetylene. They find that the effect of the medium can be accurately reproduced by
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classical electrostatic methods and that it leads to very large percentage reductions in a and y. Semiempirical AM 1 calculations of the dipole moments and polarizabilities of benzalazine and benzopyran derivatives have been carried out by Kodaka et al.249as part of a study of their liquid crystal properties. Reis et al. report theoretical studies of the urea25oand benzene25' crystals. Their calculations start from MP2 ab initio data for the frequencydependent molecular response functions and include crystal internal field effects via a rigorous localfield theory. The permanent dipolar fields of the interacting molecules are also taken into account using an SCF procedure. The experimental linear susceptibility of urea is accurately reproduced while differences between theory and experiment remain for x ( ~ ) . Hydrogen bonding effects, which prove to be small, have been estimated from a calculation of the response functions of a linear dimer of urea. Various optoelectronic response functions have been calculated. For benzene the experimental first order susceptibility is accurately reproduced and results for third order effects are predicted. Overall results and their comparison with studies of liquid benzene show that for compact nonpolar molecules environmental effects on the susceptibilities are small. Szostak et al.252have made calculations of the Phyperpolarizability of 2nitroaniline and 2 nitrophenol and for their radical ions and suggest that the latter might have a role in the nonlinear optical response. A localized orbital model of the response of the barium titanate crystal has been developed by Khatib et al.253Lacroix et al.254have performed INDO calculations on crystal structures for a highly polarizable zwitterionic merocyanine dye.
5.3.7 New Theoretical Developments.  In the preceding sections, even where new variations of computational methods have been used, the papers have involved applications of computer modelling to particular molecules or series of molecules. In this section a few publications are described that are concerned with novel approaches which have not yet reached the stage where a body of data from applications is available. Nakano and Y a m a g ~ c h i ~ ' ~are  ~ ~developing ' a method based on numerical solutions of Liouville's equation to describe the frequencydependent response of an assembly of dipoles to an electric field. The aggregates are of a size such that retardation effects are significant. One novel result is that there is a sharp change of the polarizability, reminiscent of a phase transition, that occurs as the intensity of a nearresonant field is increased. Hattig259*260 gives expressions for the frequency dispersion of the P and yhyperpolarizability components in terms of the derivatives of the hyperpolarizability with respect to the frequency arguments (see also the comment by Bishop261). Gorling et al.2629263 have been concerned with the development of new forms of timedependent DFT using coupled KohnSham equations. The theory should be suitable for the calculation of the response functions. Itskowitz and
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Chemical Modelling: Applications and Theory, Volume 1
Berkowitz264 have formulated a DFT method in terms of perturbations of atomic densities. Andrews et al.265describe a diagrammatic technique for treating the effects of permanent dipoles on the optical behaviour of systems with a response dominated by two energy levels. Hosoya266has discussed implications of graphtheoretical techniques and the CoulsonLonguetHiggins atomatom polarizability in an analysis of nelectron networks. The internal energy as a state function of the polarizability volume and the diamagnetic susceptibility has been discussed by deVi~ser.~~’
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211. A. Saxena, J. D. Kress, R. L. Martin, W. Z. Wang, 2.Shuai and A. R. Bishop, Synth. Metals, 1999, 101,257. 212. E. K. Dalskov, J. Oddershede and J. M. Bishop, J. Chem. Phys., 1998,108,2152. 213. W. Werncke, M. Pfeiffer, A. Lau, W. Grahn, H. H. Johannes and L. Dahne, J. Opt. SOC.Am. B, 1998,15,863. 214. T. Hamada, J. Chem. SOC.Faraday Trans., 1998,94,509. 215. D. A. Silva, C. P. deMelo, B. Kirtman, T. T. Toto and J. Toto, Synth. Metals, 1999, 102,1584. 216. P. Pretre, L. M. Wu, A. Knoesen and J. D. Swalen, J. Opt. SOC.Am. B, 1998,15,359. 217. H. Nanavati, P. Desai, A.S. Abhiraman, Comp. and Theor. Polym. Sci.. 1999, 9, 1089. 218. R. Dworczak, W. M. F. Fabian, D. Kieslinger, G. Gann and H. Junek, Dyes and Pigments, 1998,36,45. 219. C. Eckart, Phys. Rev., 1935,47, 552. 220. D. M. Bishop, M. Hasan and B. Champagne, J. Chem. Phys., 1995,103,4157. 221. J. M. Luis, M. Duran, J. L. Andres, B. Champagne and B. Kirtman, J. Chem. Phys., 1999,111,875. 222. B. Kirtman, L. M. Luis and D. M. Bishop, J. Chem. Phys., 1998,108, 10008. 223. D. M. Bishop, J. M. Luis and B. Kirtman, J. Chem. Phys., 1998,108,10013. 224. V. E. Ingamells, M. G. Papadopoulos and M. G. Raptis, Chem. Phys. Lett., 1999, 307,484. 225. B. Champagne, Chem. Phys. Lett., 1998,287, 185. 226. B. Champagne, T. Legrand, E. A. Perpete, 0. Quinet and JM. Andre, Coll. Czech. Chem. Comrnun., 1998,63, 1295. 227. B. Champagne, E. A. Perpete, T. Legrand, D. Jacquemin and JM. Andre, J. Chem. SOC.Faraday Trans., 1998,94, 1547. 228. D. M. Bishop, B. Champagne and B. Kirtman, J. Chem. Phys., 1998,109,9987. 229. 0.Quinet and B. Champagne, J. Chem. Phys., 1998,109, 10594. 230. E. A. Perpete, B. Champagne, JM. Andre and B. Kirtman, Theochem.  J. Mol. Struct., 1998,425, 115. 231. S. Lefebre and T. Carrington, Chem. Phys. Lett., 1998,287,307. 232. M. H. Cho, J. Phys. Chem. B, 1999,103,4712. 233. M. Rumi, G. Zerbi and K. Mullen, J. Chem. Phys., 1998,108,8662. 234. D. Jonsson, P. Norman, K. Ruud, H. Agren and T. Helgaker, J. Chem. Phys., 1998, 109, 572. 235. C. E. Moore, B. H. Cardelino, D. 0. Frazier, J. Niles and X. Q. Wang, Theochem. J. Mol. Struct., 1998,454, 135. 236. Z. A. Wan, J. M. Dong and D. Y. Zing, Comp. Theor. Phys., 1998,30,361. 237. Y. Luo, Chem. Phys. Lett., 1998,289,350. 238. Z. Gamba, Phys. Rev. B, 1998,57,1402. 239. H. J. Huang, G. Gu, S. H. Yang, J. S. Fu, P. Yu, G. K. L. Wong and Y. W. Du, J. Phys. Chem. B, 1998,102,61. 240. R. Cammi, B. Mennucci and J. Tomasi, J. Phys. Chem. A , 1998,102,870. 241. R. Cammi, M. Cossi, B. Mennucci and J. Tomasi, J. Mol. Struct., 1997,437,567. 242. P. N o m a n , P. Macak, Y. Luo and H. Agren, J. Chem. Phys., 1999,110,7960. 243. B. Champagne, B. Mennucci, M. Cossi, R. Cammi and J. Tomasi, Chem. Phys., 1998,238,153. 244. M. Cho, J. Phys. Chem. A , 1998,102,703. 245. R. H. C. Jansen, J. M. Bomont, D. N. Theodoru, S. Raptis and M. G. Papadopoulos, J. Chem. Phys., 1999,110,6463.
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246. M. Nakano, S. Yamada, S. Kiribayashi and K. Yamaguchi, Synth. Metals, 1999, 102,1542. 247. M. G. Papadopoulos and A. J. Sadlej, Chem. Phys. Lett., 1998,288,377. 248. B. Kirtman, C. E. Dykstra and B. Champagne, Chem. Phys. Lett., 1999,305,132. 249. M. Kodaka, S. N. Shah, T. Tomohiro and N. K. Chudgar, J. Phys. Chem., 1998, 102,1219. 250. H. Reis, M. G. Papadopoulos and R. W. Munn, J. Chem. Phys., 1998,109,6828. 251. H. Reis, S. Raptis, M. G. Papadopoulos, R. H. C. Jansen, D. N. Theodorou and R. W. Munn, TheoChem. Accounts, 1998,99,384. 252. M. M. Szostak, B. Kozankiewicz, B. Wojcik and J. Lipinski, J. Chem. SOC.Faraday Trans, 1998,94,3241. 253. D. Khatib, H. Chaib and W. Kinase, Physica B, 1999,269,200. 254. P. G. Lacroix, J. C. Daran and P. Cassoux, New J. Chem., 1998,22, 1085. 255. M. Nakano and K. Yamaguchi, Int. J. Quantum Chem., 1998,70,77. 256. M. Nakano and K. Yamaguchi, J. Phys. Chem. A , 1998,102,6807. 257. M. Nakano and Y. Yamaguchi, Bull. Chem. SOC.Jpn., 1998,71,1315. 258. M. Nakano and K. Yamaguchi, Chem. Phys. Lett., 1998,288,25. 259. C. Hattig, Chem. Phys. Lett., 1998,296,245. 260. C. Hattig, Mol. Phys., 1998,94,455. 261. D. M. Bishop, Mol. Phys., 1998,94,989. 262. A. Gorling, Int. J. Quantum Chem., 1998,69,265. 263. A. Gorling, H. H. Heinze, S. P. Ruzankin, M. Staufer and N. Rosch, J. Chem. Phys., 1999, 110,2785. 264. P. Itskowitz and M. L. Berkowitz, J. Chem. Phys., 1998,109,10142. 265. D. L. Andrews, L. C. D. Romero and W. J. Meath, J. Phys. B, 1999,32,1. 266. H. Hosoya, Theochem.  J. Mol. Struct., 1999,462,473. 267. S . P. de Visser, PCCP, 1,749.
2 Atomic Structure Computations BY T.E. SIMOS
1 Introduction
In atomic structures many problems are analyzed in a system of differential equations of the Schrodinger type of the form:
+‘ [
y”(x) =
V(X)  E ] y ( x )
where one boundary condition is y(0) = 0 and the other is specified at x = 00 and is dependent by the properties of the problem. There is a real need for the above problems to have methods which solve them both efficiently and reliably. In equation (1) the function W(x)= + V(x) is denoted as the efective potential, for which W(x)+ 0 as x + 00; E is a number denoting the energy. In the present review we will analyze the numerical methods used for the numerical solution of the above equation (1) or for the efficient solution of systems of equations of the form (1). In Section 2 we will investigate the methods with coefficients dependent on the
q
frequency of the problem (i.e. on the quantity
n
1 V ( x ) E ). In the same
section numerical tests for these methods are presented. In Section 3 we describe the basic theory for the construction of numerical methods with constant coefficients. In Section 4 we analyze the methods with constant coefficients. In Section 5 we present the variablestep methods which are used for the solution of the above problems. Pstable methods of high exponential order are described in Section 6 . In Section 7 we present the matrix methods. RungeKutta and RungeKuttaNystrom methods for the solution of the above problems are presented in Section 8. In Section 9 we present a new method for the solution of the two dimensional eigenvalue Schrodinger equation. Numerical illustrations for the methods with constant coefficients and variablestep ones are presented in Section 10. Finally in an Appendix a computer algebra program for the construction of an exponentiallyfitted method produced in Section 2 is presented. Chemical Modelling: Applications and Theory, Volume 1 0The Royal Society of Chemistry, 2000
2: Atomic Structure Computations
39
2 Methods with Coefficients Dependent on the Frequency of the Problem 2.1 Exponential Multistep Methods.  For the numerical solution of the initial value problem y(') = f ( x , y ) , y G ) ( A )= 0 , j = 0, 1, . . . ,r  1
multistep methods of the form k
k
i=O
i=O
can be used over the equally spaced intervals {xi}f=o E [ A ,B] and h = Ixi+l i = O , ..., k  1. The method (2) is associated to the operator
 xi[,
where z is a continuously differentiable function. Definition 1. The multistep method (2) is called 'algebraic' (or 'exponential') of order p if the associated linear operator L vanishes for any linear combination of the linearly independent functions 1, x, x2, . .. ,x P + ~  ~ (or exp(vox), exp(vlx), . . . ,exp(v,+,lx) where vi, i = 0, 1, . . . ,p r  1 are real or complex numbers).
+
Remark 1. (see refs. 1 and 2) If vi = v for i = 0, 1, . . ., n, n 5 p operator L vanishes for any linear combination of
+ r  1 then the
exp(vx),xexp(vx),x2exp(vx), . . . ,~exp(vx),exp(v,+,lx). Remark 2. (see refs. 1 and 2) Every exponential multistep method of order p corresponds, in a unique way, to an algebraic multistep method of order p (by setting vi = 0 for all i). Lemma 1. (For proof see refs. 1 and 3.) Consider an operator L of the form (3). With v E C, h E R, n 2 r if v = 0, and n 2 1 otherwise, then we have ~ [ xexp(vx)] " = 0 , m = O, 1 , .
. . , n  1, L[Y exp(vx)] # 0
(4)
if and only if the function cp has a zero of exact multiplicity s at exp(nh), where if v # 0, and s = n  r if v = 0, q(w) = p(w)/log' w  ~ ( w )p(w) , =
s =n
Chemical Modelling: Applications and Theory, Volume I
40
Proposition 1. (For proof see refs. 2 and 4) Consider an operator L with = 0, j = 0,1,. . . ,k 5 L[exp(fvi~)]
p+r1 2
then for given ai and p with ai = (l)raki there is a unique set of bi such that bi = bki. 2.1.I The Derivation of ExponentiallyFitted Methods for General Problems. 
Consider the construction of an exponentiallyfitted multistep method (2) which k exactly integrates the set of functions {exp(f~jx)]~=,. We will use this for the numerical solution of the general problem (1). From Lemma 1 we obtain the equations. p [exp(fv,h)]  (fv,h)'o[exp(fvjh)] = 0
or equivalently k
E[aiexp(ztv,h)  (f~jh)'biexp(fvjh)] = 0, j = 0, 1,. . . , n
(7)
i=O
where n l k and ai, bi, i = 0, . . . ,k are the coefficients of the multistep method (2). We investigate here the case where k is a positive number. Then, from Proposition 1 we have a set of k equations:
We let a k = 1, which is the convention adopted for all families of known multistep methods. Then (45) and (46) give the following system of equations: k/2 1 2
C aisinh[(ii)w,]
k/2 1 +akpwJ
i= 1
= 2sinhr+),
C aicosh[ (ii ) ~ j ]+ak/2 
for r = 1,3,5,.. .
k/2 1 2
WJ
i= 1
= 2cosh(y),
for r = 2 , 4 , 6 , .. .
where wj = Vjh a n d j = 0, 1, . . . ,k. We now prove that the system of equations either (i) has a unique solution
2: Atomic Structure Computations
41
when wj # *wj, or (ii) leads to so called 'indeterminate' expressions of the form when w j = &wj for some i andj. Let X ( w) and Y(w) (w = vh) be the matrices of the unknown coefficients in the systems of equations (9) and (10) respectively. Consider case (i). In order to make the matrices X ( w) (or Y(w))singular then their columns would have to be linearly dependent. The elements in a row consist of terms like coshMw,, sinhNwj and powers of wj. Then multiple angle hyperbolic functions can be expressed in terms of powers of cosh Mwj, sinh Nxj and their products. These with powers of wj form a linearly independent set of functions. Therefore, the columns cannot be linearly dependent. Hence, in this case det X(w)# 0 (or det Y(w) # 0). Thus the system of equation (9) and (10) has a unique solution. Consider case (ii). Here we simply have two rows of the matrix of coefficients the same and hence det X(w) = 0 (or det Y(w) = 0). Similarly, we have the right hand side of two of the equations in (9) or (10) the same so that the numerator determinant which is formed when a column of X(w)(or Y(w)) is replaced by the right hand column will also give two identical rows. Hence, the numerator determinant is 0. In these cases L'Hospital's rule must be used to evaluate these expressions. In the present review we investigate the case r = 2.
(8)
2.1.2 ExponentiallyFitted Methods.  Exponential fitting is a fruitful way for developing efficient methods for the solution of the Schrodinger equation. For the solution of the Schrodinger equation Raptis and Allison' have derived a Numerovtype exponentiallyfitted method. The numerical results obtained in their paper indicate that these fitted methods are much more accurate and efficient than Numerov's method for the solution of the Schrodinger equation. Exponential fitting has been the subject of great activity in recent decades, since the paper of Raptis and Allison. Ixaru and Rizea6 showed that for stiff oscillating problems of the Schrodinger equation it is generally more efficient to derive methods which exactly integrate functions of the form
{ 1, x, 2,. .. ,9,exp(fvx), xexp(&vx), ... ,X" exp(kvx)}, where v is the frequency of the problem, than to use classical exponential fitting methods. The reason for this is shown in error analysis presented in several papers of Simos (see for example ref. 2). In the following sections we present all the known exponentiallyfitted methods, which are categorized as follows: (i) Linear Multistep methods (i 1) Twostep methods (i2) Fourstep methods (ii) PredictorCorrector method (iil) Fourth order methods (ii2) Sixth order methods In each case a Table with numerical properties of the methods is given.
Chemical Modelling: Applications and Theory, Volume 1
42
2.1.3 Linear Multistep Methods 2.1.3.1 Twostep Methods.  The exponentiallyfittedmethods of this category are of the form
Using (12) a group of exponentiallyfitted methods has been produced. These are found in: ( i ) Raptis and Allison.' This integrates exactly:
*w x )}
{ I ,x, x 2 ,x3,exp(
(13)
(ii) Ixaru and Rizea.6 This integrates exactly:
{ 1, x, exp(fw x ),x exp(fw x )1
(14)
(iii) Raptis. lo This integrates exactly:
It is instructive to note that Numerov's method integrates exactly { 1, x, x2, x3, x 4 , 2 ] . 2.1.3.2 Fourstep Methods.  One family of the exponentiallyfittedmethods of this category is of the form
Using (16) a group of exponentiallyfittedmethods has been produced:
(0 Raptis.' This integrates exactly:
(i0 Raptis.12This integrates exactly:
(iiz') Simos.l 3 This integrates exactly: ,
{exp(fwx),xexp(fwx), 2 exp(fwx), x3exp(fwx)}
(19)
2: Atomic Structure Computations
43
Another family of the exponentiallyfitted methods of this category is of the form Yn+2
+ QOYn+l + QOYn1 + yn2
= h2(bo($+2
+ Y;J + bl ($+I + y;,)b2y;) (20)
Using (20) a group of exponentiallyfitted methods is produced: (i) Simos.14 This integrates exactly:
(ii) Simos.14This integrates exactly:
{ 1,x, x2,x3,exp(fwx),x exp(fwx) }
(22)
(iii) Sirnos.l4 This integrates exactly:
(iv) Simos.14This integrates exactly:
2.1.3.3 Hybrid Sixth Algebraic Order Methods.  The exponentiallyfitted methods of this category are of the form yn+1
+ Qlyn + Yn1
= h2 (bo(Y;+,
+Y;l)
+ bl (y;+i +":$) +
b24
(25)
where
Using (26) and (27) another group of exponentiallyfitted methods has been produced: Raptis and Cash. l 5 This integrates exactly:
{ I , x, 2,x3,x4,xs,exp(fwx)} Cash, Raptis and Simos. l6 This integrates exactly:
Chemical Modelling: Applications and Theory, Volume I
44
{ 1,x,x2,x3,exp(fwx),xexp(fwx)} (iiz)
(29)
sir no^.^ This integrates exactly:
(iv) Simos.8 This integrates exactly:
Error analysis for most of these methods are given in Simos.' 2.1.4 PredictorCorrector Methods 2.1.4.1 Implicit Fourth Order Methods.  Simos17has considered the following method:
Using (32) an exponentiallyfitted method has been produced (see reference 17, which integrates exactly: {exp(wx), xexp(wx),2 exp(wx), x3exp(wx)}
(33)
Thomas, Mitsou and Simos" have considered the following family of methods:
Using (34), another group of exponentiallyfitted methods has been produced (see ref. 18). These integrate exactly the following functions: Case I
Case I1
{ 1,x, x2,x3,x4,x5,exp(fwx), xexp(fwx)}
(36)
Case 111
{ 1,x,X2,x3,exp(~wx),xexp(fwx),x2exp(fwx)}
(37)
2: Atomic Structure Computations
45
Case IV
{ 1,x,exp(fwx),xexp(fwx),x2exp(fwx),x3 exp(fwx)}
(38)
Simos” has derived the final exponentiallyfitted method in this group which integrates exactly: Case V
{ exp(fwx), x exp(fwx), x2exp(fwx), x3 exp(fwx), x4exp(fwx)}
(39)
Simos and Williams2’ have considered the following methods
Using (40) a group of exponentiallyfitted methods has been produced (see ref. 20). These integrate exactly: Case I
Case I1
{ 1 , x , ~ ~ , x ~ , x ~ , x ~ , x ~ , ~ ~ , e ~ p ( ~ w x ) , x e x p(42) (~wx)} Case I11
{ 1,x, x2,x3,x4,x5,exp(fwx), x exp(fwx), x2exp(kwx)}
(43)
Case IV
{ l,x,x2,x3,exp(fwx),xexp(fwx),x2exp(~wx),x3 exp(fwx)}
(44)
Case V
{ 1,x,exp(kwx),xexp(fwx),x2exp(~wx),x3 exp(*wx),x4exp(fwx))
(45)
Chemical Modelling: Applications and Theory, Volume 1
46
Case VI
2.1.4.2 Explicit Fourth Order Methods.  Simos21 has derived the following family of explicit exponentiallyfitted methods: Fn+l = 2Yn  Yn1
+ h2 Y ,
If
Using (47), which integrates exactly, an exponentialfitted method has been produced (see ref. 21): {exp(fwx), xexp(fwx), x2 exp(*wx),x3 exp(fwx)}
(48)
In Table 1 we present the basic properties of the fourth algebraic order twostep methods. 2.1.4.3 Sixth Order Methods  Fourstep PredictorCorrector Methods. Simos and Mitsou22have considered the following family of methods:
yn+2 2y,+1 + 2y, + 2Y,l + Y n  2 = h2[4I(Y::2
+Y
L)

(49)
Using (49), a group of following exponentiallyfitted methods which integrates exactly has been produced (see ref. 22): Case I
Case I1
2: Atomic Structure Computations
47
Table 1 Properties of the twostep fourth algebraic order exponentiallyfitted methods. S = (H2: H = sqn, q = I,2,. . .). The quantities m and pare dejined by ( I I ) . A.O. is the algebraic order of the method. Int. Per. is the interval of periodicity of the method. TMS = Thomas, Mitsou and Simos.18Si Wi = Simos and Williams.2oI = Implicit. E = Explicit Method
A.O.
I/E
Int. Per.
I. E. F.
Numerov’s method Derived by Raptis and Allison5 Derived by Ixaru and Rizea6 Derived by Raptis” Method of SirnosL7 Method of TMSCase I** Method of TMSCase II** Method of TMSCase III** Method of TMSCase IV** Method of Simos” Method of SiWiCase I* Method of SiWiCase 11* Method of SiWiCase III* Method of SiWiCase IV* Method of SiWiCase V* Method of SiWiCase VI* Method of Sirnos*’
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
I I I
(096) (0,oo)  s
1 , x, 2 ,x3, x4, x5 rn = 0 , p = 3 m=l,p=l rn = 2 , p = 0 rn = 3 , p = 0 m =0,p = 7 m=l,p=5 rn = 2 , p = 3 rn=3,p=1 rn=4,p=O rn = 0 , p = 9 rn=l,p=7 rn = 2 , p = 5 rn = 3 , p = 3 rn=4,p=1 rn = 5 , p = 0 rn = 3 , p = 0
I
I I I I I I
I I I I I I E
(0,oo)s
(0,oo)  s (0,oo) 
s
(0,oo)  S (0,oo)S (0,oo) (O,m)S (O,m)S (0,oo) (O,oo)S
s
s (0,oo)  s (0,m)  s (O,oo)S (0,oo)  s (0,oo)  s
Case I11
sir no^^^
has considered the following family of fourstep predictorxorrector
methods
Using (53) another group of exponentiallyfitted methods which integrate exactly has been produced (see ref. 23): Case I (54)
Case I1 (55)
Chemical Modelling: Applications and Theory, Volume I
48
Case I11
{ 1,x,X2,x3,exp(fwx),xexp(fwx),xZexp(fwx)}
(56)
sir no^^^ has also considered the following fourstep predictorcorrector method:
Using (57) another exponentiallyfitted method which integrates exactly, {exp(fwx),xexp(fwx),x' exp(fwx),x3exp(fwx),x4exp(fwx)}
(58)
has been produced (see ref. 24). Hybrid Methods. Thomas and Sirnos*' have considered the following hybrid predictor 0, it follows that for the values of parameters given above and based on (182), the interval of periodicity of the method developed above is equal to (0,9.53). Simos66 has considered the following family of fourth algebraic order methods
where A')
Tn+l = h + l
+ ah2&  f n + l )
h  1 = ynl
+ a h 2 & , f n P 1 )
=h.This family of methods is an extension of that
produced by
Chemical Modelling: Applications and Theory, Volume 1
96
Simos and M o ~ s a d i sThe . ~ ~parameters ak, k = O( 1)5 are computed in order to obtain methods with minimal phaselag. The parameter a is obtained in order the Pstability property be satisfied. Based on the above, Pstable families of methods with phaselag of order 12(2)16 are obtained. For the same family of methods we can see (after an analysis we have made for this review) that
1 A(s)=l+s2 12
5a5 +s
5a4a5
4
3
6
6
s +
s~~+ 40a,a2a3a4a5
 20a2a3a4as 3
3
 80aQa,a2a3a4a5 3
S
14
lOa3a4as Sa 3 S
12
 80aaOa1a2a3a4a5 3
S
16
and, based on the theory of phaselag analysis for symmetric twostep methods described in Section 3.1, we have that the phaselag of the family of methods is given by
1 =a, 864
q=
5 + 3 a 3 a4 a5 
5 +a,a, 72
1 48384

1
5 36
a3a4a5
13 7257600 10 3 a 2 a 3 a 4 a 5
1 1 1 q = 4354560 a, +a4a5+a 24192
+
1 47900 160
5 a a +za2a3a4a5
216 20 17 + ya1a2a3a4a5 104613949440 1 1 1 1 a a a aaaa =574801920a5 2177280a4as 12096 108 5  a1a2a3a4a5  ~40a ~ a , a ~ a , a+, a , 19 9 20922789888000
'
(186)
97
2: Atomic Structure Computations
1 1 1 104613949440a’ + 287400960 04% + 1088640a a a 1 1 10 +6048 a a a a + gqa,a2a3a4a5 + 7a0a,a2a3a4a5 40 1 aa a a a a a 3 5261321375744000 1 1 1 aaa =  25 107347865600  52306974720  143700480 1 1  1 a a a a 3024 aIa2a3aqaS  27 a 0 a1a 2a3a4a5 544320 10 23 + 7aaOa,a2a3a4as + 1824676506 132480000
’
&=
’
‘
’
’
’
5 7 5 It is easy for one to see that for a1 = &, a2 = 308, a3 = 400, a4 = 252’ a5 =&, a. = 455 30404’ a the phaselag is equal to t =  6010699403;896d72q6s18. Since to have for a method a nonempty interval of periodicity the following condition must hold (see Theorem 1 of Section 3.1): A(s) f B(s) > 0, it follows that for the values of parameters given above and based on (185), the interval of periodicity of the method developed above is equal to (0,9.77).
=a
4.1.1.2 Sixth Algebraic Order Methods.  sir no^^^ has constructed the following family of hybrid sixth algebraic order predictorcorrector methods:
1 n+
2
n+2
98
Chemical Modelling: Applications and Theory, Volume I
The values of parameters a and b are obtained in order to construct Pstable methods with minimal phaselag. First the parameter b is determined in order to have phaselag of order eight and then the condition of parameter a in order that the Pstability property is satisfied is obtained. Based on the above procedure Simos has proved that all the methods of the above family with a > are Pstable methods with phaselag of order eight. In the same paper then a method Simos has proved that if parameter a is equal to  247 56% with phaselag of order 10 with an interval of periodicity equal to (0,8.16) is obtained. Simos68has developed the following threeparameter family of methods:
The parameters ai i = 0(1)2 are obtained in order that Pstable methods with minimal phaselag be constructed. First the parameters a1 and a2 are determined in order to have phaselag of order ten. Then the condition for the
2: Atomic Structure Computations
99
parameter a0 is obtained in order to have Pstable methods. The detailed analysis presented in the paper68leads to the summary that all the methods of a2 = a0 > 0.0558593 are Pstable methods with the family with a1 = phaselag of order ten. This family of methods is an improvement of the family obtained by Simos in ref. 67. For the same family of methods we can see (after analysis we have made for this critical review) that
a,
&,
and based on the theory of phaselag analysis for symmetric twostep methods described in Section 3.1, we have that the phaselag of the family of methods is given by
+
+ T1s8+ T2s10 T3s12+ . .. 1 247 To =   a 2 + 72 201600 1 1 907 T I = a2 + 41a2  864 36 7257600 1 1 1 76 1 T2 = 25920 a2  432a1a2 4 0Oala2 41 7 1072000 36 1 1 1 1321 T3 = a2 4 a1a2  a~ala2 16144128000 1451520 12960 432
t = Tos6
a,
It is easy for one to see that for a0 = a1 = 8892, 83 a2 = &,the phaselag is equal to t = 5 2 3 $ ~ ~ ~ ! 2 2 0Since 0 0 s 1to2 have . for a method a nonempty interval of periodicity the following condition must hold (see Theorem 1 of Section 3.1): A(s) f B(s) > 0, it follows that for the values of parameters given above and based on (185), the interval of periodicity of the method developed above is equal to (0,8.78). Simos6’ has constructed the following seven free parameter family of hybrid sixth algebraic order predictorcorrector methods
Chemical Modelling: Applications and Theory, Volume 1
100
The value of parameters ai i = 0(1)6 are determined in order to obtain families of methods with minimal phaselag. The value of parameter a is obtained in order to determine Pstable families of methods. As a result of the above: 1 . All the methods of the family with ai = 0, i = 0(1)4 and a5 and a6 given in ref. 69 and a < 0.01894989013671875 are Pstable methods with phaselag of order ten. This method is equivalent to the Pstable method with phaselag of order ten developed by Simos in ref. 74. 2. All the methods of the family with ai = 0, i = O( 1)3 and ak,k = 4( 1)6 given in ref. 69 and a < 0.02151 107788085938 are Pstable methods with phaselag of order twelve. This method is equivalent to the Pstable method with phaselag of order twelve developed by Simos in ref. 74. 3. All the methods of the family with ai = 0, i = O( 1)2 and ak,k = 3( 1)6 given in ref. 69 and a < 0.02270050048828125 are Pstable methods with phaselag of order fourteen. 4. All the methods of the family with ai = 0, i = 0, 1 and ak,k = 2(1)6 given in ref. 69 and a < 0.02328338623046875 are Pstable methods with phaselag of order sixteen. 5. All the methods of the family with a0 = 0 and ak,k = 1( 1)6 given in ref. 69 and a < 0.023554992267578125 are Pstable methods with phaselag of order eighteen. 6. All the methods of the family with ak, k = 0(1)6 given in ref. 69 and a < 0.0051 1659000294149are Pstable methods with phaselag of order twenty.
For the same family of methods we can see (after an analysis we have made for this review) that
3
A(s) = 1 + s2 20
1 + s4 60

18a6 +8a (a  a5)a6ss s6 + 288 18
2: Atomic Structure Computations
B(s)= 1s27 20
s41
60
101
a +'6
s6
36
+
( a  a5Ia6 s8 18
and, based on the theory of phaselag analysis for symmetric twostep methods described in Section 3.1, we have that the phaselag of the family of methods is given by t = q s 6 + q s 8 + qsIO+ qs'2 1 1 247
+ 7y4+ q s ' 6 + TgSl8 + qsZ0+ &s22+...
&=aa,+72 72 201600 1 1 1 907 1 a a, + aa,  a5a6 7257600 =864 864 36 36 1 1 1 1 q =2 5 9 2 0 a + E a 6 Eaa6 1 1 76 1 aa,a, +a a a + 171072000 18 18 1 1 1 q =1451520a 1451520a6 1 1296daza6 1 1 1 1 a a a + aa4a5a6 ?a3a4a5a6 + TEaasa, 216 ti 9 1321 16144128000 1 1 1 1 aa, +=  130636800a + 130636800 725760 1 1 1 1  aa,a, +a,a,a, aa4a5a6 +  a aaa 108 108 ti 6480 6480 2 2 251  ~ a a 3 a 4 a 5+a?a2a3a4a5a6 6 + 271724544000
'
725760
Chemical Modelling: Applications and Theory, Volume 1
102
'
1 1 1 = 17244057600 a  17244057600a6 + 653 18400 1 1 1 a5a6+aaa 653 18400 362880 362880 ti 1 1 1 +3240 aaqaSa,a a a a +5qaa3a4asa6 3240 1 4 4  z a 2 a 3 a 4 a 5 a+6aa2a3a4a5a6  ?a,a2a3a4asa6 9 3163 4477 18440960000
'
1 1 1 =3138418483200a+ 3138418483200a6 8622028800aa6 1 1 1 + 8622028800 aa5a6+ aaa  32659200 32659200 ti 1 1 1 aa4asa6+ a a a a 1620 aa3a4aSa6 ti 181440 181440 1 1 1 +1620a2a3a4asa6 aa 27 2 a3 a4 a5 a6 + 2 a7 1a 2a3a 4a5a6 8
8
617 15798064988160000 1 1 1 q= 753220435968000 a  753220435968000 + 1569209241600aa6 1 1 1 aaa 156920924 1600 asa6+4311014400aasa64311014400 ti 1 1 1 + 16329600 aa4a5a6a a a a +90720 aa3a4aSa6 16329600 1 1 1 a a a a a + =aa2a,a4a5a6  ~ a , a 2 a 3 a 4 a s a 6 90720 ti 2 2 16 + aa,a2a3aqaSa6  ~ a o a , a 2 a 3 a 4 a+s a~6a a O a , a 2 a 3 a 4 a 5 a 6 27 27011 165294224673177600000 1 1 &=230485453406208000 a + 230485453406208000 1 1 aa, + 376610217984000 3766 10217984000 1 1 aasa6+ aaa 784604620800 784604620800 ti
 aa 9 1a 2a 3a 4 a5 a6 + 5aOa,a2a3a4asa6 +
2: Atomic Structure Computations
103
1 1 1 aa4a5a6 + 2155507200 aaaa aaaaa 2155507200 8164800 ti + 1 a a a a a  1 a a a a a a + 1 a a a a a a 8164800 45360 45360 I 1 4  1 aa a a a a a +aoa,a2a3a4a5a6 aa 27 0 a1a2 a3 a4 a5 a6 405 I 405 3239 + 6047255377516953600000 
’
‘
It is easy for one to see that for a0 = 0.017128829582464151, a1 = 0.014491228321 1536582 a2 = 0.01541 19729064557206, a3 = 0.016033283320052376 a4 = 0.0171675573982214128, a5 = 0.0191526242929924581 15 a6 = 0.0071879343310922252637, a = 0.095402220045377939550 the phaselag is equal to t = 0.702402381015s22.Since to have for a method a nonempty interval of periodicity the following condition must be hold (see Theorem 1 of Section 3.1): A(s) f B(s) > 0, it follows that for the values of parameters given above and based on (192), the interval of periodicity of the method developed above is equal to (0,9.80). 4.1.1.3 Eighth Algebraic Order Methods.  Simos and Tsitouras7’ have developed the following family of methods. In order to determine the coefficients of the method, a constraint optimization problem is solved. In this problem the objective function is the function of the local truncation error while the constraints are: (1) the required order of each layer, (2) the Pstability property and (3) the minimal phaselag property. The solution of the constraint optimization problem is the specific method of the paper which is a Pstable eighth algebraic order method with phaselag of order eight:
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Chemical Modelling: Applications and Theory, Volume I
In Table 7 we present the properties of the methods developed in this section. 4.1.2 Methods with Non Empty Interval of Periodicity.  4.1.2.1 Fourth , ~ ~ and T ~ u g e l i d i s ~ ~ Algebraic Order Methods.  Simos and M o u ~ a d i s Simos and sir no^^^ have considered the following family of methods:
Table 7 Properties of Pstable methods. A.O. is the algebraic order of the method. P.L.O. is the phaselag order, Int. Per. is the interval of periodicity of the method. N.0.S. is the number of steps of the method. All the methods are Implicit. We note that the modijications of methods included in this Table are developed in this review. Method
A.O.
Derived by Simos and Raptisa 4 Derived by Simos and rap ti^^^ 4 Derived by Simos and Mousadd5 4 Derived by Simos and M o ~ s a d i s ~ ~ 4 Derived by Simos and Mousadis6’ 4 Modification of Method of Simos and M o ~ s a d i s ~4 ~ Method of Simos66 4 Method of Simos66 4 Method of Simos66 4 Modification of Method derived by Simos66 4 Family of Methods of sir no^^^ 6 Method derived by Simos i d 7 6 Family of Methods of Simos6* 6 Modification of the Family of Methods derived 6 by Simos68 Derived by sir no^^^ 6 Derived by Sin10s~~ 6 Derived by S i m ~ s ~ ~ 6 Derived by sir no^^^ 6 Derived by sir no^^^ 6 Derived by sir no^^^ 6 Modification of the Family of Methods derived 6 by sir no^^^ Derived by Simos and Tsitouras7’ 8
P.L.0
N.0.S
8 10 8 10 12 14 12 14 16 18 8 10 10 12
2 2 2 2 2 2 2 2 2 2 2 2 2 2
10 12 14 16 18 20 22
2 2 2 2 2 2 2
8
2
Int. Per.
2: Atomic Structure Computations
105
j i + l=ay, + (1a)yn+1 +h2[bfn
 ( ; 4 f n + l ]
7"l=ay,
(+~).L]
+(la)yn, +h2[b/.
(195)
The value of parameters a, b and ci, i = 1(1)3 are determined in order to obtain the method with minimal phaselag. For the method of Simos and Mousadis7' c1 = c2 = 0 and the other coefficients are determined in order to produce a method with phaselag of order ten and with an interval of periodicity equal to (0, &%), For the method of Simos and T ~ u g e l i d i sc1~ ~ = 0 and the other coefficients are determined in order to produce a method with phaselag of order twelve and with an interval of periodicity equal to (0,2(2772)J).This method is an improvement of the method produced by Simos and M o u ~ a d i sFinally, . ~ ~ for the method sir no^^^ the coefficients are determined in order to produce a method with phaselag of order fourteen and with an interval of periodicity equal to (0,29.0974). This method is an improvement of the method produced by Simos and T ~ u g e l i d i s . ~ ~
4.1.2.2 Eighth Algebraic Order Methods.  Consider the following family of eighth algebraic order methods:
9 n+
= (25~,+~ 1 128
+ 205yn 15Yn1  37Yn2)
Chemical Modelling: Applications and Theory, Volume 1
106
[f
+64
n+a
+$
, i=4(1)0
nt,]
For ai = 0, i = 4(1)0 the method obtained is the method of Allison, Raptis and S i m ~ s , which ~' was the first hybrid method of algebraic order eight in the literature. The method has an interval of periodicity equal to [0, 12.27741. J a m i e ~ o nmakes ~ ~ some fruitful comments on the above formula concerning specific cases of the potential. If the free parameters aj, i = 4(1)0 are defined in order to minimize the phaselag then we have the method obtained by sir no^.^' We have the following cases:
(1) the method where aj = 0, i = 0(1)3 and a4 = has phaselag of order ten and an interval of periodicity equal to (0,29.1895). (2) the method where aj = 0, i = 0(1)2, a3 = and a4 = has phaselag of order twelve and an interval of periodicity equal to (0, 32.368 1). (3) the method where ai = 0, i = 0, 1, a2 = a3 =  & and
e,
107
2: Atomic Structure Computations
a 4 =  has phaselag of order fourteen and an interval of periodicity equal to (0, 34.9295). 451037 (4) the method where a 0 = 0, a1 = *, a 2 = 3434886000’ a 3 = 233 147 1932 and a4 = has phaselag of order sixteen and an interval of periodicity equal to (0,37.0284). 451037 (5) the method where a 0 =  l l ~ ~ ~ ~ $ ~ ~a1& =  , , 3532875:04, a 2 = 3434886000’ 233 and a 4 = has phaselag of order eighteen and an a 3 = 1471932 interval of periodicity equal to (0,72.8593).
Next, the following formula is considered
Yn  s
486
yn+q
1 9 2 +3fiynl(7fi+27Js) ~~
= [fiyn+,(7J?27&)+ 1
1
=
[1680qYn+1(9q7 5%’
+98q3 +51)3360~,(9q’ 5Q6 +9Q4 51)
171360 + 168Oqy,,,(9q75Q’
+98q351)h2q[7fn+,(261q7 1318q’ +459q4
+ 1057q3 3570q2+ 31 11) + 16&(747q6  4801q‘ + 9409q2 5355) +7fn,(261q7 1318q’ 459q‘ +1057q3 +3570q2 3111)
1
yn9
=
171360[1680qYn+,(9q7 56q’ + 98q3 51) 3360yn(9q8 56q6 + 98q‘
 51) +
+ 1680qyn,(9q7 56q’ + 98q3+ 51)h2q[7fn+,(261q7 1318q’  459q‘ + + 1057q’ + 3570q2 31 11) + 1wn(747q6 4801q‘ + 9409q2  5355) + 7fn,(261q7 1314’ + 459q‘ + 1057q3 3570q2 + 31 11)
Chemical Modelling: Applications and Theory, Volume 1
108
L
9, = yn + w4h2
with to =
t, =
406q2 151 3360(q2
9(42q2 13)
=
2324q2 255 2940q2 ’ 17
7840(73q2)’” = 56q2(q21)(3q2 7)’
wheref,,o =fn =A%,m). We consider the following cases: 1. If wo = w1 = w2 = w3 = w4 = 0 and q = ,/1223;$3209, the method has phaselag of order ten and an interval of periodicity equal to (0,28.7979). This method has been developed by sir no^.^^ 2. In the second case w1 = w2 = w3 = w4 = 0. The rest of the free parameters are defined in order to minimize the phaselag. As a result of the above, the method with wo = 0.0012658825222778819 and q = 1.24832627500665355549 has phaselag of order twelve and an interval of periodicity equal to (0,32.5779). This method has been developed by sir no^.^^
2: Atomic Structure Computations
109
3. In the third case wo = w1 = w2 = 0. The rest of the free parameters again are defined in order to minimize the phaselag. As a result of the above, the method with w3 = 0.008649462218303991659, w4 = 0.00186967764609313898 and q = 1.3036449113997327910153 has phaselag of order fourteen and an interval of periodicity equal to (0,62.7458). This method has been developed by sir no^.^^ 4. In the fourth case w2 = w3 = 0. The rest of the free parameters again are defined in order to minimize the phaselag. As a result of the wo = 0.0040863354486420366668, above, the method with ~2 = 0.0017983875880914431686 W I = 0.0074098409728733872451, and q = 1.2971977400853795112 has phaselag of order sixteen and an interval of periodicity equal to (0,9.86). This method has been developed by Simos.80 5. In the fifth case w3 = 0. The rest of the free parameters again are defined in order to minimize the phaselag. As a result of the above, the method with wo = 0.0049366748476923968708, ~1 = 0.00088531522247464098382, ~2 = 0.00659934013904141 10758, ~3 = 0.0017508795455034756109 and q = 1.2928357211311266263 has phaselag of order eighteen and an interval of periodicity equal to (0,9.87). This method has been developed by Simos." 6. In the sixth case the free parameters again are defined in order to minimize the phaselag. As a result of the above, the method with wo = 0.0047547870 149343227822, ~1 = 0.0068899408394688203923, ~3 = 0.0060423 177822287246136, ~2 = 0.0013044843045696810489, w4 = 0.0017176357369357489068 and q = 1.2897596572555966535 has phaselag of order twenty and an interval of periodicity equal to (0,9.87). This method has been developed by Simos.80 4.1.2.3 Tenth Algebraic Order Methods.  Very recently Simos8' has developed a tenth algebraic order implicit method of the form
+ h2('Sfn+2
+ '6fn+l + ' 7 f n + '8fn1
+' 9 f n  2 )
(199)
The coefficients of the above method are defined as functions of a free parameter s, in order for the algebraic order of the method to be equal to ten. The free parameter of the method s is defined in order for the phaselag of the method to
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Chemical Modelling: Applications and Theory, Volume 1
be minimal (i.e. of order fourteen). The interval of periodicity of the obtained method is equal to (0,9.1672). 4.2 Explicit Methods.  4.2.1 Fourth Algebraic Order Methods.  Simos82has
derived an explicit version of Numerov’s method. This version has three extra layers of the form:
The values of parameters a and b are determined in order to satisfy the minimal phaselag property. As a result of the above a method with phaselag of order eight and with an interval of periodicity equal to (0,21.48) is produced. 4.2.2 Sixth Algebraic Order Methods.  sir no^^^ has considered the following three free parameters family of explicit sixth algebraic order methods:
wherefn,O =fn. The values of parameters a, bi, i = 1,2 are determined in order
2: Atomic Structure Computations
111
to satisfy the minimal phaselag property. As a result of this, methods with phaselag of order eight, ten and twelve and with an interval of periodicity equal to ( 0 , 2 1 4 , (0,9.55) and (0,30.69) are produced. sir no^^^ has derived the explicit version of the sixth algebraic order method developed in ref. 67. Here the method does not have the first layer. The value of parameter a is determined in order to satisfy the minimal phaselag property. As a result of this, the family of methods with a = has phaselag of order eight and interval of periodicity equal to (0,21.48).
6
4.2.3 Eighth Algebraic Order Methods.  sir no^^^ has derived the explicit version of the eighth algebraic order method derived by Simos in ref. 77. Here the parameters are ai, i = 1(1)3. The free parameters ai, i = 1(1)3 are defined in order to minimize the phaselag. As a result of the above, the method where a1 = 0, a2 = 3644 9o0315.and a3 = has phaselag of order twelve and an interval of periodicity equal to (0, 12.9394). Also, the method where 122 158423 198943 a1 =  117312708240, a2 = and a2 = &has phaselag of order fourteen and an interval of periodicity equal to (0, 12.6756). Another explicit version of the eighth algebraic order method developed by sir no^^^ has been produced by Simos.86 In this version there are two additional layers in the explicit predictors. The form is given by

plus layers of the form:
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As a result of the above, the method where b=0.003149214873243 and l:;g has phaselag of order twelve and interval of periodicity a =  = 167310 equal to (0, 16.886). Simos and T o ~ g e l i d i shave ~ ~ derived the explicit version of the eighth algebraic order method developed by sir no^.^^ The free parameter q is defined in order to minimize the phaselag. As a result of the above, the method where q= has phaselag of order ten and interval of periodicity equal to (0,9.5301). Another explicit version of the eighth algebraic order method developed by sir no^^^ has obtained by Simos.88 In this version there is an additional layer in the explicit predictors. The form of the predictors (which give the explicit modification) is equivalent to (202). The free parameters q and a are defined in order to minimize the phaselag. As a result of the above, the method where q = 2.39883862783148341 and a = 0.00056130045827fifi 0.01248637074 has phaselag of order twelve and an interval of periodicity equal to (0,14.4576). Finally, an explicit version of the eighth algebraic order method developed by sir no^^^ has produced very recently by sir no^.^^ In this last version there are additional layers in the explicit predictors. The form of the predictors (which give the explicit modification) is analogous to (202) with the difference that here we have a layer of the form
4
+
w,
The free parameters are defined in order to minimize the phaselag. As a result of the above, the method where q = a1 =  2232 135016889608844951 1 19550772148704 and a 2 =  6 3 ~ ~ ~ : & o has phaselag of order twelve and an interval of periodicity equal to (0,30.840). The method where q = a2 =  135016889608844951 60591 139 832226518490491616940548753 a1 = 9601 11935794392243080438779104’ 22321 19550772148704 and a3 =  634915233420 has phaselag of order fourteen and an interval of periodicity equal to (0,9.852).
Finally,
the
669181021029000720042812225304894859
method
al = 975495312453608180345283208197941400480~
 22321 135016889608844951 19550772148704’
w,
where a4
148155
q = G . 6059 1 139
=  634915233420’
has phaselag of order fourteen and an interval of periodicity equal to (0,37.152). Recently, Simos and Tsitourasgo have developed an eighth order explicit method which has the form: a3 =
832226518490491616940548753 a2 = 9601 11935794392243080438779104
2: Atomic Structure Computations
113
The coefficients of the method are defined in order to have the maximal algebraic order method and the minimal phaselag. As a result an eighthorder method with phaselag of order fourteen is produced. The interval of periodicity is equal to (0,8.35). In Table 8 we present the properties of the methods with nonempty interval of periodicity described above.
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Chemical Modelling: Applications and Theory, Volume I
Table 8 Properties of numerical methods with nonempty interval of periodicity. A.O. is the algebraic order of the method. P.L.O. is the phaselag order, Int. Per. is the interval of periodicity of the method. IIE is an indication for implicit or explicit methods Method
A.0
P.L.0
IIE
Int. Per.
1Derived by Simos and Mousadis7' 1Derived by Simos and T ~ u g e l i d i s ~ ~ 1Derived by sir no^^^ 1Derived by Allison Raptis and Simos7' 1Derived by Simos7+ 1Derived by sir no^^^ 1Derived by sir no^^^ 1Derived by sir no^^^ 1Derived by sir no^^^ 1Derived by sir no^^^ 1Derived by sir no^^^ 1Derived by sir no^^^ 1Derived by Sirnosgo 1Derived by Sirnosgo 1Derived by Simos" 1Derived by Simos" 1Derived by Simosg2 1Derived by Simosg3 1Derived by Simosg3 1Derived by sir no^^^ 1Derived by sir no^*^ 1Derived by Simosg5 1Derived by Simos8' 1Derived by Simosg6 1Derived by Simos and Tougelidisg7 1Derived by Simosg8 1Derived by S i m 0 8 ~ 1Derived by Simosg9 1Derived by Simosg9 1Derived by Simos and Tsitourasgo
4 4 4 8 8 8 8 8 8 8 8 8 8 8 8 10 4 6 6 6 6 8 8 8 8 8 8 8 8 8
10 12 14 8 10 12 14 16 18 10 12 14 16 18 20 14 8 8 10 12 8 12 14 12 10 12 12 14 16 14
I I
(0, (0,28.095) (0,29.097) (0, 12.277) (0,29.190) (0,32.368) (0,34.930) (0,37.028) (0,72359) (0,28.798) (0,32.578) (0,62.746) (0,9.86) (0,9.87) (0,9.87) (0,9.167) . (0,21.48) (0,21.44) (0,9.55) (0,30.69) (0,21.48) (0, 12.939) (0, 12.676) (0, 16.886) (0,9.530) (0, 14.458) (0,30.840) (0,9.852) (0,37.152) (0,8.35)
I I
I I I I I I I I
I I I I E E E E E E E E E E E E E E
rn)
5 Variablestep Methods Simos" has derived a modification of the method of Simos.82Here an extra layer is presented. The values of parameters a, b and c are determined in order to satisfy the minimal phaselag property. As a result of the above a method with phaselag of order ten and with an interval of periodicity equal to (0,31.70) is produced. Based on the above method and the method of Simos82a variablestep procedure is developed. sir no^^^ has considered two modifications of the Pstable method of sir no^.^^ In the first modification an extra layer in the top of the algorithm is used. The values of parameters a, b, c are obtained in order to produce a Pstable method with minimal phaselag. As a result of the above, all the methods of the family
2: Atomic Structure Computations
115
with c < 0.0191 are Pstable methods with phaselag of order ten. In the second modification two extra layers in the top of the algorithm are used. The values of parameters a, b, c and dare determined in order to satisfy the minimal phaselag and Pstability properties. As a result of the above, all the methods of the family with a < 0.0239 are Pstable methods with phaselag of order twelve. Based on the above methods a variablestep procedure is developed. sir no^^^ has considered families of sixth algebraic order methods. These families are based on the formula
jj
I
n
2
= (3yn+, 1
52
+ 207, + 2 9 7 4 ) + (41fn+l h2 4992
 682fn  271fn1)
In the first family layers of the form
are held in the top of the algorithm. The value of parameter a is determined in order to satisfy the minimal phaselag property. As a result of the above, a method with phaselag of order eight and with an interval of periodicity equal to (0,26) is produced. In the second family an extra layer of the form Yn
= Yfl  bh2Cf;,+l 2fn +f,,)
follows the layers (207). The values of parameters a, b are determined in order to satisfy the minimal phaselag property. As a result of the above, a method with phaselag of order ten and with an interval of periodicity equal to (0,26) is produced. Based on the above methods a variablestep procedure is developed. Avdelas and Simosg3 have considered embedded, automatically defined, families of methods. The first family of methods is a generalization of the method of Simos.82Here, we have predictors of the form:
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Chemical Modelling: Applications and Theory, Volume I
where k is the number of layers and b is the number of families. Based on theorems fully proved and described in ref. 93, the parameters of the above method ak,b, k = O(l)b are determined automatically in order to satisfy the minimal phaselag property. We note that this method is the first automatically defined method (i.e.a generator of methods) in the literature. The second family of methods is a generalization of the method obtained by sir no^.^^ Here, there isn't the third layer of method of sir no^*^ and the layer before the final method has now the form:
where, as above, k is the number of layers and b is the number of families. Based on the theorems fully proved and described in ref. 93, the parameters of the above method are determined automatically in order to satisfy the phaselag property. The above families of automatically defined methods have a simple internal error control procedure and for this reason can be used in variable step mode. sir no^^^ has produced an embedded explicit Numerovtype fourth algebraic order method of the form:
The free parameters of the above families of methods are determined in order to have methods with minimal phaselag. So, for the above families of methods we have that: 1 . q = 0. For the value of parameter a1 =  & we have a method with phaselag of order eight and an interval of periodicity equal to (0,8.5 1). 2. q = 1. For the value of parameters a2 =  &,a1 = &we have a method with phaselag of order ten and an interval of periodicity equal to (0,8.97). 3. q = 2. For the value of parameters a3 = &,a2 = $, a1 = & we
2: Atomic Structure Computations
117
have a method with phaselag of order twelve and an interval of periodicity equal to (0,9.31). 4. q = 3. For the value of parameters a4 = &,a3 = &, a2 = & and al =we have a method with phaselag of order fourteen and an interval of periodicity equal to (0,9.53). 5. q = 4. For the value of parameters a5 = &, a4 = m 1 a3 = 49 2640 a2 = 1668859 096 a1 = we have a method with phaselag of order sixteen and an interval of periodicity equal to (0,9.66). 6. q = 5. For the value of parameters a6 =  &, a5 =  $, a4 =  2640 49 a3 = 1096 668857 a2 = a1 = we have a method with phaselag of order eighteen and an interval of periodicity equal to (0,9.75).
lzt5
2iz&
2iz:i0,
6 Pstable Methods of High Exponential Order
Simos9’ has considered the following family of methods
Based on the theory fully described in refs. 95 and 96 two Pstable methods of exponential and phaselag order eight and ten are produced. Based on these methods a new variablestep algorithm is introduced. Based on the above family of methods Avdelas and Simos9’ have given formulae for the coefficients of the Pstable method of exponential order 2m + 2 for arbitrary m. We note that the coefficients are determined in the computer automatically. The new approach is called generator of methods. This was the first generator of Pstable methods in the literature. We also note that the term generator is first introduced in this paper. sir no^^^ has derived another family of Pstable methods of exponential order 2m + 2 for m = 3( 1)6. The methods here are of hybrid type and have the form:
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So, we have that for:
a Pstable method of exponential order 8 is produced. For the values 13
1
1 1740 ’
=
al,o
81 a2,1 = 5120’
29 1 al’l = 6804’ a2,o = 10240 ’
1 13608 ’
= 
a=
8 63 ’
a Pstable method of exponential order 10 is produced. For the values 13 ao = 1968 ’
47 al =  66 ’ a=
b l = 2, 1
al,o =  405607 
8 33 ’
a2 = 
1 ao,o =  6888 ’
533 4224 ’
bo = 1, 1 ao,1 = 3444 ’
41 1 13 a2,o =   a2,1 = 20280 ’ 51 168 ’ 1968 ’
al?l =
2
1 22386 ’
a3,o = 
a3,1 = 105 ’
a Pstable method of exponential order 12 is produced. Finally, for the values 17 156
10
,55
ao=,
a1 =
b l = 2,
a=
78
1439 11440 ’
1 6160 ’
ao,1 = 
3 a2,o = 566840 7 25 a3,o =  3398164’
14171 a371 =1699082
bo = 1,
5
’
a4’o
207216 7 =
a Pstable method of exponential order 14 is produced.
2809 a2,1 = 850260’ 11033 a4,1=5 18040
119
2: Atomic Structure Computations
7 Matrix Methods for the OneDimensional Eigenvalue Schrodinger Equation In this section we consider the problem of determining the eigenvalues (which correspond to the physical bound states) of the special one dimensional or radial form of Schrodinger equation in the form
where y is a wave function, V(x) is a potential and E is an energy eigenvalue. Equation (213) is solved subject to y(x) + f o o ; y will also be subject to a normalization condition
JZ
y'dx = 1
When V(x)is an even function the solutions may be of even parity y (  x ) = y(x) or of odd parity y(x) = y(x). In this case we may limit the range of (x) to [0, 003. When V(x) is of a more general form we must then use the full range [oo, 001.
7.1 Methods of Discretization.  We consider particular discretizations of (213) applied to the more general form
which results in an eigenvalue equation of the generic type B' Ay = h2Ey. The matrices involved are generally not symmetric. The matrices are transformed by balancing, reducing to Hessenberg form and finally using the QR algorithm to determine the eigenvalues. There are many published routines available to do this (see ref. 99). 7.1.1 Methods which Lead to a Tridiagonal Form of the Matrix A .  Fack and Vanden Bergheloohave used the following method of discretization
which is equivalent to
and leads to a tridiagonal (not necessarily symmetric) form for A and B = I. The method is of algebraic order 2, phase lag order 2. The interval of periodicity is (034).
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7.1.2 Methods which Lead to a Pentadiagonal Form of the Matrix A .  Fack and Vanden Bergheloohave used the following method of discretization:
which is equivalent to
and leads to a pentadiagonal (not necessarily symmetric) form for A and B = I. The method is of algebraic order 4, phase lag order 4. The interval of periodicity is (o,$)= (0,533). 7.1.3 Methods which Lead to a Heptadiagonal Form of the Matrix A .  Flack and Vanden Bergheloohave used the following method of discretization:
which is equivalent to
and leads to a heptadiagonal (not necessarily symmetric) form for A and B = I. The method is of algebraic order 6, phaselag order 6. The interval of periodicity is (0, = (0,6.04).
g)
7.1.4 Numerov Discretization.  Flack and Vanden Berghe"' have used the well known Numerov method of discretization which leads to a tridiagonal (not necessarily symmetric) form for A and B. The method is of algebraic order 4, phaselag order 4. The interval of periodicity is (0,6). 7.1.5 Extended Numerov Form.  Flack and Vanden Berghe"' following method:
have used the
which leads to a pentadiagonal (not necessarily symmetric) form for A and B.
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121
The method is of algebraic order 6, phaselag order 6. The interval of periodicity is (0, y) = (0,6.67). 7.1.6 An Improved FourStep Method.  Simos and Williams’03have introduced the following fourstep method (with one free parameter), based on the method developed by Henrici3 and Simos and Williams: *02
The method is of algebraic order 6 . It has been proved that for a =& the phaselag is of order 8. The interval of periodicity is equal to 6 231Jm) = (0,7.26).This method is rather more complicated in its matrix form. The eigenvalue equation now takes the form
(0, +)
Ay = h2EBy
+ h4E2Cy
(224)
By introducing the variable w = h2Ey, the above equation can be transformed into the standard eigenvalue problem
7.1.7 An Improved ThreeStep Method.  SimoslMhas considered the following three step method:
The coefficients are determined in order to have minimal local truncation error. It has been proved that for
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~3
175 256 ’
= d3 = 
~4
35 512’
525 1024’
= d2 =  c5 = dl = 
219 14 7 cg = do =  ao = a6 =  al = a5 = 7940 17865’ 1024’
(227)
228 36121 bo = bl = 1014 a2 = a4 =  a3 =397 7146 ’ 397 ’ the method is of algebraic order eight. The above method leads to heptadiagonal discretization described in Section 7.1.3.
7.1.8 An Improved Hybrid FourStep Method. following fourstep method:
sir no^'^' has considered the
The coefficients are determined in order to have minimal local truncation error. It has been proved that for
35 c1 = d7 = 45 4096 ’ 32768 ’
C o = dg = 
c3 = d5 = 735 4096 ’ c6 = d2
a0
735
= 8192 9
~4
~2
44 1 8192 ’
= ds = 
11025 16384 ’
2205 4096 ’
= d4 =  ~5 = d3 = 
c7 = dl
63 cg = do =  45 4096 ’ 32768 ’
=
664 3587 a1 =a7 == ag = 58922640 ’ 409185 ’
(229)
123
2: Atomic Structure Computations
the method is of algebraic order ten. The above method leads to ninediagonal discretization. 7.2 Discussion.  The above methods have been presented in order of accuracy (and phase lag order). In sir no^'^' extensive numerical testing has been carried out, which indeed shows that the method based on (228) and (229) is the most accurate.
8 RungeKutta and RungeKuttaNystrom Methods for Specific Schrodinger Equations Simos and Williamslo6 have considered the following RungeKutta methods, based on the well known classical fourthorder RungeKutta method. The free parameters of the above methods are defined in order to minimize the phaselag, and are given by:
The methods are given in Tables 9, 10 and 11. Simos, Dimas and Sideridis'07 have constructed a RungeKuttaNystrom method with phaselag of order 8 and with an interval of periodicity equal to (0,9.114). This method is given in Table 12. Based on the results presented in the relative papers and based on some numerical tests made for this review, the most efficient RungeKutta method for specific Schrodinger equations is the one developed by Simos and Williams'06 with seven stages while the RungeKuttaNystrom method developed by Simos, Dimas and Sideridislo7 gives similar results in accuracy and computational efficiency. Table 9 Modijied fivestage Explicit RungeKutta method of order four derived by Simos and Williams106
1
I I
1/6
1/3
1/3
1/6
c0,O
cO.1
c0.2
c0.3
c0.4
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1 24
Table 10 ModiJied sixstage Explicit RungeKutta method of order four derived by Simos and
Table 11 ModiJied sevenstage Explicit RungeKutta method of order four derived by Simos and Williams'06
%
0 1/6
% 1/3
1/3
1/6
c0,O
cO.1
c0.2
c0,3
c0.4
cI,O
cI,1
cl.2
cl,3
c1,4
C1,S
Table 12 RungeKuttaNystrom method with phaselag of order 8 0.25475295159 0.03244953299 0.50540316962 0.03292284493 0.09479333659 0.19014504913 0 0.30985495135 1 0.10022791553 0.18458008559 0.19318290696 0.02200909191 0.16323603847 0.01892388531 0.65237178035 0.16546832871
9 Two Dimensional Eigenvalue Schrodinger Equation Recently, Konguetsof, Avdelas and Simoslo8have developed a generalization of Numerov's method for the numerical solution of the problem a2u a2u +ax2 a y =2(V
E)u
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125
which is the eigenalue Schrodinger equation in two dimensions. In the above equation the quantity V denotes the potential in two dimensions. The generalized Numerov method produced in this paper has the form
where h, and h, are the steps of discretization in two dimensions respectively. The above method gives more efficient results compared with the classical five point formula developed by Hajj.”’
10 Numerical Illustrations for the Methods with Constant Coefficients and the VariableStep Methods 10.1 Methods with Constant Coefficients.  We consider the numerical integration of the Schrodinger equation (121) in the wellknown case where the potential V(x)is the WoodsSaxon potential (122). As in the previous numerical experiments we take the domain of integration as 0 5 x 5 15. We consider the Schrodinger equation (121) in a rather large domain of energies, i.e., E E [ l , lOOO]. The problem we consider here is the socalled resonance problem. We note that for the modified WoodsSaxon potential and for the boundstates problem the results are similar.The integration is based on the description made in the previous paragraphs. For comparison purposes we use the following procedures:
Method MI: Pstable fourth order method Simos and Raptka Method MII: Pstable fourth order method of Simos and M o ~ s a d i swith ~~ phaselag of order twelve. Method MIII: Pstable fourth order method of Simos66with phaselag of order sixteen. Method MIV: Pstable sixth order method of sir no^^^ with phaselag of order eight. Method MV: Pstable sixth order method of Simos68with phaselag of order ten. Method MVI: Modification of sixth order method of Simos.68This method has been produced in this review and has phaselag of order twelve. Method MVII: Pstable sixth order method of S i m 0 8 ~with phaselag of order twenty .
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1
2
3
4
EigenergiaEn, n=1(1)4
Figure4 Error Err of the computed eigenvalues given by: Err = loglo (Eaccurare  Ecompuredl. The eigenvalues are computed using h = 1/16. The non existence of a value indicates that the corresponding maximum absolute error is larger than I
Method MVIII: Modification of sixth order method of sir no^.^^ This method has been produced in this review and has phaselag of order twentytwo. Method MIX: Eighth order method of Simos and Tsitouras7' with phaselag of order eight. Method MX: Eighth order method of sir no^^^ with phaselag of order eighteen and interval of periodicity equal to [0,72.86]. Method MXI: Eighth order method of sir no^^^ with phaselag of order fourteen and interval of periodicity equal to [0,62.75]. Method MXII: Tenth order method of Sirnos.'' The numerical results obtained for the twelve methods were compared with the analytic solution of the WoodsSaxon potential. Figure 4 shows the maximum absolute error in the computation of all resonances E,,n = 1(1)4, for step length equal to h = The nonexistence of a value indicates that the corresponding maximum absolute error is larger than 1.
8.
10.1.1 Remarks and Conclusion.  The most accurate fourth order Pstable method with constant coefficients is the Pstable method proposed by Simos66 with phaselag of order sixteen. The most accurate sixth order method with constant coefficients is the modification of the family of sixth order methods of sir no^^^ with phaselag of order twentytwo developed in this review. The most accurate eighth order method with constant coefficients is the eighth order method of sir no^^^ with phaselag of order eighteen and interval of periodicity
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2: Atomic Structure Computations
equal to [0,72.86]. Finally, the most accurate method with constant coeficients is the tenth algebraic order method developed by Simos.glA very accurate method is also the modification of the family of sixth order methods of sir no^^^ with phaselag of order twentytwo developed in this review. 10.2 VariableStep Methods.  10.2.1 Error Estimation.  It is known from the literature (see for example refs. 110 and 111 and references therein) that the local truncation error (L.T.E.) is based on the algebraic order of the method and that there are many methods for the estimation of the LTE in the integration of systems of initialvalue problems. The local error estimation algorithm used for our illustration is based on the fact that when we have a method with local error of higher (algebraic or phaselag or exponential or Bessel and Neumann) order then the approximation of the solution for the problems which have a periodic or oscillating solution is better. Denoting the solution obtained with higher (algebraic or phaselag or exponential or Bessel and Neumann) order method as and the solution obtained with lower (algebraic or phaselag or exponential or Bessel and Neumann) order method as yf+l, we have the following definition: Definition 12. We define the local error estimate in the lower order solution yf+l by the quantity
Under the assumption that when h is sufficientlysmall, the local error in yf+l can be neglected compared with that in We assume that the solution is obtained using the a higher (algebraic or phaselag or exponential or Bessel and Neumann) order method and the solution y;+l is obtained using a lower (algebraic or phaselag or exponential or Bessel and Neumann) order method. If the local phaselag error is bounded by acc and the step size of the integration used for the nth step length is h,,, the estimated step size for the ( n + 1)st step, which will give a local error bounded by acc, must be
$+,
where q is the order of the local error (i.e. the algebraic or phaselag or Bessel and Neumann approximation order). Following ref. 112, we have considered all step changes to halving and doubling. Thus, based on the procedure developed' l 2 for the Local Truncation Error, the step control procedure which we use for the Local Error is
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128
if L.E.E < acc, hn+l = 2hn if lOOacc > L.E.E 2 acc, hn+l = h n if L.E.E 2 100acc,
hn+l
hn =  and repeat the step. 2
(235)
It is known that the local error estimate is obtained to the lowerorder solution. This is applied, also, in our case of the local error estimate, i.e. the local error estimate is obtained for yf+l. However, if this error estimate is acceptable, i.e. less than the bound acc, we consider the widely used local extrapolation technique. Thus, although we are controlling an estimation of the local error in the lowerorder solution yk+,, we use the higherorder solution y z I at each accepted step.
10.2.2 Coupled Diflerential Equations.  There are many problems in theoretical physics and theoretical chemistry, atomic physics, physical chemistry, quantum chemistry and chemical physics which can be transformed to the solution of coupled differential equations of the Schrodinger type. The closecoupling differential equations of the Schrodinger type may be written in the form
[$+k:
li(li +1) X2
Y;i
1
YO
N
= C~rn~rnj m= I
for1 s i s N a n d r n # i . We have investigated the case in which all channels are open. So we have the following boundary conditions (see ref. 1 13 for details): y O = O at x = O I
yii a kixjll(kix)b;,+[$KOkixn,l(tix) 2
(237)
where j l ( x ) and nl(x) are the spherical Bessel and Neumann functions, respectively. We note here that the methods with large stability intervals or the Bessel and Neumann fitted ones can be used for the closed channels also. Based on the detailed analysis developed113and defining a matrix K‘ and diagonal matrices M , N by:
2: Atomic Structure Computations
129 I
.1
Mu= kixil, (k,x)q, No = kixn,, ( k ,x)e, we find that the asymptotic condition (237) may be written as:
We note here that one of the most used methods for the numerical solution of the coupled differential equations arising from the Schrodinger equation is the Iterative Numerov method of Allison.’ I 3 It is easy for one to see that a real problem in theoretical physics, theoretical chemistry, atomic physics, quantum chemistry, physical chemistry and molecular physics which can be transformed to closecoupling differential equations of the Schrodinger type is the rotational excitation of a diatomic molecule by neutral particle impact. Denoting, as in ref. 113, the entrance channel by the quantum numbers ( j , f ) , the exit channels by (j’,!‘), and the total angular momentum b y J = j + I = j ’ + I ’ , w e f i n d that
where
[
kjlj = ?! E + jh2j ( i ( j + 1)  j ‘ ( J ‘ ti2
+ 1))
1
E is the kinetic energy of the incident particle in the centerofmass system, Z is the moment of inertia of the rotator, and p is the reduced mass of the system. Following the analysis of ref. 113, the potential V may be written as
and the coupling matrix element is given by
where thef2 coefficients can be obtained from formulas given by Berstein et kjy is a unit vector parallel to the wave vector kjlj and Pi, i = 0 , 2 are
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Chemical Modelling: Applications and Theory, Volume 1
Legendre polynomials (see ref. 114 for details). The boundary conditions may then be written'13 y,=Oat x=O
where the scattering S matrix is related to the K matrix of (237) by the relation
The calculation of the cross sections for rotational excitation of molecular hydrogen by impact of various heavy particles requires the existence of the numerical method for stepbystep integration from the initial value to matching points. In our numerical test we choose the S matrix which is calculated using the following parameters: 2P = 1000.0,P = 2.35 1, E = 1.1 li2 I Vo(x) = x12  x 6 ' V2(x) = 0.2283Vo(x)
As described'13 we take J = 6 and consider excitation of the rotator from the j = 0 state to levels up t o j = 2,4,6 giving sets of four, nine and sixteen coupled
differential equations, respectively. Following Berstein' l 5 and Allison' l3 the reduction of the interval [0,00] to [0,xo] is obtained. The wavefunctions then vanish in this region and consequently the boundary condition (243) may be written as
For the numerical solution of this problem we have used (1) the well known Iterative Numerov method of Allison,' l 3 (2) the variablestep method of Raptis and Cash,'12 (3) the variablestep method of sir no^,^^ (4) the variablestep exponentiallyfitted method of Simos and Williams,26 (5) the variablestep eighth algebraic order method developed by sir no^,^^ (6) the explicit variablestep method developed87by Simos and Tougelidis, (7) the variablestep Pstable sixth algebraic order method,69 (8) the embedded Pstable fourth algebraic order method,66 (9) the variablestep Bessel and Neumann fitted method developed by Simos and (10) the variablestep (of exponential orders eight and ten) Pstable method developed by sir no^:^ (1 1) the generator of various exponential order Pstable methods developed by Avdelas and sir no^:^ (12) the embedded sixth algebraic order method developed by
2: A tomic Structure Computations
131
Table 13 RTC (real of computation (in seconds)) to calculate IS12 for the variablestep methods (1)(20). acc = hmax is the maximum stepsize
Iterative Numerov' l 3 Variablestep Method of Raptis and Cash112 Variablestep Method of Simos7' Variablestep Method of Simos and Williams26 Variablestep Method of sir no^^^ Variablestep Method of Simos and Tougelidiss7 Variablestep Method of sir no^^^ Variablestep Method of Simos66 Variablestep Method of Simos and Williams4* Variablestep Method of Simos9' Generator of Methods of Avdelas and sir no^^^ Embedded Methods of Avdelas and sir no^^^ Variablestep Method of Simos' Variablestep Method of SimOssl Variablestep Method of S i m ~ s ~ ~ Variablestep Method of Thomas and Simos2'
4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16 4 9 16
0.014 0.014 0.014 0.056 0.056 0.056 0.056 0.056 0.056 0.48 0.48 0.48 0.056 0.056 0.056 0.48 0.48 0.48 0.224 0.224 0.224 0.48 0.48 0.48 0.48 0.48 0.48 0.112 0.112 0.1 12 0.224 0.224 0.224 0.224 0.224 0.224 0.112 0.112 0.112 0.224 0.224 0.224 0.112 0.112 0.112 0.112 0.1 12 0.112
3.25 23.51 99.15 1.55 8.43 43.32 1.05 5.25 27.15 0.24 0.96 5.04 0.35 1.38 6.5 1 0.27 1.48 6.3 1 0.30 1.42 6.43 0.28 1.55 6.37 0.18 0.92 5.32 1.43 8.22 40.13 0.83 6.19 12.33 1.03 6.9 1 14.05 0.72 3.20 14.35 0.30 1.54 7.42 0.52 1.88 8.04 0.18 0.85 4.82
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Table 13 continued ~~
Method
hmax
RTC
0.448 0.448 0.224 0.896 0.896 0.448 0.448
0.14 0.85 4.90 0.04 0.35 2.58 0.11
9 16 4
0.448 0.448 0.48
0.60 3.05 0.15
9 16
0.48 0.48
0.90 3.20
N
Variablestep Method of S i m o ~ ~ ~ Variablestep Method of S i m o ~ ~ New variablestep Method developed in this review (exponentiallyfitted) New variablestep Method developed in this review (constant coefficients)
Avdelas and sir no^,^^ (13) the exponentiallyfitted variablestep method developed by Simos,’ (14) the variablestep phasefitted method developed by sir no^,'^ (15) the variablestep Pstable method developed by sir no^,^^ (16) the exponentiallyfitted variablestep method developed by Thomas and Sirnos?’ (17) the variablestep Bessel and Neumann fitted method developed by Sirnos:j (18) the variablestep Bessel and Neumann fitted method developed by Simos,44(19) the new exponentiallyfitted variable step method based on the new exponentiallyfitted tenth algebraic order method developed in Section 2.1.6 and the method of sir no^,^' and (20) the new variablestep method developed using the modification of the family of methods derived by Sirn0s,6~ which is obtained in Section 4.1.1.2 (equations (192) and (193)) and the modification of method produced by Simos,66 which is developed in this review. In Table 13 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. We note that in this Table N indicates the number of equations of the set of coupled differential equations. 10.3 Remarks and Conclusion.  The most efficient variablestep method for the solution of coupled differential equations arising from the Schrodinger equation is the variablestep Bessel and Neumann fitted method of Simos.44 Efficient variablestep method for the solution of the above problem is also the variablestep Bessel and Neumann fitted method of sir no^.^^ Efficient methods also are the new exponentiallyfitted variable step method which is produced based on the new exponentiallyfitted tenth algebraic order method developed in Section 2.1.6 and the method of sir no^,^' the exponentiallyfitted variablestep method developed by Thomas and Sirnos:’ the new variabestep method which is developed using the modification of the family of
2: Atomic Structure Computations
133
methods derived by Sim0s,6~which is obtained in Section 4.1.1.2 (equations (192) and (193)) and the modification of method produced by Simos,66which is developed in the review and the variablestep exponentiallyfitted method of Simos and Williams.42 It is obvious that the critical properties for efficient variablestep procedures are (1) Bessel and Neumann fitting for scattering problems, (2) exponentiallyftting, (3) Pstability and (4) minimal phaselag.
Appendix YOThe first part of the programme consists of the calculation of the % matrix elements which form the coefficients of the system of equations. YOThe second part of the programme, as this has been explained in section YO2.1.2, consists of the iterative application of the L’Hospital’s rule % for the calculation of the solution of these equations that make up the YOcoefficients of the family of new tenth algebraic order exponentially YOfitted methods for the numerical integration of the Schrodinger type YOequations. Y O
YOWe note here that: The coefficients are YOc2, c3, c4, c5 for the basic formula. % aO, a 1, a2, a3, a4 and
YObO, bl, b2 for the appoximations. % O h
YOThe elements of the matrices for the solution % of the system of equations are determined.
YO parl: =  2 * ~ * 2 + 4 * c o s h ( w ) ~ 2 * ~ * 2 ; par2: = 2*wA2*cosh(w); par3: = 2*cosh(w/2)*wA2; par4:=wA2; par5 = 7936/13*cosh(w)7988/13+ 4*cosh(w)^2; parl0: = subs(w = w0,parl); parl 1: = subs(w = wl ,par 1); parl2: = subs(w = w2,parl); parl3: = subs(w = w3,parl); par20: = subs(w = wO,par2); par2 1:= subs(w = w 1,pad); > par22: = subs(w = w2,par2); > par23: = subs(w = w3,par2); > par30: = subs(w = wO,par3); > par31: = subs(w = wl,par3); > par32: = subs(w = w2,par3); > par33: = subs(w = w3,par3); > par40: = subs(w = wO,par4); > par4 1: = subs(w = w 1,par4); > par42: = subs(w = w2,par4);
> > > > > > > > > > >
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Chemical Modelling: Applications and Theory, Volume I
> par43: = subs(w = w3,par4); > par50: = subs(w = wO,par5); > par5 1: = subs(w = w 1,par5); > par52: = subs(w = w2,par5); > par53: = subs(w = w3,par5); > with(lina1g); YO
YO
YOThe Denominator YO YO > matd: = array( 1..4,1..4,[[par lO,par2O,par30,par40],[par11,par21,par3 1,pa > r4l],[par 12,par22,par32,par42],[par 13,par23,par33,par43]]); Y O
YOThe equations will be solved by an application of Cramer’s rule. % From the theory of exponentiallyfitted methods it has been shown YOthat 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. The theory shows that for % % We must find
Yo the 6th derivative w.r.t w0, the 4th derivative w.r.t wl, YOand the 2nd derivative w.r.t w2. > denl: =det(matd); > denl: = diff(denl,w0$6); > den 1: = diff(den 1,w 1$4); > den 1:= diff(den 1,w2$2); Y O
% We substitute the values w0 = wl = w2 = 0 and w3 = w YO
> > > >
denl: = subs(w0 = 0,denl); den 1: = subs(w 1 = 0,den 1); denl : = subs(w2 = 0,den 1); denl: = subs(w3 = w,denl);
YO YOFor the following matrices we repeat the steps described above. YOThese will determine the appropriate numerators. YO
YOCalculation of coefficient c2. % > > matc2: = array( 1..4,1..4,[Lpar50,par20,par30,par40],[par5 1,par2 1,par3 1,p > ar4l],[par52,par22,par32,par42],[par53,par23,par33,par43]]); > dc2: = det(matc2); > dc2: = diff(dc2,w0$6); > dc2: = diff(dc2,w1$4); > dc2: = diff(dc2,w2$2); > dc2: = subs(w0 = O,dc2); > dc2: = subs(w1 = O,dc2); > dc2: = subs(w2 = O,dc2);
2: Atomic Structure Computations > dc2: = subs(w3= w,dc2); > c2: = combine(dc2/denl); %
YOTaylor series expansion of the coefficient c2. %
> c2t: = convert(series(c2,w= 0,24),polynom); Y O
YOCalculation of coefficient c3. YO > matc3: = array( 1..4,1..4,[[parlO,par50,par3O,par40],[parll,par51 ,par3 l,p > ar4l],[par12,par52,par32,par42],[parl3,par53,par33,par43]]); > dc3: = det(matc3); > dc3: = diff(dc3,w0$6); > dc3: = diff(dc3,w 1$4); > dc3: = diff(dc3,w2$2); > dc3: = subs(w0 = O,dc3); > dc3: = subs(w1 = O,dc3); > dc3: = subs(w2 = O,dc3); > dc3: = subs(w3 = w,dc3); > c3: = combine(dc3/den1); Y O
YOTaylor series expansion of the coefficient c3. YO
> c3t: = convert(series(c3,w= 0,24),polynom); Y O
% Calculation of coefficient c4. Y O
> matc4: = array( 1..4,1..4,[[par 1O,par20,par50,par40],[par11,par2 1,par5 1,p > ar4 l],[par 12,par22,par52,par42],[par13,par23,par53,par43]]); > dc4: = det(matc4); > dc4: = diff(dc4,w0$6); > dc4: = diff(dc4,w 1$4); > dc4: = diff(dc4,w2$2); > dc4: = subs(w0 = O,dc4); > dc4: = subs(w1 = O,dc4); > dc4: = subs(w2 = O,dc4); > dc4: = subs(w3 = w,dc4); > c4: = combine(dc4/den1); YO
Yo Taylor series expansion of the coefficient c4. YO > c4t: = convert(series(c4,w= 0,24),polynom); YO Yo Calculation of coefficient c5. % > matc5: = array( 1..4,1..4,[[par 10,par20,par30,par50],[parll,par21,par3 1,p > ar5 l],[par 12,par22,par32,par52],[par13,par23,par33,par53]]); > dc5: = det(matc5); > dc5: = diff(dc5,w0$6); > dc5: = diff(dc5,wl$4);
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> dc5: =diff(dc5,w2$2); > dc5: = subs(w0 = O,dc5); > dc5: = subs(w1 = O,dc5); > dc5: = subs(w2 = 0,dcS); > dc5: = subs(w3 = w,dc5); > c5: = combine(dc5/den1); %
Taylor series expansion of the coefficient c5. YO > c5t: = convert(series(c5,w= 0,24),polynom); % % We repeat the above procedure for the computation of the coefficients YOof the approximations defined by the equations (87) and (89).
YO YO
% Calculation of the coefficients aO, a l , a2, a3, a4. % O h
> par 1:= 2*cosh(w); > par2: = 1; > par3: = 2*wA2+ 4*wA2*cosh(w)*2; > par4: = 2*wn2*cosh(w); > par5 = w2; > par6: =2*cosh(w/2)4*~osh(w)~2 + 2; > par 10: = subs(w = w0,par 1); > par1 1: = subs(w = w1,parl); > parl2: = subs(w = w2,parl); > parl3: = subs(w = w3,parl); > parl4: = subs(w = w4,parl); > par20: = subs(w = wO,par2); > par21: = 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 = wl,par4); > par42: = subs(w = w2,par4); > par43: = subs(w = w3,par4); > par44: = subs(w = w4,par4); > par50: = subs(w = wO,par5); > par51: subs(w= wl,par5); > par52: = subs(w = w2,par5); > par53: = subs(w = w3,par5); > par54: = subs(w = w4,par5); > par60: = subs(w = wO,par6);
2: Atomic Structure Computations > par61:= subs(w = wl,par6); > par62:= subs(w = w2,par6); > par63:= subs(w = w3,par6); > parW.= subs(w = w4,par6); > > matdl:= array( 1..5,1..5,[[par10,par20,par3O,par40,par50],[parll,par2l,p > ar31,par4l,par5l],[par12,par22,par32,par42,par52],[par13,par23,par33,p > ar43,par53],[par14,par24,par34,par44,par54]]); > den2:= det(matd1); > den2:= diff(den2,w0$8); > den2:= diff(den2,w 1$6); > den2:= diff(den2,w2%4); > den2:= diff(den2,w3$2); > den2:= subs(w0 = O,den2); > den2:= subs(w1 = O,den2); > den2:= subs(w2 = O,den2); > den2:= subs(w3 = O,den2); > > mata0:= array( 1..5,1..5,[~ar60,par20,par30,par40,par50],[par61,par21,p > ar31,par41,par5l],[par62,par22,par32,par42,par52],[par63,par23,par33,p > ar43,par53],[par64,par24,par34,par44,par54]]); > daO:= det(mata0);
daO:= diff(daO,w0$8); daO:= diff(da0,w 1$6); > daO:= diff(daO,w2$4); > daO:= diff(daO,w3$2); > daO:= subs(w0 = 0,daO); > daO:= subs(w 1= 0,daO); > daO:= subs(w2 = 0,daO); > daO:= subs(w3 = 0,daO); > aO:= combine(daO/den2); > >
> > matal:= array( 1..5,1..5,[[parl0,par60,par30,par40,par5O],[parll ,par61,p > ar31,par4l,par5l],[parl2,par62,par32,par42,par52],[par13,par63,par33,p > ar43,par53],[parl4,par64,par34,par44,par54]]); > dal:= det(mata1); > da 1:= diff(da 1 ,w0$8); > da1:= diff(da 1,w1$6); > da1 : = diff(da 1,w2$4); > dal:=diff(dal,w3$2); > dal : = subs(w0 = 0,da1); > da1:= subs(w 1 = 0,da1); > dal:= subs(w2 = 0,dal); > da 1:= subs(w2 = 0,da1); > da 1 := subs(w3 = 0,dal); > dal:= subs(w4 = w,dal); > al:= combine(dal/den2); > a1t:= convert(taylor(a1,w = 0,24),polynom); > > mata2:= array(l..5,1..5,[[par10,par20,par60,par40,par50],[par11 ,par2l,p
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> ar61,par41,par5 l],[par 12,par22,par62,par42,par52I,[parl 3,par23,par63,p > ar43,par53],[par14,par24,par64,par44,par54]]); > da2: = det(mata2); > da2: = diff(da2,w0$8); > da2: = diff(da2,w 1$6); > da2: = diff(da2,w2$4); > da2: = diff(da2,w3$2);
> da2: = subs(w0 = O,da2); > da2: = subs(w1 = O,da2); > da2: = subs(w2 = O,da2); > da2: = subs(w3 = O,da2); > a2: = combine(da2/den2); > > > > >
mata3: = array( 1..5,1..5,[[par10,par20,par30,par60,par50],[parll ,par21,p ar3 1,par6 1,par5l],[par12,par22,par32,par62,par52],[parl3,par23,par33,p ar63,par53],[parl4,par24,par34,par64,par54]]); da3: = det(mata3); > da3: = diff(da3,w0$8); > da3: = diff(da3,wl$6); > da3: = diff(da3,w2$4); > da3: = diff(da3,w3$2); > da3: = subs(w0 = O,da3); > da3: = subs(w1 = O,da3); > da3: = subs(w2 = O,da3); > da3: = subs(w3 = O,da3); > a3: = combine(da3/den2); > > mata4: = array( 1..5,1..5,[[parlO,par20,par30,par40,par60],[parll ,par2l,p > ar3 1,par4 1,par6l],[par 12,par22,par32,par42,par62],[parl3,par23,par33,p > ar43,par63],[par 14,par24,par34,par44,par64]]); > da4: = det(mata4); > da4: = diff(da4,w0$8); > da4: = diff(da4,w1$6); > da4: = diff(da4,w2$4); > da4: = diff(da4,w3$2); > da4: = subs(w0= O,da4); > da4: = subs(w1 = O,da4); > da4: = subs(w2 = O,da4); > da4: = subs(w3 = O,da4); > da4: = subs(w4 = w,da4); > a4: = combine(da4/den2);
% Y O
% Calculation of the coefficients bO, bl, b2. Y O YO
> par 1:= 2*sinh(w);
2: Atomic Structure Computations > par2: = 4*wA2*cosh(w)*sinh(w); > par3: = 2*wA2*sinh(w); > par4: = 2*sinh(w/2)4*cosh(w)*sinh(w); > parl0: = subs(w = w0,parl); > parll:=subs(w=wl,parl); > par 12: = subs(w = w2,par 1); > par20: = subs(w = w0,par2); > par2 1:= subs(w = w 1,par2); > par22: = subs(w = w2,par2); > par30: = subs(w = w0,par3); > par31:=subs(w=wl,par3); > par32: = subs(w = w2,par3); > par40: = subs(w = wO,par4); > par41: = subs(w = wl ,par4); > par42: = subs(w = w2,par4); > > matd2: = array( 1..3,1..3,[[parlO,par20,par30],[parl l,par2l,par3l],[par12 > ,par22,par32]]); > den3: = det(matd2); > den3: = diff(den3,w0$3); > den3: = diff(den3,wl$l); > den3: = subs(w0= O,den3); > den3: = subs(w1 = O,den3); > den3: = subs(w2= w,den3); > > matb0: = array(1..3,1..3,[Lpar40,par20,par30],[par41,par21,par3 l],[par42 > ,par22,par32]]); > dbO: = det(matb0); > dbO: = diff(dbO,w0$3); > dbO: = diff(dbO,wl$l); > dbO: = subs(w0 = 0,dbO); > dbO: = subs(wl= 0,dbO); > dbO: = subs(w2 = w,dbO); > bO: = combine(dbO/den3); > > matbl: = array(l..3,1..3,[[parlO,par40,par30],[parl l,par41,par3l],[parl2 > ,par42,par32]]); > dbl: = det(matb1); > dbl:=diff(dbl,w0$3); > dbl: = diff(dbl,wl$l); > dbl: =subs(wO=O,dbl); > dbl: = subs(wl= 0,dbl); > dbl: = subs(w2= w,dbl); > bl: = combine(dbl/den3); > b 1t: = convert(taylor(b1,w = 0,24),polynom); > > matb2: = array( 1..3,1..3,[[par 10,par20,par40],[par11,par21,par4l],[parl2 > ,par22,par42]]); > db2: = det(matb2); > db2: = diff(db2,w0$3);
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> db2: = diff(db2,wl$l); > db2: = subs(w0 = O,db2); > db2: = subs(w1 = O,db2); > db2: = subs(w2 = w,db2); > b2: = combine(db2/den3); > b2t: = convert(taylor(b2,w = 0,24),polynom);
References 1. T. Lyche, Numerische Mathematik, 1972, 10,65. 2. T.E. Simos, Doctoral Dissertation, National Technical University of Athens, Athens, Greece, 1990. 3. P. Henrici, ‘Discrete Variable Methods in Ordinary Differential Equations’, John Wiley and Sons, New York, USA, 1962. 4. A.D. Raptis, Bull. Greek Math. Soc., 1984,25, 113. 5. A.D. Raptis and A.C. Allison, Comp. Phys. Commun., 1978,14, 1. 6. L.Gr. Ixaru and M. Rizea, Comp. Phys. Commun., 1980,19,23. 7. T.E. Simos, Comp. Phys. Commun., 1992,71,32. 8. T.E. Simos, J. Comp. Math., 1996, 14, 120. 9. T.E. Simos, Phys. Lett. A , 1993,177,345. 10. A.D. Raptis, Computing, 1982,28,373. 11. A.D. Raptis, Comp. Phys. Commun., 1983,28,427. 12. A.D. Raptis, Comp. Phys. Commun., 1981,24, 1. 13. T.E. Simos, J. Comp. Appl. Math., 1990,30,251. 14. T.E. Simos, IMA J. N. Anal., 1991,11,347. 15. A.D. Raptis and J.R. Cash, Comp. Phys. Commun., 1987,44,95. 16. J.R. Cash, A.D. Raptis and T.E. Simos, J. Comp. Phys., 1990,91,413. 17. T.E. Simos, Modern Phys. Lett. A , 1997,12, 1891. 18. R.M. Thomas, T.E. Simos and G.V. Mitsou, J. Comp. Appl. Math., 1996,67,255. 19. T.E. Simos, J. Chem. Inf. Comp. Sci., 1997,37, 343. 20. T.E. Simos and P.S.Williams, Comp. Chem., 1997,21,403. 21. T.E. Simos, Helv. Phys. Acta, 1997,70,781. 22. T.E. Simos and G.V. Mitsou, Comp. Math. Appl., 1994,28,41. 23. T.E. Simos, J. Comp. Appl. Math., 1995,58, 337. 24. T.E. Simos, Mol. Simulation, 1998,20, 285. 25. R.M. Thomas and T.E. Simos, J. Comp. Appl. Math., 1997,87,215. 26. T.E. Simos and P.S. Williams, Comp. Chem., 1998,22, 185. 27. T.E. Simos, Comp. Phys.s, 1998,12,290. 28. T.E. Sirnos, Int. J. Modern Phys. A , 1998,13,2613. 29. T.E. Simos, Appl. Math. Comp., 1999,98, 185. 30. T.E. Simos, Int. J. Modern Phys. C, 1998,9,271. 31. T.E. Simos, Comp. Chem., 1998,22,467. 32. J.P. Coleman and L.Gr. Ixaru, I M A J. Numerical Anal., 1996,16, 179. 33. A.D. Raptis and T.E. Sirnos, BIT, 1991,31,160. 34. U. Ananthakrishnaiah, J. Comp. Appl. Math., 1982,8, 101. 35. T.E. Simos, J. Comp. Phys., 1999,148,305. 36. T.E. Simos, J. Math. Chem., in press.
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37. L.Gr. Ixaru, G. Vanden Berghe, H. De Meyer and M. Van Daele, Comp. Phys. Commun., 1997,100,56. 38. L.Gr. Ixaru, H. De Meyer, G. Vanden Berghe and M. Van Daele, Comp. Phys. Commun., 1997,100,71. 39. L.Gr. Ixaru, Comp. Phys. Commun., 1997,105, 1. 40. L.Gr. Ixaru and B. Paternoster, J. Comp. Appl. Math., 1999,106,87. 41. T.E. Simos and A.D. Raptis, J. Comp. Appl. Math., 1992,43,313. 42. T.E. Simos and P.S. Williams, Comp. Chem., 1997,21, 175. 43. T.E. Simos, Mol. Simulation, 1999,21, 191. 44. T.E. Simos, Comp. Phys., 1998,12,635. 45. T.E. Simos, Int. J. Comp. Math., 1991,39, 135. 46. T.E. Simos, Int. J. Comp. Math., 1992,46, 77. 47. T.E. Simos, Appl. Math. Comp., 1992,49,261. 48. T.E. Simos, Jpn. J. Ind. Appl. Math., 1993, 10,289. 49. T.E. Simos, Math. Comp. Simulation, 1993,35, 53. 50. T.E. Simos, J. Comp. Appl. Math., 1994,55, 125. 51. T.E. Simos, Int. J. Quantum Chem., 1995,53,473. 52. T.E. Simos, Int. J. Modern Phys. A, 1995, 10,2431. 53. E. Hairer, S.P. Norsett and G Wanner, ‘Solving Ordinary Differential Equations I’, SpringerVerlag, Berlin, Germany, 1987. 54. L.Gr. Ixaru, ‘Numerical Methods for Differential Equations and Applications’, Reidel Publishing Company, Dordrecht, Boston, Lancaster, 1984. 55. J.W. Cooley, Math. Comp., 1961, 15,363. 56. J.M. Blatt, J. Comp. Phys., 1967, 1, 382. 57. J.D. Lambert and I.A. Watson, J. Inst. Math. Appl., 1976, 18, 189. 58. R.M. Thomas, BIT, 1984,24,225. 59. T.E. Simos and P.S. Williams, London Guildhall University, Department of Computing, Information Systems and Math., Working Paper 97/01, 1997. 60. P.J. van der Houwen and B.P. Sommeijer, SIAM J. Numerical Anal., 1987,24,595. 61. J.P. Coleman, I M A J. Numerical Anal., 1989,9, 145. 62. T.E. Simos, Comp. Math. Appl., 1993, 26,43. 63. T.E Simos and P.S. Williams, London Guildhall University, Department of Computing, Information Systems and Math., Working Paper 95/02, 1995. 64. T.E. Simos and A.D. Raptis, Computing, 1990,45, 175. 65. T.E. Simos and G. Mousadis, Mol. Phys., 1994,83, 1145. 66. T.E. Simos, Int. J. Quantum Chem., 1997,62,467. 67. T.E. Simos, Comp. Phys. Commun., 1993,74,63. 68. T.E. Simos, Comp. Chem., 1996,21, 125. 69. T.E. Simos, Phys. Scripta, 1997,55,644. 70. T.E. Simos and Ch. Tsitouras, J. Comp. Phys., 1997,130, 123. 71. T.E. Simos and G. Mousadis, Comp. Math. Appl., 1995,29,31. 72. T.E. Simos and G. Tougelidis, Comp. Chem., 1996,20,397. 73. T.E. Simos, Int. J. Modern Phys. C, 1996,7,33. 74. T.E. Simos, Comp. Phys., 1993,7,460. 75. A.C. Allison, A.D. Raptis and T.E. Simos, J. Comp. Phys., 1991,97,240. 76. M.J. Jamieson, J. Comp. Phys., 1999, 149, 194. 77. T.E. Simos, Comp. Phys. Commun., 1997,105, 127. 78. T.E. Simos, Int. J. Modern Phys. C, 1996,7,825. 79. T.E. Simos, J. Math. Chem., 1997,21,359. 80. T.E. Simos, Can. J. Phys., 1998,76,473.
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T.E. Simos, Int. J. Modern Phys. C, 1998,9, 1055. T.E. Simos, Appl. Numerical Math., 1991,7,201. T.E. Simos, J. Comp. Appl. Math., 1992,39, 89. T.E. Simos, Math. Models Methods Appl. Sci., 1995,5, 159. T.E. Simos, Int. J. Theoretical Phys., 1997,36,663. T.E. Simos, Int. J. Quantum Chem., 1998,68,191. T.E. Simos and G. Tougelidis, Comp. Chem., 1997,21,327. T.E. Simos, Comp. Mat. Sci., 1997,8,317. T.E. Simos, Comp. Phys. Commun., 1999,119,32. Ch. Tsitouras and T.E. Simos, Appl. Math. Comp., 1998,95, 15. T.E. Simos, Appl. Math. Lett., 1993,6,67. T.E. Simos, Appl. Math. Comp., 1994,64,65. G. Avdelas and T.E. Simos, Comp. and Math. with Applications, 1996,31,85. T.E. Simos, Comp. Chem., 1998,22,433. T.E. Simos, J. Comp. Phys., 1993, 108, 175. J.M. Franco and M. Palacios, J. Comp. Appl. Math., 1990,30, 1. G. Avdelas and T.E. Simos, J. Comp. Appl. Math., 1996,72,345. T.E. Simos, Comp. Math. Appl., 1998,36, 51. W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, ‘Numerical Recipes in C: The Art of Scientific Computing’, Cambridge University Press, Cambridge, 1992. 100. V. Fack, and G. Vanden Berghe, J. Phys. A: Math. Gen., 1985,18,3355. 101. V. Fack, and G. Vanden Berghe, J. Phys. A: Math. Gen., 1987,20,4153. 102. T.E. Simos and P.S. Williams, London Guildhall University, Department of Computing, Information Systems and Math., Working Paper 96/01, 1996. 103. T.E. Simos., P.S. Williams, J. Comp. Appl. Math., 1997,79, 189. 104. T.E. Simos, Can. J. Phys., 1996,75,325. 105. T.E. Simos, J. Comp. Appl. Math., 1998,91,47. 106. T.E Simos and P.S. Williams, Int. J. Modern Phys. A , 1996, 11,473 1. 107. T.E. Simos, E. Dimas and A.B. Sideridis, J. Comp. Appl. Marh., 1994,51,317. 108. A. Kaonguetsof, G. Avdelas and T.E. Simos in ‘International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA’99)’, ed. H. Arabnia, in press. 109. F.Y. Hajj, J . Phys. B: Atomic Mol. Phys., 1982, 15,683. 110. L.F. Shampine, H.A. Watts and S.M. Davenport, SIAM Rev., 1975,18,376. 11 1. G. Avdelas and T.E. Simos, Technical University of Crete, Applied Mathematics and Computers Laboratory, Technical Report 1/1995. 112. A.D. Raptis and J.R. Cash, Comp. Phys. Commun., 1985,36,113. 113. A.C. Allison, J. Comp. Phys., 1970, 6, 378. London, 114. R.B. Berstein, A. Dalgarno, H. Massey and I.C. Percival, Proc. Roy. SOC. Ser. A , 1963,274,427. 115. R.B. Berstrein, J . Chem. Phys., 1960,33,795. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99.
3 Atoms in Molecules BY P. L. A. POPELIER, F. M. AICKEN AND S. E. O’BRIEN
1 Introduction
1.1 What Is AIM?  The theory of “Atoms in Molecules” (AIM) is an interpretative theory which aims to recover chemical insight from modern highresolution electron densities. These densities may be of experimental origin or derived from ab initio wave functions. AIM defines two important cornerstones of chemistry: the atom and the bond. There is a need for such a theory in view of the widening gap between chemical insight and currently accepted and taught, on one hand, and the vast evergrowing body of (highresolution) crystallographic and ab initio data, on the other. Indeed, most chemists still think in terms of the Lewis model of the 1900s (e.g. octet rule), the HeitlerLondonPauling Valence Bond model of the 1930s (e.g. resonance), or the HundMulliken Molecular Orbital of the 1960s (e.g. Mulliken charges). Of course many of these early concepts have been scrutinised and their limitations are well documented but a complete, coherent and consistent theory to bridge the gap between modern solutions of the Schrodinger equation and chemical insight is still elusive. However, an excellent candidate to fulfil that purpose is AIM. This theory is often mistaken to be another atomic population analysis, rather than an extensive and profound theory rooted in quantum mechanics.2 Being a novel paradigm3 it has gained slow acceptance although it has been incorporated as a vital part of a recent textbook on the chemical bond4 aimed at undergraduates. The theoretical community has focused most of its attention on the energy and its derivatives with respect to nuclear motion, i.e. forces, force constants, etc. If one accepts that eigenvalues and eigenfunctions are on a par as solutions of an eigenvalue problem such as the Schrodinger equation, then why is it that the electron density does not enjoy the same status as the energy? After all the electron density p is immediately derived from the wave function, which is an eigenfunction, and the energy is in fact an eigenvalue. This imbalanced view is corrected by the development and application of AIM, a theory that recognises and reveals the wealth of information hidden in the electron density and its derived functions.
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From a superficial point of view AIM seems to consist of two subtheories: one based on the topology of p and the other on the topology of the Laplacian of p, denoted as V2p. The latter theory was developed in an attempt to analyze electron pair localisation and takes up about one fifth of Bader's 1990 monograph summarising AIM.' The impression that AIM encompasses two separate parts is a consequence of historical developments and its present incomplete state. Indeed, if one views AIM as an interpretative theory which aims to retrieve chemical insight from modern ab initio wave functions then both parts of AIM could be further integrated. For example, a systematic study of the gradient vector field of V2p, at the same depth as that of p is technically possible but has not been carried out yet. Physical quantities could then be integrated over basins in V 2 p . In the longer term AIM might be expanded into a complete topological theory of many if not all chemically relevant scalar fields. For example, any of the following fields, p(r), V2p(r),the kinetic energy densities K(r) and G(r), might be integrated over basins of another field, or any scalar field evaluated in a critical point of another field. It is often tempting to invoke molecular orbitals (MO) in the extraction of chemical information from ab initio calculations, such as in an AIM inspired definition of bond order,6 but AIM traditionally strives at orbitalfree interpretations. Ideally, any definition of a chemical concept should exist for any representation of the electron density, whether given on a grid, in terms of a basis set expansion or even from experimental measurement. Currently AIM is used in many different areas of research, many of which we now sum up with examples of references relating to that area. AIM studies have been made in relation to highresolution cry~tallography,~'~ surface science,' * solid state physics,I2 biological ~ h em istry ,'~ organometallic ~ h e mis try ,'~ noble gas ~he m i s t r y, '~physical organic chemistry,I6 transition metal complex chemistry, l7 boron chemistry," lithium chemistry" and other areas. Clearly, AIM has been applied in a wide range of areas of chemical research but to our knowledge applications in supramolecular chemistry are currently lacking. The theory of AIM has been helpful in the reinterpretation of many concepts such as bond energy,20hydrogen bonding,2' the VSEPR the LigandClosePacking (LCP) nucleophilic addition,24atomic volume,*' strain energy,26 chemical rea~tivity,~' acid and base promoted hydrolysis,28 polarisabilities,29 molecular similarity,30y3'atomic charges,32 semiempirical valence electron distribution^,^^ nonbonded charge concentrations ( l ~ n e  p a i r s ) , ~ ~ (hyper)c~njugation,~~ the characterisation of atomic interaction^,^^ Fcentres,37 electron d elo ~ alisatio n a,~r ~ m a t i c i t y ,origin ~ ~ of molecular dipole moments,40structural ~tability,~' magnetic properties42and others. Many of the aforementioned concepts have been rigorously redefined within the context of AIM, or the theory has at least shed new light on the current mainstream definitions and posed new challenges. 1.2 Scope.  The current contribution reports on the literature on "Atoms in Molecules" in one year from June 1998 to June 1999. Since this is the first time that AIM is discussed in a Specialist Periodical Report we cannot refer to
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previous reports covering the period from the inception of the theory (1972) until June 1998. Instead we offer a brief historical perspective, highlighting AIM’S roots, and treat the development of AIM over the last three decades via references to review articles and selected key research papers. A measure of the impact of AIM is the number of references to Bader’s 1990 book, which expounds the theory and applications of AIM.’ At the time of writing we found 906 references in the primary literature to this seminal book since it was published. The distribution of these references on a yearly basis is given in Figure 1. Except for the curious reduction in 1995 the number of references has grown steadily, perhaps asymptotically. It should be mentioned that a substantial number of entries are referring to AIM only in a tangential way, i.e. in connection with the topology of a scalar field. For example, one paper43 referred to Bader’s book only because it presented CPs in the electrostatic potential V(r) (i.e. where VV(r) = 0) without making use of AIM or discussing any of its applications. Of course in this review we have selected only those papers that make restricted or extensive use of AIM, either at the application or at the development side (see disclaimer, Section 9). This selection encompasses 90 papers produced by more than 60 different groups worldwide. The dilemma one faces in writing a report of this kind is whether to categorise the papers in terms of chemical classes and activity (e.g. organic, inorganic, biological, surface science, etc.), or methods, concepts and techniques (e.g. algorithms, NMR, Xray, Hbonds, reaction paths, etc.). We have opted to give priority to method and concepts and have grouped papers accordingly. For 200
!
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Number of References 100
50
0
II
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Year
Figure 1 A histogram of the number of literature references per year to Bader’s book “Atoms in Molecules. A Quantum Theory” (1990)
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example, a paper which focuses on biochemical compounds, but with emphasis on hydrogen bonding, will be grouped under the section hydrogen bonding (Section 6) first and then under the subsection biochemical compounds (Subsection 6.9). Also, within each section, priority has been given to concepts and methods. For example, in the section “Chemical Bonding” (Section 5 ) , compoundoriented subsections only appear after “5.1 Theory”, “5.2 LCP”, “5.3 Hypervalency”. Although there is a degree of arbitrariness in this division, it is necessary to structure papers in a clear way in view of their mixed and overlapping content. 1.3 The Roots of AIM.  The theory of AIM was pioneered by the Bader group at McMaster University (Canada) in the early seventies. This theory emerged from studies that interpreted the electron densities of early computergenerated ab initio wave functions as they became available in the sixties, e.g. from the theoretical chemistry group in Chicago. Early work of the Bader group in the sixties focused on the HellmannFeynman theorem, in particular its use in calculating molecular energies44and understand chemical binding.45 Detailed analyses of the electron density distributions in simple molecules such as water,46 ammonia47 and hydrogen fluoride48 were the subject of following work. Subsequent studies addressed the forces operative in homonuclear diatomic molecules49 and the nature of the chemical bond,50 in particular the ionic bond.51 Further contributions concentrated on the relaxation of the molecular electron distribution and the vibrational force constant.52y53In summary, in the sixties, prior to the actual development of AIM, the Bader group was predominately preoccupied with the intricate relationship between the electron density and the energy and its derivatives with respect to nuclear coordinates. This relationship was certainly an important theme in that decade as made evident by research by Hohenberg, Kohn and others culminating in the HohenbergKohn theorems published in 1964.54 Prior to the actual paper marking the birth of AIM55 Bader, Beddall and Cade published a proposal for the spatial partitioning of a molecular electron density.56957This method was restricted to linear molecules and partitioned them via planes perpendicular to the molecular axis. The partitioning planes were positioned at a point where the electron density reached a minimum along the internuclear axis. An immediate concern arising in the construction of the early atomic fragments was that they were not virial fragments.57 It was known to Bader in 1969 that there are (at least) two types of kinetic energy densities, K(r) and G(r), which yield different kinetic energies58when integrated over arbitrary portions of space. These kinetic energies become identical when the respective energy densities are integrated over whole space and over special portions of space called virial fragments. It was only in 1972 that a spatial partitioning of molecular electron distributions was publishedss that did produce virial fragments. This spearheaded the development of the theory of AIM. Although AIM is often described as the topological analysis of the electron distribution it is important to realise that its roots lie in a primary concern to obtain atomic fragments with a welldefined energy. In other words,
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the emergence of AIM was driven by the quest for quantum mechanically meaningful fragments in molecules rather than by the discovery of topological objects in the electron density, such as the bond critical point (BCP) and the interatomic surface (IAS). In 1974 Daudel, Bader and coworkers published59a paper on the electron pair in chemistry. They presented calculations that show that the most probable partitioning of a system is the one that localises pairs of electrons in welldefined spatial regions or loges. Although logepartitioning never became part of the AIM theory its influence led to an extensive study of the pair density and the Fermi hole density.60 About a decade later the electron pair localisation problem was revisited using the Laplacian of p, a simple but informationrich function, which would provide a physical basis22for the VSEPR theory.61 1.4 The Development of AIM.  Readers uninitiated to AIM may wish to consult a didactic and introductory work to the theory62because it is beyond the scope of this report to review AIM'S development comprehensively. A recent introductory text by Bader can be found in the Encyclopedia of Computational Chemistry.63 For a formal ab ovo treatment of the theory two more advanced and classic reviews64165 are recommended, of which the more recent puts emphasis on applications. An early and informal review66was written in 1985 predated by a decadeolder thoughtprovoking review67 airing the young but important AIM results on the definition of an atom and a bond. Rather than exhaustively reviewing all contributions to AIM up to the current literature survey we merely summarise the chronological development of AIM as it reached a mature stage in the early eighties. We pick up the thread of the previous section at the point where the Bader group realised how to define a chemically appealing and conceptually economic virial fragment in a molecule. In 1973 the virial partitioning method was applied to secondrow diatomic hydrides and a year later to thirdrow hydrides6* In the following five years attention was mainly directed towards the quantum mechanical justification of the proposed virial fragments or AIM atoms. The first publication relating to this question discussed sufficient conditions for fragment virial theorems69 and for an alternative partitioning method introduced by Parr and co~orkers.~' A subsequent paper in this series of contributions focused on the development of the quantum mechanics of a subspace, with particular emphasis on a variational treatment.7* The following doublet of papers deepened the formulation of subspace mechanics. In the first paper Srebrenik and c o  ~ o r k e r obtained s~~ a variational solution of the Schrodinger equation with the zeroflux surface requirement serving as a variational constraint. Using a variational statement of a timedependent hypervirial theorem they described the time dependence of subspace averaged properties. In the second paper the authors73 introduced Schwinger's quantum action principle to obtain a quantum mechanical description of a subspace and its properties. In this paper the subspace quantum mechanics governing AIM is rederived using a further level of abstraction introduced by Schwinger for ordinary full space quantum mechanics.74
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Meanwhile the Bader group published a paper7’ which defined bond paths in terms of molecular electron densities. These considerations, the formulation of subspace quantum mechanics and the ensuing construction of a theory of molecular structure in terms of the topology of p culminated in an impressive quadruplet of papers entitled “Quantum topology of molecular charge distributions’,. In the first paper76the primary concepts of descriptive chemistry, of an atom in a molecule with its own set of properties, and of a chemical bond were introduced. In the second paper,77 Thom’s catastrophe theory78 was applied to molecular structure change and stability, while the third paper79 summarised and streamlined the mechanics of an atom in a molecule, work that would later be extended to include a variety of atomic theorems.” The fourth and final paper in this series discussed the relation between the topological and energetic stabilities of molecular structure, defining the structural diagram and the two mechanisms of molecular structure change (conflict and bifurcation).” The then established topological theory of molecular structure was subsequently the subject of two reviews,659s2ones2 of which shared Feynman’s enthusiasm for the importance of the atomic hypothesis. The Laplacian of the electron density, which forms an integral part of AIM, has been the subject of a rather independent line of development. It became clear in the mid seventies that localised pairs of electrons are not evident in the topological properties of p and the pair density does not, in general, define regions of space beyond the atomic core.6oHowever, it turned out that there is remarkable but not perfectly faithful mapping between the CPs in V 2 p and the electron pairs of the Lewis model. The Laplacian is a simple function with known shortcomings,83784 but has been successful as a physical basis2238’for the VSEPR model of Nyholm and Gillespie.61786Functions more sophisticated than V 2 p have been constructed to study electron pair localisation, for example Becke’s electron localisation function (ELF)83 and the LennardJones funct i ~ nThe . ~ Laplacian ~ turns out to be homeomorphic in many cases to ELFS8 and performs well in the characterisation of bonding and nonbonding (or “lone”) electron pairs for reasons that are not properly understood. The Laplacian continues to be used in the characterisation of molecules (e.g. metallo~enes,~~ disiloxanesgO),in particular in connection with exceptions to the VSEPR but its scope is wider. The Laplacian is proportional to an energy density (or pressure) expressing the balance between the kinetic energy density G(r) and the potential energy density V(r).36 Furthermore a Laplacian complementarity principle has been f o r m ~ l a t e dwhich, , ~ ~ in a way, parallels the Lewis complementarity defined in the context of Lewis’s generalisation of acidity and basicity and their neutralisation. A few dramatic examples of this principle appear in the prediction of the angle of nucleophilic attack to carbonyl carbons from crystallographic data,94the angle of attack in Michael addition compared with potential energy surface calculation^^^ and the reasonable prediction of a large number of hydrogenbonded complexes involving HF.96 Finally, under some conditions the Laplacian is able to point out the preferred sites of protonation, demonstrating its usefulness in both electrophilic and nucleophilic attack. It was shown that the preferred orientation (cis or trans) of the proton
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with respect to the substituent in a dozen carbonyl compounds (XCHO) is generally indicated by the relative magnitude of maxima in the Valence Shell Charge Concentration (VSCC) at the positions of the lone pairs of the oxygen atom. In summary, the Laplacian is a simple function but rich in information.
1.5 Software.  Over many years the Bader group has produced a suite of programs called AIMPAC, which is freely available via http://www/chemistry. mcmaster.ca/aimpac. The most complex algorithms appear in the atomic integration, which where designed by the applied mathematician BieglerKonig et al. in the early eighties. The first algorithm97 used the elegant idea of formulating the atomic integration in natural coordinates introducing a complicated Jacobian. Unfortunately this algorithm proved to be inappropriate because of the highly curved gradient paths near BCPs. As a result the concomitant program OMEGA became obsolete and was replaced by the more successful program PROAIM,98 which approximates the interatomic surfaces by triangulation. AIMPAC has been modified and streamlined on several occasions during the nineties, by Laidig, Keith, Popelier, Heard and others. An independent program to perform a topological analysis of p and its Laplacian has been developed by Popelier since 1991, called MORPHY. In contradistinction to AIMPAC this program is a single entity, not a suite, entirely written by one person except for a few subroutines. The first version called MORPHY 1.099was released in 1996 and deposited in the CPC Program Library, Queen's University of Belfast, N. Ireland, and the QCPE Library in Bloomington, Indiana, USA. The more recent version, called MORPHY98, is also able to perform atomic integration according to two new complementary but integrated algorithms. One algorithm'00 employs an analytical expression'" for the interatomic surfaces and the other'02 determines the intersection of an integration ray without explicit knowledge of the interatomic surface. This program also contains a robust and highly automated but flexible CP localiser based on the eigenvector following method. '03 MORPHY98 is available for a fee from http://www.ch.umist.ac.uk/morphy.Its features and benefits, as well as manual, can be inspected from this web site. A free demo version with full functionality but restricted dimension is freely available from this web site without time limitation. Finally, since 1994 the commercial program GAUSSIAN provides an AIM option based on work by the Cioslowski group. '04 This program is available via http://www.gaussian.com. In our experience the GAUSSIAN94 atomic integration subroutines can provide erroneous results (or even crash), an uncritical interpretation of which may unjustly damage the trustworthiness of AIM. However, important bugs seem to have been corrected in GAUSSIAN98. 2 Theoretical
2.1 Open Systems.  In his Polanyi Award Lecture Bader"' asks why there are atoms in chemistry. He states that Dalton's bold assumption that atoms
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retained their mass and identity in chemical combination had to await its justification by Rutherford's nuclear model of the atom. The evolution of chemistry led to the realisation that atoms also exhibit characteristic additive properties, enabling the recognition of their presence in a molecule. According to Bader the theoretical vindication of the model of a functional group as the carrier of chemical information had to await the work of Feynman and Schwinger. Their generalisation of physics leads to a unique definition of an atom as an open quantum system and accounts for the shortrange nature of the forces that identify a given group in any environment. His lecture demonstrated that the proper open systems predicted by the quantum action principle define the atom and that this definition accounts for the retention of an atom's chemical identity. In a second contribution Baderlo6 refines the previous question and asks whether there can be more than a single definition of an atom in a molecule. Arguments are presented that the theoretical definition of an atom in a molecule or of a functional grouping of atoms that derive from experimental chemistry must be unique. Bader argues that definitions based on the orbital model or, as recently proposed, in terms of domains defined by isovalued density envelopes fail. They fail for a number of reasons, among them being their incapacity to enable a quantum mechanical description of the atomic or group properties. According to Bader chemistry is concerned with the observation and measurement of properties. Definitions that do not predict the measurable, additive properties found for atoms in molecules fail to recover the essence of the atomic concept and can play no operational or predictive role in chemistry. He concludes that atoms exist in real space and that their form determines their properties. Bader states that there is but a single definition for an atom, free or bound, that meets this essential requirement. 2.2 Molecular Similarity and QSAR.  In a first contribution on the design of a practical, fast and reliable molecular similarity index PopelierIo7 proposed a measure operating in an abstract space spanned by properties evaluated at BCPs, called BCP space. Molecules are believed to be represented compactly and reliably in BCP space, as this space extracts the relevant information from the molecular ab initio wave functions. Typical problems of continuous quantum similarity measures are hereby avoided. The practical use of this novel method is adequately illustrated via the Hammett equation for para and metasubstituted benzoic acids. On the basis of the author's definition of distances between molecules in BCP space, the experimental sequence of acidities determined by the wellknown CJ constant of a set of substituted congeners is reproduced. Moreover, the approach points out where the common reactive centre of the molecules is. The generality and feasibility of this method will enable predictions in medically related Quantitative Structure Activity Relationships (QSAR). This contribution combines the historically disparate fields of molecular similarity and QSAR. In a second contribution on quantum molecular similarity O'Brien and Popelier'" investigate the relation between properties in BCP space and bond
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length. In their contribution the authors critically examine the dependence of properties in BCP space on equilibrium bond length, a topic which has been extensively investigated in the literature. For that purpose they have designed a data set of 57 molecules yielding 731 BCPs. They confirm the existence of local linear relationships provided the bonds vary little in their chemical surroundings. Such relationships break down completely for larger subsets of BCPs encompassing a wider variety of bonds. The patterns observed in the global picture show so little correlation that one may safely conclude that BCP properties cannot be trivially recovered or even predicted by knowledge of bond length alone. Meurice et a/.lo9 compare benzodiazepinelike compounds using AIM’S topological analysis and genetic algorithms. Four compounds within a set of ligands for the benzodiazepine receptors are characterised by their electron density maps at different resolution levels and reconstructed from calculated structure factors. The resulting complex threedimensional density maps are first simplified into connected graphs using AIM. Then, an original genetic algorithm method, GAGS (Genetic Algorithm for Graph Similarity search), is developed and implemented in order to compare the connected graphs. Finally, the best solutions of the algorithm are expressed in terms of functional group superimpositions. The GAGS analysis is applied to different resolution levels of the electron density maps and the resulting models are compared in order to assess the influence of the resolution on the resulting pharmacophore models. 2.3 Electron Correlation.  Cioslowski and Liu”’ discuss the topology of electronelectron interactions in atoms and molecules, introducing what they call the “correlation cage”. The concept of the correlation cage provides new insights into electron4ectron interactions in atoms and molecules. The cage constitutes the domain in the space of interelectron distance vectors R within which correlation effects are substantial. Its shape and size are entirely determined by the topological properties of the electron intracule density I(R), thus avoiding any references to illdefined “uncorrelated” quantities. Integration of observables related to I(R) over the correlation cage affords quantitative measures of electron correlation. The number of strongly correlated electron pairs M c o r r { I ) , their electronelectron repulsion energy Wcorr{ I ] , and the cage volume Vcorr{I ] that characterises the spatial extent of electron correlation are functionals of I(R). The ratio K { I ) of I(0)Vcorr{I]and Mcorr{I},which measures the strength of shortrange correlation effects, is small for systems such as Hand closer to one for those with weaker correlation effects.
2.4 Transferability.  Graiia and Mosquera’” focused on the effect of protonation on the atomic and bond properties of the carbonyl group in aldehydes and ketones. They studied the protonation in a series of aldehydes and ketones, R1(C=O)R2 (with R1,R2 = H, Me, Et, Pr and Bu) using AIM to examine the atomic and bond properties of the C=O group and its relationship to the energy involved in the protonation process. Based on the results, aldehydes, methyl ketones and the remaining dialkyl ketones exhibit three different types of
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behaviour. From the many graphs and tables in this paper a few conclusions can be drawn: protonation results in small differences in the properties of the C=O bond; while the atomic charge of C0 hardly changes, the proton bonded to 0 exhibits a high positive charge after protonation; and atomic contributions to the total energy depend on molecular size. In work on a closely related data set Graiia and Mosquera112revisit carbonyl groups in aldehydes and ketones in connection with transferability using AIM. The authors calculated the atomic and bond properties of the carbonyl group of a series of 42 aldehydes and ketones from HartreeFock (HF) 63 1 + + G**//63 1G* wave functions. They found that the quantities Pb, the intraatomic dipole moments p ( 0 ) and p(C), the volume v(C), and A3 differ between aldehydes and ketones. These quantities can be said to be transferable within each of these series, with the exception of formaldehyde. Graiia and Mosquera considered the populations N(0) and N(C), the firstorder electronic charge moments rl(O), rl(C), v(O),the internuclear distance R, the distance from 0 to the BCP denoted by r, &b and Hb as transferable dividing them into three groups: aldehydes, methyl ketones, and ketones of greater length. Both the total and potential energies varied in accordance with molecule size and, therefore, cannot be considered transferable properties of C=O. However, the molecular energies can be reproduced extremely accurately by means of a group contribution model which distinguishes the classic fragments: H aldehyde, C=O, CH2, and CH3. This reproduction stems from the complementary variation that the fragments' energies undergo throughout the series of compounds. At their level of integration accuracy for the oxygen atom, none of the integrated properties are affected by the value of L(R). However, for C the population and the first moment depend linearly on L(R), preventing the above properties from being used directly in the analysis of transferability. 2.5 Multipoles.  Chipot et af.l13performed a statistical analysis of distributed multipoles derived from molecular electrostatic potentials. They described a simple way to obtain distributed electric multipoles on a selection of sites in a molecule. The method is based on a statistical analysis of the multipole components, aimed at reproducing the electrostatic potential created by the molecular charge distribution on a grid of points around the molecule. Applications to HF, H20, CF4, CC14, CC12F2, CH30H, H2CO are presented to illustrate this novel approach.
2.6 Molecular Dynamics.  Soetens et af.'14 performed a molecular dynamics simulation of liquid CC14with a new polarisable model based on the Topological Partitioning of Electrostatic Properties (TPEP) method. This novel method is rooted in AIM and on Stone's work"59116 and was developed very recently.29*' l 8 It consists of dividing the molecular volume into disjoint regions usually centred around the nuclear sites. A multipole expansion on each site for the electron density response function leads to multicentre multipolar polarisabilities, which include chargeflow terms as well local and nonlocal atomatom dipoledipole (and higher) polarisabilities. It was found 17y1
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before29that distributed polarisabilities defined within AIM remain remarkably stable with respect to basis set extension. This is a consequence of the fact that these polarisabilities have not been obtained by partitioning in the Hilbert space of basis functions, but in real space. The van der Waals parameters of the potential of Soetens et al. have been derived from ab initio interaction energies of selected dimer structures. The computed thermodynamical and dynamical properties are in good agreement with the experimental values.
2.7 Partitioning.  An alternative partitioning scheme of molecular density into Their minimal deformation criterion, atoms is proposed by Rico et previously proposed for the partition of the molecular density into atomic contributions, is updated and extended. The authors claim that the decomposition of the molecular density p(r) into atomic densities, p(r) = C ApA(r) is intrinsically arbitrary. As a consequence the above equation has an infinite number of solutions. The relation between their method and AIM is not discussed.
3 The Laplacian
3.1 Alternative Wave Functions.  Further advances in the understanding and application of the Laplacian have been made, in particular its use in conjunction with the CNDO method. Results for CH4, CH3Cl, CC14, H2S and PH3 show that the CNDO electron density can be constructed such that the topology of  V2p obtained from fullelectron ab initio calculations is qualitatively reproduced. Following their method Sierraalta et al. I2O also evaluated the topology of the Laplacian of the spin density (p,(r)pa(r)) using CNDO calculations in a study of modelled catalysts. The location of the CPs was associated with the most reactive sites on a NiS surface in order to predict the adsorption of C and the most convenient orientation of H2 for dissociation on Mo3S 14H4. Bader et ~ 1 . resolve ' ~ ~ an apparent contradiction arising from the use of effective core potentials (ECP) in transition metal atoms. The recent experimental determination of the geometry of Ti(CH3)2C12shows it to be inconsistent with the VSEPR model, a result not uncommon for molecules containing transition metal atoms. The VSCCs that appear as maxima in L(r) = V2p(r), provide a physical basis for the VSEPR model of molecular geometry for main group molecules. The same model accounts for the geometry of transition metal molecules provided the VSCCs are formed within the outer shell of the core of the metal atom, as defined by the shell structure of L(r). This observation appears to be in conflict with calculations for Ti(CH3)2C12, showing that its geometry can be predicted using an ECP for the metal atom, a procedure that would appear to preclude the presence of core distortions. The apparent contradiction is resolved by distinguishing between the definition of the core using L(r) and one based on the orbital model. As a result the suggestion that
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the core distortions in metal atoms arise from the distortion of the density of the 3s23p6outer core orbitals cannot be sustained.86 3.2 Relation to Bohm Quantum Potential.  In 1997 Levit and Sarfatti'22 noted the great similarity between the plots of the Laplacian of p and the Bohm quantum potential Q in the case of H20. They realised that Q is identical to Hunter's oneelectron potential (OEP),'23 which is defined as  V 2 ~ / ( 2 @ . Hamilton, however, visually compared the Bader Laplacian and the Bohm quantum potential for molecules containing atoms from beyond the third row and concluded that for these molecules the two distributions can be qualitatively different'24 (Figure 2). In a reply to Hamilton's study Levit and Sarfatti argued that for all molecules studied all topological features (maxima, minima and saddles) appearing in the Laplacian also appear in the Bohm quantum p ~ t e n t i a l , but ' ~ ~the converse is not always true. 3.3 Protonation.  The Bohm quantum potential Q has been applied (introduced as the oneelectron potential (OEP), which is identical to Q) in the context
\
.._____
Figure 2 Contour plots of (a) V 2 pfor MgOH; (b) Q for MgOH; (c) V 2 pfor CaOH and (d) Q for CaOH. Solid contours are negative (Reproduced by permission of Elsevier)
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of proton affinities. It has been known for some time’26 that proton affinity correlates linearly with  V 2 p at the nonbonding (“lonepair”) maximum in  V 2 p for a series of alkylated amines. Now Chan and Hamilton127have shown that proton affinities of a series of alkyl amines and posphines also exhibit a linear correlation with valence minima in the distribution of the OEP. Polyoxometalates are relevant to catalysis, analytical chemistry, biology, medicine and material science. Their properties are related to their high charges and the strong basicity of the oxygen surfaces. The first ab initio calculations on hexametalate anions were reported by Maestre et a1.,I2*who studied the cis and trans forms of Nb2W40!c (Figure 3) and several isomers of the protonated anion Nb2W4OI9H3. Although these calculations are expensive they are superior to Xray determinations, which often suffer from disorder problems. The central oxygen ( 0 9 ) has been described as an 02species inside a cage, an observation that is compatible with the high AIM atomic charge of  1.59e. Net charges of + 3.12e and + 3.46e are assigned to the Nb and W respectively, confirming the high ionic character of the metaloxygen interaction. The authors find that the relative protonation energy of an oxygen site appears to be strongly correlated to the corresponding minimum in the electrostatic potential and to the Bader charge of the protonated oxygen atom. Furthermore it was observed that the net charges confirm that protonation induces a low electronic perturbation to the cluster. Finally the authors use the Laplacian of p to prove that the polarisation of the oxygen atoms is small and hence consistent with their high negative charge.
P b
M
b or V
( M = Nb or V ) , Figure 3 Schematic representation of the polyoxometalate M2 W&, a niobotungstate. The speckled spheres are tungsten nuclei, and the small open spheres oxygens (Reproduced by permission of the American Chemical Society)
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4 Electron Densities from Highresolution Xray Diffraction
4.1 State of the Art.  In a paper entitled "Chargedensity analysis at the turn of the century",'29 Coppens states that Xray diffraction is now used in an increasingly routine fashion for the measurement of electron densities in solids. In addition to representing the detailed electron distribution, the results can be used to derive electrostatic moments of molecules and dorbital populations of transition metal atoms. A topological analysis of the experimental density yields quite reproducible results when related molecules are compared, but differences with HartreeFock theory remain, in particular for more polar bonds. Accurate experimental electronic properties can now be obtained in only one day with synchrotron radiation and a chargecoupled device area detection technique. Recently spectacular electron densities were acquired on DLproline monohydrate at 100 K.130 The accuracy of the data is comparable or even superior to the accuracy obtained from a 6week experiment on DLaspartic acid with conventional Xray diffraction methods. A data acquisition time of one day is comparable to the time needed for an ab initio calculation on isolated molecules. This technique renders larger molecular systems of biological importance accessible to electron density experiments. The impact of the rapid collection of accurate data should not be underestimated. Indeed, if properly allocated a dedicated synchrotron source could now routinely produce accurate experimental densities at a dramatically increased rate. 4.2 Comparison between Experimental and Theoretical Densities.  Systematic comparisons between theoretical and experimental densities provide an ample testing ground for the validity of the Schrodinger equation, although agreement is still only fair rather than excellent with current computational schemes and technology. Nevertheless the new generation of rapid highresolution Xray studies furthers crystallography beyond its current status of a generator of molecular geometries. From routine Xray studies one cannot obtain chemical insight beyond that of the nuclear geometry. With the recent advent of, in principle, routine highresolution crystallography chemical insight may be retrieved from the electron density. Although this insight (e.g. the covalent character of a hydrogen bond) has traditionally been obtained using deformation den~ities'~' more workers are turning to AIM because it eliminates the necessity for an (arbitrary) reference density. One of the advantages of AIM is that it can be used on experimental densities. Although a fully streamlined and wellmaintained software interface between an AIM analyzer and a crystallographic package still does not exist, AIM serves an increasing number of crystallographers in their interpretation of experimental electron densities. In the period reported on here about two dozen papers appeared using AIM in the context of Xray and neutron diffraction. The topological analysis is rarely invoked to obtain atomic properties, such as the charge, probably due to technical difficulties and computational cost. However it is frequently used to characterise bonds in terms of their covalent and ionic character, albeit not always in a critical manner.
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The theoretical and experimental electron densities of bis(diiminosuccinonitrilo)nickel, Ni(disn)Z (Figure 4) has been compared by Hwang and Wang.132 In their work they show a rare comparison between the calculated and experimental gradient vector field (Figure 5). A quantitative description of the chemical bond in Ni(disn)2 is made in terms of the topological properties of electron densities. The asphericity in p around the Ni atom is observable from the Laplacian of p with density accumulation in the dndirection but density depletion in the da(NiN)direction. On the basis of the topological properties at BCPs, the bonding between Ni and the imino nitrogen atom is classified as mainly a closedshell interaction but with some covalent character. The bonds within the ligand (disn), are all shared interactions. The authors claim that the bond order is reflected in the density at the BCPs. For many topological properties the agreement between theory and experiment is reasonable but fairly large discrepancies are found in V2pb, the Laplacian evaluated at the BCP. Another study focusing on the comparison between theoretical and experimental densities is that of Tsirelson et al. on Mg0.'33 Here precise Xray and highenergy transmission electron diffraction methods were used in the exploration of p and the electrostatic potential. The structure amplitudes were determined and their accuracy estimated using ab initio HartreeFock structure amplitudes. The K model of electron density was adjusted to Xray experimental structure amplitudes and those calculated by the HartreeFock model. The electrostatic potential, deformation density and V 2 p were calculated with this model. The CPs in both experimental and theoretical model electron densities were found and compared with those of procrystals from spherical atoms and ions. A disagreement concerning the type of CP at (i $,,0) in the area of low, nearuniform electron density is observed. The authors noted that the topological analysis of p in crystals can be related with a closepacking concept. In the introduction of their extensive paper on the topological analysis of the experimental electron density of DLaspartic acid at 20 K Flaig et state that the extent to which topological properties are reproducible depends not only on the experimental conditions but also on the interpretation of the data. They also claim that proper treatment of the diffraction data needs increasing
Figure 4 Molecular geometry and atomic labeling of bis(diiminosuccinonitrilo)nickel, Ni (disn) 2 (Reproduced by permission of the American Chemical Society)
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Figure 5 Gradient vectorJield of p for Ni(disn)z from (a) experiment and (b) calculation (Reproduced by permission of the American Chemical Society)
support from theory, next to the opposite requirement, the urgent need for experimental verification of the theoretical results. This view is compatible with the increasing incorporation of theoretical calculations into the interpretation of experimental work in areas other than crystallography. The epoch in which calculations lagged behind experiment has been replaced by an era of symbiosis between theory and experiment. The paper of Flaig et a1.'34discussed here is part of a systematic study on the 20 naturally occurring aamino acids, which have been mainly studied by
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conventional techniques at room temperature or under liquid nitrogen cooling. Despite the importance of this class of compounds very few studies have been performed on the determination of their experimental charge distribution. In the study on DLaspartic acid special attention was paid to proper thermal convolution, the treatment of the hydrogen atoms and chemical constraints in the leastsquare refinement procedures. The density and its Laplacian extracted from the data are analyzed in terms of the topological properties of covalent bonds and nonbonded interactions. The results are compared to those calculated at HartreeFock level and to those obtained experimentally for analogous molecules. There are three main conclusions: the theoretical density of an isolated molecule is topologically equivalent to that extracted from the crystal through modeling the diffraction pattern. In other words, the number and types of CPs in p found for the energyoptimised stationary system in the experimental geometry are the same as obtained for the molecule being in thermal equilibrium in the crystalline state. Secondly, the locations of the BCPs can differ markedly between the theoretical and experimental densities, especially in polar bonds. The value of the density at the BCP typically agrees well between theory and experiment, but this is because p usually possesses a flat minimum between the atoms sharing the electrons, according to Flaig et al. Since theory and experiment can lead to considerably different CP locations in polar bonds, the concomitant AIM population analysis may give different results as well. Finally the effect of the crystal field on the Laplacian was detectable. In summary, Flaig et al. join other author^'^^'^^^ in stressing that theoretical and experimental densities are not comparable because of their different nature. They conclude that the extent to which fine details in the gradient vector field of p affect integrated atomic properties should be subject to further studies. It is curious that this careful study (and others) do not include electron correlation in their theoretical calculations, although Zadovnik et al. 137 do. Keeping up with their research program to perform highresolution diffraction experiments on amino acids Flaig et al.13*also studied LAsn, DLG~u, DLSer and LThr with the fast synchrotron method.13' Their comparative electron density determinations in the group of 20 naturally occurring amino acids focused on the reproducibility and transferability in the vicinity of the C, atom. At the same time the authors support computational work'399140and prove experimentally the transferability of electronic properties and submolecular fragments from this class of compounds onto larger systems like oligopeptides. The observed variations for a given bond range from 15% for p and 719% for V2p, indicating a high degree of reproducibility and transferability. Although integrated AIM properties were not determined, the similarity of the topology of chemically equivalent bonds found in this study yields an indirect proof for the transferability of atomic and group properties. Another comparison between theoretical and experimental densities can be found in the topological analysis of 2amino5nitropyridinium dihydrogen posphate carried out by PuigMolina et al.141 The BCP properties of the total experimental density agree fairly well with ab initio HartreeFock calculations
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for the isolated ions. The analysis of the hydrogenbond BCPs shows that the crystal hydrogen bond framework involves four anions and one cation. All the hydrogen BCPs display small positive v2pb values, consistent with ionic closedshell interactions between the participant atoms. Zadovnik et al.137 included electron correlation in the calculated electron density of urea and compared it with the experimental density obtained by highprecision singlecrystal Xray diffraction at 148 K. The displacement parameters agree quite well with results from neutron diffraction. Orbital calculations were carried out at the HartreeFock level, DFT/LDA and DFT/GGA level. The agreement between experimental and theoretical results is excellent, judged by the deformation density and the structure factors. The agreement with respect to the results of the topological analysis was only fair. Densityfunctional calculations seem to yield slightly better results than HartreeFock calculations. The authors conclude that there is a semiquantitative agreement between the experimental and theoretical (i.e. topological) characteristics of the intramolecular and intermolecular interactions. A further theoryexperiment comparison invoking AIM is made in the area of minerals by Rosso et al.’42BCP properties calculated for representative Si5OI6 moieties of the structure coesite are compared with those observed and calculated for the bulk crystal. The values calculated for the moieties agree with those observed to within 5% on average, whereas those calculated for the crystal agree to within 10%. As the SiOSi angles increase and the SiO bonds shorten, there is a progressive buildup in the calculated electron density along the bonds. This effect is accompanied by an increase in the curvatures of P b (both perpendicular and parallel to each bond) and in v2pb. Whereas Flaig et al. scrutinise the treatment of crystallographic data in an attempt to understand the discrepancy between theory and experiment Rosso et al. hold responsible possible problems in Xray data recording, basis set limitations, the exclusion of relativistic effects and improper representation of correlation. In their search for homoaromatic semibullvalenes Williams et al. 143 investigated the Xray electron density of 1,5dirnethyl2,4,6,8semibullvalenetetracarboxylic dianhydride in the temperature range of 123 K to 15 K. This paper encompasses another extensive application of AIM to experimental densities showing excellent agreement with the HartreeFock electron density. The region between the C2 and Cg carbons revealed a “normal cyclopropyl abond” for this bisannelated semibullvalene, while a very sparse density was found between the C4 and C6 carbons. These results confirm that in the solid state the semibullvalene is not homoaromatic. Topological equivalence of experimental and theoretical densities is shown to be evident although agreement for the v2pb values is not excellent (in contradistinction to the P b values). This discrepancy is claimed to be caused by the fact that the used basis set (63 1G**) cannot furnish a satisfactory representation of the Laplacian in polar bonds.
4.3 Hydrogen Bonding.  Three studies focused on the features of hydrogen bonding drawn from experimental electron densities using AIM tools. The first study investigated the intramolecular hydrogen bonding in benzoylacetone (or
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lphenylbutane1,3dione)by means of a 8 K Xray and 20 K neutron diffraction experiment. Madsen et showed that from the neutron data we ascertain that the hydrogen, between the two oxygens in the ketoenol part of the molecule, is asymmetrically placed in a large flat potential well. In Figureo6 we see that the hydrogen bond hydrogen nucleus is slightly closer to 0 2 (1.25 A) than to 0 1 (1.33 Evidence for extensive ndelocalisation in the ketoenol group was obtained from the use of multipolar functions and topological methods. The multipole populations show that there are large formal charges on the oxygens and the enol hydrogen, which impart polar character to the hydrogen bond. This effect is also evident in the Laplacian and in the electrostatic potential calculated from the Xray data. It is found that the hydrogen position is stabilised by both the electrostatic and covalent bonding contributions at each side of the hydrogen atom. The authors note that the electronic nature of the hydrogen bond cannot be inferred solely from the distance between oxygen atoms of the hydrogen bond. They base their argument on a comparison with a similar ketoenol system found in citrinin. Recently much attention has been devoted to the detailed mechanism by which the class of enzymes called serine proteases work. These enzymes catalyze the ubiquitous and paramount cleavage of peptide bonds and all have the socalled catalytic triad (HisAspSer) in common. A number of studies have suggested that a low barrier hydrogen bond (LBHB) is involved in the reaction mechanism as a partial proton transfer between His and Asp (NH.O). In
A).
Figure 6 Contour plot of L(r) vs. V2p(r) in the plane of the ketoenol group in 1phenylbutane1,Edione. The contours are drawn at logarithmic intervals of 1.0 x 2" eJAS.The dotted line is the zero contour, solid lines are positive and broken lines negative contours. Thefirst two positive and negative contours are omitted for clarity. BCPs are marked with solid in circles (Reproduced by permission of the American Chemical Society)
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order to investigate the electronic structure of short NH bonds Overgaard et carried out a combined lowtemperature (28 K) Xray and neutron diffraction study of the cocrystal between betaine, imidazole and picric acid. This complex serves as a model for the catalytic triad (biomimetic) (Figure 7). The data show that all the hydrogen atoms are localised at the nitrogen atoms, and that none are involved in LHBHs. The absence of LHBHs is supported by the topological analysis, which shows that the O..H interactions are predominantly electrostatic interactions (V2pb > 0) between closedshell atoms. This study demonstrates that short, strong hydrogen bonds do not have to be lowbarrier. The Espinosa hydrogen bond energy estimate'46 is strictly valid for electrostatic interactions (based on a large number of normal hydrogen bonds) and breaks down for the present hydrogen bonds. The authors are performing highlevel ab initio calculations on their model to gain further insight about the catalytic triad. A new type of hydrogen bond was discovered in the midnineties by Richardson et al. called the dihydrogen bond,'47 in which the acceptor atom is a (hydridic) hydrogen atom. A few years later it was shown by P ~ p e l i e rthat '~~ the dihydrogen bond complied with all the hydrogen bond criteria previously proposed by Koch and Popelier2' based on observations of the (H3BNH3)2 moiety. Popelier's study also showed that the boron in this complex is very positive (q(Q) = 2.15) which is diametrically opposed to the Mulliken charge (q = 0.26) quoted by Richardson et a ~ , thereby ' ~ ~ rendering their explanation of the geometry of the complex incompatible with AIM. s
06A
.
0
d
Figure 7 Cocrystal of betaine, imidazole, and picric acid (ORTEP drawing at 95% probability level based on neutron diflraction data at 28 K ) . Hydrogen atoms have been omitted for clarity. A short CH... 0 interaction exists between C8BH8B... 0 7 A , which is important for the crystal packing (Reproduced by permission of Angewandte Chemie, VCH)
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Figure 8 Molecular geometry of HMn( CO)4PPh3shown with 60% probability ellipsoids based on neutron difraction rejinement. Dashed line indicates the CH. .HMn hydrogen bond (Reproduced by permission of the American Chemical Society)
Abramov et al.149 carried out a low temperature neutron and highresolution Xray diffraction study on another system, cisHMn(C0)4PPh3, that contains a dihydrogen bond. The hydride ligand H(l) (Figure 8) is nucleophilic in nature and makes a short contact (2.1 with an electrophilic ortho phenyl hydrogen. The electrostatic component of the Ha+Hainteraction energy is calculated to be 23.9 kJ mol' from the experimental data. This electrostatic evidence coupled with the geometry and the identification of an H . .  H bond path in p strongly supports the characterisation of this interaction as an intramolecular CH.. HMn hydrogen bond. Both the deformation density and the topological study clearly illustrate the adonor nature of both the HMn and PPh3Mn interactions, and the adonorlnacceptor nature of the MnCO bonds. The topological study further confirms the decrease in the CO bond order upon coordination to the metal. This method demonstrates for the first time that the metalligand bonds have a significant dative covalent component, although they show characteristics of a closedshell interaction.
A)
4.4 Organic Compounds.  The electron deficient and multicentre nature of BB
prompted a topological and CC bonds in 8,9,10,12tetrafluoroocarborane study on its electron distribution obtained by highresolution Xray diffraction at 120 K. The detailed analysis by Lyssenko et al.'*O of p and its Laplacian at the BCPs revealed unexpected features in the bonding pattern, i.e. the positive value of the Laplacian in the homopolar CC bond and the negative values for the BC bonds (Figure 9). These data were compared with corresponding ab initio calculations on small deltahedral boranes and carboranes. The authors
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I
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1
.
.
Figure9 Laplacian of the electron density through the centre of the icosahedron of 8,9,10,12tetraJZuoroocarboranein the plane, containing two carbon and boron nuclei. Logarithmic scale, positive contours are dashed (Reproduced by permission of the American Chemical Society)
concluded that the electronwithdrawing effect of the fluorine atoms causes a considerable redistribution of the electron density in the molecule. In particular this is reflected in the shift of Pb from the more electronrich CC bonds to the BC bonds. Deformation density maps showed as well that p is essentially delocalised over the surface of the cage and locally depleted in its centre. All BB and BC bonds in the polyhedron are characterised by significant bending, which is evident in shifts of their BCPs from the straight lines between the respective nuclei. PuigMolina et al. 14' compared the theoretical and experimental electron density in the nonlinear optical material 2amino5nitropyridinium dihydrogen phosphate, 2ASNPDP. The experimental p was determined from Xray diffraction data interpreted in terms of the Hansen & Coppens pseudoatom formalism. The BCP properties of the total experimental electron density agree fairly well with HartreeFock calculations for the isolated ions. The analysis of the
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hydrogenbond CPs shows the crystal Hbond framework to involve four anions and one cation. All the Hbond CPs show small positive V2pb values, consistent with ionic closedshell interactions between the participant atoms. Bianchi’ 5’ studied the experimental electron density study of 4cyanoimidazolium5olate (Figure 10) at 120 K obtained from accurate Xray data. The interpretation chemical of the data was aided by a multipole model (“POP”, Stewart152),where p is represented by an expansion in terms of rigid pseudoatoms, an ab initio HartreeFock calculation on the isolated molecule and AIM. The topological analysis reveals the presence of three HBs with ionic character that contribute to stabilizing the crystal structure. The title compound can be described as the juxtaposition of two separated nconjugated systems, linked to each other by two nearly single CN bonds (NlC5 and N3C4). The Mulliken atomic charges and molecular dipole moment derived from the HartreeFock calculations are close to those obtained from the multipole model. However, the authors admit that if a partitioning were carried out by means of IASs a more significant set of atomic populations would be obtained. Finally we mention the study of AguilarMartinez et al. who synthesised and analyzed the substituent effects on the redox properties of 19 compounds of 3’(meta) and 4’(para) substituted 2((Rphenyl)amine}1,4naphtalenediones in acetonitrile. ‘53 Beside an UVvis analysis and a voltammetric study the authors performed semiempirical (PM3) and DFT calculations (B3LYP with double[
03
Figure 10 (a) An ORTEP drawing of the mo>ecular structure of 4cyanoimidazolium5olate at 120 K (ellipsoids are drawn at the 50% probability level). (b) The same molecule in the crystal, connected by hydrogen bonds (Reproduced by permission of International Union of Crystallography)
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split valence basis set). They claim that V 2 p is very sensitive to electronic delocalisation, which allows them to differentiate effectively between the two quinone carbonyls. In six compounds two unexpected bond paths were detected. One corresponds to the hydrogen bond between the amine proton and a quinone carbonyl. The second unexpected CP corresponds, according to AguilarMartinez et al., to an interaction line expressing steric crowding. This interpretation is controversial in the light of a paper by Bader’” which we discuss below. Each interaction generates one ring critical point (RCP) but with different local Hessian eigenvectors. In the case of the RCP associated with the hydrogen bond, the major ellipticity axis is practically parallel to the ring plane. In the RCP generated by the alleged HH steric repulsion, however, the axis is perpendicular to the ring plane. 4.5 Transition Metal Compounds.  Macchi et al. performed the first experimental electron density study of a nligand q2coordinated to a metal atom.lS6 They considered this work as an “experimental test” of the DewarChattDuncanson (DCD) bonding formalism and expected that their work would provide information concerning the ncomplex versus the metallacycle dichotomy. The authors claim that the successful application of AIM to the experimentally determined p has been the most important step in the coupling of Xray studies and theoretical chemistry. Their paper reports on the determination of an accurate electron density of crystalline bis( 1,5~yclooctadiene)nickel, Ni(COD)2 (Figure 11) by Xray diffraction at 125 K.
Figure 11 ORTEP view of the Ni(COD)* molecule; thermal ellipsoid for nonH atoms are drawn at the 50% probability level while H atoms are idealised (Reproduced by permission of the American Chemical Society)
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The overall bonding picture (Figure 12) that emerges from the AIM analysis is in agreement with the DCD model. The NiC bond paths are inwardly curved, which is interpreted as odonation, but the bond paths are well separated, which is a sign of nbackdonation. The topology supports a ncomplex with a concave ring structure intermediate between a Tshape and a convex ring. The former would imply electrostatic interaction and the latter a fully covalent model. Previous theoretical work on Ni(C2H4)n (n = lqiS7 demonstrated that the DCD approach is substantially correct. The authors note that an AIM analysis provides information that goes beyond a simple geometrical approach, which cannot address separately the two main bonding effects, or an orbital study, which cannot define a clear picture of the interactions (here represented by bond paths). The same authors (Macchi et al.) investigated metalmetal (MM) and metalligand (ML) bonds in C O ~ ( C O(AsPh& )~ (Figure 13) via deformation densities and an AIM analysis from an accurate Xray electron density at 123 K.15*The existence of a true MM bond remained controversial partially due to the interpretation of rather noisy deformation density maps. It was recognised
C
Figure 12 The complete bond path for the nsystem. Note the eight lines that link Ni to each C(sp2) atom (Reproduced by permission of the American Chemical Society)
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Figure 13
View of the molecular geometry of Co2(C0),5 (AsPhJ)2;ellipsoids are drawn at the 50% probability level (Reproduced by permission of the American Chemical Society)
that the major weakness of the deformation density approach is caused by the choice of a proper promolecule. This choice is particularly delicate when the total density between the two atoms is small or when atoms have more than halffilled shells (a problem which was first observed with the F2 molecule). The authors claim that AIM offers a better and less ambiguous theoretical understanding of MM interactions. They conclude that the “expected” lack of charge accumulation in the deformation density map is “contradicted” by the presence of a BCP and a bond path linking the two Co atoms. The major conclusion of their paper is the experimental proof of the presence of a genuine, covalent CoCo bond. However, their conclusion is not reached in a straightforward manner because based on the Laplacian (Figure 14) one might conclude that the CoCo interaction is not shared. Also, the electron density at the BCP between the two cobalt atoms is small. Based on the observations made in F2 and invoking the functions H(r)Is9 [H(r) = V(r) + G(r)] and G(r)/p(r) Macchi et al. conclude that “the CoCo is far from the closedshell limit and we do not find any reason for not considering it a shared interaction as suggested by common chemical sense.” However, the first topological analysis of the experimental electron density in a binuclear metal complex concludes that the metalmetal interaction is unshared. In their highresolution Xray diffraction study at 120 K of M I I ~ ( C O )(Figure ~~ 15) Bianchi et a1.160 quote a positive value of V2p at the
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Figure 14 The distribution of L ( r ) in theplane containing C o l , C02, As1 and C1 (contour levels at 2.0 x lo", 4.0 x lo", 8.0 x lo" e l 2 , n =  2 , 1, 0, k I ; the solid lines represent positive values) (Reproduced by permission of the American Chemical Society)
Figure 15 An ORTEP plot (30%) of the eniire molecule of Mn2(CO),0. The molecule has crystallographic C2 symmetry with the twofold axis passing through the middle of the MnMn bond (Reproduced by permission of the Royal Society of Chemistry)
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MnMn BCP and observe that the density is contracted towards each Mn nucleus. It should be mentioned that the deformation maps showed no evidence of significant density accumulation in the MM bonding region. Scherer et a1.161 embarked on a topological analysis of the experimental and theoretical electron densities in EtTiC13(dmpe) (dmpe = 1,2bis(dimethylphosphino)ethane) in order to pin down agostic interactions in a precise way. Agostic interactions are of particular interest in organotransitionmetal chemistry in view of their role in CH activation, but reliable ways of detecting and characterising these interactions are still at a premium. Work by Popelier and logo the ti^'^ suggested to employ AIM as a means to identify agostic interactions. Scherer et al. agree that some agostic interactions may be identified solely on the basis of electron densities. Nevertheless they are concerned about the fact that a Ti..H BCP representing an agostic bond may not be detectable if the agostic interaction is weaker than in the compound they studied. Therefore the authors propose the nonlinearity of the TiC, bond as a more robust criterion of Pagostic interaction. An interdisciplinary study on transition metal coordinated Al(X)L2 and Their work Ga(X)L2 fragments has been carried out by Fischer et describes the new intermetallic systems by means of elemental analysis, IR, Raman, NMR, mass spectroscopy, singlecrystal Xray diffraction and ab initio calculations at MP2 level. The Laplacian is used to characterise the WA1 bond in the calculated model compound (CO)5WA1H (Figure 16), as well as other bonds occurring in the systems under study. The area of electron concentration in the A1 atom represents the lone pair of electrons. The electron concentration at A1 in (C0)SWAlH becomes deformed and shifted toward the nbonding region as a result of AlW formation. A comparison of the electron concentration in the WA1 bonding region of (CO)5WAlH(NH3)2and (CO)5WAlHshows that in the former the concentration is more in the nbonding region along the WA1 bond path. Along similar lines one concludes that there is stronger W + A1 back donation in (CO)5WAlH. 4.6 Minerals.  Ivanov et performed a hybrid study (static deformation density and AIM) study on the highresolution Xray diffraction electron density of topaz. The electron deformation density, positive values of V2pb and the net atomic charges indicate a closedshell type interaction in the SiO4 polyhedra. Anion valenceshell charge depletions are revealed and it is found that maxima in V2p are displayed towards the closepacked plane owing to the mutual repulsion of the anion valence shells. The relationship between p's topology and the closepacking concept is discussed. Shifts of the CPs from the internuclear vectors reflect the strain in the structure. Another application of AIM can be found in the work of Kuntzinger et al.,lM who determined the highresolution Xray electron distribution of scolcite (CaA12Si3010.3H20).The densities on the Si0Si and SiOA1 bridges have been characterised using v2pb values. The Si0 and A 1 4 bond features are related to the atomic environment and to the SiO(A1,Si) geometries.
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171
Contour diagram of V’p in a plane through (CO)5WAlH calculated from an MP2 ab initio wave function. Dashed lines indicate charge depletion, solid lines charge concentration. The solid lines connecting the atomic nuclei are the bond paths, those separating the nuclei are intersection of the zeroflux surfaces with the plotting plane (Reproduced by permission of the American Chemical Society)
5 Chemical Bonding
5.1 Theory.  Recently bond paths in certain systems have been interpreted as indicative of a “repulsive interaction” rather than a bonded interaction. 1659166In response Bader rebutted this proposal stating that a bond path is the universal indicator of bonded interactions. He argues that the “attractor interaction line between nonbonded is an oxymoron making clear that they restrict their definition of bonding to the Lewis model of the electron pair. Bader reasons that Cios€owski et af. use subjective judgements of relative bond strengths to determine when a bond path represents a bond or a repulsive interaction. For example, Bader deduces that the bond path responsible for the formation of the C12 dimer and for the cohesive energy of solid chlorine is described by them as a repulsive interaction when present in perchlorocyclohexane . Bader’s main argument is that there is a homeomorphism between p and the virial field,’67which determines the system’s potential energy. As a result every bond path is mirrored by a virial path, along which the potential energy density is maximally negative, i.e. maximally stabilising with respect to any neighbouring line.
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5.2 Ligand Close Packing (LCP)Model.  In an educational contribution Gi1lespiel6* proposes an alternative method of interpreting simple fluorides using AIM. For example, he criticises the textbook explanation of the electronic structure of BF3 by means of resonance canonicals containing a double bond (invoking socalled “backbonding”). Because atomic charges, which were previously held to be small, can now be obtained through use of AIM, Gillespie proposes that some “covalent” molecules are in fact predominantly ionic. Other evidence comes from the FF interligand distances, which are remarkably constant and independent of coordination number. According to Gillespie such molecules are best regarded as a central cationlike nucleus with a closepacked arrangement of anionlike ligands surrounding. Size determines whether crystal structures are possible or not. Finally, the description of these molecules as ionic does not violate the octet “rule”. In a second contribution Gillespie et a1.’69discuss the bonding and geometry of OCFT, ONF3, and related molecules in terms of the Ligand Close Packing (LCP) model. Inspired by AIM this model is an extension and refinement of a model proposed by Bartell in the sixties,17’ which recognised the importance of nonbonded interactions. The authors reinvestigated the nature of the bonding in these and some related molecules by analyzing their calculated electron density distributions. The results show that the bonding in the series OBF:, OCFF, ONF3 ranges from predominately ionic in OBFZ to predominately covalent in ONF3. Also the interligand distances are consistent with the close packing of the ligands around the central atom. The difficulties of trying to describe the bonding in these molecules in terms of Lewis structures are discussed. In a final contribution Gillespie et al. 17’ further criticise Pauling’s suggestion to interpret the lengths of XF and X0 bonds in terms of multiple character. This multiple character would result from backbonding, after “correction” of the observed bond lengths for polarity on the basis of the socalled Schomakershow that there is no Stevenson equation. In a recent paper Robinson et justification for the purely empirical SchomakerStevenson equation and that there is little convincing evidence for the supposed doublebond character in molecules such as BF3 or SiF4. In their study Gillespie et al. have surveyed the experimental data for 0x0, hydroxo, and alkoxo molecules of Be, B, and C and have shown that the intramolecular interligand distances for a given central atom are remarkably constant and independent of coordination number and of the presence of other ligands. AIM charges for a large selection of molecules of this type have shown that these molecules are predominately ionic. The authors suggest that the bond lengths and geometries of these molecules can be best understood in terms of a model in which anionlike ligands are closepacked around a cationlike central atom. 5.3 Hypervalency. Dobado et used AIM to focus on chemical bonding in hypervalent molecules such as Y3X and Y3XZ (Y = H or CH3;X = N, P or As; Z = 0 or S). The nature of the P0 bond in particular has been extensively reviewed and explained in terms of a combination of two different descriptions, R3P+O and R3P==0. The former structure obeys the octet rule but requires
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backdonation from oxygen to phosphorus. Three plausible models were contrasted with the AIM analysis: (i) one abond and two nbackbonds (negative hyperconjugation), (ii) one abond and three nbackbonds, and (iii) three a (banana) bonds. The topological analysis [based upon Pb, V2pb,&b and populations N ( a ) ] is consistent with a highly polarised abond with bond strength dependent on the electrostatic interactions. The bonding nature is close to model (ii) but no xback donation can clearly be found. Note that Dobado et al. do not reach the same farreaching conclusion as Gillespie et al. against the validity of the concept of negative hyperconjugation. 169 The quantitative and qualitative AIM results obtained from the different wave functions (MP2 and B3LYP) were in strong agreement, suggesting that the Bader analysis was independent of the theoretical method by which the densities were generated. In a second related study Dobado et al.'74 revisited bonding in hypervalent molecules involving only chalcogen (0,S, Se) substituents. They applied AIM to Y 2 X Z and Y 2 X Z 2 (Y = H, F or CH3; X = 0, S or Se; Z = 0 or S) compounds. The topological analyses (based upon the Pb, v 2 P b , &b and E b , the local energy density, and the atomic charges) clearly displayed the dependence of the bonding properties on the central atom. In particular, when the central atom is 0, the main electron charge concentration remains in the surroundings of the central atom, yielding a very weak coordinate bond. On the other hand, bonding to the central S and Se is consistent with a model of a highly polarised abond, its strength depending mainly on electrostatic interactions. In other words, no evidence was found for double bonding, which has so far been the conventional way to describe the interaction in these systems. The equilibrium geometries were optimised at B3LYP and MP2(full) level using the 631 1 + G* basis set. C h e s n ~ t ' ~produced ' an ab initio NMR and AIM study of the PO bond in phosphine oxides and reached the conclusion that posphines are best pictured as R3P+O, a result to be contrasted with that of Dobado et al. Chestnut found that ab initio NMR calculations on the effect of correlation on phosphorus shielding in the phosphine oxides clearly suggest the absence of conventional multiple bonding in the PO bond. AIM studies that yield AIMbased localised MOs indicate one highly polarised abond plus strong backbonding of the oxygen norbitals, a picture consistent with a number of prior investigations. While it has been argued that the strong character of the PO bond in the phosphine oxides is highlighted best by the R 3 P = 0 formula, the present study indicates that the situation is better pictured as R3PfO.  In their outspoken and upbeat study on the 2norbornyl cation Werstiuk and M ~ c h a 1 1apply ' ~ ~ AIM to solve the controversy on the exact nature of this species. Since 1949 this cation has been described by many terms, such as equilibrating classical, abridged, edgeprotonated or facecentred species, cornerprotonated nortricyclene and ncomplex. The authors conclude that the 2norbornyl cation is not a nonclassical, abridged species but a ncomplex, based on their AIM analysis. Similarly, there is nothing nonclassical about the 2bicycl0[2.2.2]octyl cation. The study concludes that no
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“new bonding principle” is required to explain observed bonds. The studied cations have an unstable topology and a small movement of any atom results in a change in structure. Cho and Park’77 performed an ab initio study (including BLYP, BP86, MP2 and various basis sets) on the bonding nature of the NN bonds in 1,2,5Bond properties including Pb trinitroimidazole and 1,2,4,5tetranitroimidazole. are strongly dependent on the equilibrium bond length. Thus, accurate prediction of geometric parameters is of particular importance in deriving reliable bond properties. A substantial difference in bonding properties is observed when electron correlation is included. According to CHELPG charges the NN bonds of both compounds appear to have a significant ionic nature, and the 1nitro group bears a considerable positive charge. This group also exhibits attractive electrostatic interactions with 0 atoms of adjacent nitro groups. Significantly long NN bond lengths calculated at MP2 and DFT level imply a strong hyperconjugation effect, which may explain the ease with which these compounds form a salt. Glaser et have been interested in deamination reactions and their relation to modifications of DNA. Small aliphatic and aromatic diazonium ions can play a role in DNA alkylation. In their work the authors focus on the anomalous behaviour of dediazonation reactions for which the dual substituent parameter relations yield reaction constants of opposing sign. Glaser et al. computed AIM quantities based on the total electron density, which is an observable, as well as the 0 and 71 components of the atomic populations. While these components are not “observable” in the strict quantum mechanical sense, the concept of CT/Zseparation is a successful one and it is central to all models invoking dative and backdative bonding. The main conclusion is that the classical tool of nelectron pushing is not sufficient to provide a correct account of the electronic structures. This conclusion affects the way we ought to think of the majority of donoracceptor substituted conjugated dyes and nonlinear optical materials. In particular, the analysis resolves the apparent paradox that the amino group can function as an electron donor even though it is negatively charged.
5.5 Transition Metal Compounds.  Dobado et ~ 1 . used l ~ AIM ~ to study the electronic properties of seven isomers of threecoordinated copper(1) thiocyanates, calculated at MP2 and B3LYP level using the 631 1 + G* basis set. The results indicate that in the gas phase Nbonding is preferred to Sbonding. The coordination bond between the Cu(1) cation and the donor atoms is strongly polarised, almost ionic. The charge depletion around the Cu(1) cation is in accordance with sp2 hybridisation. Moreover, the canonical form for the noncoordinated as well as Scoordinated thiocyanates is mainly SCN, whereas the Nbonded thiocyanates have also N=C=S contribution. Jansen et al.18’ examined the metalmetal bond polarity in heterobimetallic complexes containing TiCo and ZrCo bonds. The concept of bond polarity, which is used in main group chemistry in the context of electronegativity, is less rapidly applied in a quantitative way to bonds between transition metals. The
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increased level of sophistication required in a meaningful discussion of bond polarity in MM' bonds necessitates elaborate quantum chemical methods. The authors reported the synthesis and Xray structure of {CH3Si{SiMe2N(4CH~C,H,)},MCO(CO)~(L)) (M = Ti or Zr; L = CO, PPh3 or Ptol,) and performed DFT calculations (BLYP, B3LYP) on the curtailed model systems (Figure 17) (H2N)3TiCo(C0)4 and ( H ~ N ) ~ T ~  C O ( C O ) ~ ( PThe H ~ )authors . provide a thorough theoretical analysis of the electronic structure of the bimetallic bond, using AIM charges and bond orders, natural population analysis (NPA), charge decomposition analysis (CDA), and the electron localisation function (ELF). Both the orbitalbased NPA and CDA schemes and the essentially orbitalindependent AIM and ELF analysis suggest a description of the TiCo bond as being a highly polar covalent single bond. The combination of AIM and ELF is employed for the first time to analyze metalmetal bond polarity and appears to be a powerful theoretical tool for the description of bond polarity in potentially ambiguous situations. Boehme and Frenkingl8*applied AIM to characterise bonds in complexes containing copper, silver and gold, warranting the use of relativistic ECPs and large valence basis sets (at MP2 level). Their work on Nheterocyclic carbene,
Figure 17
Views of the two model structures [(H2N)3TiCo(C0)4Jand [(H2N)3TiCo (CO)j(PH3) J optimised by DFT calculations (Reproduced by permission of the American Chemical Society)
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silylene, and germylene complexes of MC1 (M = Cu, Ag or Au) evaluated the quantities Pb, V2pb and Hb. The metalligand bonds have a strong ionic character, which comes from the Coulomb attraction between the positively charged metal atom and the oelectron pair of the donor atom. The covalent part of the bonding shows little nbackbonding from the metal to the ligand. The aromaticity of the Nheterocyclic ligands is slightly enhanced in the metal complexes. Pavankumar et al. 18* carried out extensive ab initio calculations on the important anticancer drug cisdiaminedichloroplatinum(I1) (cisplatin), comprehensively reporting on its structure, bonding, electron density, and vibrational frequencies. Although the discussion of the bonding in the title compound was centred around MO concepts, the authors used AIM to complete their analysis. The charge density and the Laplacian of charge density of cisplatin were calculated to determine its bonding relationships. Pavankumar et al. were unable to locate a BCP between C1 and H (Figure 18) that they expected on the basis of the contour lines. The last 15 years have seen phosphaalkynes (P = CR) transformed from chemical curiosities to versatile synthetic building blocks. The compounds have been shown to undergo headtotail or headtohead dimerisations at transition metal centres to form 1,3 and 1,2diphosphacyclobutadiene complexes. Howard and Jones'83 accomplished an ab initio study on the analogous
Figure 18 Contour plot of the electron density of cisplatin using the MP2/6311 + + G(2d,2pd) basis set with an ECP on Pt. The numbers indicate the positions of the BCPs in cisplatin: 1 is the PtCl BCP, 2 is the PtN BCP, and 3 is the NH BCP (Reproduced by permission of Wiley)
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diarsacyclobutadienes, on which there had been no reports until recently. They report MP2/63 1lG* and MP4SDQ/63 11G* calculations for the various possible conformers of the simple 1,2 and 1,3diarsacyclobutadienesin order to determine their geometries, the nature of the bonding, and the relative stabilities of different isomers. These calculations revealed some qualitative differences between the analogous 1,2 and 1,3diphosphacyclobutadienes.The authors use AIM to characterise AsC and AsAs bonds. Howard and Jones conclude that the AsAs BCP appearing in one system's transition state cannot be interpreted as a bond because it does not occur in a stable equilibrium Atomic charges were determined with both the NPA and AIM schemes, which give remarkably similar values for As. 5.6 Minerals.  Soscun et ~ 1 . " present ~ the first investigation of topological properties of p of a series of hydroxyl acid sites in zeolites at ab initio level. The zeolite acid sites were modelled by using the following molecular clusters: silanol H3SiOH and the clusters H3SiO(H)AlH3, (OH)3SiO(H)A1(OH)3and H3SiO(H)Al(OH)2SiH3. The calculations showed that the frequency of the OH vibrational modes of acid sites is linearly related to Pb at the OH BCPs. These results indicate that pb(OH) can be used as a tool for interpreting the structural and electronic features of the zeolite hydroxyl groups. This is not possible using the Mulliken charge of H in the OH bond, which gives a poor correlation with the frequency. From the v2pb values of the bonds of the acid sites the authors conclude that the zeolite structure is dominated by a network of Si0 and A10 ionic interactions. The 0H bonds are characterised as covalent bonds, with different extents of charge concentration. ~ AIM in the area of the physics and The work of Gibbs et ~ 1 . " invokes chemistry of minerals. The topological properties of p for more than 20 hydroxyacid, geometry optimised molecules with SiO and GeO bonds with 3, 4, 6 and 8coordinate Si and Ge cations were calculated. Electronegativities calculated from the BCP properties indicate that the electronegativity of Ge ( f 1.85) is slightly larger than that of Si (f1.80) for a given coordination number. The electronegativities of both atoms increase with decreasing bond length. With an increase in Pb the quantities v2pb, and Ai (i = 1, 2 or 3) of each bond increase with decreasing bond length. The covalent character of the bonds is assessed, using BCP properties and electronegativity values calculated from the electron density distributions. A mapping of the (3, 3) CPs of the valence shell concentrations of the oxide anions for bridging SiOSi and GeOGe dimers reveals a location and disposition of localised nonbonding electron pairs that is consistent with the bridging angles observed for silicates and germanates. The BCP properties of the SiO bonds calculated for representative molecular models of the coesite structure agree with average values obtained in Xray diffraction studies of coesite and danburite to within 5%. In a second mineralogical contribution Gibbs et u1.'86 advocate to their community the use of the Laplacian in delineating those regions of a mineral surface that are potentially susceptible to electrophilic and nucleophilic attack.
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They compute Laplacian and bond critical point properties of sulfide bonds containing first and secondrow main group Mcations and compare them with oxide bonds. The Laplacian maps of the distributions show that the VSCC of the sulfide anion is highly polarised and extends into the internuclear region of the MS bonds, coalescing with the VSCCs of the more electronegative firstrow cations. On the other hand, maps for a corresponding set of oxide molecules show that the oxide anion tends to be less polarised and more locally concentrated in the vicinity of its valence shell, particularly when bonded to secondrow Mcations. A search for extrema in V2p reveals maxima in the VSCCs that can be ascribed to bonded and nonbonded electron pairs. The different and distinctive properties of sulfides and oxides are examined in terms of the number and the positions of the electron pairs and the topographic features of the Laplacian maps. The evidence provided by p and its topological properties indicates that the bonded interactions in sulfides are more directional, for a given Mcation, than in oxides. The Pb value and the length of a given MS bond are reliable measures of a bonded interaction: the greater the accumulation of p and the shorter the bond, the greater its shared (covalent) interaction. A final mineralogical paper by Feth et al.lS7focuses on bonded interactions in nitride molecules in terms of BCP properties and relative electronegativities. This study discusses an AIM analysis, which is very similar to the one in the previous study, but now the MN bonds are compared with MO bonds in oxides. 5.7 Solid State.  In their work on ionic materials Pendas et a1.'" enthusiastically embrace AIM as the definitive frame to assign an accurate meaning to geometric solids. The authors have turned to AIM because they believe that many of the unsatisfactory characteristics with the ionic radius concept stem from its rather slippery definition in quantum mechanics. They paid particular attention to the concept of ionic radius, in relation to the shapes of ions in crystals, and to the various correlations among atomic properties such as electronegativity and deformability. Using a simple model to fit their results to a theoretical frame, Pendas et al. show that ionic bonds display properties in complete parallelism to those known in covalent bonds. This enables the unambiguous definition of the strength of an ionic bond, which is found to correspond to Pauling's bond valence. Topological families of ions are discussed and pbond length relationships mentioned. According to the authors the development of physics and chemistry of the solid state has been intimately linked to the vague concept of atomic size, but using AIM the traditional concept of ionic radius has been illuminated. AIM uncovers the complexity and richness of atomic or ionic shapes in the crystalline state. Figure 19 shows an example of a prototypical ionic basin for one of the three topological families in the I32 phase of alkali halides. 5.8 Compounds of Atmospheric Interest.  Alcami and C ~ o p e r "performed ~ ab initio calculations on neutral bromine oxide and dioxides and their correspond
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2211 (NaCI) Figure 19 Prototypical ionic basinfor one of the three topologicalfamilies in the B2phase of the alkali halides. The small basin corresponds to the cation and the larger one to the anion (Reproduced by permission of the American Chemical Society)
ing anions in view of their importance in atmospheric chemistry. Both singleconfigurationbased methods { MP2, QCISD, and QCISD(T)} and multiconfigurationbased methods (CASSCF and CASMP2) have been used. A topological analysis of p (Pb, v2pb, Hb, &b) shows that the nature of the BrO bond is very different within OBrO and BrOO. The authors use V2pb and H b to gauge the covalent character of bonds.
5.9 Van der Waals Complexes.  Rayon and Sordo’” report on the nature of the interaction in donoracceptor van der Waals complexes. A6 initio calculations at the MP2/&31G** level have been carried out on BH3..C0,BF3..C0, BH3.s.NH3 and BF3..NH3. They found a good correlation between charge transfer, as measured by a method based on the expansion of the MOs of a complex in terms of the MOs of its fragments, and the corresponding bond lengths. Additional consideration of the energy gaps between the frontier orbitals involved in the charge transfer allows for rationalisation of the strength of the donoracceptor bonds. However, analyses based on natural bond orbitals and energy density at the BCP suggest that no correlation exists between charge transfer and bond strength.
6 Hydrogen Bonding 6.1 Reviews.  In a recent review Alkorta et ~ 1 . ~discuss ’ ~ nonconventional hydrogen bonds (HB), which are much more exotic than the once unconventional CH.O hydrogen bond. Their survey includes HBs such as CH.C, where isocyanides, CO or carbanions and zwitterions act as HB acceptors.
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Furthermore the authors describe: C...HC bonds, involving highly reactive carbenes, and similar systems encompassing silylenes and carbynes. They discuss HB systems that contain n acceptors and dihydrogen bonds, and even suggest socalled inverse HB complexes, where the H atom provides electrons and another nonhydrogen atom accepts them as in LiH..LiH for example. Alkorta et al. expand on the work presented in the authoritative review by Hibbert and E l m ~ l e y and ' ~ ~ embrace the HB criteria proposed by Koch and Popelier2' as a more accurate and precise perspective in the understanding of currently exotic HBs. 6.2 Relationships.  Espinosa et al. 146 investigated the relationship between the energetic properties of the HB interaction and the topological overlapping of the electronic clouds at the H...O BCP. Their study involves a total of 83 XHO (X = C, N, 0) HBs, which have been described in terms of the following topological properties: v 2 p b , &b, G b , v b , li( i = 1, 2 or 3), where G is a kinetic energy density, V the potential energy density, and A an eigenvalue of the Hessian of p. Espinosa et al. show that the kinetic energy density of the electrons around the BCP is proportional to the curvature of p at the BCP (Figure 20a), and that the potential energy density of the electrons around the BCP is linearly related to the negative curvatures at the BCP (Figure 20b). The topological variation of the curvatures at the BCP, and therefore changes in the H...O overlapping, are related to the onset of the repulsion between the electronic clouds of the basic and acidic atoms. In their study Alkorta et a1.'93 look at the additive properties of P b in HB complexes. A large variety of hydrogenbonded complexes (protic, hydric and protichydric) leads to the conclusion that P b is an additive property when it is expressed relatively, or more precisely as a dimensionless quantity. The authors propose an equation, involving P b at covalent bonds in the isolated molecules, which probes the additivity.
6.3 Cooperative Effect.  The ab initio study of Parra and Zeng'94 focused on the cooperative effect in mixed dimers and trimers of methanol and trifluoromethanol. Cooperativity is the enhancement of of the dipole moment in a HB complex compared to the sum of the dipole moments of the constituents. The authors interpreted the value of P b for the C0 and 0H bonds in four dimers and four trimers in terms of bond strength. A weakening of the 0H bond is seen to be favourable for hydrogen bonding. Another indication of cooperativity is noticed in the strengthening of the HB in the trimer as opposed to the dimer. Their AIM observations provide a consistent picture in support of the cooperative effect. Another study which used AIM in the context of the cooperative effect was performed by Masella and Flament.19' In their ab initio computations at the MP2 level on five dimers and five cyclic trimers, drawn from water, ammonia, and formaldehyde, they evaluated the density only at the HB BCPs. The authors mainly use AIM to detect HBs but do not characterise them via topological properties. Instead the cooperative effect is described in terms of geometry
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250 200
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Figure 20 (a) Phenomenological behaviour of G b versus the positive curvature of p at the HB BCP. The solid line corresponds to the linear Jitting Gb = 15.3 A3, the correlationfactor being 0.98. (b) Phenomenological behaviour of vb versus the sum of the negative curvatures of p at the HB BCP. The solid line corresponds to the linearfitting Vb = 35.1 (Al i 12) the correlationfactor being 0.9.5 (Reproduced by permission of Elsevier)
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changes and shifts in vibrational frequencies. Masella and Flament claim that their results exhibit the great incidence of cooperative effects on the properties of XH ...Y interactions, which are of importance to understand the properties of biochemical systems. 6.4 Bifurcated Hydrogen Bonds.  Rozas, Alkorta and Elguero studied’96 the nature of bifurcated HBs or threecentred HBs. They point out that the term bifurcated HB can designate two entirely different situations (Figure 21). The threecentred HBs the authors investigate explain a large number of biological structures. They are commonly used by biochemists and biologists to account for certain interactions in biomolecules, such as zwitterionic amino acids, where they were first seen in 1939. The authors chose different families of compounds: monomers with intramolecular threecentred HBs, dimers with a HB donor and a molecule with two HB acceptor groups, and trimers with one HB donor and two HB acceptors. All the systems were optimised at the B3LYP/&31G* level, and, in the case of the complexes, the interaction energies were evaluated and corrected with the BSSE. The study of Rozas et al. relies on the HB criteria proposed by Koch and Popelier*’ and proves the existence of bifurcated bond paths and distinguishes four different types of interactions based on the degree of symmetry and magnitude of the two P b values. Therefore, looking at the geometry, electron density, and energy results, the nature of these HBs as threecentred interactions has been confirmed.
6.5 Lowbarrier Hydrogen Bonds.  Schiott et al. investigated the electronic nature of lowbarrier hydrogen bonds (LBHBs) in enzymatic reactions and published their work in two very similar articles. ‘977198 The intramolecular hydrogen bond in their model system, benzoylacetone (Figure 22), has been studied with highlevel ab initio .HartreeFock and density functional theory methods. The transition state (doublewell potential) for intramolecular hydrogen transfer was located with the barrier estimated to be about 8 kJ mol’, consistent with a LBHB. Upon addition of the zeropoint vibration energies to the total potential energy, the internal barrier vanished, overall suggesting that the intramolecular hydrogen bond in benzoylacetone is a very strong HB,
H.
/ X
=*
:A
\
.o.=*
H
1
2
Figure 21 Diflerent configurations to which the term bifurcated has been applied. The term “threecentred interaction” only corresponds to configuration 1 (Reproduced by permission of the American Chemical Society)
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Figure 22
183
19 A
Optimised structures for the two hydrogenbonded cisPketoenol isomers at the B3LYP/6311G(d,p) level of theory (Reproduced by permission of the American Chemical Society)
estimated at 67 kJ mol'. To determine the type of bonding in the O...H...O region the authors looked at the classical triplet (Pb, v 2 p b , ~ b ) .In general, very good agreement between the theoretical and Xray electron density is found, except for &b in the polar c0 bond. The authors take up the suggestion of Cheeseman et al.'99that, in the case of heteroatomic bonds having a large charge transfer, &bis not a sensitive indicator of ncontributions. Rather Cheeseman et al. suggest looking at the ellipticity over the entire bond. Schiott et al. conclude that there is only weak ndelocalisation over the 0H bonds and view the O..O..Hsystem in benzoylacetone as a 3centre, 4electron abond with considerable polar character. Therefore, the HB gains stabilisation from both covalency and from the normal electrostatic interactions found for long, weak HBs. Based on comparisons with other systems having shortstrong hydrogen bonds or LBHBs, it is proposed that all shortstrong and LBHB systems possess similar electronic features in the
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hydrogenbonded region, namely polar covalent bonds between the hydrogen atom and both heteroatoms in question. 6.6 Dihydrogen Bonds.  Recent theoretical and experimental studies on transition metal complexes involving a new type of interaction EHSHX (where E is a transition or alkali metal or boron and X is any electronegative atom or group) have stimulated much interest. Kulkarni et ~ 1systematically . ~ ~ studied the dihydrogen bonds in the complexes and dimers of complexes involving the main group elements (LiH, BH3, AlH3 with HF, H20, NH3 and their dimers) using ab initio calculations at the MP2 level. The H...H bond energy in (BH3HF)2,(BH3H20)2,and (A1H3H20)2is analogous to the conventional moderate or weak hydrogen bond. The bonding features of these complexes and their dimers are analyzed via the quantities Pb, v2pb and the ellipticity &b. The decomposition analysis of interaction energies of dimers reveals the predominance of electrostatic contributions followed by charge transfer and polarisation. 6.7 Very Strong Hydrogen Bonds.  In their contribution on very strong HBs ~ ~ ’ that the dimerisation energies of posphinic acid (PA) Gonzalez et ~ 1 . conclude and its dimethyl derivative (DMPA) are the highest reported so far for neutral homodimers in the gas phase (order 100 kJ mol’). To the authors’ surprise there is a complete lack of theoretical studies on these dimers, which is why they performed highlevel ab initio calculations on these systems. Gonzalez et al. find that for O.Hthe Pb values can be as high as 0.054 a.u., although they reported 0.046 a.u. for the formic acid dimer, which is significantly larger according to the authors. 6.8 Organic Compounds.  Kovacs and Hargittai202studied the potential energy hypersurface of 2trifluoromethylresorcinol and 2,6bis(trifluoromethyl)phenol at HF/631G** and MP2/631G** levels. The global minimum is stabilised by two HBs in 2trifluoromethylresorcinol and by one in 2,6bis(trifluoromethyl)phenol. The authors provide a list of detailed geometrical changes in these molecules due to the formation of the O H    FHBs compared to their “parent” compounds, phenol, trifluoromethylbenzene, resorcinol and metabis(trifluor0methy1)benzene. It is concluded that all the HBs in the title compounds can be characterised by a BCP between the interacting F and H nuclei and by a bent shape of the bond path. No atomic (integrated) properties are calculated in this work. According to Alkorta et d 2 0 3 radicals are poor HB acceptors and the strength of the HBs qualitatively correlates with the molecular electrostatic potential (MEP) minimum of the isolated radicals. They studied the ability of carbon radicals to act as HB acceptors using three well established ab initio methods, i.e. B3LYP, MP2 and QCISD. The complexes formed by four radicals { H3(d), C2H4(t), (CH3)2C(t) and (CH3)3C(d)) with four standard hydrogen bond donors {H, H20, HCN and H3N) were studied and their geometry, interaction energy, and electronic properties were analyzed within
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the AIM framework. Atomic properties indicate that HBs complexes involving radicals behave differently from other HBs formed between neutral molecules. In another contribution204 Alkorta et al. focused on nine strained hydrocarbon compounds, five simple hydrocarbons and HCN, each forming a complex with ammonia. The strained compounds have been selected on the basis of an increasing number of three and fourmembered rings. The ability of the strained hydrocarbons to act as HB donors has been explored at B3LYP level using the 631G* and 631 1 + + G** basis sets. The characteristics of the HB formed were correlated with geometrical parameters (number of three and fourmembered rings, bond angles, and HB bond distances), electronic characteristics of the complexes and isolated monomers ( p b , atomic charges, and dipolar magnetisation), and other properties (gasphase acidities and atomic volume and energy). For example, an excellent linear relationship is found between p at the HB BCP and the value of the Laplacian there. The results have been rationalised on the basis of a simple strain model and compared with nonsaturated hydrocarbons with donor CH groups. In their work on threemembered heterocycles Alcami et al.*05 aimed to find out if stabilisation takes place by cation association to the zwitterionic tautomers of the studied compounds. The authors chose Li+ as a suitable reference acid and include species derived from aziridine, diaziridine, triaziridine, oxirane, dioxirane, oxadiaziridine and dioxaaziridine in their survey using high level ab initio methods. The effects of the tautomerisation on the bonding characteristics of the different parent compounds were investigated mainly by means of the Laplacian of p.
6.9 Biochemical Compounds.  AIM has been invoked in a study in the field of molecular biology206where Luisi et al. attempt to understand why opposing exocyclic amino groups in duplex DNA (Figure 23a,b) may form close NH...N contacts. In order to comprehend the nature of such interactions, the authors examined the CSD database of highresolution crystal structures of small molecules. They also performed ab initio calculations on model complexes, which indicate that the hydrogenamino contact is improved energetically when the amino group moves from the conventional geometry, where all atoms are coplanar with the base, to one in which the hydrogen atoms lie out of the plane and the nitrogen is at the apex of a pyramid, resulting in polarisation of the amino group. Only a few AIM criteria are invoked to characterise an NH..N HB (Figure 23c,d) in the orthoaminopyridine dimer. The authors speculate that the amino group can accept HBs under special circumstances in macromolecules, and that this ability might play a mechanistic role in catalytic processes such as deamination or amino transfer. Epibatidine, an alkaloid discovered in 1974, has attracted considerable interest because it appears to be the first compound exhibiting analgesic activity as a selective and potent nicotinic receptor agonist. Campillo et al.*07performed a theoretical study on the conformational profile of epibatidine and its protonated form (Figure 24) using molecular mechanics, semiempirical and ab
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N
\ H,
(4 Figure 23 (a) and (b) Schematic representation of a basepair showing the propeller twist operation. (c) A possible case for pyramidalisation of the amino group in the deanine5methylcytosine contact, based on calculations. (d) An exaggerated schematic representation of pyramidalisation in aminobenzene, to illustrate the H B to the nitrogen lone pair that is proposed to occur in (c) (Reproduced by permission of Academic Press) 1’
Figure 24 Neutral (forms A and B ) and protonated epibatidine with numbering (Reproduced by permission of Elsevier)
initio methods. The stability of the minima has been explained using the AIM methodology. In particular, the intramolecular HBs, although weak as shown by the low P b values, are able to explain the stability of the A conformers versus the B conformers. The authors discovered a second interaction, denoted as H . ..H and traditionally associated with repulsive interactions. However, a model calculation of the ammonium/benzene complex, with one of the ammonia hydrogens pointing towards a benzene hydrogen, yielded an optimised geometry with H..H distances close to those observed in the protonated epibatidine. This model calculation indicates that the epibatidine H . . . H interactions can be attractive, according to Campillo et al. HernandezLaguna et aL208 examined the theoretical proton affinities of histamine, amthamine and some substituted derivatives. The internal HBs
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occurring in some conformers of these important biological molecules have been characterised with the help of an AIM analysis. 6.10 Compounds of Atmospheric Importance.  Berski et al.209applied AIM to molecules of atmospheric interest, such the dimers of hypofluorous acid, (HOF)2, which was studied at CCSD(T)/&3 1 1 + + G(2d,2p)//MP2/631 1 + + G(2d,2p) level. The authors noticed that Mulliken charges for the atoms in HOF are not realistic, even at MP2/6311+ + G(3df,3pd) level, because the oxygen’s atomic charge turns out to be more negative than that of fluorine. The CHELP method210 also fails to be compatible with expected electronegativity scales. However, the net charge on F according to AIM is 0.23e and 0.15e for 0. Berski et al. found two structures, a cyclic dimer (planar, C2h), which is only 1.5 kJ mol’ higher than the linear dimer (Cl). The authors introduce the ELF in a comprehensive way and show that it reveals a shift in electron density from the F0 bonds to the regions of the free, valence electron pairs located at the fluorine atoms in the cyclic dimer (Figure 25). The cyclic structure which assumed planar (C2h)geometry was found to be unstable (transition state) within a dielectric medium.
S.O&!
I
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4.00 6.00 4.00 4.d200 1.00
Figure 25
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om
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Twodimensional representation of the ELF map for the (HOF)2 in the cyclic conformation. Contour lines are spaced at 0.05 au; distance is measured in Bohr on both axes (Reproduced by permission of the American Chemical Society)
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7 Reactions
7.1 Organic Compounds.  CalvoLosada et aZ.21 report a detailed investigation of the reaction path for the thermal rearrangement of 3,4dihydrolaHazirine[2,3c]pyrrol2one to yield the cyanoketenefonnaldimine complex, carried out at MP2/631G* and B3LYP/631G* level (Figure 26). The authors used AIM to characterise bonds in 2Hazirine, in particular the ellipticity of the C3N bond. The authors designate the C3N single bond as a moderate bonding interaction intermediate between a typical shared and closedshell interaction. CalvoLosada et al. continue their AIM analysis with the BCP properties and fitted bond orders of the azirine intermediate. From the eigenvalues of the Hessian of p they conclude that the electronic rearrangement through the transition state skeleton should be interpreted as a preferential n accumulation of the electron density. Episulfonium ions (ESI) are threemembered sulfurcontaining cationic species that have been postulated as intermediates in many reactions. Mechanisms of the reaction of 1alkoxy2~(arylsulfanyl)alkyl halides with different nucleophiles in the presence of a Lewis acid appear to be more complex because of the possibility of forming either ESIs or oxonium ions (10s) as intermediates (Figure 27). Dudley212 reported equilibrium structures and harmonic vibrational frequencies of two model compounds related to the intermediates of nucleophilic attack on lalkoxy2(arylsulfanyl)alkylhalides 0
 N2
z
0
0
N+R, trans iii
N
R2 azirine
cyanoketeneformaldimine
complex Figure 26 The thermal rearrangement of 3,4dihydro1 aHazirine(2,3c]pyrrol2one to yield the cyanoketeneformaldimine complex (Reproduced by permission of Wiley)
Figure 27 Reaction scheme involving the Ialkoxy2sulfanylethan1yl cation (Reproduced by permission of the American Chemical Society)
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and 2(arylsulfany1)pyranosylhalides at H F and MP2 level. Mulliken and AIM atomic charges in the model ions were compared with corresponding sites in oxonium and episulfonium. The studies of the model ions suggest that the intermediate is more like an oxonium ion than an episulfonium ion, although the sulfur is critical to the stereoselectivity. The authors observed that a typical positive formal charge on the oxygen atom ought to be assigned a fairly large negative charge according to the calculations. Not only are the partial charges contrary to the standard view but also the C0 bonds in the studied structures may not be “classical” Q and bonds. Formamide is the simplest amide containing a prototype HNC=O linkage, and is therefore frequently used as a model to understand proton exchange processes in peptides and proteins. The aim of the work of Luna et aZ.213 was to investigate whether the isomerisation of formamide to its tautomers, formamidic acid and (aminohydr0xy)carbene can be catalyzed by association with closedshell transitionmetal monocations such as Cu +. The structures, relative stabilities, and bonding characteristics of complexes of formamide, formamidic acid, and (aminohydroxy)carbene with Cu+ have been investigated through the use of highlevel density functional theory (DFT) calculations. The values of Pb, v2Pb and Hbwere invoked to investigate the bonding features of the complexes. The AIM analysis show that the electron densities at the NCu, 0Cu and CCu BCPs are almost an order of magnitude larger than the typical values (0.0120.038) found in ionic linkages. Fang and Fu214invoked AIM as a suitable tool for studying whether or not cyclisation has occurred in the cycloaddition between a ketene and an imine. The reaction of a ketene with pyridine for example would give rise to a ketenepyridine ylide.
Scheme 1 Reproduced with permission of Elsevier
The formal positive charge on nitrogen in the ylide is not compatible with Mulliken or AIM charges, which agree in sign for the stationary points in the examined reaction. However, overall there is charge transfer (about 0.3e) from the pyridine moiety to the ketene moiety in the ylide, leaving a net positive charge on the pyridine, and confirming that the ylide has some zwitterionic character. The quadruplet (Pb, v2pb, &band Hb) was used to extract information about the nature of the bonds, for example in the ylide (Figure 28) The last paper mentioned in this subsection is not a study of a reaction profile, but a contribution which cannot be categorised in any other section of this report, other than the current one, in view of its relevance to the general understanding of organic addition reactions. Halonium ions have played vital roles as intermediates in organic chemistry. Bridged species such as the bromonium ion were proposed in 1939 to explain the stereochemistry of Br2
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Figure 28 Molecular graphs and Laplacian distribution in the stationary point corresponding to the ylide (Reproduced by permission of Wiley)
addition to alkenes. Such ions have only been observed directly after the development of experimental conditions for stabilising cations in superacids. Damrauer et aL2” present a computational study of halonium ions of cyclohexene and cyclopentene. Figure 29 shows the atomic AIM charges for some geometries of C6HloX+ (X = F, C1 or Br). According to the authors it seems clear that the tradeoff between halogen electronegativity and socalled backbonding effects is very finely tuned in these systems. It is striking that in both cation series, whether for open or bridged cations or for 1,2 or 1,4bridges, that the F atoms have quite negative charges, the chlorine atoms are nearly neutral and the bromine atoms are more positive still. A detailed analysis suggests that electronegativity is quite dramatic and that even in the 1halo cation the CF bond is still polarised C+F. Inspection of the electron distributions suggests that there are electrostatic and size effects that dominate the stability of the cations.
7.2 Inorganic Compounds.  Hamilton2I6 characterised the proton transfer in the isoelectronic species HOT and HF: . Electron densities were calculated at the QCISD/6311+ + G(2d, 2p) level for the nonlinear equilibrium geometry and the C2” saddle point and linear saddle point geometries. AIM is applied to partition p into its atomic components and atomic and molecular properties are calculated. These quantities are used to characterise the proton dynamics as similar to internal rotation. Kulkarni and Koga217performed a similar study to the previous one from the point of view of AIM, i.e. the characterisation of bonds in reaction schemes. This time the ( p b , V 2 p b and ~ b triplet ) was used in the mechanistic investigation of samarium(II1)catalyzed olefin hydroboration reaction via ab initio calcula
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0.58
0.06
0.31
Figure 29 Atomic AIM charges for the C 6 H ~ ~ X( X+ = F, Cl or Br) some geometries obtained at MP2 level (Reproduced by permission of the American Chemical Society)
tions. In their work the bonding features of all stationary structures on the reaction path are obtained from the topological analysis of the corresponding electron density distributions. The authors note that the small core ECP they employed leads to correct topological behaviour, for it is known that the omission of considerable core density may lead to severely corrupted topologies. A reaction mechanism involving a transition metal was studied by Decker and Klobukowski2'* who investigated the role of the acetylene ligand from a density functional perspective in M(C0)4(C2H2) (M = Fe, Ru or 0s). Recent kinetics experiments have shown that the rate of CO substitution in complexes of the type M(C0)4(C2R2) is accelerated by factors of 10210'3 over their respective pentacarbonyl complexes. These substitution reactions have been shown to be dissociative in nature and show a marked metal dependence on the rate. The origin of the increased reactivity of these alkyne complexes was
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studied with BLYP, using both ECP and allelectron basis sets, in conjunction with Frenking’s charge decomposition analysis (CDA) scheme and AIM. By using the CDA scheme the nature of the acetylene ligand was characterised in both the reactant and the unsaturated dissociation product. Acetylene was found to act as a twoelectron donor in the reactant complex and as a fourelectron donor increasing the stability of the otherwise 16electron unsaturated dissociation product. The predicted structural changes along with the results of the AIM analysis fully support the CDA findings.
8 Conclusion Since the early seventies the theory of AIM has grown and gained acceptance in a wide community of users, ranging from mineralogists to enzyme crystallographers. Although controversies occasionally arise on some aspects of AIM the theory has reached the stage where it is used as a practical tool by crystallographers and experimental (synthetic) groups with access to an ab initio package. AIM’S use to characterise bonds is a prominent application but discussions based on atomic populations and structural stability feature as well. Already now, over a hundred research papers appear yearly that devote extensive or even exclusive space to AIM. Provided continuous investments into properly interfaced, welldocumented, wellmaintained, affordable and userfriendly software are made and provided computer hardware keeps improving at accelerated pace AIM has the potential to become a prime method of extracting chemical information from the electron density, a fundamental quantity that has hitherto been almost neglected in chemistry.
9 Disclaimer
The primary set of references for this manuscript was retrieved from the Science Citation Index, accessed via BIDS (Bath Information and Data Services, University of Bath, UK). The completeness of the first automated selection of papers depends on the accuracy of the information present in the Science Citation Index, the determination of the nontrivial publication time window (June 1998 to June 1999) set by our publisher, and the accuracy of the reference to Bader’s 1990 monograph in the original paper. All papers appearing in readily obtainable journals were screened in full and included if they were of sufficient interest from the point of view of AIM. When in doubt it was decided to exclude a possible paper in favour of other papers with higher AIM content. This choice was determined by space restrictions and the readability of the report. We apologise to authors for the possibly erroneous omission of their work, but believe that we have provided a rather complete and unbiased reflection of the research activity in the AIM area.
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185. G. V. Gibbs, M. B. Boisen, F. C. Hill, 0. Tamada and R. T. Downs, Phys. Chem. Min., 1998,25, 574. 186. G. V. Gibbs, 0. Tamada, M. B. Boisen and F. C. Hill, Am. Min., 1999,84,435. 187. S . Feth, G. V. Gibbs, M. B. Boisen and F. C. Hill, Phys. Chem. Min., 1998,25,234. 188. A. M. Pendas, A. Costales and V. Luana, J. Phys. Chem. B, 1998,102,6937. 189. M. Alcami and I. L. Cooper, J. Chem. Phys., 1998,108,9414. 190. V. M. Rayon and J. A. Sordo, Theochem.J.Mol. Struct., 1998,426, 171. 191. I. Alkorta, I. Rozas and J. Elguero, Chem. SOC.Rev., 1998,27, 163. 192. F. Hibbert and J. Elmsley, Adv. Phys. Org. Chem., 1990,26,255. 193. I. Alkorta, I. Rozas and J. Elguero, Theochem.J.Mol. Struct., 1998,452,227. 194. R. D. Parra and X. C. Zeng, J. Chem. Phys., 1999,110,6329. 195. M. Masella and J. P. Flament, J. Chem. Phys., 1999,110, 7245. 196. I. Rozas, I. Alkorta and J. Elguero, J . Phys. Chem. A, 1998,102,9925. 197. B. Schiott, B. B. Iversen, G. K. H. Madsen, F. K. Larsen and T. C. Bruice, Proc. Nut. Acad. Sci. USA, 1998,95, 12799. 198. B. Schiott, B. B. Iversen, G. K. H. Madsen and T. C. Bruice, J . Am. Clzem. SOC., 1998,120, 12117. 199. J. R. Cheeseman, M. T. Carroll and R. F. W. Bader, Chem. Phys. Lett., 1988,143, 450. 200. S . A. Kulkarni, J. Phys. Chem. A, 1998,102,7704. 201. L. Gonzalez, 0. Mo, M. Yanez and J. Elguero, J. Chem. Phys., 1998,109,2685. 202. A. Kovacs and I. Hargittai, Theochem.J.Mol. Struct., 1998,455,229. 203. I. Alkorta, I. Rozas and J. Elguero, Ber. BunsenGesell.Phys. Chem. Chem. Phys., 1998,102,429. 204. I. Alkorta, N. Campillo, I. Rozas and J. Elguero, J. Org. Chem., 1998,63,7759. 205. M. Alcami, 0. Mo and M. Yanez, Theochem.J.Mol. Struct., 1998,433,217. 206. B. Luisi, M. Orozco, J. Sponer, F. J. Luque and Z. Shakked, J. Mol. Biol., 1998, 279, 1 123. 207. N. Campillo, J. A. Paez, I. Alkorta and P. Goya, J. Chem. SOC.,Perkin Trans 2, 1998,2665. 208. A. HernandezLaguna, Z. CruzRodriguez and R. Notario, Theochem.J. Mol. Struct., 1998,433, 247. 209. S . Berski, J. Lundell, Z. Latajka and J. Leszczynski, J. Phys. Chem. A, 1998, 102, 10768. 210. L. E. Chirlian and M. M. Francl, J. Comp. Chem., 1987,8,894. 21 1. S. CalvoLosada, J. Joaquin Quirante, D. Suarez and T. L. Sordo, J. Comp. Chem., 1998, 19,912. 212. T. J. Dudley, I. P. Smoliakova and M. R. Hoffmann, J. Org. Chem., 1999,64, 1247. 213. A. Luna, J. P. Morizur, J. Tortajada, M. Alcami, 0. Mo and M. Yanez, J. Phys. Chem. A, 1998,102,4652. 214. D. C. Fang and X. Y. Fu, Theochem.J. Mol. Struct., 1998,455,59. 215. R. Damrauer, M. D. Leave11 and C. M. Hadad, J . Org. Chem., 1998,63,9476. 216. I. P. Hamilton, Ber. BunsenGese1ls.Phys. Chem. Chem. Phys., 1998, 102,298. 217. S. A. Kulkarni and N. Koga, Theochem.J. Mol. Struct., 1999,462,297. 218. S. A. Decker and M. Klobukowski, J. Am. Chem. SOC.,1998,120,9342.
4 Modelling Biological Systems BY RICHARD I. MAURER AND CHRISTOPHER A. REYNOLDS
1 Introduction
“All philosophy is either trivial or false”.’ The same is probably true of molecular modelling applied to biology since the size and complexity of interesting systems rarely permits all aspects of a problem to be studied to high accuracy. At the same time, the availability of modelling software makes it relatively easy to produce wrong results  see for example ref. 2 on the validation of molecular dynamics simulations. However, it would be premature to dismiss all modelling on biological systems, just as it would be wrong to dismiss all philosophy. Richardson’s intention was that we ask, “What can we learn from this?” and indeed molecular modelling can provide key insights that are not readily available by any other technique. One of the unique characteristics and opportunities in modelling biological systems is the need to interact with protein sequence data. New sequences are being produced at an enormous rate through the human, and other, genome projects and these would contain a wealth of information  if only we knew how to interpret them. Another characteristic of modelling biological systems that isn’t so apparent for other areas of modelling is the wide range of methods that may need to be applied to address a particular problem. For example, modelling a receptor may involve multiple sequence alignment, secondary structure prediction, homology modelling, geometry optimisation, quantum mechanical studies of the ligands, molecular dynamics simulations and docking. Here therefore, one would ideally address a wider range of topics than is possible in this review. We trust our readers will be forgiving of our fleeting coverage and many omissions. We begin the review with a brief account of Gprotein coupled receptors since these receptors, perhaps more than any other system, highlight the problems and rewards that can occur in modelling biological systems. We follow this with some recent work on proteinprotein interactions since here sequencebased approaches have given excellent preliminary results in an area that was perhaps previously dominated by traditional potentials. Increasing computer power has enabled us to review some exciting simulations on the early stages of protein ~~~
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folding (the first 1 p),something that would have been unthinkable a few years ago. DNA simulations, reviewed in the fourth section, is another area that has had a boost recently, primarily through the increasing tendency to treat long range electrostatics through the particle mesh Ewald method; the disadvantages of this method are therefore discussed in the DNA section. In the 1980s, free energy simulations were seen as the exciting development that was transforming the area. Ten years later they remain in the realm of the specialist as they are very difficult to carry out. However, in our review we are able to note some interesting new developments, particularly where several molecules are treated at once. We present two large sections, one on continuum methods, because these are proving to be both cost effective and accurate, and one on hybrid methods, because these are essentially the only realistic way to study reactions. Finally, we present some relevant CarParinello calculations as this method is just beginning to be applied to biological systems. 2 GProtein Coupled Receptors
There has been considerable interest in homology modelling of membrane proteins, particularly Gprotein coupled receptors and ion channels, because of their pharmaceutical importance. This is fraught with difficulty because to date relatively few Xray crystal structures are available for membrane proteins. For the Gprotein coupled receptors, the starting point is usually the low resolution rhodopsin cryoelectron microscopy structure of Unger3 or the acarbon template of Baldwin4derived from this on the basis of an extensive and diligent analysis of experimental data and amino acid substitution patterns. Note that rhodopsin is a Gprotein coupled receptor while bacteriorhodopsin, for which there is an Xray structure, is not. As viewed from the extracellular side of the membrane, the rhodopsin helices 17 are arranged counterclockwise but the modelling community was divided as to whether this pattern was followed by all Gprotein coupled receptors. The majority of authors favoured the counterclockwise arrangement (e.g. refs. 5 and 6 ) but a significant number favoured the clockwise Superficially, this could be taken as a strong statement that the models are wrong and that the modellers should not have tackled such problems. In some senses some of the modellers were wrong because as far as we are aware the new models ( 1998 + ) are all counterclockwise. However, the authors following the clockwise models showed considerable insight in identifying experimental data inconsistent with the counterclockwise model, namely that transmembrane helix 7 in an ideal ahelical conformation cannot be reconciled with a number of important interactions between helix 7 and the surrounding transmembrane helices. The jury is still out as to the correct structure for helix 7, with debate surrounding the conformation at the (N/D)P of the NPXXY motif. We have proposed that a 3residue section of a 310helix is sufficient perturbation to satisfy the experimental data" while an elegant study by Konicka et al. involving database searching, homology modelling and Monte Carlo simulations in torsional space has
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suggested that helix 7 contains an Axx turn and a flexible hinge.12 It is unlikely that both suggestions are correct but both studies highlight the need to consider the conformational properties of this region of the helix  a point that has been missed by the experimentalists. Having alluded to the problems in modelling such systems it is also important to note some of the successes. Firstly, while the best experimental structure for helix 7 in rhodopsin is essentially straight, there is a small distortion near the extracellular end of the helix (the other end to the NPXXY motif). This was p r e d i ~ t e das ' ~ early as 1986. Secondly, computer simulations on the conformational changes occurring during receptor activation are in good agreement with experiment (including fluorescence and spinlabelling studies) in identifying helices 3, 5 and 6 as the main regions where changes occur (see refs. 11 and 14 and references therein). Thirdly, there is a growing body of evidence that dimerisation is important for Gprotein coupled receptors and that the dimerisation is linked to activation  see ref. 14. Ideas on Gprotein coupled receptor dimerisation have been acredited" to research in 1992 but actually go back much earlier. Nevertheless while dimerisation was largely being ignored in the mid 1990s, the first modelling reference to dimersI6 appeared in 1996. This was followed by detailed analysis of the dimerisation interface using molecular dynamics simulations on receptor d i m e r ~ " ~and ' ~ analysis of the amino acid patterns at the dimer interface. The latter approach used both correlated mutation analysis and the evolutionary trace m e t h ~ d . ' ~ ~Thus, ' ~  ' ~while the jury is still out on the exact role of receptor dimerisation in Gprotein activation, this is certainly an area where modelling has been playing a leading role. There are many other interesting articles on modelling Gprotein coupled receptors but these will be reviewed elsewhere in a dedicated Gprotein coupled receptor review.2o 3 ProteiwProtein Docking
3.1 Traditional Docking Approaches.  One of the continuing themes in recent studies of proteinprotein docking experiments is that shape complementarity is important.21922A successful application of shape complementarity to the binding of human growth hormone to its receptor is reported by Hendrix et Despite the conformational changes that generally occur on binding (to main chain and side chains), ensuring that the general mode of binding is inducedfit rather than lock and key, it appears that the recognition can often be treated as lock and key to a first a p p r ~ x i m a t i o nBlind . ~ ~ trials have shown that proteinprotein docking approaches may able to identify the interaction sites but not necessarily the detailed side chainside chain interactions.25y26Using multiple copies of key side chains and an explicit treatment of solvation using Langevin dipoles, Jackson et al. have been able to report improvements on rigid docking.27 Some problems were experienced for interactions involving Lys, Arg and Glu due to the difficulties of generating suitable rotamer libraries for these amino acids with long side chains. It was noted that the solvation
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treatment did indeed help to steer these charged amino acids towards the correct binding site when several options were available. More improvement was noted for the proteaseinhibitor systems than for the antibodyantigen systems, probably because the former have a greater lock and key character. 3.2 Sequencedbased Approaches to Docking.  The proteaseinhibitor work shows that general docking approaches are reasonably efficient within a rigid body framework (i.e. redocking monomers from their complex geometry), but they become more unreliable and intractable when all possible degrees of freedom are allowed to vary. Recently, however, PazosZ8and Lichtarge et ,1.293 1 have laid the foundations for sequencebased approaches to docking which take as their starting point a multiple sequence alignment. Pazos has exploited the observation that correlated mutations seem to accumulate at protein interfaces to correctly dock 21 heterodimers. His approach was to generate several thousand randomly docked orientations and to select the best docked structures based on an accululation of correlated mutations at the protein interface (Figure 1). Lichtarge has essentially built on work by Livingstone and Barton3' to develop the evolutionary trace (ET) method for predicting functional sites on the surface of proteins. The basic assumptions of the ET method are that within a multiple sequence alignment the protein family retains its fold, that the location of the functional sites is conserved, that these sites have distinctly lower mutation rates and that this lower mutation rate is punctuated by mutation events that cause divergence. In particular, Lichtarge was able to identify the contact sites in both SH2 and SH3 domains of proteins. The essence of the approach involves determining the conserved in class residues (Figure 1) and plotting these on a spacefilling model of the protein (Figure 2). For SH2 domains, the ET method had a specificity and sensitivity of about 90% and 70% respectively (the figures for SH3 domains were lower). In our hands, both of these methods have helped to support proposals as to the location of the dimerisation interface in GPCRs, namely between helices five and six.1'714318~'9 4 Simulations on the Early Stages of Protein Folding
The protein folding problem  the ability to predict a protein fold from its sequence  is one of the major prizes in computational chemistry. Molecular dynamics simulations of solvated proteins is currently not a feasible approach to this problem. However, Duan and Kollman have shown that a 1 ps simulation on a small hydrated protein, here the 36 residue villin headpiece, is now possible using a massively parallel super computer.33The native protein is estimated to fold in about 10100 ps and so the simulation can only be used to study the early stages of protein folding. Nevertheless, starting from an extended structure the authors were able to observe hydrophobic collapse and secondary structure formation (helix 2 was well formed, helices 1 and 3 were partially formed and the loop connecting helices 1 and 2 was also partially
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 receptor type  receptor subtype r n  conserved LFVFFFIIILLLLLLAAFF~L  random hydrophobic
(A)PPPPPPPPPPPP
1111122221111112222233
m  correlated ~ ~ ~ I I I I V V V V V V V V WV Wcorrelated TTTTTVVVVTTTTTTTTTTTTT  Correlated E’FF~~~YYYLLLLLLIIIIILL  conserved in class (c)
(D)
a1 aa a1 ab a1 ad
Figure 1 (A) Part of a multiple sequence alignment of nine a and 13 p receptors where the a receptors have two subtypes designated 1 and 2 and the /,? receptors have three subtypes designated I , 2 and 3. Hypothetical residues are shown for eight sequence positions, which are not necessarily sequential. The first position contains a conserved Trp, the second contains random hydrophobic residues. The residues at the third andfourth positions are correlated since, whenever a Val mutates to an Ile there is a corresponding Phe to Asn mutation. Since Val and Phe are conserved in the a receptors and Ile and Asn are conserved in the /Ireceptors, these are also conserved in class residues. The residues labelled (b) are also correlated and conserved in class. The residues denoted (c)are conserved in class but not correlated as no corresponding mutations were noted elsewhere (they do, however, correlate with the receptor subtypes). (B)The grey ovals represent two docked proteins. Here, correlated residues (black circles) accumulate at the interface in the correctly docked structure when compared to ‘normal’ residues (black squares). Here Pazo 3 Xd descriptor is positive. ( C ) Correlated residues do not accumulate at the interface in an incorrectly docked structure and so Xd is negative. (D) A phylogenetic tree denotes the relationship between the receptors and is used to determine the conserved in class residues
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The SH2 domain, plotted onto a space filling model of phosphotransferase (pdb code Isps). The residues contributing to thefunctional site are shown in shades of grey. We acknowledge Mark K. Dean for redetermining this picture using the method of Lichtarge et al.3' The picture is based on results for 46 kinase sequences
formed). Such simulations raise the possibility of comparisons between experimental and simulation studies of protein folding. A similar study was performed by Daura et al. on a Pheptapeptide (which forms a stable lefthanded 31helix)for up to 50 ns in 962 methanol molecules.34 Starting from an extended structure, the 31helixformed in less than 10 ns and the interconversion between folded and unfolded states took considerably less ( 0.05 ns). Structures generated during the simulation were classified as folded or unfolded on the basis of their root mean square deviation (rmsd) from the experimental structure (the criterion was < 0.15 nm). A crude estimate of the free energy of folding was made from the relative populations of the folded and unfolded states; this was 3.5 kJ mol at 360 K. Interestingly, the simulation did not randomly sample conformational space but rather spent most of the time sampling just a small proportion of the conformations that had been identified. Thus, of the 321possible conformations ( 10") only 310 separate conformations were identified by the root mean square deviation criterion. Of these 310, the system spent 50% of the time sampling just 5 (at 360 K). These observations suggest that the search problem in peptide folding is surmountable. Rather than attempt to simulate the actual folding process, other approaches by the same group have been geared towards explicit simulation of

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intermediates in the folding process. Thus Smith et al. have performed simulations designed to mimic the molten glob~le,~’ which is a compact partially folded form of a protein with a nativelike fold. To mimic the molten globule Astate of lysozyme, which has been much studied experimentally and which forms at low pH in the absence of calcium, the aspartate, glutamate and histidine side chains were protonated, as was the Cterminus, and the side chains were given nonnative like conformations. The simulations yere performed in a periodic box of 5582 SPC water molecules with a 14A electrostatic cutoff. Good agreement with experiment was found on size (radius of gyration) and secondary structure content. It was concluded that the ensemble of conformations generated probably resemble the more native like members of the full conformational ensemble for a molten globule but nevertheless the simulations were beginning to address the nature of these folding intermediates. Further insights into the ensemble of conformations defining the nonnative states have also been obtained through molecular dynamics simulations on peptide fragments from l y ~ o z y m eKey . ~ ~ features to emerge included persistent structural features and local clusters of hydrophobic side chains. 5 Simulations on DNA
A clear illustration of the advantages of the particle mesh Ewald (PME) method for treatment of long range electrostatic interactions in simulations is described by Cheetham et aZ.37Here it was found that fully hydrated DNA and RNA retained their structure for up to 1 ns without the use of constraints or nonphysical changes to the forcefield provided that the PME method was used. On the other hand, if the electrostatic interactions were merely truncated at 9 as is common procedure, then severe distortions in the structures were observed after about 200 ps. This clearly illustrates the importance of PME in simulations of highly polar systems and such studies have led the way to many high quality simulation studies on DNA. For example, both Miaskiewicz et al.38and Spector et af.39have used long simulations of hydrated DNA containing cissyn or 64 dithymidine lesions (which result from UV radiation) to study the molecular details of these local distortions. Good structural agreement with NMR experiments was obtained and now the way has been opened to study the interaction between these structures and endonucleases and related repair enzymes whose role is to eliminate the potential carcinogenicity of these lesions. The TATA boxbinding protein has been the subject of many studies as the conformation of the DNA (referred to as TADNA) is distinctly different to BDNA. In some respects it is more similar to ADNA and so Pardo et gradually changed the torsional angles of a double stranded ADNA dodecamer to those required by TADNA and showed that the transition can occur smoothly. Although the particle mesh Ewald technique was used, hydrogen bond constraints were required to prevent the base pairs from slipping during the bending process. This observation suggested that the role of Phe190 in the TATA boxbinding protein is to prevent this slippage. The interaction between
 A,
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DNA and the TATA boxbinding protein has also been studied e ~ p l i c i t l yThe .~~ TATA boxbinding protein can distinguish between T:A and A:T base pairs despite their similarity in the minor groove (the protein interface is primarily to the minor groove of the DNA). The origin of this preference was identified through a combination of ab initio calculations and PMF simulations. The ab initio calculations involved the four base pairs in the TA step interacting with the four key amino acids from the protein. The PMF simulations were on the conformational transition of selected tetramers of different sequence (e.g. GTAT) from A or BDNA to TADNA. Firstly, steric clashes between bulky hydrophobic residues and the NH2 group of guanine are responsible for selectivity against guaninecontaining base pairs. More importantly, steric clashes between the methyl groups of the thymine in AA or TT stretches select against AA and TT while hydrogen bonding between A and T bases in the AT step selects for this step.
5.1 Particle Mesh Ewald.  The disadvantages of the particle mesh Ewald techniques must be appreciated, however, particularly in the light of the recent popularity of the method. The method imposes periodic boundary conditions on the system. While this is a lesser error than the use of a cutoff to truncate the electrostatic forces in highly polar systems, it certainly does not facilitate the study of dilute systems as the simulations effectively correspond to those of a concentrated solution. Consequently, Resat and M ~ C a m m o nhave ~ ~ developed an algorithm to extrapolate the results of a simulation with periodic boundary conditions and cutoff's to the infinite dilution limit. The correction is generated by analysing sample configurations from the start and end points of a free energy perturbation simulation. The technique is based on the generalised reaction field approach, which gives the free energy of a distribution of N point charges, qi, inside a cavity of radius, R,, in a dielectric continuum. The consequences of the Ewald and related methods were assessed by implementing the PoissonBoltzmann method using both Ewald periodic boundary conditions and nonperiodic boundary condition^.^^ Although the method was only tested on a few very simple systems, some distinct energy artefacts were observed due to perturbation by the periodic images. Often these artefacts can cancel, resulting in only small perturbations. There are cases where this cancellation may not happen and so Ewald techniques must be used with care. 6 Free Energy Calculations Free energy simulation methods heralded an exciting development during the 1980s but it soon became clear that much work had to be done to use them properly. Van Gunsteren has long been an advocate of the need to ensure that the molecular dynamics simulation time is longer than the relaxation time for the process being studied.2 This is certainly true for free energy simulations where conformational changes can give rise to slow convergence. In a modifica
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Simulation involves copies of key groups and reduced torsional barriers Multiple a Additional grown, groups barriers
zb
+GI
lo wered
Multiple p
A%
1
A%
Singlea
b
Additional groups removed, barriers restored
SingleP
Figure 3 The basic locally enhanced sampling method for determining free energy diflerences through a thermodynamic cycle
tion of the locally enhanced sampling (LES) method, applied to the anomeric effect in glucose, Simmerling et al. have shown that the free energy of the mutation of the anomeric OH (to hydrogen) converges an order of magnitude more quickly because of the enhanced conformational sampling.44 The preferred scheme, LES2, is shown in Figure 3. Here, on going from a single ct conformation to a multiple o! conformation, five copies of each OH group and the C H 2 0 H group were grown (AGI). The torsional barriers were reduced in the new (artificial) system to aid sampling for the multiple ct to the multiple p transformation (AG2). Finally, the system was restored to a single p conformation (AG3). The overhead on the AG1 and the AG3 stages was relatively low. The thermodynamic cycle ensures that the desired free energy can be determined (AG4 = AG1 + AG2 + AG3); it should also ensure that the final result is not influenced by the artificially low barriers in the second transformation since the system is finally restored to its correct form (AG3). (The final scientific conclusion was that the anomeric effect added to the force field favours the cc anomer but that the improved hydrogen bonding in solution favours the p anomer; this system was also studied using the CarParinello method  see Section 9.) It is tempting to focus on electrostatic effects when modelling highly charged systems such as DNA. Indeed, in a free energy perturbation study of the doubly charged netropsin binding to the minor groove of DNA, the electrostatic contribution was an order of magnitude greater than the van der 20 kcal mol '). Waals contribution4' (about  235 kcal mol versus However, the electrostatic free energy of netropsin binding to DNA and to water were virtually identical and so most of the calculated free energy of binding (  10 kcal mol') can be attributed to the van der Waals terms. Clearly, the electrostatic effects will have an important orientational role, and they will not always be as energetically neutral as in this case. However, these results should clearly influence our qualitative thinking on the energetics of drug binding. Similar conclusions have emerged from PoissonBoltzmann binding studies despite the focus on electrostatics in such methods  see Section 7.3.

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Free energy calculations on proteins are usually carried out within a molecular dynamics framework and so it was encouraging to see the relative binding affinities of four benzaminide inhibitors with trypsin determined using Monte Carlo simulation^.^^ The protein backbone was fixed but sampling of side chains was allowed. One of the significant aspects of this article is the generation of closed thermodynamic simulations to justify the simulation protocol. 6.1 Free Energy Calculations from a Single Reference Simulation.  The free energy perturbation formula
AF(1) = kBZln(exp
 [H(R) H(0)]/kBt)o
(1)
enables free energy differences, AF, to be obtained from molecular dynamics (or Monte Carlo) simulations. Here kB is the Boltzmann constant, T is the temperature and H(0) is the Hamiltonian of a reference system while H(1) is the Hamiltonian of the system of interest and R is a coupling parameter linking the reference system to the system of interest. In the netropsinDNA study (see above) the system of interest is netropsin (1 = 1) and the reference state ( R = 0) represents complete annihilation of netropsin and so AF(1) gives the free energy when the system is run in water and the free energy of binding can be obtained when the simulation is also run in the presence of DNA. However, this simulation was extremely costly in terms of CPU time because such a large change had to be run over many intermediate windows. To avoid this CPU expense, attempts have been made to calculate free energy changes from a single reference simulation; an interesting approach has been followed by Liu et al.47 Using a 300 ps simulation of pmethylphenol as the single reference state they obtained values of 15.8 and  7.2 kJ mol for the changes to pchorophenol and pcyanophenol. Both results' agreed with the thermodynamic integration results to within 2.5 kJ mol'. However, in simulations where the methyl group was mutated to methoxy, to a dummy atom (which had no interactions with the solvent) or when an additional methyl group was grown, errors of up to 18 kJ mo1l were obtained. Clearly, using the basic perturbation formula approach with a single physical reference state does not work when atoms are created or annihilated. The proposed solution to this problem involves the use of a nonphysical reference state containing softcore interaction terms, Vij.
where rii, cij and ai, have their normal meaning in a LennardJones function.47 The parameter a describes the softness of the interaction since the energy now approaches a finite (rather than infinite) value as rii approaches zero. The softcore coupling scheme uses
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so that at A = 0, Vii reverts to the correct LennardJones potential. With softcore potentials in the single reference state, good simulations of the above free energy changes were obtained. (A thermodynamic cycle was used to relate say both pchlorophenol and pmethylphenol to the nonphysical reference state so that the free energy difference for the mutation from pchlorophenol to pmethylphenol could be obtained.) The reason for these good results is that the solvent molecules can diffuse into and out of the softcore regions thus extending the configurational space accessible relative to the physical p methylphenol reference state. We are not aware of many developments based on this work but it is extremely promising as the perturbations for which free energy estimates are required can be chosen after the single reference simulation has been run and can be analysed by postprocessing the saved trajectories.
6.2 Multimolecule Free Energy Methods.  Empirical scoring potentials permit the rapid assessment of how a potential drug binds to its environment. These methods use approximate energy functions and some success has been achieved in blind trials (typically, the wellrespected methods were successful in about 25% of the tests and partially successful in another 50°/0.4852The welldocked solutions, however, were not always the highest ranked, showing that there is much scope for improvement in the scoring potentials). However, these methods are not based on rigorous physical principles and their success is inevitably limited. Free energy methods on the other hand have the potential to be accurate if good parameters are used and a reliable protocol is followed. They are, however, impractical in a drug design context because the CPU requirement is too large. A middle way may be the emerging multimolecule free energy methods. A recent example illustrating the essence of the approach describes the binding of fluoro and chloromethane guests to the tennis ball dimer host.53A molecular dynamics simulation is performed on a series of hosts within the guest. Thus, molecular dynamics is used to sample coordinate space. Only one of the guests is considered as a real molecule and this interacts fully with its environment; the other guests are considered as ghost molecules. Their interactions are fully recorded but do not affect the trajectory of the host or the other guests (though the guests may be restrained to each other by a harmonic potential to improve sampling). After say 1 ps of molecular dynamics, a chemical Monte Carlo step is performed in which a new real molecule is chosen at random and is accepted or rejected according to the usual Metropolis criteria. If it is accepted, it becomes the new real molecule and the old one becomes a ghost; if it is rejected, the old molecule continues to be the real one. The velocities are randomly reassigned after every Monte Carlo move. This
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process is repeated until the probability of observing each ligand converges. The probabilities of observing each ligand can be used to determine the relative free energies. However, this does not take into account the free energy of solvation of each ligand. Consequently, the free energy of solvation can be used as an energy offset in the Metropolis acceptance criteria and then the observed probabilities can be used to determine the relative free energy of binding. Results of two simulations were reported, one with four guests and one with nine. The method appears particularly good at ranking compounds, since the binding order emerges before the free energies converge. The method has also been adapted and applied to TIBO analogues, which are nonnucleoside inhibitors of the enzyme HIV1 RT.54 In this application, the offsets were iteratively adjusted to ensure even sampling of all the drugs  the offsets then give the binding energy. The offsets were iteratively adjusted after either 125 or 500 Monte Carlo moves and either 30 or 150 iterations were required depending on the set of ligands run (each containing eight ligands). Generally, the ranking produced by the Chemical Monte Carlo/molecular dynamics was in good agreement with experiment. Some discrepancies were observed and these may be due to the sampling problems that would arise if the ligands bind in different modes, given that the simulations were run for a maximum of 560 ps. It was suggested that multiple copies of protein side chains may help to alleviate this problem. Finally, it was noted that the PoissonBoltzmann method, with a surface area correction, gave equally good results and that the two methods were complementary because of the different approximations involved. (It was also noted that the limited free energy perturbation/thermodynamic integration simulations performed on this system gave disappointing results despite the higher computational cost .) Other examples of similar multimolecule work are given in refs. 55 and 56, but interestingly the origin57of this approach could be traced to 1976. Another example where continuum methods are competitive or even superior to the free energy perturbation method was given by Best er af. where the Generalised Born method and the free energy perturbation method were both used to calculated octanolwater partition coefficient^.^^ The unsigned error for the Generalised Born method was 0.5 log P units as opposed to 1.01 for the free energy perturbation method. 6.3 Linear Response Method.  The binding of nonnucleoside TIBO analogues to HIV1 RT has also been studied5’ using the increasingly popul!r linear response method (or linear interaction energy method) associated with Aqvist er af.60Here the interaction energy is given by
Where Ubound represents the interaction energy between the drug and the enzyme and Uunbound is the corresponding term in solution. The superscripts elec and vdw refer to the electrostatic and van der Waals components and the
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average is taken over configurations generated using molecular dynamics, or in this case Monte Carlo simulations. The additiona! term noting changes in the solvent accessible surface area (SASA) was not in Aqvist’s original method. As for many such methods, the intramolecular energies are usually not taken into account, and indeed their inclusion is often detrimental. This is essentially a semiempirical method as the coefficients a, /? and y are fit to experiment. (In some of the early publications it appeared that a value of a = 0.161 was reasonably transferable and that a value of B = 0.5 could be justified from the Born model of ion solvation.) The method is therefore more useful in evaluating potential modifications to known enzyme inhibitors rather than in de novo predictions of binding. In the TIBO analogues study, the van der Waals term was attractive but uncorrelated with activity while the electrostatic term was repulsive but correlated with activity. The surface area term essentially did not change across the series and so was effectively a constant that could be added to make the overall binding energy negative. After refitting of a, /3 and y , the rms error was of the order of 1 kcal mol’ and the theoretical compound ranking was similar to the experimental ranking which is comparable with the errors observed in similar applications.61962 Analysis of the errors in the fitting is likely to show that errors of less than 0.5 kcal mol cannot occur except by chance.62 6.4 Free Energy Perturbation Methods with Quantum Energies.  Previously, free energy perturbation studies of enzyme reactions were primarily carried out using the empirical valence bond method of W a r ~ h e l A . ~ couple ~ of studies addressing this problem are referenced here.64*65In particular, we note that Wood et al. analysed water as a hydrated solute and used quantum (density functional) energies. By generating a classical trajectory and using only 50 quantum energies they were able to assess how well the classical potentials performed and by using different size clusters were able to assess in which region, e.g. inner shell, second shell, good performances were obtained. They were also able to suggest improvements to the potentials using adjustment cycles.

.
6.5 Force Fields.  We have discussed force fields in the free energy section because free energy simulations provide one of the most stringent tests of parameters and because several of the articles discussed here involved free energy simulations. Force fields have been gradually refined over the last 1020 years, sometimes with help from free energy simulations, but their basic form, as used in biological applications, has not changed greatly. Polarisation is probably the most important effect that is neglected, and a few examples of its use have been reported but we are not aware of any applications where polarisation has played a key role in obtaining high quality results. Indeed, in some cases polarisation (induction) resulted in poorer structures66  but that may be because current force fields include polarisation implicitly and explicit polarisation would result in double counting.67968 Ref. 66 described a free energy perturbation study of alkali cation extraction by crown ethers. Hostguest systems are not biological but they do provide important systems for developing
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relevant theoretical models. One of the important conclusions to emerge from this work was that the extraction process occurs in an organic solvent mixed with water molecules. This conclusion is probably also relevant to simulations on lipid bilayers. Ref. 66 described the traditional implementation of induction effects through induced dipoles involving expensive dipoledipole interactions. At least two alternative approaches have been described recently; the first is the fluctuating charge model, based on work by Rappe and G ~ d d a r d This .~~ method adds a relatively low cost ( 10%) for including polarisation into molecular dynamics simulations. Some recent applications and developments are discussed in Section 8 on hybrid methods. The other is based on a method of representing atomic induced induced dipoles as a set of induced charges on the atom plus its neighbours.68This approach can be extended to a variety of other methods such as hybrid QM/MM methods or PoissonBoltzmann methods7' and yet essentially remains within the same framework as the widely used potential derived charges. Moreover, extensions beyond the atomic charge level to charges plus dipoles are relatively straightforward and efficient in this f r a m e ~ o r k .Such ~ ~ high level calculations are not normally carried out on biological systems, often because they can only be implemented at the expense of more important issues (such as simulation time or incorporation of solvent). However, the early indications are that they may be important in determining subtle binding effects, such as the subtype specificity of epinephrine and norepinephrine to /I1, /I2 and /I3 subtypes of the adrenergic receptor.'1y71 Sometimes the absence of polarisation is given as the reason for less than optimum agreement with experiment; the free energies of hydration of amines are a prime example. However, Rizzo and Jorgensen have generated OPLS charges and other parameters for amines by fitting to experimental data for pure liquids and hydrogen bond strengths from ab initio calculation^.^^ These new parameters give excellent results for solvation energies in both water, as would be expected, and in chloroform. Polarisation effects are also discussed in Section 8.2 on hybrid quantum mechanicalmolecular mechanical models. Henchman and Essex have shown that OPLSlike charges can be generated from molecular electrostatic potentials using restraints. Their method has advantages over the similar RESP charges commonly used by Kollman and the charges have performed extremely well in free energy simulations of hydration.73974 Ref. 74 also includes a comparison between free energy perturbation and linear response theory  see Section 6.3.
7 Continuum Methods Electrostatic interactions play an important role in many aspects of biology, such as protein structural stability, enzyme function, gene expression, ion transport, and proteinprotein interactions. Consequently, the number of publications studying these interactions continues to grow every year. Continuum methods are by far the most common approach to studying electrostatic interactions. In continuum methods, the solute is usually represented as a
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constant dielectric material immersed in a solvent of higher constant dielectric. Atoms are usually represented by point charges at coordinates garnered from published Xray crystal or NMR solution structures. Various methods exist within the family of continuum methods. Those based on finite difference solutions of the linear (or nonlinear) PoissonBoltzmann equation are by far the most n u r n e r o u ~Others . ~ ~ ~approaches ~ include: 0
0
0
methods that use the Generalised Born (this method is computationally inexpensive and so is growing in popularity). microscopic and semimicroscopic methods that use the Langevin equation, termed “PDLD” and “PDLD/S”. Here the solvent and protein atoms are modelled as polarisable point dipoles and the interactions between the solvent and a protein are modelled explicitly. In the semimicroscopic counterpart, electrostatic interactions are scaled by an effective “protein dielectric” to represent the parts of the model that are not modelled explicitly.81*82 a selfconsistent method based on LorentzDebyeSack theory, using a sigmodial screened coulombic PO ten tial.83
The relative basic advantages and disadvantages of each method have been discussed previously. For a comprehensive review of methodology before the annual period of this review the interested reader is directed to reviews by Allewell and O b e r ~and i ~ Warshel ~ and Pap a~ y a n .~ ’ 7.1 Parameter Dependence.  The accuracy of any model is only as good as the parameters used to describe it. As such, the constant need to redevelop and tighten the accuracy of the present parameters is evident. Hendsch et a1.86 studied the dependency of continuum methods, particularly PoissonBoltzmann based methods, on the choice of atomic charges and radii, and also the representation of the dielectric boundary, by calculating the total electrostatic free energy contribution to the stability of 21 salt bridges. The contributions that make up the free energy are also discussed. They found that despite variation in the results, the relative relationship between the unfavourable desolvation term, resulting from the transfer of the two interacting charged groups from the solvent to the protein, and the favourable electrostatic interactions was well described by all methods. They found that the greatest variation in results occurred for the desolvation term as the electrostatic term was of a similar magnitude for all methods; this emphasised the importance of fitting parameters to this property in future developments. Also, they found that a smoothed representation of the dielectric boundary produced the smallest, and therefore best, desolvation penalties. , in most As is so often pointed out, the socalled “protein dielectric”, c e ~used continuum studies has little to do with the actual internal dielectric constant of the protein, cp, but is in fact a scaling factor that represents the contributions not explicitly included. In this respect, it is analogous to the effective protein dielectric used in the PDLDtype methods mentioned earlier. The neglect of
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protein relaxation and its consequences with regards to the size of the effective protein dielectric has been discussed by Sham et a1.** They studied chargecharge interactions between ionised residues in the reaction centre of Rhodobacter sphaeroides. Protein relaxation was taken into account by performing molecular dynamics upon the model, with ionisable groups in their charged and uncharged states, and then averaging the energy of each group over configurations generated during the molecular dynamics run. They found that when this reorganisation is neglected it necessary to use a larger value of the effective protein dielectric, E,K. The size of the effective protein dielectric, and the consequences with regards to the calculation of titratable residue pKas, is discussed in the next section.
7.2 pK, Calculations.  One of the most common uses of continuum methods is the calculation pK,s for titratable residues. Before the last two decades, the calculation of pKas was dependent upon an analytical method derived by Tanford and K i r k ~ o o d , 'where ~ ionisable residues were represented as point charges a fixed distance below the surface. By necessity, the systems studied were restricted to simple geometric shapes such as spheres or ellipsoids. Interactions between groups were calculated using distances that could be adjusted to fit experimental parameters. With the growth in computational power, the accuracy and complexity of these calculations has increased exponentially. This increase in computing speed meant that numerical methods could replace the analytical treatment, allowing a full description of the molecular surface through finite difference methods.75Full titration curves can be calculated using a variety of methods, from those based on Monte Carlo routines" to an efficient cluster method based upon the binding polynomial treatment of multiple ligand binding.89The majority of calculations make use of the formalism of Bashford and K a r p l ~ s which , ~ ~ is based upon the PoissonBoltzmann equation, though the protein dipole/Langevin dipole method of Warshel has been used successfully to predict PKaS for the last two decades.82 Despite numerous improvements in the methodology, there are still problems to be addressed, including the optimum choice of parameters. The most interesting issue relates to the choice of protein dielectric constant, as the optimum choice is sometimes coun terintuitive. Antosiewicz et al.90 implemented an extensive study of atomic charges and radii, protein dielectric ( E ~ ) ,and conformational changes on the accuracy of pKa calculations using the linear PoissonBoltzmann method. They found the use of NMR solution structures and PARSE atomic charges and radii (a set of parameters specifically designed to reproduce hydration free energies from PoissonBoltzmann calculationsg1)produced a small but significant improvement. However, by far the largest improvement came from an increase of protein dielectric from 4 to 20. Traditionally, pK, calculations use a protein dielectric of 4, (but calculated PKa shifts are greatly exaggerated with this value). Antosiewicz et al. showed that a protein dielectric of 20 gives greatly improved results. As mentioned before, this is not the actual empirical protein dielectric, E ~ but , rather a scaling factor representing the contributions to polarisation that
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are not explicitly included in the calculations. Suggested factors not included that might influence the results were sidechain conformational flexibility and neglect of protein reorganisation. Also suggested was the possibility that water molecules close to the ionisable residues might themselves have a dielectric less than that of the solvent. A similar study of parameters by Demchuk and Wade92 went further and showed that the root mean square deviation between calculated and experimental PKaS was smallest when a protein dielectric equal to that of the solvent dielectric was used. This result was also seen by Sitkoff et al.93For a small number of residues, however, a lower dielectric minimised the error further. These residues were usually buried or in an area of relative structural stability. This small group of "errorprone" residues exhibited a vshaped dependence of the root mean square deviation with the protein dielectric, with the minimum corresponding to a protein dielectric of 1520. Demchuk and Wade introduced a statistical criterion, Zrf, to predict which residues should be assigned a lower dielectric. This Zrfvalue is calculated by considering the opposing effects of the Born desolvation and Coulombic terms on the PKa shift. A cutoff value for choosing which residues should be modelled using a low or high dielectric is chosen based upon experimental results. Comparison with experimental results showed an error of only 0.5 pKa units for the whole test set when a variable dielectric constant was used. A subsequent report94 addressed the sensitivity of the method to different charges and radii, while predicting the PKa of Lys73 in Class A Plactamases, since it has been proposed that this residue might play a role in the catalytic cycle as a proton abstractor (see also Section 8.5). Results showed that although the Zrfcriteria needed to be redetermined for use with different parameters, results were in good agreement with experiment whether an OPLS or CHARMml9 unitedatom parameter set was used. The best calculations on Lys73 predicted a downward shift in its pKa, but this shift was not enough to allow the residue to play a part in the catalytic cycle as a proton abstractor. Recent PKa calculations in the presence of bound substrate uphold this c o n c l ~ s i o n . ~ ~ The search for a more realistic representation of the protein dielectric, which means determining those factors whose neglect necessitates the use of a high protein dielectric, has prompted much study. Gibas and S ~ b r a m a n i a m ~ ~ showed that the addition of one explicitly modelled water molecule for each titratable site, while having very little effect on the overall quality of the cakulated pKas for the test set, did improve slightly certain individual PKaS. Bashford and Beroza and Case,98 Alexov and Gunner,99 Zhou and Vijayakumar"' and most recently van Vlijmen et al.'" have all undertaken the task of explicitly modelling sidechain flexibility. Bashford and You systematically generated 36 conformers for each site via molecular dynamics, restricting the changes to the local region only. The results show that although a slight improvement is gained, a protein dielectric of 20 still produced the best results. Zhou and Vijayakumar went further and weighted each conformer according to the partition function, relative to the first chosen conformer. Results again showed a slight improvement when using a protein dielectric of 4, but less than the improvement gained by using a protein dielectric of 20. Beroza and Case
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modelled only two different conformations, the Xray crystal or NMR structure conformation and a conformation where the group experienced maximal solvent accessibility. Improvement was similar to that obtained by Bashford and You but took less computational effort. Other interesting results from PKa calculations include studies by Carlson et al.,''* who calculated the pK,s for all titratable groups in the enzymeligand complex of Da1anine:Dalanine ligase of the ddlb gene from E. coli (DdlB). Special attention was given to the ionisation state of the ligand. It was found that the pK, of Dala2 is shifted up by over 5 units, implying that the ligand is bound to DdlB in the zwitterionic state not in the free base form as previously suggested. Comparisons are made with the depsipeptide ligase from Vancomycinresistant cascade A (VanA), which is also seen to catalyse the production of Dala2, but which preferentially binds Dlactate. The difference in binding affinity is attributed to the ionisation characteristics of the two ligands. Lipscombe et al. conducted a joint molecular dynamics/PoissonBoltzmann study on the role of Glu246 from Saccharomyces cerevisiae yeast chorismate mutase in stabilising the transition state during the catalytic isomerisation of chorismate to prephenate. They found that the pKa of Glu246 was shifted towards a more alkaline pH such that the residue is likely to remain protonated at the pH corresponding to maximum enzyme turnover. The parallel molecular dynamics calculation supports this result, showing that a protonated Glu246 leads to a stable structure with Glu246 involved in hydrogenbonding to the transition state structure. This result agrees with Xray experiments, but brings to question the results of an alternative computational study. lo4 (This alternative study, where a transition state analogue was docked into yeast chorismate mutase, suggested that the Glu246 would be unprotonated and an interposing water molecule would mediate its interaction with the transition state analogue. No water molecules are seen in either the crystal structure or in the M D study by Lipscombe et al.) 7.3 Binding Studies.  The factors that influence binding between proteins and proteins or ligands, and in DNA, have been the subject of much study using continuum approaches. Misra et al.'05 investigated the binding free energies of the K I repressor to DNA, using the nonlinear PoissonBoltzmann method and a protein dielectric of 24. The results showed that longranged coulombic interactions were responsible for the strong affinity of the protein for the DNA. However, the overall electrostatic contribution to the binding free energy opposes binding, the major unfavourable component being the desolvation of the phosphate groups. Hydrogen bonding was shown to make only a minor contribution, with hydrogen bond desolvation being balanced by reformation in the proteinDNA complex. After comparison of all contributions to the binding free energies, the major driving force was seen to be the entropic release of water between the DNA and protein surface upon complexation. The energy resulting from the burial of the hydrophobic interfacial surfaces outweighed all other individual unfavourable contributions. The whole study paints a picture of ligandDNA binding that is forced by nonpolar interactions, and through
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which specificity is controlled by electrostatic energies that weaken binding. A study of ligand binding to cAMP (adenosine 3'5'monophosphate) dependent protein kinases by Hunenbuerger et a1.1°6 have shown an almost identical picture to the previous study of ligand binding. The nonpolar interactions being controlled by how well the ligand fits in the cavity. The results hold significance for the design of synthetic models intended to mimic the effects of ligands that bind to proteins and DNA. (Similar conclusions were obtained using molecular dynamicsbased free energy simulations of the interaction between netropsin and DNA  see Section 6). Other PoissonBoltzmann binding studies have come to the same conclusions. A number of interesting results are referenced involving studies of MHC class 1 proteinpeptide, homeodomainDNA, and anthracycline antibioticsDNA interactions and the interested reader is directed to follow these for more information. 107109
7.4 Protein Folding and Stability.  There have been a number of interesting studies on the contributions of salt bridges to the stability of proteins. Luo et al.' lo used the computationally efficient Generalised Born treatment, plus a recently developed a technique' called 'mining minima', to include conformational sampling in a study of the strength of salt bridges in solvent and in helical peptides. The technique is performed in four steps. 1. A local energy minimum in a potential energy well is found using a Monte Carlo routine to sample random dihedral angles of a nonsymmetrical nature. The current lowest energy conformation is stored and the routine repeated in the region of the low energy conformer using smaller angle sizes. The process is iterated to a convergence threshold. 2. The extent of the potential energy well is determined by taking successive samples of the local region, the hypershell, to find nearby low energy conformations. 3. The total free energy of the potential energy well is determined by a Monte Carlo integration over all dihedrals in the potential energy well and a correction is added to take into account those conformations not sampled in the region. 4. A new minimum is then obtained, using the same routine as step 1 but with the condition that no previously sampled conformations are allowed. The total free energy is determined by cumulatively adding the total energies of all potential energy wells until the free energy change is less than a convergence threshold.
The method uses different radii for small and large molecules. Despite this, the results compare favourably with experiment (to within a few kJ molI). The method predicts a gain of 2.0 kJ mol only in stability for solventexposed salt bridges. The study suggests that the strongest salt bridges are between phosphotyrosine and arginine; some of these phosphotyrosinearginine interactions are twice as strong as any other saltbridge studied.

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It has been noted that salt bridges make little contribution to protein stability at room temperature. The total gain of 2 kJ mol determined in the previous reference is a good example of this point. The reason for this is that the large unfavourable desolvation term is always equal or greater than the favourable electrostatic interactions between charged groups and the rest of the protein. However, it has been suggested that salt bridges play a crucial role in promoting hyperthermostability in proteins, as those proteins that exhibit hyperthermostability have a greater than average number of salt bridges. This seemingly contradictory idea was studied by Elcock.l12 In a previous study, it was shown that the desolvation free energies of charged side chains are more adversely affected by increasing temperature than are the desolvation free energies of hydrophobic groups. l 3 They parameterised their solvation model to reproduce the temperature dependence of the hydration free energies of each of the 20 standard amino acids. In this current study,' l 2 the earlier results' l 3 were used to study the stability of salt bridges at high temperature. It is shown that the desolvation penalty associated with the hydration energy is reduced as temperature is increased. Consequently, the destabilising effect of saltbridges at room temperatures may be mitigated at high temperature, providing a suggestion for the increase in salt bridges seen in hyperthermophilic proteins. Elcock and McCammon' l4 have also studied the electrostatic contribution to the stability of halophilic proteins, found in conditions of extreme salt concentration. The model makes a number of approximations that limit the study. Firstly, in studying the stability of the protein in the folded state, they have had to make assumptions about the unfolded state. The pK, calculation requires the foreknowledge of the pK,s of titratable groups in the unfolded state. However, no such information is available and so they made the assumption that the behaviour of the pK,s of titrating groups is equivalent to the groups being at an infinite distance away (i.e. noninteracting). This approximation is not always correct and residual interactions can lead to pK,s significantly shifted from their model compound values. Secondly, they have accounted only for the electrostatic aspects of stability, neglecting the increased strength of hydrophobic interactions at high salt concentrations and the related saltingin and saltingout effects of the dissolved ions. Despite these approximations, they have shown that electrostatic interactions between the charged groups constituting the salt bridge are always destabilising and that allowing titratable residues to ionise favours the unfolded protein. They suggest that the increase in ionisable residues seen in halophilic proteins is therefore an adaptation to prevent aggregation in high salt concentrations and not to increase the stability of the folded protein. Schaefer et al.' l 5 present a method of calculating absolute electrostatic free energy differences between conformers to aid in the study of the pH dependence on protein stability. As mentioned in the article, most electrostatic calculations neglect the electrostatic energy of the unfolded protein, assuming all sites possess their standard pK,s and titrate independently in the unfolded protein. The method calculates the electrostatic energy of a chosen conformer relative to a reference state, which in this case is the protein at pH = 00, when the protein is

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completely unprotonated. Because this state is completely unprotonated, there is only one charge state to consider and so the electrostatic energy may be evaluated completely. The free energy of this reference state is then added to the calculated relative energy of the chosen conformer. Titration curves at different pH values are then determined using a Monte Carlo routine. The same is performed for a model of the denatured state. In this study, a completely linear (extended) conformation of the protein (to maximise the solvent accessible area) is used to represent the unfolded protein. The advantage of the reference state is to allow residual interactions between charge groups in the unfolded protein, which could significantly shift their pKas from that of the model compound, to be taken into account. Stability curves, relative to pH, are then obtained by integrating the equivalent titration curves for the conformers and the unfolded protein. Results are compared to those gained using the “null” model, where the selfenergy difference between protonated and unprotonated sites is the same in the protein as it is in a model compound. The calculation was performed on hen egg white lysozyme, and the calculated stabilities compare well with experiment. Interestingly, the comparison to the null model shows a significant difference in calculated absolute energies, suggesting that there are important interactions in the unfolded protein between titrating sites that are ignored in normal calculations.
7.5 Solvation and Conformational Energies.  The calculations of solvation and conformation energies, along with the calculation of pKas, are two of the easiest ways to evaluate the current state of a continuum method. The breadth and depth of published experimental results provides a wealth of data to test the accuracy of a model. Merz and Gogonea116 have used the calculation of solvation free energies to test a new approach that combines semiempirical quantum mechanical (QM) calculations with continuum electrostatic calculations to allow fully quantum mechanical calculations of proteins to include solvent effects. Recent improvements in linear scaling algorithms allow fully quantum mechanical calculations to be performed on small proteins in the gas phase. While such calculations have their obvious advantages, they fail to predict accurately the solvation free energy of proteins whose natural environment is in solution. The approach utilises the divideandconquer approach of Yang et af.’l7 In quantum mechanical calculations, the most prohibitive part for calculations on large systems is the diagonalistion of the Fock matrix after each SCF calculation. This requires N3 operations, where N is the number of basis functions used. For systems where N > 1000, which includes most proteins of real biological interest, this diagonalistion becomes impractical. The divide and conquer approach partitions the system into N overlapping subsystems and the diagonalistion of the Fock matrix is replaced by a series of diagonalisations for each subsystem. By splitting the system into subsystems, contributions from basis functions outside the subsystem are ignored, an approximation justified by the sparcity of the Fock matrix. The N*N Fock matrices for each subsystem are therefore reduced to a smaller W *W matrices. This makes each diagonalisation cheaper by a factor of (WIN).If the system is large enough then the collective
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diagonalisation of each subsystem's Fock matrix becomes less expensive than for the single system Fock matrix. Merz and Gogonea have tested their method by calculating the solvation free energies for 29 neutral organic molecules, 36 small ions, the NacetylMmethylamide derivatives of the 20 essential amino acids, DNA bases and nucleotides, and for seven proteins ranging from 46 to 275 residues. Their results compare well with experiment and with results published by Cramer and Truhlar using their SM5.2R solvation model.' l 8 The calculated solvation free energies of the proteins however were 815% bigger than those obtained previously by York et al.' l 9 The stated reason for this is the approximate nature of the nonelectrostatic term, which involves the calculation of the surface area. It is suggested that a more accurate description of this term would give better results. However, the calculation of the biggest protein took only 21 hours on a standard SGI workstation. The possibility of parallelisation is evident. The method suggests that fully quantum mechanical calculations on large proteins that include solvent, and that take a reasonable time, are just around the corner. Dominy and Brooks'20 have parameterised a Generalised Born model, proposed by Qui et a1.,l2' for use with the CHARMm peptide, protein and nucleic acid allhydrogen and polarhydrogen forcefield. The method was parameterised to minimise the difference between the electrostatic solvation free energy, as calculated by the Generalised Born method, and a finite difference PoissonBoltzmann method. The method was tested by calculating the electrostatic solvation free energies of a test set of small proteins and nucleic acids. The results were found to compare extremely well with the corresponding calculated electrostatic solvation free energies, as calculated by the finite difference PoissonBoltzmann equation. Inclusion of the method in molecular dynamics simulations led to the generation of average structures comparable to those from a simulation of a small 56residue protein in explicit water and the crystal structure, showing the model's effectiveness in replacing explicit solvent.
7.6 Redox Studies.  The redoxBohr effect, i.e. the observed coupling of the protonation states of titratable residues with redox potential, was investigated by Baptista et al. Previous studies have been concerned primarily with the effect of redox state on observed P K , S . ' ~ ~ Baptista  ' ~ ~ et ~ 1 . attempted l ~ ~ to simulate simultaneously the binding equilibrium of protons and electrons. It was pointed out that ignoring the full description of the binding equilibrium of protons and electrons introduces two errors in the calculated energies of the system. These are Firstly, the calculated energy is affected by the creation of partially occupied sites. Secondly, by introducing these extra empty/occupied sites, the entropy of the system increases; this is termed occupational disorder in this work. The study primarily attempts the simultaneous weighting of protonation states and redox potentials in the Monte Carlo sampling routine. The method
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successfully determines for the first time the full reduction order of the heme groups in cytochrome c3 from Desulfovibrio vulgaris Hildenborough. It does, however, fail to predict the positive cooperativity seen between hemes 1 and 2. The reason is possibly the neglect of the allosteric conformational change seen upon reduction. The contribution to the redox potential and to the pK,s due to occupational entropy is determined and found to be in the region of 20 mV or 0.5 pK, units.

7.7 Additional Studies.  A number of studies have addressed the nature of the electrostatic profile of ion channels.’2c129In particular, Adock et al.127using a model built by Sansom,’28 have calculated the pK,s of the “rings” of positive and negative side chains around the mouths of the nicotinic acetylcholine receptor and a related glycine receptor. Results show that side chains at the mouth of the extracellular side may not be fully protonated, whereas those in the intermediate pore lining and at the mouth of the intracellular region appear to be fully ionised. Inclusion of the other domains of the respective proteins have shown a complex interplay between the pore lining and extramembrane regions that govern the electrostatic potential experienced by an ion travelling through the channel. The effect of the electrostatic interactions on the catalytic mechanism of dihydrofolate reductase (DHFR) from E. coli has been undertaken by Cannon et al.l3’ The study of two enzyme+ofactor complexes and two enzymecofactorsubstrate complexes shows that Asp27 plays a major role in catalysis, and that the formation of 4hydroxypterin is the more likely intermediate than 4oxopterin, which is found to be more stable in vacuo, in the conversion of dihydrofolate to tetrahydrofolate. Also, a structural motif for polarising the cofactor for catalysis, involving three weak bases at positions 44, 76, and 98, is reported, and a careful examination shows the same motif in 20 other DHFRs.
8 Hybrid QM/MM Calculations One of the major challenges to theorists when studying chemical reactions in biological systems is size. Systems of real biological interest, such as enzymes, typically contain hundreds of atoms. When solvent effects are modelled explicitly, this number can rise to thousands. Hybrid quantum mechanical/ molecular mechanical (QM/MM) methods currently provide the most useful approach for studying chemical reactions in such systems. Molecular mechanics (MM) methods have long been used to study the structural and energetic properties of biological systems. They have been utilised in the study of protein folding, proteinprotein and proteinsubstrate interaction, and membrane dynamics. Their advantage comes in the speed with which the classical potential, represented by force fields, can be evaluated. As computational speed increases, these methods become increasingly popular. However, most force fields are parameterised against ground state systems and are unable to give an adequate description of the formation and breaking of chemical bonds.
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In quantum mechanical (QM) methods, the electrons are described explicitly. They require little or no parameterisation and offer the potential of high accuracy when compared with other methods. These methods are computationally expensive. The advances made in linear scaling algorithms, for example the work pioneered by Yang et and by Merz et L Z ~ . , "offer ~ much hope. However, the study of chemical reactions in biological systems is, at best, prohibitive when using semiempirical methods, and, at worst, completely unfeasible when using ab initio methods. Hybrid methods are therefore a way of combining the computational speed of MM methods with the high accuracy and flexibility of QM methods. This section of the review focuses on these hybrid methods. A brief description of the methodology follows and then a review of the methodology used currently is given. Some recent applications of QM/MM methods are then described.
8.1 Methodology Developments.  In hybrid methods, the system is partitioned into two or more sections, as seen in Figure 4. One region contains the chemically active elements of interest and is described by quantum mechanics. The second section is assumed to change very little and is described by a classical potential. A possible third section consists of solvent, and could be included in the molecular mechanics region, or studied using some other technique such as continuum methods. The total energy of the system is then given by equation (5) below
where ETOTis the total energy of the system, EQMis the quantum mechanical energy, EMM is the molecular mechanical energy and E Q M I M is Mthe interaction between the two regions. The question of how to represent the boundary between the two regions is hotly debated and will be discussed later. The first reported use of this methodology is credited to Warshel and Levitt131who used it to study the catalytic mechanism of lysozyme. Despite being the first, the model includes a high level of sophistication. The model allowed polarisation of
=
Solvent mgion

\
Quantum Mechanical
\ ENYZME~
Molecular Mechanical mgion (Protein Enviroment)

I 0 l/
Figure 4 Schematic diagram of the different regions involved in hybrid QMfM M calculations
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the QM region by allowing the core Hamiltonian to be perturbed by the electric field of the MM region. It also included polarisation of the MM region by using polarisable point dipoles in place of atomic charges. The model also included solvent effects. Instead of a problematic link atom that became popular much later, they pioneered the use of hybridised orbitals to represent the link between the QM and MM regions. In spite of the improvements to date, this study still stands as one of the most sophisticated applications attempted. Since that publication by Warshel and Levitt, QM/MM methods have been implemented using semiempirical, 311 33 ab initio and density f ~ n c t i o n a l , ' ~ ~ ' ~ ~ C a r  P a r i n e l l ~ ' and ~ ~ ~valence ' ~ ~ bond13* methods. In particular, Thiel, whose QM/MM methods are based upon an MNDO/MM formalism, has published MNDO parameters that include the treatment of dorbitals. Parameters for Na, Mg, Al, Si, P, S, C1, Br, I, Zn and Cd have been p ~ b 1 i s h e d . lThe ~ ~ parameters for Mg were implemented by Hutter et al.13' in a QM/MM study of the photosynthetic reaction centre of Rhodobacter sphaeroides and found to give accurate ionisation potentials and heats of formation. Though MNDO/d parameters for the first row transition metals are proving somewhat more difficult to obtain,14' an unpublished set has been implemented in the commercial software p a ~ k a g e ' ~MND094. ' This affords the opportunity for the study of transitionmetal containing enzymes using semiempirical methods. The MNDO/d method would give a considerable decrease in the computational time required to treat such systems and offers the chance to study larger QM regions containing transition metals than has previously been attempted. The inclusion of transition metals could prove vitally important for systems such as nickeliron hydrogenase and bacterial photosynthetic reaction centres. For a more detailed description of the hybrid QM/MM methodology, the interested reader is directed to the excellent articles of Eichinger et al.'36 and Bakowies and Thiel.'42 In addition to the above, articles by Friesner and B e a ~ h yand ' ~ ~Lyne et al.'34 give good reviews of QM/MM methods before the annual period of interest of this review.
'
8.2 The Models.  Bakowies and Thiel presented a three level hierachical representation of the QM/MM coupled systems. This representation gives a good description of the state of current models employed in QM/MM calculations. The hierarchy is as follows.
1. Model A: The simplest model. Model A is a purely mechanical embedding of a quantum mechanical core in a molecular mechanical environment. No electrostatic perturbation of the quantum region by the classical region is included. All interactions between the QM region and the MM region are treated using molecular mechanics. The next two models can be considered as consecutive improvements on this model. 2. Model B: Includes a calculation of the electrostatic QM/MM interactions from the QM electrostatic potential and the MM atomic charges. Additionally, the effect of the MM atomic charges is included in the core Hamiltonian. This is analogous to allowing polarisation of the QM region
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by the external field generated by the MM region. The majority of QM/ MM methods employed stop at this level of sophistication. 3. Model C : Finally, the MM region is allowed to polarise in the presence of the QM region. This is the highest level of sophistication and few methods include this effect. The neglect however, as Bakowies and Thiel mention, results in an unsatisfactory asymmetrical description of the nonbonded interactions between the two different regions and the calculation of absolute energies is impossible. Model A, although simple, still produces reasonable results if one is merely interested in a qualitative approach. However, for the calculation of absolute energies, and for the study of reaction mechanisms that involve charge rearrangements, such as hydride and proton transfer steps, the neglect of the effects covered by models B and C can at best produce large errors and at worst make the results completely meaningless. The polarisation of the quantum region is implemented by including the effect of the atomic charges in the core Hamiltonian. This leads to a perturbed density matrix and a perturbed energy for the QM region. This technique has barely changed since the first implementation by Warshel and Levitt. A better description of the electrostatics of the MM region has produced the greatest improvement in this area. In the study by Eichinger et al.'36 the effect of polarisation on the QM region is implemented using a novel hierarchical combination of multipoles and Taylor expansions to calculate the electrostatic interactions over a grid outside the QM region. The effect is to scale the electrostatic interactions as the distance of the interacting group from the grid point increases, thus producing a more accurate description of the longrange electrostatics. The calculated electrostatic potential is then added to the density functional calculation. A similar approach is used in calculating the electrostatic potential in the MM region under the influence of the electric field generated by the QM region. The hierarchical method is used to calculate the electrostatic potential at the boundary of the QM region and is then included in the calculation of the electrostatic interactions in the MM region. Thus, the effects of polarisation in both regions are included. However, such thorough considerations of the electrostatics of the MM region and their effects on the QM region are rare, as most methods merely rely on the atomic charges of the preferred force field to accurately describe the electrostatics. Where the polarisation of the QM region has received minimal consideration, the polarisation of the MM region has been the subject of intense research. While it is true that few published applications include a description of the polarisation of the MM region, the computational effort required being prohibitively large, many articles discussing its implementation have been published. Excluding the above method of Eichinger et al., there exist two popular methods for including polarisation of the MM region into the calculation. The first method is based on atomcentred polarisable dipoles; the second, the fluctuating charge method, is based on the idea of electronegativity equalisation. The atomcentred polarisable dipole method was first suggested
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by Warshel and Levitt in their groundbreaking paper. The induced dipoles are added to each MM atom, in addition to the usual atomic charges; these dipoles are defined by equation (6), pj
= aj Ej
where aj is the atomic polarisability of atom j , and Ej is the electric field at atom j ; this includes the electric field generated by the QM region. Luzhkov and W a r ~ h e l ' ~later ~ ' ~refined ~ this model for both ground and excited states, presenting the results in a study of solvent effects. However, as Thompson points out, 146 the calculation of analytical gradients, essential for geometry optimisations, required that the QM wavefunction be converted to atomcentred charges to avoid the computationally expensive calculation of the derivative integrals of the oneelectron matrix elements that make up the QM electrostatic potential. This introduced inconsistencies between the forces and the total energy as the external field felt by the MM region, due to the atomcentred charges derived from the QM wavefunction, is inconsistent with the external field felt by the QM region, due to the point polarisable dipoles. Thompson and S ~ h e n t e r ' have ~ ~ ' implemented ~~~ an approached based on this early work but which uses a slightly different approach to solving the analytical gradients. It solves the inconsistencies by converting the atomcentred dipoles into sets of six point charges positioned along the Cartesian axes. These charges are then included in the Hamiltonian and the integrals involving the induced dipoles that would arise from this slightly different approach are avoided. Thus, the method retains the full QM wavefunction and the MM dipoles, allowing for a consistent treatment of the forces and total energy. The second method of fluctuating charges is based on the theory of electronegativity equalisation, proposed by Rappe and G ~ d d a r dand ~ ~continued by Rick et al. 148 In a fluctuating charge model, the electrostatic energy of an atom is cast in the form of a quadratic function of its charge. The total electrostatic energy of a set of these atoms can then be calculated from the sum of the individual energies and the interaction energies between atoms. The electronegativity (per unit charge) of an atom, x, is the derivative of the energy with respect to its charge. The charges of the system are then determined subject to the constraints that all electronegativities are equal and that the total charge on the system is conserved. In a hybrid QM/MM system, the calculation of the system energy is now dependent on the selfconsistent calculation of the electrostatic energies for the coupled regions, i.e. the electrostatic energies of both regions are calculated successively until the change in energy is below a chosen threshold. The calculation of the QM electrostatic energy, however, requires the repeated diagonalisation of the Fock matrix, and so scales as N3 where N is the number of basis functions used. This method is therefore computationally expensive, but does afford a simple and meaningful way of including polarisation of the MM region. In addition, the methodology offers itself to the linear scaling techniques pioneered by Yang et al. (see section on Continuum Methods). Field149has published an excellent review of fluctuating
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charge models before 1997; included in the article is a clear and concise derivation of the fluctuating charge model.
8.3 The Link Atom Problem.  As mentioned earlier, the question of how to describe the boundary between the quantum and classical regions is hotly debated. Following the example given by Bakowies and Thie1,’42consider a bimolecular system X + + Y . The question of how to partition this system is trivial. X + may be treated quantum mechanically and Y may be treated classically, or vice versa. However the partitioning of a covalently bonded, unimolecular system, XY, is more difficult as none of the obvious fragments, X + + Y  , X’ + Y’, or X + Y + accurately describe the electron distribution of either X or Y as part of the whole system XY. In 1986, Singh and K ~ l l m a n ’ ~developed ’ the idea of “Link Atoms”. These atoms, usually hydrogen atoms, were used to cap the quantum mechanical region and satisfy the free valencies created by the breaking of the covalent bonds between the different regions. As Bakowies and Thiel point out, this corresponds to the chemically intuitive description of the charge density of a fragment being part of a whole molecule. These link atoms are invisible to the MM region but are included in the QM calculation. The bond between the QM boundary atom and the MM boundary atom was treated by a classical potential. The use of link atoms, however, perturbs the wave function and changes the forces on nearby nuclei in the QM fragment, through the addition of extra degrees of freedom, and can result in the double counting of certain interactions. As such, link atoms introduce a significant error into the total energy of the system and this must be corrected. As Gao et al. point out however, the arbitrariness of these corrections to the total energy throughout various QM/MM methodologies introduces inconsistencies in the reported total energies and forces in the QM/MM systems being studied. In addition to these problems, the vibrational frequencies calculated using this method are worse than those calculated from a purely MM calculation. 13‘ The search for solutions to the problems created by the use of link atoms has been the subject of many publications. Field et al.’ 32 constrained the link atom to an equilibrium position between the two boundary atoms, thus removing some of the degrees of freedom. This is the method used most commonly in today’s methods. Eichinger et a1.13‘ suggest that the ideal link atom should not only correct the errors introduced by the addition of the link atom but also restore the mechanical properties of the broken bond. With a view to creating an accurate method with which to study the photochemical reaction of retinal in bacteriorhodopsin, Eichinger et al.13‘ used a QM/MM system incorporating density functional methods based on planewaves and have attempted to address the above problems. Termed “SPLAM” (Scaled Position Link Atom), the method constrains the link atom to the line of the broken bond. The atom is also constrained to an equilibrium position between the two formerly bound atoms. Thus, the method is analogous to that of Field et al. 132 However, unlike Field et al., this equilibrium position is not an energy minimum for a CH bond but is
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scaled so that the stretching force of the CH bond closely resembles the CC bond it is replacing. This is done via equation (7):
where TCH is the scaled bond length, rEH > is the reference CH bond length, kcc and ~ C are H harmonic force constants, rcc is the corresponding expected CC bond length, and r& is the reference CC bond length. A correction is added to account for the extra stiffness in the shorter CH bond. They have tested the method by calculating the vibrational frequencies of ethane and a small Schifl's base related to retinal and comparing them with those calculated using a purely QM method. The frequencies are well described. Those frequencies corresponding to the QM region are virtually identical to those obtained from the purely QM calculation. The frequencies corresponding to the boundary region are intermediate between the purely QM calculated frequencies and those previously calculated using a purely MM method. This is a large improvement on the previous methods where those frequencies calculated from hybrid QM/MM methods in the boundary region were seen to be worse than those obtained from a purely MM calculation. Perhaps though, a large influence on the success reported in the article is the systematic correction by the authors of the errors introduced by the link atom to the final energy. These errors, such as the correction for a now polar bond in the QM region, are often neglected by other QM/MM methods. As Gao et al. point out,'50 the best solution to the link atom problem is not to use the link atom at all. In their early work, Warshel and Levitt131 introduced the idea of using hybridised orbitals to represent the unsatisfied valencies left by the broken covalent bond. Building on this idea, ThCry et al.'" proposed a method, called the Local Self Consistent Field method (LSCF), where the atomic orbitals of the QM boundary atom were transformed into hybrid orbitals that were strictly localised along the lines of the bonds to the QM atom. The charge density on those hybrid bonds between the QM and MM atom is frozen and excluded from the QM calculation, acting only as external point charges on the QM fragment. The orbitals are obtained from initial QM calculations on small molecules with a similarity to the system being studied, and are considered transferable to the larger system. Gao et a1.150expanded on this work, presenting a method for modifying the semiempirical parameters used in the LSCF procedure such that the parameters could be considered as transferable as any other semiempirical parameter. The semiempirical parameters were modified to mimic the bonding properties of the full QM system. Thus, the need to reparameterise for each new system is removed. So far, Gao et al. have only parameterised for the AM1 method. In addition to the hybridised orbital approach, Zhang et al.'52have proposed a pseudobond approach. The boundary atom of the MM region is replaced by a single free valence carbon atom, Cps.Each C,, has 7 electrons, a nuclear charge of 7 and an effective core potential parameterised to mimic an sp2 hybridised orbital between it and the QM boundary atom, thereby forming a sort of
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“pseudobond”. The C,, atom then takes part in the QM calculation. The total charge on each MM boundary atom was set to zero, along with some atoms two bonds away, to counteract overly large electrostatic interactions at the QM/ MM boundary. They have tested the method on small molecules and on succinamide and found the resulting energies reproduce those from a full QM calculation well. 8.4 Miscellaneous Improvements.  In addition to the specific problems addressed above, a number of interesting papers have been published on QM/ MM systems. Turner et have presented a method for performing systematic searches for transition states and saddle points. Termed ‘GRACE’, the method is based upon the partialrational function optimiser. The optimiser follows one eigenvalue to a maximum while simultaneously minimising all other eigenvalues. The Hessian is then recomputed. The above steps are repeated until a transition state is reached. At each step, the eigenvalue of the new Hessian closest to the direction of the previous maximum eigenvalue is chosen as the eigenvalue to follow. The GRACE method noninvasively interfaces popular QM and MM codes to allow a fully flexible search engine. The QM region is optimised as above. For each QM step, a number of MM steps are performed in order to minimise the total potential energy and to keep the gradient of the MM part at zero, or as close to it as computational accuracy allows. Therefore, the whole system is described only by the degrees of freedom of the QM region, but the method allows the inclusion of the whole protein when searching for a saddle point. GRACE also includes options to determine intrinsic reaction coordinates, necessary to fully characterise a saddle point. Turner et al. have tested their method by searching for, refining and fully characterising six different and nearly degenerate transition states for the solvated enzymesubstrate+ofactor complex from lactate dehydrogenase. The starting structures came from six different points on a classical molecular dynamics run. Lactate dehydrogenase catalyses the interconversion of pyruvate to Llactate. Two points came from this and a previous Firstly, the barrier for the reaction was calculated at 188 kJ mol’. A similar QM/MM study by Gready et presented an energy barrier of just 81 kJ mol’. It is unclear as to whether the inclusion of the whole protein contributed to the discrepancy as other factors are involved. More interestingly, though, is the observation of six nearly degenerate transition states all of which differ considerably in the position of the residue sidechains at the active site, but which all contain some identical and obviously important elements. This observation lends credence to the view that there exists a family of nearly degenerate transition states involved in any enzymatic reaction, whose structures retain important interactions, but whose less important residue sidechains assume different positions. This is a view upheld by other publications, both t h e ~ r e t i c a land ’ ~ ~experimental. 157 Bentzien et ~ 1 . have ’ ~ used ~ a QM/MM approach to calculate the free energy surfaces for an enzyme reaction mechanism, in particular the nucleophilic attack step in the catalytic reaction of subtilisin. The method uses an empirical valence bond’58mapping potential as a reference for the generation of ab initio
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structures, which are then used to direct a molecular dynamics run. Finally, the QM/MM energies of structures from throughout the molecular dynamics run are calculated. The corresponding reaction in solution is calculated and the free energy difference is determined. The free energy surface is then generated. The method requires the calibration of the empirical valence bond parameters for use in the mapping potential and an automated process for their generation is presented. The method also uses link atoms to define the region between the QM and MM regions. As Bentzien et al. point out, the advantage of considering the free energy of a reaction as the difference in free energy between the reaction in the enzyme and in solution means that the errors introduced tend to cancel. The free energy surfaces are compared and while they show some encouraging results, the method obviously still requires additional work such as the inclusion of polarisation in the MM region. 8.5 The “Onion” Approach.  In addition to the usual hybrid methodology described above, Morokuma et al. have taken a slightly different “onion” approach to combining calculations at different levels of theory. In the ONIOM 59 (Own Nlayered Integrated molecular Orbital molecular Mechanics) method, and its early predecessors IMOMM (Integrated Molecular Orbital Molecular Mechanics) and IMOM0161 (Integrated Molecular Orbital Molecular Orbital), the system is composed of N layers of different theory, with the size of the system being increased at each new level. For a system consisting of two levels of theory, the total energy of the system is then given by equation (8)
’
ET~= T E(High, model)  E(Low, Model)
+ E(Low, Real)
(8)
where E(High, Model) is the energy of a model part of the system treated at a high level of theory, E(Low, Model) is the corresponding system treated at a lower level of theory, and E(Low, Real) is the energy of the whole system treated at the lower level of theory. The equation bears some similarities to the techniques of Pople in considering the effects of both higher levels of electron correlation and larger basis sets simultaneously without explicitly carrying out the higher level calculations with the larger basis sets by using energy expressions such as aMP2/631G**]
=
aMP2/631G*]  qHF/631G*]
+ qHF/631G**] (9)
More sophisticated examples of such expressions are used in Pople’s G1, G2 and G3 theories. 162164 In the “onion” approaches, the calculated structure of the lower level is used as a constraint upon the higher level of theory. The breaking of covalent bonds between the model and the real system is treated by replacing the corresponding atom in the real system with a hydrogen atom in the model system, much like the link atom method. The coordinates of the new
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hydrogen are, however, dependent upon the coordinates of the replaced atom. In the published methods, this is seen by freezing the bond lengths of the two interchanging atoms to reference values, and by keeping any bond angles and dihedral angles involving the interchanging atoms identical. In the ONIOM method, this idea is extrapolated to include three different levels of theory, and essentially the method could be used to address N levels of theory as the name suggests. Despite the obvious simplicity of the method, its uses in the literature have been restricted to organic and organometallic chemistry, and it has yet to find favour with those modelling biological systems.
8.6 Applications.  The following is a by no means exhaustive selection of some the interesting results obtained in the last couple of years using hybrid QM/MM methods. 8.6.1 NickelIron Hydrogenase.  Amara et al.'65have attempted to study the active site in nickeliron hydrogenase in various redox states. Their QM/MM model uses density functional methods for the QM region, and the CHARMm all atom force field for the MM region. They have not considered polarisation of either the QM or the MM region. The method also uses link atoms to treat the boundary between regions. Due to the complexity of the system, they were forced to reduce the size of the system to 300 MM atoms and 30 QM atoms. The transition metals were treated using an effective core potential to represent to the represent the core electrons. They also calculated the pK,s of the titratable residues prior to the simulation, something that is neglected in most other QM/ MM applications. The model performed well despite the simplicity of the system. The results suggest that Cys530 acts as the general base in the mechanism and that a hydride might also be involved in the reaction as a bridging hydride is seen in the transition structure. However, no sulfur radical was observed in the reaction. In contrast to a gasphase calculation by Pavlov et the iron atom was seen not to be redox active; it was stabilised by the three nonprotic ligands in a lowspin state. The three ligands were seen to be noninterchangeable. Pavlov et al. previously studied a model of the active site in which it was observed that one of the cyanide ligands leaves the binding pocket to bridge the bimetallic centre in an unusual bridging mode. 8.6.2 PLactam Hydrolysis.  Pitarch et al.'67 have studied the hydrolysis of plactams by plactamase, with particular interest in the acylation step. This same enzyme has also been the subject of pK, calculations (see Continuum Methods section). The catalytic mechanism proceeds by nucleophilic attack by the hydroxyl group of Ser70 on the ketocarbon atom, followed by acylation, and then hydrolysis, and reprotonation of Ser70. The proton acceptor is unknown, as is the proton donor to the plactam nitrogen atom. Three possible reaction mechanisms are present. The first involves Glu166 as the general base whose role is to assist in deprotonating Ser70. It is the favoured mechanism but suffers from the fact that Glu166 is a considerable distance away and would be required to move a fair distance to take part in the reaction mechanism. The
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second involves indirect deprotonation of Ser70 via Lys73 and Ser130. This is unlikely, however, in view of the pK, studies mentioned previously. The third and final route involves a concerted mechanism where Ser70 itself protonates the plactam nitrogen. In this study, the concerted mechanism was investigated. Transition structures were obtained via GRACE for the blactamasepenicillate complex. The following conclusions were reached. Firstly, an energy barrier was obtained for the concerted mechanism and was found to be prohibitively high, suggesting that a concerted mechanism was unlikely. Secondly, Glu 166 was found to move close to Lys73 when the Lys was protonated, thus placing Glu166 close enough to act as the general base and lending credence to the proposal of a protonated Lys73. Thirdly, in apparent contradiction to the results obtained above that propose a protonated Lys73 in the catalytic mechanism, the structures obtained when Lys73 was deprotonated were found to give the best agreement with the crystallographic structure. 8.6.3 Bacteriorhodopsin.  The reaction centre of bacteriorhodopsin, which assists with the photoisomerisation of retinal from the all transconfiguration to the cisconfiguration via rotation about the C13C14bond, has been studied by Humphrey et a1.16* They present a threeexcited state model of the phototransformation. Using the density evolution method'69 that combines a QM density matrix to a classical mechanics trajectory, thereby allowing a quantum dynamical simulation, Humphrey et al. modelled the three lowest electronic states. They found that after excitation to the first excited state, strong coupling between the first and second excited states allows the system to overcome a slight energy barrier to photoisomerisation of about 13 kcal mol Also, they noticed that after this first crossing point, at a rotation of about 30°, rapid rotation follows to the second crossing point where the system crosses back to the ground state, at about 90°, and it is this crossing that determines the quantum yield. The results show that neglect of the second excited state would mean that the system is unable to proceed over the energy barrier to complete isomerisation. The results compare well with published experimental results.
'.
8.6.4 The Bacterial Photosynthetic Reaction Centre.
 The bacterial photosynthetic reaction centre in Rhodobacter sphaeroides and Rhodobacter viridis has been studied by Hutter et aZ.'39 Despite the high level and sophisticated studies of this system, most of the previous work was performed on the unmodified Xray coordinates. Hutter et aZ. point out that molecular properties are dependent on the geometry and that the Xray coordinates are poorly resolved. The result is an error of nearly 120 kcal mol' in the single point quantum mechanical energies between the four bacteriochlorophylls. In an attempt to correct this, they implemented an unusual optimisation technique. Firstly the QM region was optimised under the influence of the electrostatic field of the MM region, but the MM region was kept frozen. Next, the MM region was optimised while the QM geometry was frozen, and finally the QM region was reoptimised again while keeping the MM region frozen. The result of this routine was a reduction in the error of the calculated ab initio energies between the four bacterio
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chlorophylls to less than 5 kcal mol'. An interesting point of note that also came from this study is that of the dimerisation energy for the important bacteriochlorophyll dimer. A QM study by Sakuma and Ka~hiwagi'~' suggested that the dimerisation energy was in the region of 21 kcal mol' and that dimerisation was actually unfavourable. The study was performed on the Xray coordinates. Hutter et al. have found that the dimerisation energy of the optimised structures was reduced from 21 kcal mol' to 3 kcal mol' for Rhodobacter viridis, and when calculated using density functional methods instead of semiempirical methods, this energy was further lowered to  12 kcal mol'. These results not only show the importance of a properly optimised structure on the calculation of QM energies, but also show that the bacteriochlorophyll dimer is held together by mutual attraction and needs no further stabilisation from the protein environment. 8.6.5 Other Studies.  In addition to these studies, other QM/MM studies
include the study of catalysis by HIV p r o t e a ~ e , ' ~the ~  mechanism '~~ of carbonic a n h y d r a ~ e , ' ~tyrosine ~.'~~ ph~sphatase,'~~~'~~ citrate synthase"' and the mechanism of serine protein kinases.18' 9 CarParrinello Calculations
An alternative to molecular orbital or density functional calculations with atomcentred basis sets is the CarParinello method. Typically, the valence electrons are represented using a plane wave basis set and the core electrons are represented using pseudopotentials. Nonlocal density functional methods can be implemented using the generalised gradient approximation as in traditional orbitalcentred density functional approaches. An advantage of plane waves in a density functional calculation is that the forces can be calculated efficiently. However, the calculations require the system to be periodic and so to date the vast majority of applications have been to material science where the systems are naturally periodic. However, by using the supercell approximation in which a molecule is repeated periodically and surrounded by large vacuum regions it is possible to treat molecules of biological importance. The applications to date have not been extensive but a couple are presented here because the technique is likely to have a growing influence on modelling biological systems. Using this approach Segall et al. have studied both the conformation of acetylcholine"* and the interaction between cytochrome P450 and its substrate or substrate analogues. 183185 For acetylcholine, the bond lengths were reproduced well (they converged to within 1.3% and were within 1.6% of experiment). On the negative side problems were encountered bectuse convergence of the energy with respect to the supercell size (typically  8 A for acetylcholine) could not be obtained and this restricted the convergence of the energy between different conformations to & 5 kcal mol Another difficulty arises because of the overall charge of + 1 on acetylcholine and this was handled through a post hoc correction to the energy. The cytochrome P450 studies are significant
',
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because most previous studies on cytochrome P450 used semiempirical methods. The Fecontaining heme group, ligands and several amino acids were included in the calculations. The studies to date have focused on the change in spin state of the iron in the ligand free system (low spin), in the substrate bound system (high spin) where water is displaced and in the substrateanalogue bound system (low spin) where water is not displaced. Removal of water from the substrateanalogue bound system did not facilitate the switch to high spin and so alternative mechanisms were explored. The spin state was found to be linked to the FeS bond length to the axial cysteine residue in that artificially lengthening the bond induces a transition to high spin. The periodicity requirement is not such a problem for aqueous solutions. The a//?equilibrium in glucose has been visited many times through both ab initio calculations and molecular dynamics simulations (including a relatively novel approach described in Section 6 above). The real power of the CarParinello method lies in the opportunities it gives for ab initio molecular dynamics and so for the first time the anomeric effect has been investigated in solution using molecular dynamics without any empirical potentials. 186 Three simulations were performed on /?glucosein the gauche  ,aglucose in the gauche  and aglucose in the gauche + conformations. Fiftyeight water molecules were included in the periodic cell (a total of 198 atoms). The BLYP combination of exchangecorrelation functionals were used as these had previously given good results in simulations of water, in combination with pseudopotentials. A time step of 0.145 fs was used in a simulation of total length 6 ps, of which the last 3 ps were used for data analysis. Experimentally it is found that the aconformer is more stable in the gas phase due to the anomeric effect but the /?conformer is more stable in solution. The ratio is about 36:64% corresponding to about 0.35 kcal mol'. The total simulation length was too short to study conformational changes or to calculate the free energy difference between the two anomers but was sufficiently long to analyse the water structure. For the panomer it was seen that the water molecules flow more freely, i.e. in a more disordered manner, around the anomeric site while for the aanomer the water molecules were bound more tightly and in a more ordered manner. This may well underlie the stability of the flanomer in aqueous solution (see also Section 6).
Acknowledgements
We would like to acknowledge the EPSRC for support (RIM),
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5 Relativistic Pseudopotential Calculations, 1993June I 9 9 9 BY PEKKA PYYKKO AND HERMANN STOLL
1 Methods
1.1 Introduction.  Relativistic effects for heavyelement compounds are at the focus of much current theoretical work, and the interest in these phenomena is rapidly growing still; see the Bibliographies on the Relativistic Theory of Atoms and Molecules'3 (a subset of which, dealing with relativistic pseudopotential calculations since 1993, is appended to this article). While the influence of spinorbit splitting on atomic spectra has been known since the early days of quantum mechanics, the impact of scalarrelativistic effects on valence properties of molecules has been widely acknowledged only within the past two decades. As an example of such effects, consider spectroscopic constants of diatomic gold compounds. as monitored by selfconsistentfield (SCF) calculations using nonrelativistic (NR) and scalarrelativistic pseudopotential^:^'^ bond lengths & are typically reduced by 0.150.40 due to relativity, while changes of dissociation energies D, are between 0.8 eV and + 1.9 eV; e.g. for AuH, 80% (Re)and 30% (De)of the difference between N R SCF and experiment is recovered by the relativistic contributions, which gives an indication of the relative importance of relativistic and electroncorrelation effects. A quantitative theoretical description of heavyatom compounds requires high computational effort. Working with a relativistic DiracCoulomb or DiracCoulombBreit Hamiltonian
A
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(where c is the velocity of light, G, /3 are the 4 x 4 Dirac matrices, p'i is the momentum operator for electron i, Vl denotes the external potential due to nucleus 1 with charge 21, and atomic units are used throughout) leads to fourcomponent wavefunctions describing electronic and positronic states. Basisset requirements for the two upper components (which are the large ones for bound electronic states in molecules) are already high due to the large number of electrons per atom and to the increasingly higher orbital angularmomentum quantum numbers 1 of occupied valence shells in heavy atoms; an additional complication arises due to the need of fulfilling a kineticbalance condition, e.g.
in the simplest case (where 5 stands for the Pauli spin matrices and for twocomponent basisset spinors), which means that even higher 1values are needed for the description of the smallcomponent spinor, xs, than for the largecomponent one, xL. Moreover, as in the nonrelativistic case, electroncorrelation effects have to be taken into account, and again additional difficulties arise for heavy atoms since (a) the excitation spectrum is usually very dense (especially for d and $elements) and (b) the coupling between relativistic and correlation effects often cannot be neglected. Thus, accurate correlated fourcomponent calculations, e.g. of the coupledcluster (CC) type, albeit available nowadays for atoms6 and molecules,798are currently restricted to compounds with two heavy atoms at most; they play a very useful role for benchmark purposes, but do not provide a viable route for larger systems. There have been many attempts to get rid of both the positronic degrees of freedom and the lower (small) components of the wavefunction in a relativistic treatment of heavyatom compounds, the most successful of which are the DouglasKrollHess (DKH)9' and the ZeroOrder, FirstOrder, etc. regularHamiltonian approximation^'^'^ (ZORA, FORA, ...). Using these approaches, twocomponent and  after averaging out spinorbit coupling  even onecomponent calculations are possible. However, there is no reduction in the number of electrons per atom, and the need for large oneparticle basis sets at least partially remains. A more radical point of view is taken in pseudopotential theory. The key observation is that for most properties of chemical interest only electrons residing in the outermost (valence) shells of heavy atoms are responsible, the atomic cores being to a large extent unchanged when molecules are formed from atoms. Moreover, relativistic effects  not only those of the cores but also those directly acting on valence electrons  arise in regions near the nuclei where the local kinetic energy of the electrons is high. Assuming that in both cases the relevant information can be transferred from atoms to molecules, one is led to considering a model system of n, formally nonrelativistic valence electrons interacting with fixed cores, described by a Hamiltonian like x L p S
'
5: Relativistic Pseudopotential Calculations, 1993June 1999
24 1
(4)
Here, effects on valence electrons originating in the core regions are taken care of by means of atomically parametrized pseudopotentials VTp, the central quantities of pseudopotential theory. In principle, also the mutual interaction of the cores needs parametrization, but very often the simple longrange pointcharge form used in equation (4) is sufficient (Ql = 21 nc,l, where nc,l, is the number of core electrons at site A); the latter is exact for spherical nonoverlapping and nonpolarizable cores, cf. refs. 1517 and Section 1.7 for possible refinement. Note, at this point, the analogy between the pseudopotential approach, as outlined so far, and densityfunctional theory (DFT): there, electroncorrelation effects are approximated by considering a formally uncorrelated (independentparticle) system in which correlation is only implicitly accounted for by means of potentials/functionaIs; the latter are, in the simplest case, locally transferred from the homogeneous electron gas to the system in question. Originally, pseudopotentials (PP)  other names are model potentials (MP) or effective core potentials (ECP)  were defined as onecenter/oneelectron operators which simulate frozencore  valence interaction (and relativistic effects) by means of a local (multiplicative) potential. It soon turned out, however, that a much better performance can be achieved with a semilocal (different local potentials for different angularmomentum quantum numbers I ) or nonlocal form (e.g. including projection operators onto core orbitals). Especially the nonlocal form is gaining increasing popularity nowadays due to the computational ease with which quantities like PP gradients can be evaluated. Several groups have produced pseudopotentials which retain the original allelectron (AE) nodal structure of valence orbitals. More often, however, a transformation to radially nodeless pseudovalence orbitals is implied when using PPs; since the valence interaction energy changes with this transformation, the pseudopotentials have, in this case, also to include correction terms which compensate for this change. When working with planewave basis sets (in solidstate calculations, CarParrinello studies, etc.), it is essential that the potentials be as smooth as possible; since the early 1990s, specially designed pseudoorbital transformations have been developed leading to (ultra)soft PPs satisfying these requirements. While early pseudopotentials were generated for (and used in) onecomponent (i.e., nonrelativistic or scalarrelativistic) calculations, modern PPs incorporate spinorbit (SO) terms as well, thus simulating relativistic effects in full. Finally, it is increasingly recognized nowadays that, for heavy atoms, cores may be so polarizable that they can no longer be treated as strictly frozen; corevalence correlation effects, in particular, can be of the same order of magnitude as relativistic effects. Adding corepolarization terms to the PPs efficiently remedies these shortcomings of the original pseudopotential theory  however, the pseudopotentials are then no longer oneelectron operators, and also the onecenter approximation no longer strictly applies.
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Table 1 Recent reviews on pseudopotential calculations Year
Authors
Subject
1993 1994
Kudo and N a g a ~ e ~ ~ ~ Bala~ubramanian~~~ Ermler and mar in^^^* Huzinaga** Cundari et ~ 1 . ~ ’ Dolg and S t ~ l l ~ ~ Ermler and mar in^^^^ Frenking et al.30 Gordon and C ~ n d a r i ~ ~ ’ Musaev and M ~ r o k u m a ~ ~ ’ Bala~ubramanian~’~*~~~ Frenking and P i d ~ n ~ ’ ~
Ge, Sn, Pb polycyclic compounds Ln, An compounds Molecules, clusters, surfaces “Concept of active electrons” “Heavier elements” Ln compounds Nonlocal effects Transition metal compounds Transition metal compounds Trans. metal catalysed reactions Rel. pseudopot. calculations Transition metal compounds
1995 1996
1997
Thus, pseudopotential theory has reached a degree of sophistication and versatility nowadays which makes it applicable to a broad range of problems and which, at the same time, allows for an excellent simulation of allelectron results. This is documented in a number of review articles (e.g. Krauss and Stevens (1984),’* Christiansen et al. (1985),19 Ermler et al. ( 1988),20Laughlin and Victor ( 1988),21Gropen (1988),22 Pickett (1989),23 Huzinaga (1991),24 Chelikowsky and Cohen (1992),25Payne et al. (1992),26 Kresse and Hafner ( 1994),27Huzinaga ( 1995),28Dolg and Stoll ( 1996),29Frenking et al. ( 1996),30 Cundari et al. (1996),31Balasubramanian (1998),32cf. also Table 1). The present article adds to this series by giving (a) a short introduction to the most reliable of the current ab initio PP schemes, (b) information on new developments recently published in literature, (c) a personal view on problems/questions in pseudopotential theory to be addressed in the future, and (d) a list of applications during the title period; for applications before 1993, see refs. 1 and 2 or the reviews quoted. 1.2 Model Potentials.  The theoretically simplest and most transparent strategy for generating pseudopotentials relies on a direct transfer of operators/potentials from allelectron (AE) ab initio schemes. The level of the AE “reference” approach is chosen in such a way that the operators of interest, i.e. those mediating corevalence interaction and those describing relativistic effects onto the valence shell, acquire (at least approximately) a onecenter/oneelectron form, and these operators are modelled then by suitably parametrized potentials. Pseudopotentials of the variety just characterized are usually called model potentials. The most successful and widely used variant of them are the “ab initio model potentials” (AIMP) of Huzinaga, Seijo, Barandiaran and cow o r k e r to ~ be ~ ~briefly ~ ~ reviewed in the following. For corevalence interaction, a nonrelativistic or quasirelativistic (cf. below) HartreeFock (HF) description of the core naturally leads to a oneelectron potential
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Z
VCV= r
243
+ C(2JC  K,)
(5)
c
where the sum is over all core orbitals of a given nucleus with charge 2,and J , K are the usual Coulomb and exchange operators; for nonoverlapping cores, orthogonality requirements between core orbitals from different nuclei are satisfied a priori, i.e. the V,, are strictly onecenter terms. Relativistic operators cannot be taken directly from equations (1) and (2), since a twocomponent formalism is aimed at. However, use can be made here of an (approximate) elimination of the small components originally suggested by Dirac and later adapted, in several variants, to atomic allelectron calculations by Cowan and
[email protected] and Wood and Boring.45 In essence, the central (Coulomb and exchange) potential V,b(r) for an orbital spinor with atomic quantum numbers n, 1,j = 1 f 1/2 is supplemented by a relativistic term of the form
where is the orbital energy, and pno is a suitable (usually DFTlike) approximation to Vno. Note that both equations ( 5 ) and (6) rely on an independentparticle approximation  equation ( 5 ) cannot describe corevalence correlation from the outset, and equation (6) is different for different orbital spinors it acts on. Since the model potential to be derived should be energyindependent and applicable to arbitrary wavefunctions, a further approximation is necessary for equation (6):
here Po are projectors onto atomic spinors with angularmomentum quantum numbers 1,j , and the are taken to be the operators belonging to the valence spinors of lowest energy for a given ljcombination (simply quoted as in the following). It is easy to separate equation (7) into a onecomponent scalarrelativistic part and a spinorbit

where PI = Pl,1+1/2+ P I , J  ~ and , ~ 1, s' are the orbital angularmomentum and spin operators. Replacing the enl,l1/2 and ~,1,1+1/2 in equation (6) by average values &,,I (e.g. the orbital energies of a spinorbitaveraged allelectron Cowan
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Griffin calculation in LScoupling) and doing a similar averaging for the vno, equations (8) and (9) can readily be evaluated. The steps leading to equations (8) and (9) have relevance not only for the relativistic operators but also for the corevalence interaction, equation (5): the first two terms in equation ( 5 ) are local operators anyway (i.e. multiplicative potentials), CcK , can be written as an Idependent operator, for an atom as in equations (7) and (8), and the orbitals necessary for building up Jc and K, can be taken from an LS CowanGriffin AE calculation as above. Now that the operators to be modelled have been defined, the frozencore approximation comes in, i.e. the operators are derived for a suitable atomic state in a fixed form (and not changed any more for molecular applications). The form used is tailored to the known functional form of equations (5) and (6) in the appropriate limits (r + 0, r + oo),and reads
The first two terms in equation (10) represent the corresponding ones in equation (5) (recall that Q = Z  n,, where n, is the number of core electrons), while 1, Kc and the Idependent (semilocal) terms in equation (8) are cast into a nonlocal form  a projector involving (uncontracted) onecenter basis functions in the AIMP implementation. Using large expansions in Gaussians in equations (10) and (1 1) and large primitive basis sets > (p numbers different functions belonging to a given combination of atomic I, mi quantum numbers), fitting errors can be made arbitrarily small, in principle. When working with the final model potentials of equations (10) and (1 1) in molecular applications, we are interested in valence properties and, in fact, do not need to include core orbitals in our wavefunction any more. This is so, since corevalence interaction has been accounted for implicitly in equation (lo), and the sum of energies of individual cores with fixed orbitals is a constant, anyway. However, in order to get an aujbau principle for the valence wavefunction, core orbitals 14c> have to be shifted from energies E, to energies above those of the (occupied) valence ones, and this levelshift is done by
The prefactor 3, = 2~, is more or less arbitrary in atomic calculations, but one should note that only with a prefactor 3, + 00 is an AIMP calculation really equivalent to a frozencore allelectron one, in the molecular case. One should also keep in mind that formally unoccupied core orbitals at finite energy in the (virtual) valence spectrum may lead to unphysical excitations, in ab initio CI calculations, and should be removed beforehand.
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What about basis sets in modelpotential calculations? Only valence orbitals have to be described explicitly, but since these orbitals are required to be orthogonal to the core orbitals and hence have wiggles in the core region, the primitive basis sets cannot be much smaller than in the allelectron case. (Basis functions are even neeeded for core orbitals with no (occupied) counterpart in the valence space (e.g. 4f in Pb), since otherwise orthogonalization tails of valence orbitals at neighbouring atoms cannot be described properly.) However, heavy contraction of basis functions with high exponents is usually possible (as long as enough flexibility is retained for the relativistic operators SO in particular  which mainly sample the core region). AIMP parameters and corresponding basis sets are currently available for all maingroup and d transitionmetal elements (e.g. by contacting http : // www .qui .uam.es/Data/AIMPLibs.html); data for felements are upcoming.42 Some examples may serve to demonstrate their reliability. Benchmark SCF calculations with scalarrelativistic AIMPs were compared to AE DiracHartreeFock (DHF) results in ref. 38, for Group 14 hydrides and oxides (XH4, XH2, XO). A uniform accuracy of 1 pm for bond lengths could be obtained with a 14valenceelectron space (for X = Ge, Sn, Pb), relativistic bondlength changes for the hydrides were accurate to looh, and reaction energies XH4 + XH2 + H2 could be reproduced to few kcal mol'. SO effects and electron correlation were included in benchmark calculations for the halogen compounds XH in ref. 39, using a 7valenceelectron space for X (17ve for I); in comparison to experiment, bondlength errors are of 5 1 pm again, vibrational frequencies are accurate to 1% for X = C1, .., I, and dissociation energies to few tenths of an eV. Very recent studies on atomic SO ~ p l i t t i n g s ~ ~ , ~ ~ indicate that deficiencies formerly believed to be due to the AIMPs vanish with an accurate treatment of valence correlation; for Pt, e.g., the experimental atomic spectrum up to 20,000 cmI could be reproduced with a uniform accuracy of few hundreds of cmI (Le., significantly better than with available AE fourcomponent configurationinteraction (CI) calculation^^^), using empirical spinfree data in a spinfreestateshifted SOCI AIMP calculation. It should be mentioned finally that many of the AIMP references cited above also contain a careful comparison with results of other MP (cf. e.g. refs. 5054) and ECP schemes (see the following sections). Let us close this section with some remarks on current and future development lines of the modelpotential method. Firstly, corevalence correlation is not covered by the AIMPs but is increasingly important for heavy maingroup elements, cf. Section 1.7. Fortunately, a simple remedy is available, by means of corepolarization potentials (CPP), which should be added therefore to (largecore) AIMPs for compounds containing such elements. Secondly, the quasirelativistic CowanGriffin/WoodBoring scheme, although very accurate in practice for valence properties of atoms, has some drawbacks indicated above due to its restriction to the HF level and the energy dependence of the operator equation (6). A more general approach would be to use the twocomponent relativistic operators mentioned in Section 1.1, such as the DKH one. The DKH scheme is especially suited for use with pseudopotentials: it has been shown that
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relativistic twoelectron terms can be left out, in the scalarrelativistic case,” or can be replaced by a meanfield expression, in the SO case,56without significant loss of accuracy; moreover, a onecenter approximation has been successfully tested with DKH.57*58 The DKH operators can either be explicitly used as parts * ~ could ~ of the AIMP  a line which is already being actively e x p l ~ r e d~or~they be used as reference data for fitting the model potentials equations (10) and (1 1) (similarly as equation (6)). Finally, an extension of the AIMP method should be mentioned which is currently exploited for defect states in crystals: the operators equations (9, (6) and (12) need not necessarily be taken from freeatom references, it is also possible to represent deformed atoms/ions in crystal surroundings.61 1.3 ShapeConsistent Pseudopotentials.  While with model potentials the wavefunction is (ideally) not changed with respect to the valence part of an AE frozencore wavefunction, such a change is desirable for computational reasons. The nodal structure of the valence orbitals in the core region requires highly localized basis functions; these are not really needed for the description of bonding properties in molecules but rather for the purpose of corevalence orthogonalization. The idea to incorporate this “Pauli repulsion” of the core into the pseudopotential is as old as pseudopotential theory Modern ab initio pseudopotentials of this type have been developed since the end of the seventies, cf. e.g. refs. 6468. Similarly as in Section 1.2, one starts from atomic AE reference calculations at the independentparticle level (some kind of quasirelativistic H F or fully relativistic DHF). The first step now in setting up pseudopotentials consists in a smoothing procedure for valence orbitals/spinors (“pseudoorbital transformation”). In the D H F case, to be specific, the radial part 41i(r)/rof the large component of the energetically lowest valence spinors for each fjcombination is transformed according to
withf,(r) being radially nodeless and smooth in the core region ( r < rc). The choice of rc (which may be different for different fjsets) and offj(r) is to some extent arbitrary, apart from normalization and continuity conditions for the Jli(r) (cf. Section 1.5); the agreement between Jli(r) and 41i(r)in the valence region (“shapeconsistency”), on the other _hand, is of the utmost importance for valence interactions calculated with the +li(r) to have physical significance. As a next step, a pseudopotential has to be adapted to the pseudovalence orbitals J0 defined in equation (13). The idea is to look for a oneelectron operator V p pwhich, when inserted into the valence Hamiltonian % p p , equation (4), yields the &b as H F orbitals for the chosen atomic reference state, at the original AE orbital energies EQ. One should be aware, at this point, that an atomic operator cannot be fixed by its lowest (eigen)solutions for each ljcombination  the whole spectrum would be needed, in general. However,
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restricting V p pto a semilocal form, i.e. to the form of a radially multiplicative potential V r ( r )for each Ijset, does the job of making the adjustment unique. This is so, since the problem is effectively decoupled at the HF level: for each ljcombination, a Focklike radial equation can be set up:
the first two terms on the lhs constitute the radial kineticenergy operator, while w1; is an effective valence Coulomb plus exchange potential acting on J0(r), which is built up with the set of pseudovalence orbitals, {&,}, instead of the AE ones, {4,}. With given (Pri(r) and E,, V r is the only unknown quantity in equation (14) whose radial behaviour can be determined pointwise by inversion. (The longrange behaviour is dominated by the Coulomb attraction of the core,  Q / r , where Q is the core charge, so only the shortrange part of V r needs to be determined.) Repeating this procedure for each Ijset, one can combine the V r to a common semilocal V p pas in equation (7). Relativistic effects are implicitly included in V p psince the AE reference calculation explicitly describes these effects while equation (4) does not. With a final analytical fit of the V r , e.g.
and a separation into scalarrelativistic and spinorbit potentials, V:: and Vi;, as in equations (8) and (9), the procedure is finished, in principle. One should add, though, that there are several variants. Firstly, the analytical fit is often done only after the separation into Vc: and V!$, leading to expansion parameters { A $ , n$, a$} and { A i O , nco, a;'}. Secondly, the sum over I , j in equation (15) has to be truncated in practice, and this truncation if often compensated for by the approximation V, = V L for I ? L, where V , is the scalarrelativistic potential for an angularmomentum quantum number not occupied in the core. The idea behind this approximation is that for I ? L no pseudoorbital transformation is needed, making the V, identical to the model potentials of the previous section; the latter have a much less pronounced ljdependence than the potentials incorporating Pauli repulsion to the core. Using this approximation, we have the replacement
Q
 +
r
Q + r
VL(r)
In this case, the analytical fit parameters { A ,n, a } refer to VG V Lor V:"  V L , rather than Vo, Vy". Thirdly, one has to be careful about the spinorbit potentials VFu which often contain numerical factors (such as 2/(21+ l), cf.
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equation (9)) not included in the analytical fit parameters A i o , Concluding this paragraph, we want to stress that the radial dependence of the relativistic contributions to V p p ,equation (15), is quite different from the BreitPauli form often used in AE calculations. It has been suggested sometimes6e71 to set VfO(r)= 2$$/r3, in analogy with the AE case, and parametrize 2$$ to fit valence SO splittings, but this is not really consistent with the pseudoorbital transformation, equation (1 3). Although the present approach seems to be very different from that used for deriving the model potentials, note the similarity: in a (Dirac)HartreeFock context, both methods should yield the correct (i.e. allelectron) valence orbital energies for the chosen atomic reference state, with the only difference that the orbitals/spinors are identical with the AE ones in the modelpotential case, while they are replaced by radially nodeless pseudoorbitals/spinors in the case of the shapeconsistent pseudopotentials. In both cases, however, there is no guarantee for transferability of the potentials when going from the reference state to other atomic or molecular states and/or to other (higher) theoretical levels, or even for the reference state itself when going from the lowest valence orbitals of a given lj to higher unoccupied ones not used for fitting. There are ways to influence this transferability, cf. e.g. refs. 7275 and Sections 1.5 and 1.6; note, however, that it does not only depend on the specific approach chosen, it also critically depends on the choice of corevalence separation. Let us briefly consider the latter issue at the case of the Ce atom. From chemical intuition, 4f, 5d, and 6s are the valence orbitals contributing, with varying occupation, to bonding in Ce compounds. These are the orbitals with highest energy and they are clearly separated from the lowerlying ones. However, this does not constitute a valid corevalence separation for pseudopotential theory, for the following reason: the 5s, 5p orbitals are spatially localized outside the region where 4f has its maximum and will certainly substantially relax with varying 4f occupation. Thus, the frozencore approximation underlying the use of fixed pseudopotentials will not be valid any more. This can easily be checked, in fact, by means of frozencore atomic AE calculations for the Ce atom: errors with respect to a fully variational treatment are of the order of several eV, for atomic excitation energies, with this core definition. With a 1s4d core, the situation is considerably improved  frozencore errors are reduced to several tenths of an eV. The remaining deficiencies are again nut caused by problems with a missing energetic corevalence separation (a necessary prerequisite, of course), but with the missing spatial separation between 4s4d (core) and 4f (valence). In the present approach, involving a pseudoorbital transformation, the problem is accentuated by eliminating radial nodes of valence spd orbitals (5s5d for Ce) in a region where the 4f valence orbital has its main contribution  one cannot expect the valence interaction to be invariant with rc values in equation (13) being largely different for different 1. Possible ways of solving these problems are either a reduced 1s3d core (frozencore errors of the order of eV),76or a “4fincore” p ~ e u d o p o t e n t i a l ~featuring ~  ~ ~ an openshell core averaged over states with a fixed integralfoccupancy. Once a suitable corevalence separation has been chosen, leading to reason
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ably small atomic frozencore errors, can one really forget subsequently about the role of core orbitals in pseudopotential approaches? In the modelpotential scheme of the previous section, their role is still indirectly acknowledged by means of the levelshift term equation (12) by which device their occupation in the valence wavefunction is avoided. In the present approach involving the pseudoorbital transformation equation (13), such a device is no longer necessary: it is a mathematical property of local (or semilocal) potentials that the lowest solutions (for each Ijsymmetry) must be radially nodeless, i.e. coincide with the $4(r) by construction. One should be aware, however, that the core orbitals have not altogether disappeared  corelike solutions still exist, as in the modelpotential scheme, at high energies in the virtual spectrum of the pseudovalence Hamiltonian. Although usually not discussed in the literature, this has some consequences, the most important being a too weak Pauli repulsion of valence electrons of partner atoms due to cores simulated by pseudo potential^.^^ For heavy alkali fluorides, e.g., underestimations of bond lengths by 10 pm have been found with 1valenceelectron PPs for the alkali atoms, even after introducing an overlap correction for corecore interaction.80 (Note that this deficiency has nothing to do with the frozencore approximation which is very accurately satisfied for alkali atoms, at the H F level, and which in the molecular case leads to deviations of just the opposite sign.) The only remedy in such a case is to redefine the corevalence separation and adopt a smallcore definition. Modern shapeconsistent pseudopotentials as described in this section  also often termed effective core potentials (ECP)  have been published by several groups. The most extensive tabulation is due to Christiansen, Ermler, Ross and c o  ~ o r k e r s , ~containing ~  ~ ~ DiracHartreeFock derived scalarrelativistic and SO potentials for all elements of the Periodic Table up to element 118. In some cases (e.g. d transitionmetal atoms and postd maingroup elements) two sets of potentials (smallcore/largecore) are provided; for the 4f elements the core definition (e.g. Ce4+) is probably not optimum, cf above. Quite popular are also the Hay/Wadt pseudo potential^,^^^^ comprising nonrelativistic potentials for Na to Kr, and scalarrelativistic ones (derived from CowanGriffin AE calculations in LS coupling, cf Section 1.2) for Rb to Bi and U, Np, Pu, excluding the 4felements. Again two core definitions are available in critical cases, but no SO potentials have been generated. Two further sets of pseudopotentials should be mentioned, those of the Toulouse group (cf. e.g. ref. 97) and the “compact effective potentials” (CEP) of Stevens, Basch, Krauss and c o  w ~ r k e r s ; ~in ~ the ’ ~ latter case, a fairly complete set of nonrelativistic (Li to Ar) and DHFbased scalarrelativistic pseudopotentials (K to Rn, including the 4f elements) has been published. A special feature of the Toulouse PP and the CEP is the use of an operator norm for parameter adjustment in the final analytical form of V p p ,equation (15). Consider the expression

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where J b , EQ are the prescribed pseudovalence orbitals and orbital energies for are the corresponding the atomic reference state, equation (13), while 4b, (radial) H F solutions of the valence Hamiltonian, equation (4), including the V p pto be optimized. By minimizing the IlObll, the primed quantities approach as closely as possible the prescribed ones. This way, V p pcan be determined without resorting to the inversion of the Fock equations (14). Possible advantages are twofold: firstly, orbitals J b with radial nodes can be included in the fit procedure (which is important in smallcore cases when more than one orbital of a given &combination is occupied in the valence space), and secondly, the pseudopotentials can be brought into a rather compact expansion. Another way of avoiding problems with the inversion of the Fock equations (14) in the case of orbitals with radial nodes has been suggested by Titov et al. (generalized relativistic ECPs, G R E C P S ) ; ' ~ '  'this ~ ~ work also extends the semilocal ansatz equation (15) by additional projectors, in the case of outercore and valence orbitals of the same lj. Note that all the pseudopotentials listed above (except for the GRECPs) have the same functional form, equation (15), irrespective of the details of adjustment; thus, the same program packages can be used for all of them (with different parameter sets, of course). Note also that the pseudopotentials are derived numerically (or using large basis sets); the basis sets published in connection with the PPs have been optimized a posteriori for use in molecular calculations. Thus, it is well possible to generate new basis sets, of larger or smaller size, according to the accuracy requirements of specific applications, simply e.g. by energy optimization in a valenceonly atomic calculation using the PP in question. Functions with high exponents are not necessary by construction of the potentials, polarization functions should be very similar to those of AE calculations. 1.4 DFTBased Pseudopotentials. The model potentials and shapeconsistent pseudopotentials as introduced in the previous two sections can be characterized by a HartreeFock/DiracHartreeFock modelling of corevalence interactions and relativistic effects. Now, HartreeFock has never been popular in solidstate theory  the method of choice always was densityfunctional theory (DFT). With the advent of gradientcorrected exchangecorrelation functionals, DFT has found a wide application also in molecular physics and quantum chemistry. The question seems natural, therefore: Why not base pseudopotentials on DFT rather than H F theory? This is perfectly possible, indeed, and has been worked out for model potentials (cf. e.g. ref. 107) as well as for shapeconsistent pseudopotentials (usually called normconserving PP in this context). lo'' lo In the first case, the main change is to replace the H F corevalence exchange operator in equation (5)
C
and to do a corresponding replacement in the energy expression
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25 1
here, $, and pv are the valence wavefunction and oneparticle density, respectively, and Vxc/Excare the exchangecorrelation (xc) potentials and energies of DFT, in a suitable approximation. There is a problem here, however, since Vyc is not linear in p which would make the rhs in equation (18) dependent on pc (the core density) only. Moreover, the rhs of equation (19) is not simply the valence expectation value of the xcpotential in equation (18). A (simple) solution is to neglect the overlap between pc and p,, enforcing a linear and zeroing the corevalence xcenergy. In a more approximation for sophisticated (and preferable) version one refrains from including xceffects into the model potential at all. That is no serious disadvantage: using a fixed parametrized form of pc, cf. e.g. ref. 11 1, p = pc + p, and the rhs of equations (18) and (19) can be easily evaluated on the fly, in valenceonly calculations. Similarly, for transferring the pseudopotentials of Section 1.3 into a DFT form, one starts from a (relativistic) DFT instead of a (D)HF allelectron referense calculation and, after performing the pseudovalence transformation 44 44,equation (13), imposes the conditio! that a valenceonly calculation, with the valence interaction formed from the +4 (and the corresponding density p,), yields the J0as radial KohnSham (KS) orbitals, at the original AE orbital energies EQ. The “classical” paper here is that of Bachelet et al.,’ l 2 who used the localdensity approximation of DFT (LDA) to generate relativistic pseudopotentials for nearly the whole of the Periodic Table (HPu). Special care was taken in the construction of the pseudovalence orbitals, equation (13), to make V p pnonsingular at r = 0 (cf. Section 1.5); accordingly, the functional form for the final expansion of the pseudopotential differs from equation (15) by a screening factor (effective for the core region) in the longrange Coulomb part of the potential
vyc
where erf is the error function, linearly vanishing for r +0. Unfortunately, the same difficulty arises as above for the DFT model potentials  for an accurate treatment, especially in largecore cases, a nonlinear core correction’ l 3 is  V.yc[pv],implicitly included in the pseudopotential necessary: the term Vyc[p] construction as described so far, is subtracted out and replaced by an explicit treatment relying on some suitable approximation of pc, and a corresponding treatment is applied to the corevalence xcenergy, rhs of equation (19). Again, as in the MP case, the nonlinear core correction is often neglected in routine applications. Let us finally note that, parallel to the development of improved DFT approximations (generalized gradient approximation (GGA), optimized effective potential (OEP)), pseudopotentials corresponding to these theoretical levels are desirable and indeed have been generated; cf. refs. 114 and 115 for recent examples. A code for adjustment and validation of modern norm
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conserving PPs, at various DFT levels, is available from the CPC Program Library (Belfast).’ l6
1.5 SoftCore Pseudopotentials and Separability.  From the computational point of view, the removal of the core with pseudopotentials helps to reduce basisset requirements. This is particularly true for the (either HF or DFTbased) shapeconsistent pseudopotentials of Sections 1.3 and 1.4, since the pseudoorbital transformation, equation (13), eliminates radial nodes in the valence orbitals. Details of this transformation are not overly critical when using local basis sets (e.g. Gaussiantype orbitals (GTO)), as is usual in quantum chemistry, but a careful optimization is decisive for use with the planewave basis sets of solidstate physics. In order to keep the number of plane waves to a manageable size (low “planewave cutof€”), the radial potentials V r ( r )should be smooth and nearconstant in the core region. This is not necessarily the case with the shapeconsistent pseudopotentials of quantum chemistry, which usually exhibit strong Pauli repulsion for Ijvalues occupied in the core (where the pseudoorbital transformation takes place) and strong attraction otherwise [essentially determined by the Coulomb potential of the core the pointcharge approximation to which,  Q / r , appears in equation (15)]. Thus, for use in connection with planewave basis sets, the shapeconsistent pseudopotentials of solidstate theory should be based on specially designed pseudoorbital transformations, and these transformations should not be restricted to valence orbital spinors with radial nodes only, but should be applied for all spinors occupied in the valence space. Sophisticated procedures have been devised for this purpose, in the early nineties27,117120 relying on special expansion functions for the hj of equation (13) and/or minimization of highkineticenergy Fourier components of the reference pseudovalence orbitals $o, or on smoothness conditions for the (screened) potentis1 at r = 0 resulting from the inversion of the Focklike equation (14) for 44. However, there are limitations, set by the normalization and continuity conditions mentioned in connection with equation (13): the J j should be (at least) twice continuously derivable at rc in order to avoid discontinuity of the pseudopotential, and the pseudoorbital charge contained within r, should equal that of the corresponding AE orbital (“norm conservation”). Of course, one could increase flexibility by choosing larger r, values than those normally used  the latter are typically smaller by a factor 12 than the outermost radial maximum of $lj(r). However, there is a limitation again: reducing the size of the “valence” region where pseudo and AE orbitals coincide adversely effects the transferability of the pseudopotential. Progress is still possible, though: the normconservation condition for the pseudovalence orbitals can be relaxed provided an a posteriori correction is introduced in the SCF formalism for calculating densities, valence interaction potentials/energies, etc. from these orbitals. This possibility, suggested many years ago,12’ is most easily realized by working in a (changed) metric involving the (fixed) density difference between AE or normconserving PP orbitals and the relaxed PP orbitals under discussion, cf. equation (26) below.
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Another computational issue relevant for the shapeconsistent (or normconserving) pseudopotentials of both molecular and solidstate physics is the semilocal form, involving Iprojectors on all pseudopotential centers. Integrals for semilocal potentials require expansion of basis functions around the site of the Iprojector (yielding modified spherical Bessel functions, in the case of GTO basis functions) and a subsequent a n a l y t i ~ a l ’or ~ ~numerical’24 ~’~~ radial integration. The question is whether simplifications are possible here. Let us discuss this issue specifically for the scalarrelativistic potentials V:,f introduced in connection with equation (1 5); recall that their general form is V:,f = Vloc(r) Cf Vi(r)Pl, where Vloc is some common local potential. The computational effort due to the 1projectors cannot be entirely avoided: the restriction Vf = V, for all I, 1‘, leading to a fully local pseudopotential, is generally too strong for an accurate description of corevalence interaction, relativity, and  in particular  Pauli repulsion. On the other hand, computational simplifications can also be achieved using a completely nonlocal “separable” form
+
as in equation (lo), where the projector involves suitable onecenter auxiliary functions > coupled by constants A;! (instead of rdependent quantities): only overlaplike integrals need to be evaluated when working with equation (21). In principle, a transcription to the form of equation (21) is possible for every semilocal oneelectron potential, provided the space spanned by the auxiliary basis IZm,> is nearcomplete. In fact, databases with transcribed semilocal potentials along these lines are available nowadays;’25 a library with parameters for the energyconsistent potentials of the next section, e.g., is contained in the MOLCAS program package (http://www .teokem. lu.se/ mo 1cas/). However, in particular for DFTbased applications in solidstate theory, a small expansion length in equation (21) is desirable, to keep the computational effort within manageable limits. Such a small expansion should at least include the functions
in order to be able to simulate the action of Vf on the reference pseudoorbitals I$fm, >. (Here, for simplicity, we assume that the pseudopotential has been generated in a onecomponent quasirelativistic scheme, i.e. that the I&fm, > are directly accessible from radial orbitals 4,(r) as generated in equation (13).) Indeed, the simplest separable pseudopotential form suggested by Kleinman and Bylander: 126
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per construction gives the correct answer when applied to the (lowest) pseudovalence orbitals of each lsymmetry. A more general also introduces additional orbitals in order to enlarge the space in which the semilocal and the separable PPs act identically. These orbitals may be (nonnormalizable) solutions of the radial Focklike equation (14) for arbitrary energies &,lz9 but, again for simplicity, we concentrate on bound highern SCF solutions J$nlml >of the valence Hamiltonian equation (4), with V p p= V::: In this case, the I $ n l m , > (including the lowest valence solutions denoted as > above) are orthogonal, and with the definitions
we can set up a separable PP
with improved transferability properties. In view of the fact that equation (25) explicitly contains more than one orbital per Ivalue, it seems natural to relax the condition that these orbitals should belong to one and the same pseudopotential V I .In fact, a separate pseudoorbital transformation, equation (13), can be done for each of the JnI(r),and the corresponding ( x > can ~ be~ directly ~ ~ obtained from equation (14). However, the matrix B in equation (24) (and the PP operator in equation (25)) will be no longer hermitian then, in general. This was the startingpoint for a further generalization of the formalism pioneered by Vanderbilt. 128 He observed that hermiticity can be regained by suitably modifying B and, at the same time, introducing a new metric, both changes being related to the “augmentation function”
> has an additional Abandoning the orthonormality condition for the I $ n l m , important advantage: there is no “normconservation” any longer for thefnl(r) in equation (13)  which allows for achievement of lower planewave cutoffs and explains the term “ultrasoft pseudopotentials” for the present approach. The price to be paid in DFT calculations is not high  solving generalized eigenvalue problems, and evaluating electron densities from the solutions with the help of the augmentation functions  which makes this approach very popular in current solidstate work, cf. e.g. ref. 130. At the end of this section, we want to mention a very recent a p p r ~ a c h , ’ ~ ~ which has much in common with the other schemes just described (applicability
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in planewave calculations, separable form of the pseudopotentials), but also incorporates some features of the energyconsistent pseudopotentials of the next section (simultaneous optimization of PP parameters and pseudovalence orbitals during adjustment). The aim is to add efficient realspace integration to the merits of the previous pseudopotentials. This is done by using an ansatz for V p p ,equation (Zl), essentially built up from GTOs (which have the property of good convergence in real and reciprocal space simultaneously); these functions are used to supplement a leading screened Coulomb potential (cf. equation (20)) in Vim, and they are also employed as auxiliary basis set for the nonlocal part (and the SO potential). The (rather small) parameter set has been adjusted by minimizing the difference to AE reference data, for orbital energies (not just the occupied ones only, but also the lowest virtual ones) and corresponding orbital charges within a suitably chosen atomic sphere. A fairly complete set of relativistic pseudopotentials (from H to Pu) is available. 1.6 EnergyConsistent Pseudopotentials.  The basic approximation of pseudopotential theory is the transfer of information from atoms to molecules. But this is clearly not the only approximation, with the pseudopotentials described so far. Each V y ( r ) , cf. equation (14), is extracted from orbital properties of a single atomic reference state, at an independentparticle level ((Dirac)HartreeFock or (Dirac)KohnSham). However, the atomic spectrum is not always dominated by a single state, and atomic states are not always well described by HF/DHF or KS/DKS. For transitionmetal atoms, e.g., several lowlying atomic states are usually of relevance for molecular bonding, and neither H F nor DFT (with current density functionals) is able to describe their energy differences very accurately. With regard to relativistic effects, jjcoupling as underlying the DHF/DKS schemes is not very realistic for valence shells of heavy atoms except at very high 2values. Of course, one could object that this does not really matter since the same kind of error is made in the allelectron reference calculations and the corresponding valenceonly ones defining the pseudopotentials. However, it is tempting to ask whether it is not possible to base adjustment of pseudopotentials on physical observables rather than orbital models. The idea to adjust pseudopotentials to atomic valenceenergy spectra goes back to the pioneer days of pseudopotential theory. There, pseudopotentials were generated from experimental spectra of singlevalenceelectron ions, cf. e.g. refs. 62, 132 and 133. This technique has been abandoned later, since the highly charged singlevalenceelectron ions are often rather artificial reference states and the experimental data contain corevalence correlation effects which are not well described by onecenter/oneelectron pseudopotentials (cf. Section 1.7). However, the technique has been using a multielectron fit to states of neutral atoms and nearneutral ions, and basing the fit on theoretical reference data. In the current version of the scheme, these data are derived from allelectron multiconfiguration D H F (MCDHF) calculations, each MCDHF calculation including relativistic states belonging to one or more (nonrelativistic) orbital configurations. The adjustment of the pseudopotential to
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these data is done in a leastsquares fit: starting from a physically reasonable ansatz for V p pas in equation (15), one varies the parameters of this ansatz so as to satisfy
where EfE are the valence energies of individual AE relativistic states, ErP are the corresponding energies from twocomponent valenceonly calculations using VPPin the Hamiltonian equation (4), and wi are suitable weight factors. Usually, very short expansions for the radial potentials (k 5 3 in equation (15)) are sufficient to obtain an average accuracy of eV for all states i included in equation (27). Obviously, the PP scheme just described avoids the critical points mentioned at the beginning of this section, since it is based on physical observables, can accommodate information on as many atomic states as desirable, and reference data can be extracted at any useful theoretical level. However, one could well ask whether energetic information alone is sufficient to fix the valence density of the pseudowavefunction, or even the nodal behaviour of the pseudovalence orbitals in the core region. While the answer is no, in general  a quantummechanical operator is only fixed by eigenvalues and eigenfunctions  it is affirmative in the present case. This is so, since the PP is essentially zero in the valence region (with the exception of the  Q / r term, i.e. the longrange Coulomb potential of the core which also appears in the AE case), thus providing the correct external potential for that region anyway; the only role of the PP is to fix boundary conditions for the pseudowavefunction (orbital amplitudes, e.g.) at some rc value. Whether one prescribes these explicitly or more indirectly, by energy differences between states differing in valence occupation, is a matter of choice. Regarding orbital densities in the core region: with a local radial potential for each Ijcombination the lowest valence solutions #g do not have radial nodes, and they will be smooth provided the PP itself is so, too. Again, it is a matter of choice, whether one prescribes the behaviour of the pseudovalence orbitals in that region explicitly, or more indirectly by means of a compact ansatz for the PP. It should be clear, from this discussion, on the other hand, that there are many common points between the energyconsistent pseudopotentials of this section and other PP approaches described earlier. With currently used DHF/ MCDHF reference data, e.g.,corevalence correlation effects are left out in both cases (for remedies, cf Section 1.7). Similarly, the pseudovalence transformation is involved, as an approximation, for both shape and energyconsistent PPs. It may be worthwhile, at this point, to elaborate a bit more on the latter approximation. It has long been recognized (cf. e.g. ref. 136) that the transition #g + Jri must have some influence on valence interaction, which can be compensated for the reference state(s) somehow, but cannot be fully suppressed with a simple oneelectron/onecenter PP. Valence correlation effects, e.g., rely on twoelectron exchange integrals which are affected by the change in the nodal

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pattern of the valence orbitals. A systematic investigation of the differences in valence correlation energies between AE and PP calculations has been performed in refs. 137 and 138. It is seen that absolute changes are really nonnegligible (overestimation of up to 10% with PPs), but fortunately differential correlation errors for properties such as excitation and dissociation energies are only 1 mH, on the average. Let us close this section with some practical hints to the use of energyconsistent pseudopotentials. Scalarrelativistic potentials of this type (and corresponding optimized basis sets) are available for practically the whole of the Periodic Table;4*'5~7"80,'35~'39159 see also the Stuttgart home page ( h t t p : // www .theochem. m i  s t u t t g a r t .de); in many cases also SO potentials and CPPs (cf. following section) have been generated. Since the functional form has been chosen to be the same as for the shapeconsistent PPs, these potentials can be used in connection with standard quantumchemical program packages (GAUSSIAN, TURBOMOLE, MOLPRO, ..). The accuracies of the potentials can be judged from a number of atomic and molecular benchmark studies (e.g. refs. 29, 79, 80, 137, 138, 152, 154, 159170). Typical deviations from accurate AE or experimental values are 50.01 for bond lengths, 1% for vibrational frequencies, and 50.1 eV for dissociation energies; for the halogen compounds XH and X2 (X = F, .., At), the overall performance with 7valenceelectron PPs has been shown to be comparable to that of the best available fourcomponent ab initio AE calculations. 1649 171173 . Although information on atomic densities has not been included in the fit, the potentials are also well suited for the calculation of densitydependent properties like polarizabilities ( A ~ = D 1..2% for the raregas atoms NeXe with 8vePP~"~) or van der Waals interaction in Group 12 d i m e r ~ . ' ~Relativistic ~ ~ ' ~ ~ effects are reliably described even in the extreme case of smallcore pseudopotentials for the actinides (with 2943 formally nonrelativistic valence electrons for LaLr).

A
1759176
1.7 CorePolarizationPotentials.  One of the major issues of pseudopotential theory is the question of which physical effects can implicitly be incorporated into the pseudopotentials. While the original intention only was to cover corevalence interaction, it is nowadays considered as an important asset of pseudopotentials that they are capable of describing relativistic effects in a way both economic and efficient (not only indirect effects due to relativistic changes of core orbitals, but also direct relativistic effects on valence electrons). It is natural to ask whether or not a similar thing would be possible with correlation effects. Regarding correlation among valence electrons, it is clear that the situation is not too favourable for pseudopotentials since these effects do not originate in the core region and thus are transferable from atoms to molecules only to a limited extent. Nevertheless attempts in this direction have been and are being made, cf. e.g. refs. 177179, with the idea of fitting specific dynamical atomic correlation contributions into pseudopotentials. Still, such effects can be (and usually are) covered explicitly in valenceonly calculations with current PPs. This is not possible, on the other hand, with correlation effects involving core orbitals  once the latter have been simulated by the PPs, an explicit
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a posteriori treatment is out of the question. Originally it had been hoped that such effects would be constant and cancel out for physical properties of valence electrons, but it has long been recognized now that this is not the case, especially for large, easily polarizable cores. Just for the case of the alkali atoms, e.g., which are often presented as token examples of pseudooneelectron systems in textbooks, the frozencore approximation is insufficiently accurate! What remedies do we have? The "bruteforce" device tried in pioneer days, of incorporating core and corevalence correlation effects into pseudopotentials just by fitting to experimental reference data containing these effects, does not work since the oneelectron/onecenter PP ansatz is insufficient for this purpose, cf. below. Certainly more reliable is a DFT description of core contributions to correlation effects which is possible with (and actually implied in) the nonlinear core corrections discussed in Section 1.4. Another device, which has shown excellent performance in the context of quantumchemical ab initio calculations18' and has later been adapted to PP work (cf. e.g. refs. 139, 181184), is that of corepolarization potentials (CPP)
vcpp
1
= 
2 1
uD,d:
The physical idea behind the CPPs is that core A (with dipole polarizability ~ 1 experiences an enezgy lowering due to (static and dynamic) chargeinduceddipole interactionif1 is the field generated at the site A by valence electrons i and other cores/nuclei ,u (with charges Q,); the cutoff function g1 takes care of the fact that the polarization picture breaks down when the polarizing charge penetrates core A; the parameter d1 in gA is usually semiempirically adjusted (for fixed m),in the spirit of PP theory, e.g. to the ionization potential of singlevalenceelectron atoms/ions. + Note that, due to the quadratic dependence on h, V c p p contains twoelectron and electronothercore cross terms making it more complicated than usual oneelectron/onecenter pseudopotentials. However, these terms factorize and can therefore efficiently be implemented in quantumchemical program packages  they are available, e.g., in MOLPRO ( h t t p : / / www .theochem. m i  s t u t t g a r t .de/molpro/). A distinct advantage of the CPPs is that they do not only account for corevalence correlation but also correct for frozencore errors at the SCF level (static polarization of the cores). Thus, one of the basic restrictions of pseudopotential theory can be removed! In view of this and the fact that corepolarization effects on bond lengths can be of the same order of magnitude as relativistic effects for heavy postd maingroup element^,'^'"^^ routine use of CPPs supplementing relativistic pseudopotentials is expected to become a standard tool in quantum chemistry.
)
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1.8 Concluding Remarks.  The basic aim of pseudopotential theory, as described in this review article, is directed towards getting rid of atomic cores. But what about properties sampling the core region (NMR chemical shifts or spinspin coupling constants, electric field gradients, etc.)? Can they be extracted from PP calculations? The answer to this question (and possible ways of achieving that purpose) depends on the PP approach chosen. With model potentials, the wavefunction calculated is identical, ideally, to the valence part of a frozencore AE wavefunction; it is sufficient, therefore, just to add the core orbitals (which are needed, anyway, in the valenceonly calculation, cf. equation (12)), in order to restore the full wavefunction. The situation is more complicated in PP approaches involving a transformation to radially nodeless pseudovalence orbitals. For calculating and discussing total densities, the simple device of just adding core densities, as in the MP case, may be sufficient. In a better approximation, the nonorthogonality of the pseudowavefunction to the cores is acknowledged, by performing an a posteriori GramSchmidt orthogonalization and calculating coresensitive properties from the resulting wavefunction. The theoretical foundation can still be improved by explicitly taking into account differences of atomic valence AE and PP reference expectation values, by means of augmentation functions similar to that in equation (26), and using these “corerepair” terms for correcting molecular/solidstate pseudovalence properties. 186 In a final step, one might even wish to remove the frozencore approximation  this is possible either implicitly, by using modified property operators in the CPP formalism (Section 1.7), or explicitly, by recalculating core orbitals selfconsistently in the field of the (frozen) valence wavefunction. 1879188 One of the major assets of pseudopotential theory is its versatility. Adaptation to a spectrum of diverse purposes will certainly be increasingly needed in the future. Just to mention some examples: Pseudopotentials will be tailored to the specific demands of theoretical methods (Quantum Monte Carlo, e.g., cf. ref. 189). Within conventional ab initio schemes, different pseudopotentials may be useful depending on whether SO effects are included in a twocomponent formalism from the outset (SCF),19’ or only introduced at the CI level (in a basis of determinants or correlated onecomponent wavefunctions)  with decreasing flexibility of the SO wavefunction, as much as possible of the innershell spinpolarization effects should be parametrized into the SO potentials. Pseudopotentials describing whole atoms or functional groups will be required for embedding; in this case, it may be useful to extend the set of reference data for adjustment, including e.g. scattering data or molecular/bulk properties, cf. refs. 191 and 192 for recent examples. Reference data more or less closely related to the envisaged application field should also be used when pseudopotentials of a simplified (e.g., local) form are required, for certain r e a ~ 0 n s . On l ~ ~the other hand, progress in functionality and efficiency of AE program codes will be paralleled by increasing accuracy demands on pseudopotentials, i.e. lead to development of smallcore PPs with gradually improving transferability properties (maybe, even to fourcomponent PPs eventually’5 5 ) . Coupling smallcore pseudopotentials with largecore CPPs, accounting for correlation contribu
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tions of outercore shells, may well provide a new standard for a both economic and reliable treatment of heavyatom compounds.
2 Applicationsby Element Many applications can be classified by the heavy element, or group of elements, in the compound studied. The data in Table 2 cover the present period.
3 Some Applicationsby Subject 3.1 New Species.  Pseudopotential calculations were used to predict the existence of the new species AuF,19' NUO+ 19' and the first noblemetalnoblegas chemical bonds in the species AuXe+ and x e A ~ X e + .All ' ~ ~these species were later prepared by massspectroscopic methods. 198*199 The first thermodynamically stable diatomic trication UF3+2oo or the new hydride CdHz201are further examples. A large body of pseudopotential work has been done to interpret the vibrational spectra of the matrix isolated species XeH22029203 and other covalent XeH, XeS or XeX (X halide, pseudohalide) bonded systems by the Helsinki group (see Table 2 under Kr and Xe). The matrix spectroscopy group of L. Andrews used pseudopotential methods for assignment of Zr and Hf hydrides MH,, n = 141,204 Group 4 (TiHf) nitrides and dinitrogen species,212Pb Zn and Cd oxides,206alkaline earth oxides207and isocyanides,21 T1 ~~ and Nb, Ta and Re oxides,208Mo and W nitrides and n i t r o ~ y l sIr, ~oxides210 nitrides and din it ride^.^^^ The intriguing prediction of a Hg(1V) compound, HgF4, still remains a pure prediction.21k216So do the predictions for neutral, diatomic PdXe and PtXe.2'7 The pentagonal [XeOFsI anion,218the pentagonal bipyramidal XeF7+ and TeF7 219 and the pentagonal planar IFs2 220 were rationalized using pseudopotential calculations.
3.2 MetalLigand Interactions.  Systematic studies exist on the bonding of various ligands to alkali metals up to Cs(I), to Sr(II), Ta, Group 6 metals up to W, Group 8 metals up to Os, Group 11 metals up to Ag(1) and Au(1); see Table 2 under these elements. 3.3 ClosedShell Interactions.  Numerous studies appeared during the period on closedshell interactions, notably on the metallophilic attraction221between docations, such as Ag(1) or Au(1). Look for the symbol dod0in Table 2. The attractions between compounds of the elements in Main Groups 15 and 16 were also performed.222The largest contribution was found to be the dispersion one but also the ionic contributions (virtual excitations of type A+A', BA' were found to contribute.223
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Table 2 Pseudopotential calculations, classijied by the heaviest element in the in vest igat ion Z
Symbol
Compounds
Ref.
20 22 24 28 29
Ca Ti Cr Ni cu
30
Zn
31
Ga
32
Ge
33
As
34
Se
35
Br
CaCI2+ TiF"+; n = 13 Cr2 Ni2+:Mg0.Electronic spectrum C U ~C, U ~CUH, , CUF,CuCl CUX, cu2x; x = 0Po CUM,C U ~ MCuM2; , M = SiSn (cu2 s)n (PR3)rn (Cu2Se),, (cu~Se),(PR3), CU + CH2N2 C02 + CuH(PH3)2 Ln(Cu202)L, oxyhemocyanin models Zn2 ZnH Ga2P2 GazP, GaP2, their ions Ga3P2, Ga2P3 Ga3P, GaP3 R2GaGaR2 Ge5 GeCP GeCl GeF GeF+ AS^, AS: ASH AsF AsX2, AsXl, X = Cl, Br As speciation in minerals and aqueous solution EH3 effective potential, E = NAs. Se;''; n = 2 or 3 ClSeNSeCl+,a new cation (CU2Se)n, (CUZSe)n(PR3)rn Selenoiminoquinones: Se. 0interaction HBr CaBr2+ ASH+,SeH, HBr+, BrO, BrF+, NaBr+, Brz SO splittings BrO3 PH2Br 2ndorder SO for GeH4,... Br2 KrF, KrF+, KrF2 sr2 SrAr+ MO, OMO, MO2; M = BeSr L+Sr2+; L = R3P=0,amides, pyridine OPR3 complexes of Sr2+,Sr(NO&. R = H, Me, Ph
255 256 257 258 177 178 259 260 26 1264 265 266 267 268 269 270 27 1 272 273 274 275 276 277 278 279 52 280 28 1 282 283 284 285 286 261,264 287 288 289 90
36 38
Kr Sr
290 29 1 90 292 293 294 207 295 296
Chemical Modelling: Applications and Theory, Volume 1
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Table 2 continued
39
Y
40
Zr
41
Nb
42
Mo
43
Tc
44
Ru
Y,, n = 1 4 297 YC,; n = 26 298,299 Zr3 300 Zr5 30 1 ZrF"+; n = 13 256 Zr2 + ethene, butadiene. DielsAlder 302 303,304 ZrXi; X = FI ZrN2L complexes 305 CpzZrCH; as model polymerization catalyst 306 307 (p2n2)Zr(p  q2N2)Zr(p2n2)+ H2. (p2n2) = (PH3)2(NH2)2 (NH&M=NH; M = Ti, Zr. CH4 adducts 308 C12Zr=E; E = MH2 (M = CSn), 309 MH (M = NSb), M (M = 0Te) L,Zr=NX + CH4 310
[email protected] 31 1 NbN 312 Nb,; n = 810 313 314 Nb4C4 Nb5c6 Nb6 c7 NbCO 315 MO, M02, MO;; M = Ti, V, Zr, Nb. Reactions with 316 C2H4, C2H6 317,318 [CpzNbH3] + Lewis acid A A = HB02C2H2, BF3, BH3. H2 elimination MoClS 3 19 MoC4 320 MCO, MCS; M = YMO 32 1 M02 + ethene, butadiene. DielsAlder 322 Solid MoO3. Adsorption of CO, H20 on (100) surface 323 324 M8CI2; M = YMO M02C14(PH3)4 325 MMo(02CH)4;M = Cr, Mo. Mo2&(PH3)4; X = ClI 326 MOO;, MoSi, MoOC14 327,328 M(C0)b; M = Cr, Mo 329 M002X2; X = FI 330 H/D exchange in CpMoH3(PMe3)2 33 1 Mo2C14(PH3)4 models for diphosphine complexes 332 333 [MO2C4(PH3141 L ~ M o A M o LL~ ;= PH3, C1 334 Mo(VI)H40 + PH3, Mo(VI)H202 + PH3 335 336 (NH3)2(SH)2Mo(VI)02 + H20 TcX(NR)3 337 M2(02CH)4, M2Cl4(PH3)4; M = Nb+Tc 338 RUN 339 RuCO 340 M(C0)s; M = Fe, Ru 329 Y ~ ( P X ~ ) ~ R U =alkylidenes CZ~ 34 1 MAr;; M = Nb, Rh; n = 4,6. Ar = argon 342 [ R u ( N H ~ ) ~ ] ~redox + / ~ +pair 343 RuHX(CO)(PR3)2;X = FI, OH, OPh, SPh,... 344
263
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45
Rh
46
Pd
Ru(H)2(co)4 [Ru(PP)2“H3”]+,PP = diphosphine ligand Cp2Ru~(pH)4.Anal. 2nd derivatives [Ru(H.  H)Cp(H2PCHzPH2)]+.Nuclear dynamics Fluxional TpRu(PPh3)“H2SiR3” complexes. Tp = Hydridotris(pyrazoly1)borate Ru(CO)(PH3)2 + benzaldehyde Rh4 Rh5 Rh;, n = 35 RhC RhCliRh2 + H2 Rh3 + CO Rho/+/ + CO, +N2 Rh’ + CH4 Butadiene + Rh,; n = 1 , 2 Benzene + Rh+, R h l CpRhCO + CH4 Cp*Rh(PMe3) + thiophene C02 + [RhHz(PH3)3]+ or RhH(PH3)3 RhXL + CH4 [HRh(CO)4] + CO + H2 [RhCl(CO)3]2. 88 [CPRh(C2H4)(q2c2Hdlf “Rh2 CP2(PCH2)212(PS4)I2+ M2(HNCHNH)4, M2(HNNNH)4; M = Nb+Rh RhCl(PH3)Z as activator of SiH, SiSi and CH bonds [Rh(PH3)2]+ complexes of Nalkenylamides Rh(CO),, Rh(C0)zF:Rh6( c o )16 Pd2 Pd,; n = 16 Pd,; n = 16 + S, C1 PdH MH, MO; M = Ni, Pd PdC Pd3 + H2 Pd, + H2, n = 2,3 PdMgO, PdOMg PdCO, PdCH3 Pd(C0)2, Pd(C0)2FiM(CO)4; M = Ni, Pd Pd(NO)(CH3)(PH3) + CO substitution reaction M + C2H2; M = Y+Pd MX2; M = Y+Pd; X = H, F, C1 MH2, MHX; M = Y+Pd; X = F, C1 MH, + H2, CH4; M = Y+Pd PdC12(H20),, PdHCl(H20)n CH4 + M’; M = Y+Pd Pd2 + CO Pd2, PdCu or Cu2 + CO or NO Pd2 + C6H6 adsorption model PdC1:
345 346 265 347 348 349 350 35 1 352 353 354 355 356 357 358 359 360 36 1 362 266 363 364 224 365 366 338 367 368 369 235 370 37 1 372 373,374 375 376 377 378 379 380 369 329 38 1 382 383 384 385 386 387 388 389 390 39 1
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Table 2 continued Z
Symbol
47
Ag
48
Cd
49
In
Compounds
Ref.
Pd(H)2(Cl)(NH31 Pd(H)2(NH3)2 Pd2(pBr)OLC3H5)(PH3)2, PdCl(t13C3Hs)(PH~) M=CH$; M = Y+Pd [L2MCH3]+ + ethylene reaction q3Allylpaiiadium( + ) complexes, nucleophilic attack on Pd acetate catalyzed Wacker reaction Pd acetate catalyzed acetoxylation of ethene Solid (Me4N)[M(dmit)2]2; M = Ni, Pd “dmit” = 2thioxol,3dithion4,5dithioiate Ag2 Agn Ag,+
392 393 363 394 395 396 397 398
399 179,400402 403,404 4% 405 Agn; n I9 406 Mn; M = Ru, Pd, Ag; n = 55,135,140 407 AgF 4084 10 Solid AgX; X = CiI 41 1 AgHe 412 HeAgHe 413 AgNH;/+‘ 414 MNO2; M = CU,Ag Ag(CN);; q = 1,2 415 416 Ag2L; L = NH3, CO, C2H4, H20 417 M’L; M = CU,Ag; L = H20, OH, CO Ag+L for 18 ligands 418 419 Ag(C6H6):; n = 1,2 Ag+ + benzene, other hydrocarbons 420 (Ag+), + butadiene complex, n = 1,2 42 1 422 Ag2 + olefin + 0 , 0 2 390 A& + C6H6 [A&(p4E)I2+,its diphosphine complex. E = STe 423 424 [Ag(NHCHNH)12, [Ag(dmtp)(NO3)12. 425 M(Mg13012)surface model; M = Rb, Pd, AgAg2/“AgBr7’ 426 MgO( 100)/Ag(100)and MgO( 110)/Ag(110) interfaces 427 Solid DCNQIM; M = Cu, Ag; 428 DCNQI = dicyanoquinonediimine Cd2 429 206 MO, MO2; M = Zn, Cd Cd(en), CdNH3 exciplexes. “en” = ethyienediamine 430 43 1 Cd + CH4 Cd+ + C6H6 420 M2+L;M = Zn,Cd; L = H20, OH, CO 417 432 M2+:M’O; M = Cu, Ag; M’ = MgSr. Impurities 433 InR, (InR)4; R = H, Me In2P2 434 In3P2, In2P3 435 InzP, InP2 436 In3Sb2, In2Sb3, their cations 437 InzAs;, n =  l,O, + 1 438
5: Relativistic Pseudopotential Calculations, 1993June I999
50
51
Sn
Sb
InH, InF, InCl InC1, InC13 InMH6, MBH6; M = €3In X3MD; M = A1In; X = FI; D = YH3, YX3, X; Y = NAs. Donoracceptor compl. Solid CSn. Cohesive energy SnH SnF+ SnF Sn5 Mf, M = Ge, Sn MH3 + MH4; M = SiSn MH4; M = CSn. Infrared intensities M2, CUM,CuzM, CuM2; M = SiSn MOH+, HMO+; M = CSn Sn[E(SiH3)2]2; E = N, P SbO SbH SbI 332
52
Te
53
I
Ozonelike, Sb: compounds Sb4 and isoelectronic species In3Sb2, In2Sb3, their cations Solid EM semiconductors; E = €3In; M = NSb. Cohesive energy Sb speciation in sulfidic solutions SbF SbF2, SbFi SbX2, SbXi (Sbnom)', 4 = 0, 1 CH3EH2, CH2EHi; E = NSb H2Te MF6, M = Se, Te MCl6, M = Se, Te Tech CH2EH+; E = STe CH2(EH)2; E = 0Te HTeCH2CHO [C(ER)#; E = 0Te X2Y2; X = NSb; Y = 0Te. Planar or butterfly? CH3 + CH3EH; E = S T e C6H&Te HESi(SiH3)3; E = 0Te. Photoelectron spectrum 12, I,, 13, IF I;; n = 3,5 X2; X = FI X2, HX; X = FI HX, XY, X2 complexes with NH3. X, Y = FI IF X l , XY;; X, Y = FI Solid MX; M = Li ... Rb; X = FI NanXqX = ClI; q = 0,+ 1. n = 13, rn = 12 Ca12+m' CUX; X = FI
265 79 161,439 440 441 442 443
444 445 446 447 448 449 259 450 45 1 452 453 454 455 456 457 437 458 459 460,461 462 463 464 465 466 467 468 469,470 465 47 1 472 473 474 475 476 477 478,479 480 137 164 48 1 482 483 484 485 289 486
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Table 2 continued ~
2
54
Symbol
Xe
~
~~~
Compounlis
Ref.
GaX2, GaX3, X = ClI HI, CH31 Pseudopot. vs. allelectron on Br, I compounds. G2 set C&, CXT; x = c1I CD3X; X = FI CF3X; X = ClI w3); XN3; X = H, FI ICN, INC HOX; X = FI XO;; X = C1I; n = I 4 H2CI2 isomers PI2, PIT P2If BX2, BX;, X = Br, I W3) PdCH2I' IF,, IF:AlX,; X = FI. A1 NMR shift SiX4, Six13 Si NMR shift Ti&; X = FI. Ti NMR shift NbX;, NbClsX; X = FI. Nb NMR shift IOF~ IOF;, IF7, XeFTeF,, IF7, XeF,5 are D5h XON02; X = FI CH2X'; X = FI ICH~I Reactions of iodolithioethenes C6H4X (benzyne t halide); X = FI C6H612complex RgH+; Rg = ArXe RgH; Rg = NeXe XeXq; X = FI; q =  1,0, + 1 XeO, XeS RbXe+ (Rg)2; Rg = NeXe (Rg)2; Rg = Kr, Xe RgNO+; Rg = HeXe RgAu+, RgAuRg+; Rg = HeXe RgBeO; Rg = ArXe HXeH RgF2; Rg = Kr, Xe XeF4 [XeOFsIXeHXe+ HXeCl HXeSH HKrCN, HXeCN, HXeNC
487 488 489 490,491 492 493 494 495 496 497 498 499 500 50 1 502 503 504 220 24 1 242 245,505 245 506 507 219 508 465 509 5 10 51 1 512 513 514 515 516 517 518 53 519 197 520 202,203 52 I 522 218 523 524 525 526
267
5: Relativistic Pseudopotential Calculations, 1993June 1999
55
cs
56
Ba
57
La
58
Ce
62
Sm
64
Gd
67
Ho
FXeX; X = N3,NCO,OCN F2C=CXe HXeX; X = ClI. HKrCl RgHX+; Rg = ArXe; X = ClI CsFsRg+; Rg = HeXe XeH2H20 dihydrogenbonded complex MH; M = LiCs MF, M = KCs ML; M = LiCs; L = H, Me, NH2, OH, F, Cp M,Xm; M = LiCs; X = Se, Te, n,rn 5 2 M,Xm; M = LiCs; X = As, Sb, n 5 2, m 5 4 CsRg; Rg = NeXe M+CO; M = LiCs M+L; M = LiCs; L = adenine, guanine Ba2 Ban, n = 214 Solid Ba BaLi BaRg; Rg = HeXe BaH+ Ba12+ MX2; X = H, F; M = CaBa MX2; M = Sr, Ba; X = FI MCN, MNC; M = BeBa M(CN)2, M = BeBa M2H4; M = MgBa M2+L;M = MgBa; L = adenine, guanine Hydrated M2+ + adenine, thymine base pairs; M = MgBa Lao, Lao+ LaX3, X = F, C1 LaX3; X = ClI MFi; M = Sc, Y, La Cp2 LaL Lac2 LaC,+;n = 12,13 Lac,; n = 26 Lac,; n = 28 LaC2, Lac,+ Lac!+; q = 13 MC,; n = 36; M = Y, La
[email protected] CeO
[email protected]/+' CeX,, n = 3,4; X = F, Cl NaCeCl4 Sm(II1)catalyzed olefin hydroboration Ethylene insertion into SmC bond of H2 SiCp2SmCH3 [W H 2 Oh13+ M(C&&j)2; M = Y, Gd . Gd(II1) polyamino carboxylates CF bond activation by Ln+; Ln = Ce, Ho
527 528 529 530 53 1 532 155 80 533 534 534 535 536 537 538 539 540 54 1 542544 545 546 547 548 21 1 549 550 537 551,552 553,554 555 556 303 557 558 559 560 56 1 562 563 564 565 42,54 566 567 568 569 570 57 1 572 573 574
Chemical Modelling: Applications and Theory, Volume I
268
Table 2 continued Z
Symbol
Compounds
Ref:
70
Yb
Yb2 LnO, LnF; Ln = CeYb YbF YbH,Y bF,YbO L+Ln3+, Ln = La, Eu, Yb; L = R3P=O, amides, pyridine Ln(C8Hg)z; Ln = Nd, Tb, Yb. Excited states Ln(C8H8):; Ln = Ce, Nd, Tb, Yb. Excited states LnH, LnF, LnO; Ln = La, Lu. Lanthanide contraction LnX3; Ln = LaGd, Lu; X = F,C1 LnX3; Ln = La, Gd, Lu; X = FI LnX3; Ln = LaLu; X = F, C1 LnX3; Ln = CeLu; X = FI Ln(C6H6)2, Ln = LaLU
[email protected]();M = Sc, Y, LaLu
[email protected]; M = Sc, Y, LaLu MH,; M = Zr, Hf; n = 14 MC14;M = TiHf. Vibr. freq. HfXi; X = FI. Salts MN, NMN, M(N2), M(pN)zM; M = TiHf MMe2C12; M = TiHf. Bent's rule for bond angles [MMe6I2; M = TiHf. Nonoctahedral Cp2M=E; M = TiHf; E = 0Te H2(X)MNH2 + H2M=NH + HX; M = TiHf; X = H, C1, Me, NH2, SiH3 H3MEH3, H2M=EH2; M = TiHf; E = CSn [MCp(C0)4]; M = TiHf 13CNMR shifts M(C6&)2; M = TiHf
[email protected] [email protected] MO; M = VTa TaL5; L = H, C1, Me TaF; Ta + CO Ta+ + CH4 TaC, TaC+ TaC+ TaC!; n = 713 MN,; M = Nb, Ta; n = 13 [MMe6]; M = VTa. Nonoctahedral [Ta(OH)zL'(H2)L]+;L = PH3; L = F, Br, I, CO, CN, ... M(NH2)( =NH)2; M = VTa imido adducts of CH4 C13Ta=E; E=MH2 (M = CSn), MH (M = NSb), M (M = 0Te) [CPZTaH31 [MCp(C0)4]; M=VTa 13CNMR shifts
159 78 575 576 295
71
Lu
72
Hf
73
74
Ta
w
w3
wL6; L = H, Me WXs; X = H, F
577 577 175 578 579,580 58 1,582 583 584 585,586 587 204 588 589 212 590 59 1 592 593 594 595 596 597,598 599 60 600 303 60 1,602 603 604 605,606 607 213 59 1 608 308,609 309,592 317 595 610 600 61 1
5: Relativistic Pseudopotential Calculations, 1993June 1999 WH6 WMe6 [MMeb]; M = CrW. Nonoctahedral WF4 triplet state MOzC12; M = CrW. Vibr. freq. M3X4+;M = Mo, W; X = 0, S M20!; M = CrW Solid W03 M + CO;M = Mo, W M(CO)6, M = CrW M(CO)6, M = CrW M(CO)6, M = CrW M(C0)b; M = CrW. HOMO electron density w3
75
Re
W(C0):; n = 16 WC14L, WClsL, W(C0)sL; L = acetylene, ethene [C~~MEN],; M = Mo, W; n = 16 M(C0)sCX; M = CrW; X = 0Se M(C0)sL; M = CrW; L = CO, SiO, CS, N2, NO+,CN, NC, HCCH, CCH2, CH2, CF2, H2 W(CO)sNH3 MN, NMO, MNO*; M = Mo, W MO;', MN2, MP2; M = Mo, W WC2H,f Cp2M, CpzMO, Cp2MCl(OSiH3); M = Mo, W [MCpMe(C0)3]; M = CrW 13CNMR shifts [M(CO)sL]; M = CrW; L = PH3. PX3 31PNMR shifts [Cp(CO)2M]2(pE);M = Cr, W; E = STe [WL2(pLCR)]2; L = H, Me, F, OH; R = H, F, Me Lowvalent (Fischer) carbenes [(CO),W=CH2], ... highvalent (Schrock) carbenes [&W=CH2], ...; X = FI Lowvalent (Fischer) carbynes [F(C0)4WrCH], ... highvalent (Schrock) carbenes [X3WECHI, ...; X = FI M(=NH)3; M = Mo,W imido adducts of CH4 W(OH)2(=NH) + CH4 Cl4W=E; E = MH2 (M = CSn), MH (M = NSb), M (M = 0Te) [(N3,N)WL], L = C1, PBH3, =E; E = NSb. (N3,N) = N(CH2CHzNSiMe3)3 M02X2; M = CrW; X = FBr W(C0)3(PH3)2(H2). Classical or nonclassical? M(CO),(PH3)5,, + H2; M = CrW; n = 0,3, 5 [CP~MH~]';M = Mo, W &M(HCCH); M = Mo, W; X = F, Cl vinylidene rearrangement M(C6H6)2; M = crw [W(oH)2(C&)(Ph3)] + N2H2 4e reduction HM(C0)5, M = Mn, Re ReH ReN, ReN2 M + CO; M = TcRe, Ta
269 612 613 59 1 614 588 615,616 617 618 619 620,621 622,623 624 625 610 626 627 628 629 630 63 1 209 632 633 634 595 243 635 636 637 638 308 639 309,592 640 592 64 1 642 317 643,644 596 645 646 647 213 602
Chemical Modelling: Applications and Theory, Volume 1
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Table 2 continued ~~~
76
0s
77
Ir
Addition of LRe03, L = 0, C1, Cp, to C2H4 648 Epoxidation of ethene by MeRe(O)2(02), 649 MeRe(O)(O2)2 + water M03X; M = MnRe; X = FBr, Me 592 [Cp(CO)2M]2(pE);M = Mn, Re; E = STe 635 L4ReAReL4 650 Re(0) acetylides, ReOCl;, ReOFS 651 [(c6H6)M(cO)3]+: M = MnRe 652 [MMe#; M = Tc, Re. Nonoctahedral 591 Lewis basicity of Clz(PH3)3ReNL complexes 653 Os04 588,654 Os04, OsO3F2, ..., OsFS 655 Os04(NR3 656 osco 657 Os(C0)4H2, metallacyclophanes 658 M(C0)5, M = FeOs 659 M(C0)5, M = FeOs 624 M(CO)s, M(C0)4(C2H2); M = Fe0s 660 [MnC(CO),]q; M = Fe, Rh, 0 s interstitial carbides. 237 13CNMR shifts HzM(CO)4, M = Fe0s 646 O S H ~ X ( P H ~X) ~ =;C1, I 661 MH+, M = FeOs 662 M(C0)5, M = FeOs 623 M(C0)4(C2H4), M = FeOs 663 cisOs02F4 664 MEX;; M = Mo, W, Re, 0s; E = N, 0;X = F, C1; 665 n = 35 0s2cl8,0s2C1i666 Os04catalyzed dihydroxylation of olefins 667 Cp2M; M = Fe0s 668 CpM(C0) + CH4; M = Ru, 0 s 669 Bis(dihydroquinidine)3,6pyridazine~OsO~catalyzed 670 dihydroxylation of styrene 671 [M(SiR3)H3(PH3)3]; M = Ru, 0 s [oS(NH3>4(oAC)(H2)]+ 672 [Os(NH3)4L(q2H2)]q+:HD spinspin coupling 673,674 M3(C0)9(pH)3(~3CH);M = Ru, 0 s 675 OsC12H2(PPr;)2 645 trans[LM(H2PCH2CH2PH2)2(q2H2)In+; 676 M = Ru, 0 s ; n = 1,2;L = H, CH,, F, CF,, CN, C1, Br, CO, NCH, NH3, PH3. Dihydrogen acidity O S H ~ X ~XL=~ C1I; ; L = PH3 677 OSH3(BH4)(PR3)2 678 [Os(PR3)3“Hs’]’ 679 [Os(PR3)3Hq] and other [ML,(H.  .H)] 680,681 H2 addition to Ir(PR3)2(CO)X 682684 MOC, MCO, M = Rh, Ir 685 IT3 686 IrC 687 IrO, OIrO, Ir(02), (02)Ir02, Ir20, 11202 210 112+ H2 688
27 1
5: Relativistic Pseudopotential Calculations, 1993June 1999
78
Pt
Ir+ + CH4 YIrC, YIrC2 M(CO)E, M = Cr+Fe, Tc+Ru, Hf+Ir CpM(PH3)(CH3)+ + CH bonds. M = Rh, Ir CpM(PH3) + CH bonds. M = Rh, Ir CpML; M = Rh, Ir; L = CH2, CO, SH2, PH3 CpM(C0) + CH4; M = Rh, Ir transIr(Cl)(PH3)2 + C&; X = FI Ir( trans(PH3)2)(CO)(iPr2SiO) cis[M(CO)212] + CH31 IrXH2(PR3)2; X = ClI; R = H, Me Ir(PH3)2(X) + CH4; X = H, C1 IrCiH2(PH3)3 [Ir(H)2(PR3)21+ w 3 (PH3)CPl+ Octahedral LSM(CH)=CHR alkenyl complexes. M = Ru, Re, Ir PtH. Meanfield spinorbit PtH, PtH+, Pt2, Pt2H MH; M = La, Hf+Pt PtH, PtOo/+',PtCHT. Spinorbit MXe, M = NiPt Pt,; n = 212 Pt3, Pt4 M4, M = Pd, Pt Pt,; n = 26, 13 M13; M = Pd, Pt Pt13Hn;n 5 20 ZrPt3 MO; M = NiPt PtN MXe; M = NiPt P ~ ~ APtAu u, M3 + H2; M = Pd, Pt Pt3/0, Pt4/O adsorption models PtSn + H2 PtA1, Pt/A19, Pt4A14 S/PtI2 PtL, ZrL, PtZrL, ZrPtL; L = H,CH3 PtCN, PtNCPt + co Pt(C0)3, Pt3(C0)6 M(C0)3L; M = NiPt; L = Co, SiO, ... M(CO)4, M = NiPt Pt + H2 H2 + M, M = Re+Pt PtCH2f PtO*+'+ benzene M + CH4; M = Re+Pt M, M2 reacting with H2, CH4; M = Pd, Pt CH4 + Pt' PtCH2f CH4 + Pto'+/CH4 + Pt,; n = 110
689 690 69 1 692 693 694 669 695 696 697 698,699 700 70 1 702 703 704 57 705 59 706 217 707 708 709 710 71 1 712 713 60 714,715 217 716 377 717 718 719 720 72 1 722 723,724 725 630 623,624,659 726 727 728 729 730 73 1 732,733 728 734 735
Chemical Modelling: Applications and Theory, Volume I
272
Table 2 continued ~~~
79
Au
~
Pt2 + co 736 737 Pt + C2H4 activation reaction PtCH: + NH3 738 739 Pt+L; L = glycine, formate, formamidate 740 MO+ + CH or CC bonds. M = Os, Pt [PtC12(C0)2]2 dimer: 88 interaction 224,741 PtI2 742 Pt2(PH3)4(pS)2. Complexes with Ga(III), In(III), 743 Tl(I), Pb(I1) 744 [Pt(NH3)2I2+interaction potential with H2 0 q2C60Pt(PH312 745 I3CNMR shifts in Molefin complexes, M = Cu, Rh, 746 Ag, Pt 747 C59M; M = Ir, Pt [Pt(NH3)3(adenine)12.Force field determination 748 Pt(PX3)2; X = H, F. The nature of the PtP bond 749 750 M(PH3)z + BX2BX2; M = Pd, Pt; X = H, OH 75 1 Pt(PH3)2catalyzedhydrosilylation of ethene Pt(PR3)2catalyzed alkene, alkyne diboration. Pd/Pt? 752, 753 M(PH3)2 + SiX bonds; M = Pd, Pt; X = H, Si 754 PtC12(PH3)2 + SnC12 755 Pt(H)(PH3)2 (SnC13)(C2H4) 756 757 M H ( V ” ~  C ~ H ~ ) ( PMH= ~ )Pd, ; Pt DiimineM(I1) complexes (M = NiPt), and their 758 zirconocene complexes in olefin polymerization Pt(PR3)2 + H2 759 PtL2 + HOCH3. L2 = (C0)2, (PH3)2, diphosphine 760 PtH(SiH3)(PH3) + C2H2 76 1 PtH3(PH3);. Dihydrogen complex 762 669 CpM(C0)’ + CH4; M = Pd, Pt [M(PH3)2]nC60;M = Pd, Pt; TI = 1, 2, 6 763 M(PH3)zXz; M = NiPt 663 Pt(II)(OOH) complexes 764 MM’(PH3)4; M,M’ = Pd, Pt. dlod’O 765 Ptz(dta)4X2; X = BrJ. “dta” = CH3CSy 766 [Pt2Cl~(C0)4], [Pt2C14(CO)2l2. 88bond 767 MM2 force field derived for Pt(I1) square planar 768 complexes Pt(PH3)2(R), R = olefin (C2H4, C8Hl0, ..., CllH16) 769 (X3P)2Pt=PR phosphinidenes 34 1 P ~ ( P ( ~  B u+) ~H2, ) ~DielsAlder reaction of 770 acrolein + isoprene, etc. (C2H4)MClX(NH3)3,; M = NiPt; x = 13 77 1 M[(CHNH),]R+ + ethene; M = NiPt; R = Me, Et 772 [Pt(NH3)2L2I2+;L = C1, H2O 773 774 “Carboplatin” versus “cisplatin” trans[(CH3NH2)2PtC12] + adenine, thymine 775 Au,; n 5 75 403 Auqn= 14;q=  l , O , + 1 776 MG’; M = Rh, ... Au 777,778 MH; M = CuAu. Comparison between methods 779 A u ~AuH. , Spinorbit effects 190
5: Relativistic Pseudopotential Calculations, 1993June 1999 AuF, AuF', AuF2, A u ~ F ~ MF;; M = CuE111; n = 2,4,6 AuCl MI, M2I+, M312+,M413+;[I(MCI),] (m= 1 4 ) ; M = CUAU AuI;; n = 2 , 4 RgAu+, RgAuRg+; Rg = HeXe Au=C+. Do other Au=L exist? SAu:, S(AuPH3)'. d"d" (XA;PH3)2; X H, FI, CH3, CCH, CN, SCH3 dl 0d' 0 (CIAUPH~),isomers, n = 2,4 MzSe, M2I+, M = Ag, Au H~PAuCECAUPH~ complex with CHC13 MCO; M = CUAU M(CO)+; M = Ag, Au MCO'; M = CUAU AuCO+. Accurate ClM(CO), M = CUAU M(CO)$, [M(C0)2F2]; M = CUAU M(CO), ;MCN, M(CN),; M = Ag, Au; n = 13 MR; M = CuAu; R = Me, Ph MBH4; M = CUAU Au speciation in aqueous solution Decomposition of Au(1) compounds AuSH, Au(SH) + H2S, H 2 0 Cyclic [Se,Au&, n = 5 , 6: d"d" (M&; n = 1,2; M = Ag, Au: dIodo [ X A U P H ~X ] ~= ; H,Cl P ~ ~ APtAu u, Solid MCl; M = Ag, Au. dod" M=CHl, MCH:; M = CUAU MCH;; M = La, ...Au ClM=M'R2; M = CuAu; M' = CGe MCH;; M = ScCu, La, HfAu C6H6 M+; M = CUAU C2H4 + M+; M = CUAU H20 + Au+ M+(H20),; n = 1  4 AU;; n = 115 + CH3OH [Me3PAu]+ + alkyne + alcohol [E(AuPH3)4]+;E = NAS. dl'dlo X(AuPH3)?+; X = C, N, 0, P, S; n = 16. [HC(AuPH3)4]+.d"d'O [CH2(PH2AuPH2)2CH2I2+ [M2(H2PCH2SH)2I2+;M = CUAU Rings [M2(PH2CH2PH2)2I2+,[M2(NHCHNH)2], [M2(SCHS)2], [M2&I2; M = CUAu; X = ClI [Au2(PH2CH2PH2)2]C12.[Au2Te4I2. d0do A u ~ X ~ C ~ H ~ (XP = HC ~1)I ~.;Cis has dlodlo C1,AuSMez; n = 1, 3. Inversion at sulfur [ C ~ ~ T ~ ( C I C H ) ~ ] M"tweezers"; CH~ M = CuAu
273 780,781 782 783,784 785 480 197 786 787 221,788,789 790 79 1 792 793 794 62 1 795 796 797 369 798 799 800 80 1 802 803 804 805 223 716 806 807 808 809 810 81 1 812 813 8 14 815 816 817 818 819 820 82 1 822 823 824
Chemical Modelling: Applications and Theory, Volume 1
274
Table 2 continued (Et3P)Au(2thiouracil) AuPt, A u ~AuHg' , AuPt, + H2; n = 1 , 2 [(AuPH3)6Pt(PH3)l2++ H2 Au(C2 H4) [Au(III)(C6H4S2)21Au'L; L = H20, CO, NH3,C2H4 ,...
80
Hg
ClMPH3, MPH:; M = CuAu M+L; M = CuAu; L = adenine, guanine M+L; M = CUAu; L = C2H4, N2, HCN, C2H2, N20, FCN, HCNO, HN3, HCCF, CH2N2, CH3CN, C2F2, FCNO, CH3CNO [M2(PY)(PXR)L4],[M2(CLXRhL41; M = RhPd, IrAu; X = 0Te; Y = C1, S: 88 [Au(CNHCH =CHNH)2Cl2]+, [Au(CNHCH =CHNH)2]+ carbenes AugSCH3 A u ~ ~ / HHSCH3 ~S, M16/SH, SCH3; M = Ag, AU AU38(SCH3)24 Hg2 Hg, M2; M = ZnHg M,, n 5 6, M = ZnHg Mi+, M = ZnHg. Their compounds AuHg+ HgCd HgH HgH, HgH+ MH; M = La+Hg HgX2; X = C1,CN HgX, HgX2, Hg2X2; X = H, F, CI, CH3, CF3. Hgz; q = o,+ 1 , + 2 MR2; M = ZnHg; R = Me, Ph Photodecomposition of Hg methyl complexes HgI2 HgF4 HgX,; n = 2,4; X = F, Cl MO:; M = IrHg. Extreme oxidation states (d0/2/4). H F only (HgX2)2; X = H, FI MeHg+, MezHg, MeHgX (X = ClI), MeHg(PH3)+, EMeHg(PH3)31+ RHg, RHgH Solid 2HgSsSnBrz (HgPMe),, n = 4  6 , 8 , 1 2 M(CO)i+, [M(C0)2F2]; M = ZnHg M2+L;M = CdHg; L = adenine, guanine Hydrated M2+ + adenine, thymine base pairs; M = ZnHg
825 170 826 827 828 829 813,831 199,830,832 833 537 834 835,836 837 838 839 840 84 1 163, 780, 842,843845 846 174 158,847 848 170 849 850 152 43 851 852 799 853 742 214,215 216 854 855,856 85 78 59 860 86 1 862 369 537 551,552
5: Relativistic Pseudopotential Calculations, 1993June 1999
81
T1
82
Pb
Hg' (3P1) + H2, CH4, C3H8, SiH4 FHgX + CH4 T12 Tl2, T1H. Spinorbit effects TlH TlH, TlH3 MH, MH3, M2Hb; M = In, T1 M2H2; M = BTI TIC1 TlX, T1X3 MX3, MH2X; X = FI; M = BTI TlX;, TlX,f4H20;X = C1, CN TlAr MOT; M = In, T1 M202; M = AlT1 M2O3; M = GaTI T12E$; E = Se, Te T1MTe:TlCp, TlCp, Tl2 Pt(CN)4 TlN(SiMe3)z H3MEH3; M = BTI. Uses a molecular EH3 effective potential, E = NAs Pb2 Pb5 Mg; M = Sn, Pb Pbn; n = 314 pbil4MH2, MH4; M = SiPb, MO; M = GePb MH2; M = CPb MH;, MH4, MH,; M = SiPb MH4; M = CPb PbnHz+' M4H4; M = CPb PbH2, PbH4 Aromatic or polyhedral compounds with Ge, Sn, Pb skeletons EM6; E = CPb; M = LiCs MnPbm; M = Li, Na; m = 1,4 MH4; M = Sn, Pb. Lowenergy electron scattering MH4, MH3Me; M = GePb MH;; M = SiPb MMezC12; M = CPb. Bent's rule for bond angles M3H;; M = CPb PbBr PbI MF2; M = GePb PbI2 Solid Pb(N3)2 MXT, MH2X'; M = CPb; X = FI PbO. Dipole moment Solid PbS MX2, MX;; M = GePb PbnOm;n = 1 4 ; rn = 14
275 863; 864,865 866 190 867869 870 87 1 872 873 874 875 85 1 157 208 876 877 878 879 880 225 88 1 284 882 883 884 885 886 38 887 888 160 889 890 870 891 892,893 885,894 895 896 888 590 897 898 899 900 742 90 1 875 902 903 904 205
Chemical Modelling: Applications and Theory, Volume I
276
Table 2 continued
83
Bi
84
Po
85
At
PbR,&,, PbR,X2,. Why org. Pb(IV), inorg. Pb(II)? MH3Cl; M = CPb. “Chargeshift bonding” M2Hs; M = SiPb MzEg; M = Sn, Pb; E = STe M,Pb,; M = LiK; rn = 27; n = 1 , 4 H2M0, Me2MO; M = CPb H2M=MH2; M = CPb H2M=MH2, H2MO; M = SiPb Group 14 metalloles: C4H4MH2, C4H4MH*, C4H4M2, ... MC bonds; M = GePb. H3MY; Y = H, A, ... ABCDH3 TolueneXY; complexes; X = CPb; Y = H, C1, Me H3M’MHl, H3MY bond energies, M, M‘ = CPb. 68 ligands, Y (Me2NCHz)z adducts of MCp2, M = Sn, Pb Biz M4, Mg; M = PBi Bi3+ M!+; M = Sb, Bi BiH MH; M = T h B i MH3; M = AsBi MH,, MF;; M = PBi MH5; M = PBi MF5, MH;, MF;, MH,, MF,; M = PBi BiF BiI BiN BiO BiS RM=MR; M = AsBi MF3; M = NBi MF:; M = AsBi (RhBi7)Br. 5c4e bonding MMe5; M = Sb, Bi BiOH, HBiO PMH2; M = AsBi Bi3O;, BisOt [HzEEH~]~; E = AsBi. Chain fragments Bi2H4. (C4H4Bi)2. Chain fragments. HFlevel H,M=CH,; M = NBi. Their reactions M3; M = SePo M2H2; M = Tl+Po [HEEHIz; E = SePo. Chain fragments EF;; E = SePo CUX, cu2x; x = 0Po MH; M = Tl+At X2, XH; X = FAt HX, HX+; X = FAt (HX)2; X = FAt
905 906 162 879 907,908 909 910 91 1 912 913 914 915 916 917 918 919 884 920 89 92 1 922 923 924 925 926 927929 930 93 1 274 932 933 934 935 936 937 938 222 939 940 94 1 889 222 942 178 39,943 164 39 944
277
5: Relativistic Pseudopotential Calculations, 1993June 1999
86
Rn
88 90
Ra Th
92
U
93 94
Np Pu
103 105 111
Lr Db
112 113 114 117 118
CH3X, C&; X = BrAt XF3; X = C1At XF,; X = C1At Rn2 Xe2, XeRn, Rn2 FeRg+; Rg = ArRn CoRg+; Rg = ArRn RgF6; Rg = ArRn M2+(H20),; M = MgRa; n = 16 Tho Tho+ M(CgH8)2; M = Ce,Th C12Th(CH2)PMeC)H2) UF"+; n = 13 u o + , uo;+ UO;+, HOU02+,U(OH):+ NUO+ isoelectronics, u O ~ ,U O ~  ,U F ~ , (OUO)2+L,; L = CO:, NO;, n = 3; L = F, n = 26 NUO+, NUS+ [U02&l2; X = OH, F, C1
889 945 942 946 947 948 949 942 950 153 95 1 952,953 954 200 955 956 196
198 957,958 uc14 959
[email protected] 960 96 1
[email protected]; An = Pa, U 962 Cp3AnL; An = Th, U; L = Me, BH4 OPR3 complexes of UO;+, UOz(NO3)z. R = H, Me, 296 Ph U(C8&)2. Excited States 577 963 An(NH&An = U , N p AnF6; An = U+Pu 96,958 964 (AnFb),,; n = 1,2; An = U, Pu 965 An02+;An = U, Pu. Their nitrates, sulfates 966 AnOl+; 3 An = U, Pu A n 0 +.nHzO; An = U, Pu; n = 0,46 967 PuO$ excited states 968 584 An(C6&)2; An = Th, U An(CsH8)2; An = uPu 960 AnH, AnF, AnO; An = Ac, Lr. Actinide contraction 175 151 MO; M = Nb, Ta, El05 165 MH; M = CuEll1 782 MF;; M = CuE111; n = 2,4,6 156 (El 12)H+, (El 12)F2, (El 12)F4 (El 13)H, (El 13)F 969 91 MH; M = SnEl14 970 MX2, M&, M = CEll4, X = H, F, C1 (El 17)H 969 97 1 XH; X = BrEll7 972 RgF,; n = 2,4; Rg = Xe, Rn, El 18 91,973 RgF4; Rg = XeEll8
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Because the crystalfield splittings are large, the interactions between d 8 ions are qualitatively analogous.224 Mixed cases, like the s2d interaction in T12Pt(CN)4 have been studied using pseudo potential^.^^^ A classical problem is the attraction between two raregas atoms; see Table 2 under Xe and Rn. Gradually improved results were obtained for the weakly bound Hg;! and the larger Hg, clusters; see Table 2 under Hg. For a general review of these closedshell interactions, see ref. 226. 3.4 Chemical Reactions and Homogeneous Catalysis.  Reaction mechanisms are one of the most important applications for pseudopotential calculations. For studies during the title period, see Table 2 and look for the symbol “ + ”.
3.5 Chemisorption and Heterogeneous Catalysis.
 Small cluster models of metals or ionic crystals have been used in this area; see Table 2.
3.6 Other.  Pofarizabifities,CI and hyperpolarizabilities, y of MO: species, up to 0 ~ 0 4 were , calculated by Cundari et af.227 Structures of a large amount of different molecules without a chemical Leitmotiv were studied using TroullierMartinstype pseudopotentials and DFT by Chen et af.228For a study using model potentials, see ref. 229. Molecular shapes of metal alkyls and hydrides, ML,, were discussed at Valence Bond level by Landis et al.230 Vibrational frequencies of numerous inorganic molecules were studied by Bytheway and W ~ n g and ~ ~Cundari ’ and R a b ~ . * ~ ~ N M R chemical shifts can be calculated for light elements, treated at allelectron level, while the heavier elements in the system are treated using pseudopotentials. Examples exist for 1H,233 13C,233238 27~1,241 29si 242 31p 243,244 ~ i 2 4 5and Nb.245 In particular, the spinorbitinduced “heavyatom chemical shifts” can be incorporated. 17~,235923692397240
7
7
Acknowledgements.  This study is supported by The Academy of Finland. H.S. is grateful to M. Dolg (Dresden) for a longstanding cooperation on pseudopotentials, and for critically reading the manuscript. References 1. P. Pyykko, Lecture Notes in Chemistry no. 41, Springer, Berlin, 1986. 2. P. Pyykko, Lecture Notes in Chemistry no. 60, Springer, Berlin, 1993. 3. P. Pyykko, Bibliography on Web, http: //www. csc ./lul/rtam, 1999. 4. P. Schwerdtfeger, M. Dolg, W. H. E. Schwarz, G. A. Bowmaker and P. D. W. Boyd, J . Chem. Phys., 1989,91, 1762. 5. P. Schwerdtfeger and M . Dolg, Phys. Rev. A , 1991,43, 1644. 6. U. Kaldor and E. Eliav, Adv. Quantum. Chem., 1999,31,313. 7. L. Visscher, T. J. Lee and K. G. Dyall, J . Chern. Phys., 1996,105,8769. 8. W. A. de Jong, J. Styszynski, L. Visscher and W. C. Nieuwpoort, J . Chern. Phys., 1998,108,5177.
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6 Density Functional Theory BY MICHAEL SPRINGBORG
1 Introduction
The interplay between theory and experiment is of paramount importance in understanding materials properties that in turn is important for technological applications when selecting materials with specific properties. The research in this area accordingly spans the complete range from basic research where the fundamental principles behind materials properties on the one side and their structure and composition on the other side are explored to applied research where one seeks the optimal combination of materials with predefined properties for specific applications. Depending on the properties and systems of interest one can choose different theoretical approaches for such studies. When focusing on the mechanical properties of macroscopic samples, the precise arrangement of the atoms and electrons is often of only secondary interest (although they ultimately dictate the mechanical properties) and might therefore not be considered in the models. On the other hand, when, e.g., studying the electronic properties of semiconductors or the reactivity of specific molecules, one needs to include explicitly both electronic and atomic degrees of freedom in the models. Methods based on the densityfunctional theory of Hohenberg, Kohn, and ShamIv2represent one class of methods for theoretical studies of materials properties. They are socalled parameterfree methods, indicating that in principle such methods require only the types and positions of the nuclei as input. However, it also means that everything has to be calculated, making such calculations computationally heavy. Therefore, only for the absolutely simplest systems can the statement above be considered justified, whereas for more complex systems one has to apply one or more carefully chosen approximations. Furthermore, such methods are currently not able to study processes that take more than, say, some few ns, or to describe systems with more than a couple of 1000 atoms (an exception is that of infinite, periodic solids, as well as isolated impurities in such crystals and their surfaces). During the last 12 decades densityfunctional methods have become increasingly important in chemistry (see, e.g., ref. 3 and references therein), which Chemical Modelling: Applications and Theory, Volume 1 0The Royal Society of Chemistry, 2000
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culminated with the award of a share of the Nobel Prize in Chemistry 1998 to Walter Kohn for his fundamental contributions to this development. In this chapter the current state of applications of densityfunctional theory to chemical problems will be reviewed. Since there often is some confusion about the interpretation of the results of densityfunctional calculations we shall in the following section describe the fundamental principles behind the densityfunctional calculations as well as relate them to other methods for electronicstructure calculations, most notably to the HartreeFock method and extensions thereof. Subsequently, we shall in a number of sections discuss various, often very recent, applications of the densityfunctional methods on different types of chemical problems. Since densityfunctional methods currently are very popular methods for electronicstructure calculations (e.g., any recent issue of Journal of the American Chemical Society, Journal of Chemical Physics, or Journal of Physical Chemistry contains at least one example of such and often more), it is completely impossible to give just a fair presentation of all the questions, systems, and properties that are addressed with these methods and the examples that will be discussed below cannot be considered anything but aiming at giving a flavour of the large number of such studies.
2 Fundamentals
Our main goal is that of calculating the properties of a given system consisting of M nuclei and N electrons. In principle, this can be achieved by solving the Schrodinger equation. In Hartree atomic units (me = lel = 4m0 = h = 1) this equation is in the static, timeindependent case
Here, Zj and ?k are combined position and spin coordinates of the ith electron and kth nucleus, respectively, ?i is the position of the ith electron, and &, k f k , and Zk the position, mass, and charge of the kth nucleus, respectively. The terms on the lefthand side of Eq. (1) are, in their order of appearance, the kinetic energy of the nuclei, that of the electrons, the Coulombic interaction energy between the nuclei, that between the electrons, and that between the electrons and the nuclei. Through knowledge of the complete set of solutions to this equation one can, in principle, calculate any quantity of interest. However, only for the absolutely simplest systems (i.e., for N + M 5 2) can this be achieved, and one has accordingly to resort to approximations. Among those the BornOppenheimer
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or adiabatic approximations are some of the most important ones, although, as we shall see below, these are sometimes abandoned. These assume that the electrons move so much faster than the nuclei that one can separate the motion of the two subsystems and that, when changing the structure of the system ( i e . , moving the nuclei), the electrons adjust themselves immediately to this reorganization. Mathematically it amounts to assuming that Y can be factorized into a nuclear part depending only on the nuclear coordinates and an electronic part that depends parametrically on the nuclear coordinates but functionally on the electronic coordinates. For the latter one obtains thereby an electronic Schrodinger equation
Neglecting the kinetic energy of the nuclei (Le., also the zeropoint motion), the total energy becomes
(3)
rvefunctionbase lethods.  Within the Hal LreeFockapproximation, the solution to Eq. ( 2 ) is approximated as a single Slater determinant,
2.
and through application of the variational principle one arrives at the wellknown HartreeFock singleparticle equations,
with
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Here,
is the operator for the kinetic energy and for the electronnuclei interactions, whereas
describes the classical Coulomb interaction of the electrons, with VC being the Coulomb potential, and
is the exchange interactions, with i 1 2 being the permutation operator that interchanges the arguments of the two subsequent functions. Within the HartreeFock approximation, the total electron density becomes
where the summation runs over the N orbitals with the lowest orbital energies 6;. Furthermore, according to Koopmans' theorem4 the singleparticle eigenvalues 6; can be interpreted as ionization energies and electron affinities as far as relaxation effects can be neglected. Solving the HartreeFock equations ( 5 ) yields in most cases more than N orbitals and one may accordingly construct improved approximate (socalled CI, configurationinteraction) wavefunctions
where @:$::"is the Slater determinant obtained from 0 0 but substituting the occupied orbitals t$i, t$j,    with the unoccupied ones q5b,  . Moreover, P is the total number of (occupied and unoccupied) orbitals. In that case, only the coefficients c;f::" are optimized. Within this approach the electron density becomes e.
P
P
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Here, Ji are the socalled natural orbitals, and their occupancies ii lie in the interval [O; I]. Per definition, the improvements that are obtained when replacing Eq. (4) by Eq. (1 1) are the correlation effects. These may be included either through application of the variational principle or perturbatively. In order to solve the HartreeFock equations, one usually expands the solutions in some finite set of basis functions,
where only the coefficients cpi are determined, whereas the basis functions X , are fixed. These functions are chosen partly according to tradition and partly according to convenience. The most common choices include Slatertype orbitals that have the advantage that they per construction are very similar to atomic orbitals so that their total number P can be kept small but the disadvantage that the required matrix elements are complicated to calculate. The latter drawback is to a much lesser extent shared by Gaussians that, however, have the disadvantage that more functions need to be included. For crystalline materials one often uses plane waves as basis functions with which the matrix elements can very easily be calculated, but which require many functions in order to yield accurate results. Also numerical functions are sometimes applied which lead to a relatively small P but to complications when calculating matrix elements. This is also the case for augmented waves that are either planar or spherical waves that inside nonoverlapping, atomcentered spheres are augmented with numerical functions adapted to the potential inside the spheres. Finally, we mention that the HartreeFock equations are solved iteratively. One chooses some initial orbitals, defines from these the Fock operator and solves the HartreeFock equations, which leads to a new set of orbitals, whereby one can start the whole procedure again. This is repeated until selfconsistency is reached.
2.2 Approximating the Schrodinger Equation.  With the approaches we have discussed so far one attempts to solve the exact Schrodinger equation as accurately as possible. All quantities are formulated in terms of the manyparticle wavefunctions and, consequently, these approaches are called wavefunctionbased methods. Their main disadvantages are that the equations are highly complicated and, therefore, that the solutions become more and more approximate the more complex the system of interest is. Typically, the computational efforts scale as for the HartreeFock approximation and up to N7 for the methods that include correlation effects. A completely different aspect is that the outcome of such calculations, i.e., the manyparticle wavefunction, contains much more information than is required for the calculation of any observable. For the latter, one needs usually only the dependence of the wavefunction on one or two particles, whereas that on the coordinates of the N  1 or N  2 other particles is redundant.
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As a consequence of these points it has been attempted to seek alternatives for calculating the electronic properties of a given system using quantummechanical principles. One of the first attempts in this direction was the ThomasFenni theory and improvements thereof (see, e.g., ref. 5). This theory is per construction an approximation. It applies statistical arguments in deriving an approximate expression for the total energy as a function of the electron density, whereby it is assumed that the number of electrons per volume element is large. It turned out, however, that the approximations are too crude to give accurate results except for some few cases. One may understand this by observing that many effects (e.g., the creation of chemical bonds between atoms) are dictated by rather small changes in the total energy, and that these therefore often tend to be much smaller than the inaccuracies in the approximate totalenergy function. In order to avoid these problems one may attempt to approximate only those parts of the Hamilton operator that cause the major computational problems. This was the approach of Slater6 and G a ~ p a rSlater . ~ observed that the major problem in solving the HartreeFock equations (5t(9) was related to the exchange part, Eq. (9). The Coulomb part of the electronelectron interactions, Eq. (8), appears as a multiplicative potential that depends only on the total electron density of Eq. (10) and not on the individual orbitals. Furthermore, the singleparticle part of the Fock operator, i.e., Eq. (7), is computationally simple. Since the exchange interactions, Eq. (9), are small, Slater sought an approximate descriptions of those. He suggested accordingly
whereby also these interactions appear as based on a multiplicative potential that in turn depends only on the total electron density, Eq. (lo), and not on the individual orbitals. Gaspar7argued that the potential on the righthand side should be multiplied by 3, and ultimately a more general approach with a scaling parameter a on the righthand side was devised. This was the socalled Xa method. Here, the HartreeFock equations (5) are consequently approximated as
Also the Xa method was developed as an approximation. It has been used for many years, first of all in combination with the socalled multiplescattering methods. For a given system the threedimensional space is then separated into (nonoverlapping) spheres surrounding the nuclei as well as the interstitial region. Often, one approximates also the total potential V(3)of Eq. (15) keeping only the spherically symmetric part inside the spheres and a constant value in the interstitial region, which often is a significantly more crude approximation
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than the Xa one. Such approaches have been used, e.g., for transitionmetalcontaining molecules (see, e.g., ref. 8 and references therein), but, partly due to this additional approximation, the Xa methods have in some cases obtained a less favourable reputation. 2.3 Densityfunctional Theory.  Whereas the ThomasFermi and Xa approaches were constructed as approximations to the exact quantummechanical problem of calculating the electronic properties of the system of interest, the densityfunctional theory of Hohenberg and Kohn' is, in principle, an exact theory. The theory contains two fundamental theorems. The first of those states that from knowledge of only the electron density p(?) of the ground state one can construct the full Hamilton operator of the system. We may now assume that we can solve the corresponding Schrodinger equation exactly and accordingly calculate any groundstate property of interest. This means that we in principle need to know only the electron density in the threedimensional position space, and not the full wavefunction, in order to calculate any groundstate property. Alternatively formulated this theorem states that any groundstate property is a functional of the density. The proof of this theorem is relatively simple but will not be reproduced here (the interested reader may consult ref. 9). It is based on the variational principle and uses that from the integral of the electron density one knows the total number of electrons and accordingly the kineticenergy and the electronelectroninteraction parts of the total Hamilton operator. Only the external potential (which above was only the Coulomb potential of the nuclei, but which may contain other parts, too) is unspecified, but assuming that this can be written as a sum of identical singleparticle terms, Hohenberg and Kohn proved that also this is uniquely determined within an additive constant. A small example will illustrate the strength of this theorem. We consider the N2 and CO molecules and assume that the bond lengths are identical for the two systems. They contain the same number of electrons, and by looking at the positions of the peaks of the electron density we can identify the positions of the nuclei. Furthermore, for N2 the symmetry of the electron density can help us identify this system. However, adding an electrostatic DC field parallel to the molecular bond will polarize the electron distribution, and it is no longer obvious how to distinguish between CO and N2. But the theorem of Hohenberg and Kohn states that they can be exactly identified through knowledge of only their total electron densities. This first theorem states that any groundstate property is a unique functional of the density. But most often it is used exclusively as stating that Ee is a functional of p ( 3 , although the theorem does not give any expression for this functional. Assuming, however, that we know its precise form, the second theorem states that
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where p o ( a is the true density for the system of interest and p(?) is any other density obeying
This second theorem gives accordingly a variational principle for the density functionals. Eqs. (16) and (1 7) can then be combined into
where p is a Lagrange multiplier for the constraint of Eq. (17). The main problem related to practical applications of the theorems of Hohenberg and Kohn is obvious: the theorems provide proofs for the existence of certain relations but do not give any specific forms for these. In order to obtain a practical scheme for electronicstructure calculations, Kohn and Sham2 reformulated the problem of calculating Ee from the electron density p ( 3 . They observed that parts of Ee directly can be written as functionals of p, i.e., the classical Coulomb interaction energy between the electrons and the energy related to the interaction with the external field (e.g., due to the nuclei, or to an external DC field, etc.). Furthermore, they assumed that it is possible to construct a model system consisting of noninteracting particles but with the same density and energy as the real system. In order to achieve the latter, these particles move in some external effective potential V e r ( q that is adjusted to fulfill these requirements. For this model system, Ee can, however, be written down immediately, N
Here, t+bi are the orbitals of the model system and the total density of this is given as
i= 1
which per construction equals that of the electrons. Notice that the mathematical form for this density is very similar to that obtained for the HartreeFock approximation above [Eq. (lo)], but includes correlation effects and is thus identical to that of Eq. (12) in the limit P + 00.
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For the real system we now choose to write Ee as
The second and third term on the righthand side is the Coulomb energy of the electrons and the interaction energy with external fields, respectively. Furthermore, the first term is the kinetic energy of the noninteracting particles, and the last term is a remainder that contains all unknown parts of Ee as a functional of p. The model system and the real system have the same density in their ground states. Therefore, Eq. (18) holds for E, of Eq. (19) and also for E, of Eq. (21) for the same density p ( 3 . Equating the two resulting expressions leads to an expression for the effective potential (see, e.g., ref. 9),
with
being the Coulomb potential and
being the socalled exchangecorrelation potential. Finally, it may now be observed that since the particles of the model system are noninteracting, the HartreeFock approximation is exact. For this system these equations take the particularly simple form
which are the equations that in practical applications are sought solved. Although also this approach works with wavefunctions, the singleparticle operator depends only on the density, and, accordingly, these methods are called densitybased methods. In total, through this construction one studies a model system of noninteracting particles instead of the real system of interacting electrons. But before proceeding a number of points about this approach should be discussed. First, although the theory is exact, the lack of precise knowledge about Ex,[p] means that the theory only provides an exact framework in which approximate methods can be developed. Second, it is assumed that for any groundstate electron density one can find
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315
an effective external singleparticle potential ( V e ~for ) the model system. It is an open question whether this is always possible for real systems (this is the problem of the socalled vrepresentability), although it has been shown that one may construct reasonable densities where it is not possible. '07" Third, the fact that the energy functional E&] is unknown manifests itself in Eq. (21) as the fact that Ex,is unknown. However, Ex,is only a smaller part of Ee, and, compared with the ThomasFermi approach, the final results are significantly less sensitive to approximations in Ex,. Such approximations are often introduced by first writing Ex,as
with ex, being the exchangecorrelation energy per particle. In principle, ex, in the point r'depends on p in the complete space, but often it is assumed that ex, is some function of p and some of its derivatives in that very point, i.e.,
When keeping only a dependence of cxc on p but not on the derivatives, one arrives at a socalled localdensity approximation (LDA). Such approximations are derived through reference to the socalled homogeneouselectrongas model. Here, one considers an infinite, homogeneous electron gas with a given density. For this, the exchangecorrelation energy can be calculated accurately and subsequently be fitted with some closed analytical expression as a function of its density. Finally, for the real system one determines the value of E~~ in the point r' by using the value for the homogeneous electron gas with the constant density as that of the point of interest, p(3. Ultimately, one may also include dependences on the derivatives which leads to socalled generalized gradient approximations (GGAs). Fourth, so far we have considered only spinunpolarized systems, i.e., we have assumed that the local spin polarization
vanishes identically everywhere in space. Here, pT and pr is the electron density of the spinup and the spindown electrons, respectively, and their sum equals the total electron density. If m(?) is not identically vanishing (which, e.g., obviously is the case for systems containing an odd number of electrons), the theory above has to be generalized so that all functionals depend not only on p ( 3 but also on m(3. Thereby, e.g., the localdensity approximations (LDA) are generalized to localspindensity approximations (LSDA or LSD). Fifth, the theory has been developed as a groundstate theory. Accordingly, it allows, in principle, for the calculation of any groundstate property for the system of interest. However, it can easily be generalized so that it allows for the
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calculation of any property for the energetically lowest state for each symmetry representation. l 3 Sixth, often cXc is split into one part for exchange effects and one for correlation effects,
Within an L(S)DA ex becomes identical to the Xa potential that was introduced as an approximation in the HartreeFock equations by Slater and Gaspar. Seventh, the HartreeFock equations for the model system, Eq. (25) (i.e., the socalled KohnSham equations) are very similar to the HartreeFock equations for the real system. The only difference is that the nonlocal operator of Eq. (9) has been replaced by a local operator, VXc(F)t,bk(q. And since the exchangecorrelation effects are only a minor part of the total electronic energy E,, this difference is, in most cases, so small that the resulting singleparticle orbitals [4i(qfor the HartreeFock equations and t,bi(qfor the KohnSham equations] are very similar. Hence, although from a basic point of view the KohnSham orbitals $i are orbitals for a fictitious model system, it is often a good approximation to analyse them as if they were electronic orbitals. As one example of this, we show in Figure 1 the densities for the orbitals for three closedshell atoms, both as obtained with the HartreeFock orbitals of Huzinaga and Klobukowski6’ and as obtained from a numerical solution of the KohnSham equations with a localdensity approximation for eXc. Through the doublelogarithmic presentation it can be recognized that the two sets of orbitals in fact are very similar, not only concerning the positions of the radial nodes and the regions of the maxima, but also close to the nuclei (where the total potential diverges) and far away from those. However, for the last region one may recognize a slightly different longrange behaviour of the two sets of functions. Eighth, Koopmans’ theorem, which is valid for the HartreeFock approximation, states that, as long as relaxation effects can be neglected, the singleparticle eigenvalues of the HartreeFock equations are ionization energies and electron affinities for the occupied and unoccupied orbitals, respectively. Despite the close similarity between the HartreeFock and the KohnSham equations, similar results do not exist for the latter, although related ones do. Thus, assuming that the exact exchangecorrelation functional Ex, were known, only the energy of the highest occupied orbital is the first ionization potential. 14, l 5 Furthermore, the singleparticle energies of the KohnSham equations obey’ 57l 6
i.e., they can be considered partial, orbitalspecific, electron affinities or ionization potentials, and not integral ones as is the case for Koopmans’ theorem.
6: DensityFunctional Theory
'4

m 
'4
317
0
.
0
'4
0
'4
0
I
"
'
1
4
0
'4
0
.4
0
'4
0
s
log,&)
'I
(a.u.1
Figure 1 The radial densities of various orbitals for isolated (a) He, (b)(d) Cd, and (e)(h) Rn atoms as obtained with either HartreeFock calculations (dashed curves) or LDA calculations (solid curves). The orbitals are [(a), (b), (e)]the s orbitals, [(c), (f)] the d orbitals, and (h) the f orbitals. Note the doublelogarithmic representat ion
As the ninth and final point we discuss the meaning of the parameter ,u of Eq. (18). Originally, it was introduced as a Lagrange multiplier for the constraint of fixed total number of particles. It is, however, also the chemical potential of the electrons,'
As was the case when solving the HartreeFock equations, also the eigenfunctions to the KohnSham equations are $xpanded in some basis set as in Eq. (1 3). And since the singleparticle operator h,fi depends on the density, which in turn depends on the orbitals, in this case also the singleparticle equations have to be solved selfconsistently.
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Densityfunctional methods were originally developed for treating crystalline solids, and until the middle of the 1980s almost all applications of this theory were in that field. There are, however, some few exceptions of which we mention two of the pioneering groups in the field of applying densityfunctional methods to molecular systems without applying additional approximations concerning and in Julich." the shape of the potential, i.e., those in Am~terdarn'~ 2.4 Hybrid Methods.  During the last few years, socalled hybrid methods which combine the densityfunctional and the HartreeFock methods have become increasingly popular, first of all for calculating total energies and their changes due to structural changes or chemical reactions. In this subsection we shall briefly review their foundations. The true Hamilton operator for the complete Nelectron system can be written as
Here, the terms on the righthand side are the kinetic energy, the external potential (e.g., due to the nuclei or electrostatic fields), and the electronelectron interaction, respectively. Within the KohnSham approach we consider instead a system of N particles whose Hamilton operator is
We may now consider a whole class of systems," characterized through the parameter A, each having the Hamilton operator
Thus, for ;i= 1 and A = 0 we have the two systems above, and for all intermediate values of ;i we have different systems characterized by varying strengths of the electronelectron interactions. It is now assumed that we can for all values of A so that we always have adjust the effective, local potential the same total energy and density. The assumption of the existence of this whole class of potentials is more strict than that above related to the vrepresentability and it is not clear whether it is always justified. We now split V$d into three terms,
c$
i.e., into one part due to the external potential, another due to the Coulomb
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319
potential of the electrons, and a remainder. Of these three, only the last depends on A. Denoting the exact Nparticle groundstate wavefunction for
[email protected]), "1, the HellmannFeynman theorem gives
By integrating this equation over A from 0 to 1, one obtains, after some simple manipulations and by using that all systems for different values of A have the same total density p(3, that the total exchangecorrelation energy of the system for A = 1 (i.e., of the real system of interacting electrons) can be written as an integral
with
In practical applications one has to evaluate the integral of Eq. (37). This integral is approximated. First, the integrand is split into one term due to correlation and one due to exchange. Second, each integral is approximated as a weighted sum of the integrand at the two endpoints, i.e., at 1 = 1 (the real system of interacting electrons) for which the HartreeFock approximation is applied, and at A = 0 for which a standard densityfunctional method is applied.20 Since the HartreeFock and the KohnSham orbitals are very similar, one can finally approximate the HartreeFock exchange energy with the one calculated using the KohnSham and not the HartreeFock orbitals. The parameters entering the weighting are determined by requiring that a certain given set of bond energies etc. are reproduced optimally accurately.20
3 Structural Properties
The concept of structure of a molecule or solid is closely related to the BornOppenheimer approximation. Thus, without the assumption that the nuclei are resting point charges one can no longer talk unambiguously about the positions of the nuclei or of bond lengths and bond angles, etc. From a theoretical point of view, the structure of a given system can be calculated by determining that set of nuclear coordinates {&} for which the total energy of Eq. (3) has its minimum. According to their construction, the currently applied densityfunctional methods are first of all methods for calculating the total energy accurately. Therefore, it should be expected that these methods provide accurate informa
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tion about structure. In this section we shall explore this through some few examples of molecules with covalent bonds. But first we shall discuss general strategies for determining the minimum of the total energy. 3.1 Structure Optimization.  Independent of whether a wavefunctionbased or a densitybased method is applied, E of Eq. (3) is a function of all nuclear coordinates,
arfd the problem of determining the structure becomes that of determining those { Rk} for which E has its minimum. In a typical calculation one chooses a certain set of nuclear coordinates and calculates E for this. Subsequently, one selects a new set of coordinates and calculates the corresponding new value of E. This procedure is repeated until one believes that the minimum has been located. However, for all but the absolutely smallest systems, the functional dependence in Eq. (39) is so complicated that it becomes very involved to locate the minimum. Furthermore, very many systems have more (several, many, . . .) local minima and it is in practice impossible to determine the absolute minimum. For some computational methods the only information about the functional dependence of E on the nuclear coordinates that is obtained from a given calculation is E itself.’In that case it is in practice impossible to determine the minimum and one considers often only a single or some few structural parameters (e.g., bond lengths or bond angles) that are optimized whereas all other parameters are fixed at some realistic values. A great improvement is achieved if also the forces on the nuclei are calculated. These are defined as
Then a steepestdescent method can be used in locating that (local or global) minimum that is closest to the starting geometry. This amounts to changing all nuclear coordinates simultaneously according to
where z is some fixed parameter. The calculation of the forces is the calculation of the derivative of E of Eq. (3) with respect to some nuclear coordinate, RI,, where s equals x, y , or z. The derivative of the second term of the righthand side of Eq. (3) with respect to RI, is straightforward, but this is not the case for dEJdR1,. Independent of whether a HartreeFock or a densityfunctional method is applied, one may at first assume that the singleparticle equations are solved exactly. In that case, the
6: DensityFunctional Theory
32 1
HellmannFeynman theorem provides the exact forces,
with i e f f defined in Eq. (25), and a similar equation for the HartreeFock method. In Eq. (42),
and the calculation of the matrix elements in Eq. (42) is straightforward. However, almost exclusively the solutions to the KohnSham or HartreeFock equations are expanded in a finite basis set as in Eq. (13). This means that neither the KohnSham nor the HartreeFock equations are solved exactly and, therefore, the HellmannFeynman theorem is only approximately valid. In practical calculations the deviations are so large that the forces calculated from Eq. (43) are essentially of no use. Pulay” has shown that one has to include an extra term, the socalled Pulay force,
where we have used that Len is Hermitian. It is directly seen that these corrections vanish only when the singleparticle equations are solved exactly or when basis functions that are not atomcentered are used. The latter is the case for plane waves. Finally, we mention that further corrections may have to be added to the forces when other approximations in solving the KohnSham equations are introduced. But we consider a discussion of these beyond the scope of this presentation. There are two fundamental problems related to Eq. (41). First, often very many steps are required in order to locate the minimum, i.e., the forces are often not in the direction of the position of the minimum (see, e.g., ref. 22). Instead, one may use the socalled conjugategradient method (for details, see ref. 22), which also is based on the forces but which uses not only the information obtained for one given structure in order to obtain an improved structure [as in Eq. (41)] but that obtained from more previous structures. Second, both the steepestdescent and the conjugategradient methods lead to the closest minimum independent of whether one of a lower total energy exists.
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It would be advantageous if the trajectories in the {&}space followed by the system also would allow for the total energy to increase so that the system ultimately may in end in a structure of an even lower total energy. This can be obtained through the socalled simulatedannealing method (see, e.g., ref. 22). Then, the nuclei are treated as classical particles with a given mass and a given velocity. Because of these velocities the nuclei move and the changes in their velocities are given by the forces. The velocities may be scaled so that the average kinetic energy corresponds to a given temperature and by gradually reducing this temperature ( i e . , annealing), the nuclei slowly become trapped in a lowenergy structure which might have a lower total energy than the one that would have been obtained with steepestdescent or conjugategradient methods. The methods for structure optimization that we have considered so far are all based on the approach of first selecting some structure, then to calculate the total energy and, if possible, also the forces by solving the KohnSham equations selfconsistently, and finally determining a new structure and repeat the whole approach. Car and Parrinel10~~ proposed that one might instead optimize electronic and structural degrees of freedom simultaneously, i.e., not optimizing the electronic degrees of freedom for each set of nuclear coordinates. To this end they introduced the fictitious Lagrangian
Here, a dot marks derivatives with respect to a time coordinate, and pi is a fictitious mass associated with the ith electronic orbital. Moreover, Aij are Lagrange multipliers that are introduced in order to assure the orthonormality of the orbitals. From this Lagrangian the equations of motion become
which in the static case reduce to the standard equations for the electronic and structural degrees of freedom. But in the dynamical case one may use a moleculardynamics scheme in solving these equations simultaneously for the electrons and the nuclei and have thereby an alternative way of performing structure optimization. 3.2 Examples of Structure Optimizations.  In principle, densityfunctional calculations give exact information on the total energy, for instance as function of structure, and, therefore, they should be able to give exact information on structure. However, due to the introduction of approximate density functionals
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it is not obvious how these approximations influence the outcomes of the calculations. Therefore, there exists a relatively large number of studies devoted to exploring how well such calculations can produce structural parameters. Thereby, the densityfunctional calculations are often supplemented with HartreeFock or CIcalculations as well as experimental information. Furthermore, in many cases more different density functionals are investigated. In this subsection we shall review some few examples of such studies and thereby concentrate on molecules with covalent bonds. Later, we shall consider weaker bonds as well as crystalline materials. Systems containing only lighter atoms are those that most easily can be treated with wavefunctionbased methods and, therefore, such systems are those for which the HartreeFock and CI results are expected to be most trustworthy. In a recent study, Nendel et ~ 1studied . ~ one ~ example of such systems, i.e., the bismethano[ 14lannulenes of Figure 2. They considered both socalled localized structures with strong CC bondlength alternations and delocalized structures with significantly less bondlength alternation. They optimized the structures using the HartreeFock method, a GGA function, and a hybrid method. In Table 1 we compare the calculated CC bond lengths with those of experiment. The table shows that the average CC bond length is roughly the same for all approaches with, however, a slightly larger value for the GGA and hybrid calculations and the smallest value found in the experiment. On the other hand, the HartreeFock calculations lead to a considerably larger bondlength alternation than the other methods. As we shall see below, when calculating the relative total energies of the various forms, too, the hybrid calculations are found to yield the closest agreement with experiment and the HartreeFock calculations the worst. These materials resemble the conjugated polymers for which transpolyacetylene, Figure 3, is the prototype. The fact that they in the ground state have a CC bondlength alternation is responsible for a number of their interesting properties. Thus, there is a strong coupling between the electronic and structural properties so that charge that is added to the system often is localized to regions where the bondlength alternation is modified (see, e.g., ref. 25). Therefore, it is important to be able to describe the existence of the bondlength alternation as well as the energy gain related to this. About a decade ago it
1
2
Figure 2 Structure of (left) syn and (right) anti bismethano[ll]annulene. The structure to the right is simultaneously sketched as the localized one, whereas that to the left is shown as the delocalized one. The dots mark the starting positions for the bond lengths of Table 1 (Reproduced by permission of the American Chemical Society from ref. 24)
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Table 1 Experimental CC bond lengths for the bismethanor IQ]annulenes of Figure 2 together with calculated values. The bond lengths are listed for the periphery starting at the central carbon atom marked with a dot in Figure 2. The experimental ones are for the syn isomer, whereas theoretical values are given for both the syn and the anti isomers. All bond lengths are given in A". HF, GGA, and hybrid denotes HartreeFock, DFT, and hybrid results, respectively, and L and D distinguish between socalled localized and delocalized structures. The results are all taken from ref. 24 ~
Exp. syn 1.39 1.41 1.38 1.40 1.37 1.41 1.40 1.39 1.42 1.37 1.41 1.37 1.41 1.39
~~~~
HF syn L
HF syn
HF anti
HF anti
GGA syn
GGA anti
D
L
D
D
D
Hybrid Hybrid Hybrid anti syn anti L L D
1.46 1.36 1.44 1.36 1.44 1.36 1.46 1.34 1.47 1.34 1.48 1.34 1.47 1.34
1.39 1.42 1.37 1.43 1.37 1.42 1.39 1.39 1.42 1.37 1.43 1.37 1.42 1.39
1.50 1.33 1.46 1.35 1.46 1.33 1.50 1.33 1.47 1.34 1.48 1.34 1.47 1.33
1.40 1.40 1.39 1.42 1.39 1.40 1.40 1.40 1.40 1.39 1.42 1.39 1.40 1.40
1.42 1.43 1.40 1.44 1.40 1.43 1.42 1.42 1.43 1.40 1.44 1.40 1.43 1.42
1.43 1.41 1.42 1.43 1.42 1.41 1.43 1.43 1.41 1.42 1.43 1.42 1.41 1.43
1.40 1.42 1.39 1.43 1.39 1.42 1.40 1.40 1.42 1.39 1.43 1.39 1.42 1.40
1.49 1.36 1.44 1.38 1.44 1.36 1.49 1.36 1.46 1.36 1.47 1.36 1.46 1.36
1.41 1.40 1.40 1.42 1.40 1.40 1.41 1.41 1.40 1.40 1.42 1.40 1.40 1.41
Figure 3 Structure of trans polyacetylene with (a) nonalternating and (b) alternating CC bond lengths
was argued26that densityfunctional calculations with a localdensity approximation were not capable of describing this bondlength alternation, although this conclusion subsequently was q u e ~ t i o n e dRecently, .~~ we have optimized all structural parameters for this system28by calculating the forces and using a steepestdescent approach. We found CC bond lengths of 2.520 and 2.636 a.u., CCC bond angles of 123.6', CH bond lengths of 2.065 a.u., and, finally, CCH and C=CC bond angles of 119.7' and 116.7', respectively. Experimentally, the CC bond lengths are 2.57 and 2.72 a.u., the CH bond lengths are
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2.052.10 a.u., and the CCC bond angles 123°,29330which agree well with our values. These results support the conclusion above that densityfunctional calculations can provide accurate information about carboncarbon bonds in conjugated systems. Hrovat et al.31studied the Cope rearrangement of 1,2,6heptatriene, which is shown schematically in Figure 4. As seen in this figure, the reaction processes from 1,2,6heptatriene via the intermediate 2methylenecyclohexane1,4diyl to the final 3methylene1,5hexadiene. Accordingly, a theoretical study of the reaction involves determination of three stable structures and two intermediates, which by Hrovat et al. was performed both using a CIlike approach and a hybrid densityfunctional method. Moreover, being a diradical, the intermediate may exist both as a singlet and as a triplet. In Table 2 we have collected their calculated CC bond lengths using the two approaches. From this table it is seen that, once again, the wavefunction and the densitybased methods yield very similar structural information about the (meta)stable systems, but for the transition states there are some difference, in particular concerning the longest bonds. One may repeat this analysis considering many other systems, but the general trends will not change. Therefore, we shall here only consider two further examples, this time from the other end of the periodic table. Li and Bursten3* studied the protactinocene Pa(q4C8H& which consists of the Pa atom sandwiched between the two C8H8 rings. In this case it is important to include relativistic effects, which was done by the authors. The results (see Table 3) show somewhat larger scatter for the individual parameters, although in no case there are severe deviations. Finally, Hay and Martin33 studied the actinide compounds UF6, NpF6, and PuF6. Also in this case the results (Table 4) show that the densityfunctional calculations produce results that are in good agreement with the experimental results, which also is the case for the HartreeFock methods. So far the HartreeFock results have been in good agreement with experimental information. This means that correlation effects have been of only minor importance in predicting structural information. Therefore, one might expect that the fact that the densityfunctional calculations yield information that is very similar to those of the HartreeFock calculations is due to the similarity of the KohnSham and the HartreeFock equations, and that structural parameters are in general not strongly influenced by correlation effects. There are,
1
2
3
Figure 4 Schematic representation of the Cope rearrangement of (left) I ,2,6heptatriene via (middle) 2methylenecyclohexaneI ,Idiyl to (right) 3methyleneI ,Shexadiene (Reproduced by permission of the American Chemical Society from ref. 3 1)
TS(1P)
TS(EI)
P
31
1.65 1.75 1.84 1.83
1.55 1.53 1.55 1.53 1.55 1.52 1.92 1.84 1.61 1.61
1.34 1.33 1S O 1.49 1S O 1S O 1S O 1S O 1.41 1.41 1.48 1.47 1.51 1.50 1S O 1.49 1S O 1.49 1.34 1.33 1.47 1.43 1.43 1.42 1.57 1.55 1.56 1.55 1.56 1.54
1.51 1.51 1S O 1S O 1S O 1S O 1.34 1.34 1.48 1.44 1.44 1.44
1.32 1.31 1.39 1.39 1.39 1.39 1.48 1.47 1.40 1.39 1.47 1.44
1.32 1.31 1.40 1.39 1.40 .39 .35 .34 .34 .33 .34 .34
CI hybrid CI hybrid CI hybrid CI hybrid CI hybrid CI hybrid
E
'I
C(2)C( 7 ) C(6)C( 7 )
C(5)C(6)
C(Q)C(S)
C(3)C(4)
C(2)C(3)
C( 1)C(2)
Method
Structure
Table 2 Calculated CC bond lengths in for the Cope rearrangement of Figure 4. E, 1, and P denote the educt (left in the figure), intermediate (middle), and the product (right), whereas TS(EI) and TS(IP) are the transition states for the two reaction parts. Finally, both singlet and triplet structures are considered for the intermediate, and CI and hybrid specify the method of calculation. All results are from ref. 31
4
2fL
, 3 t
5
h
2
%=1
b
2' .
% 3
0
%
g
39
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A.
Table 3 Calculated and experimental distances for P ~ ( v ~  C in ~ H ~HFS ) ~ denotes HartreeFock with Slater Xa for exchange eflects, whereas LDA is a localdensity approximation, and BL YP, BP86, and P W91 are three diflerent generalizedgradient approximations. Finally, X marks the center of the C8H8 rings. The results are all taken from ref. 32 Parameter
Exp.
HFS
LDA
BLYP
BP86
P w91
PaX PaC CC CH
1.96 2.67 1.39
1.94 2.67 1.40 1.10
1.92 2.65 1.40 1.10
2.04 2.76 1.42 1.09
1.98 2.71 1.41 1.09
1.98 2.70 1.41 1.09
A)
Table 4 Calculated and experimental distances (in between the actinides and the fluorine atoms in UF6, NpFb, and PuF6. HF denotes HartreeFock results, and LDA, GGA, and hybrid marks results from densityfunctional calculations with a localdensity approximation, a generalizedgradient approximation, and a hybrid method. The results are taken from ref. 33
HF LDA GGA Hybrid Exp.
1.98 2.00 2.04 2.0 1 2.00
1.97 2.00 2.05 2.01 1.98
1.94 1.98 2.02 1.99 1.97
however, examples that contradict this conclusion. Among those are systems containing bonds between two transitionmetal atoms, for instance the Cr2 molecule. As discussed by Jones and G u n n a r ~ s o n the , ~ ~experimental bond length of this molecule is 3.17 a.u., whereas HartreeFock calculations obtain values ranging from less than 1.5 a.u. to 1.95 a.u. Including correlation effects leads to bond lengths between 3.04 and 6.14 a.u., i.e., only in some cases in good agreement with experiment. On the other hand, localdensity calculations have given 3.173.21 a.u. Here, we have concentrated on bond lengths which are those structural parameters on which the total energy depends the strongest. But also other structural parameters (e.g., bond angles and dihedral angles) can be calculated with densityfunctional calculations and the results will show the same trends as those we have discussed here, i.e., densityfunctional calculations give in most cases structural information that is as good as or better than those of HartreeFock calculations and often as accurate as those of CI or similar calculations although, since the total energy depends more weakly on these three or four
Chemical Modelling: Applications and Theory, Volume 1
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body interactions, larger deviations may result. In particular, interatomic bonds involving metal atoms are often considerably more accurately described with densityfunctional methods than with HartreeFock methods. 4 Vibrations
The calculation of structural properties requires that the total energy as a function of structure is accurate. Calculating vibrational properties requires in addition that the second derivatives are accurate and is accordingly a more stringent test on the reliability of electronicstructure calculations. In their work on Pa($CgH&, Li and B ~ r s t e ncalculated ~~ also the vibrational frequencies of the infraredactive modes. In Table 5 we reproduce their results, which clearly show that also these are accurately calculated with the densityfunctional methods, largely independent of whether a LDA or a GGA method is used. A similar agreement was also found by Hay and Martin33 in their study on the actinide compounds. For these, we show in Table 6 their results for UF6 as a representative example. This table shows, besides confirming the results from the study of Li and Bursten, that HartreeFock calculations lead to an overestimate of the vibrational frequencies, which is a general finding. Actually, the HartreeFock frequencies are often rescaled by some factor slightly smaller than 1, whereby a generally good agreement between theory and experiment is obtained. A much more critical case is that of the umbrella vibration of ammonia NH3, which is very soft. Based on densityfunctional calculations, Aquino et al.35 calculated recently the frequency of this and obtained 0.837 cm' compared with an experimental value of 0.793 cm'. Previous theoretical values ranged
Table 5 Calculated and experimental infraredactive vibrational frequencies in em' for Pa(q8C8He),. LDA is a localdensity approximation and PW91 is a generalizedgradient approximations. The results are all taken from ref. 32 Exp.
P w91
LDA ~~
135 245 695 745 775 895 1310 2920 3005
46i 230 238 695 756 747 914 1404 3045 3046
33 22 1 240 684 772 762 896 1438 3063 3065
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Table 6 Calculated and experimental vibrational frequencies (in cmI) for UF6 from HartreeFock (HF), LDA, GGA, and hybrid calculations. The two sets of values for the hybrid calculations are due to two diflerent basis sets, with the second value being the one for the larger basis set. The results are taken from ref. 33 Exp.
HF
LDA
GGA
Hybrid
667 534 626 186 200 143
76 1 582 702 209 216 157
652 565 657 174 169 141
598 517 605 175 158 135
653,621 552,506 647,606 191,187 178, 177 150,145
from 0.94 to 1.02 cm', so that the new results demonstrate that also here the densityfunctional calculations are accurate. It should be added that they used a hybrid functional. Finally, Halls and S ~ h l e g e studied l ~ ~ a whole series of molecules for which they calculated the infrared intensities with various (HartreeFock and CIlike as well as LDA, GGA, and hybrid densityfunctional) methods. Their general conclusion is that the densityfunctional calculations, essentially independent of which type of functional was used, lead to considerably better agreement with the most accurate CI results than do the HartreeFock calculations. Although one may recognize some improvement of the hybrid functionals over the LDA or GGA approximations, it is interesting to observe that this is only small. This may be taken as a consequence of the fact that the hybrid forms have been developed first of all in order to produce accurate total energies and less for the calculation of other properties.
5 Relative Energies Both the calculation of structural properties and the calculation of vibrational properties require that the total energy is accurate when the structure is changed slightly from that of the totalenergy minimum. The situation is different when studying relative total energies. Such are relevant when comparing different structures (e.g.,isomers) of the same system as well as when studying dissociation energies which both will be considered in this section, and when considering chemical reactions which is the subject of the following section. 5.1 Dissociation Energies.  When being interested in thermochemistry, it is of ultimate importance to be able to calcuiate accurately the energies related to breaking or creating chemical bonds. Densityfunctional calculations with a localdensity approximation suffer from the problem that the bond energies as obtained with such methods are in general overestimated by values from some
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few tenths of an eV to some few eV (see, e.g., ref. 34). Often the errors are constants that only depend on the two atoms forming the bond, but it means that the relative energies of isomers differing in the number and types of bonds often are unreliable. Through the introduction of the generalizedgradient approximations and of the hybrid methods this situation has improved significantly. In order to be able to access the accuracy of a given computational scheme, Pople and coworkers have constructed the socalled G 2 set of data.37 This contains a large number of experimental heats of formation for small molecules, and through comparison with calculated values the deviations are expected to yield valid information about the computational scheme. It shall be added that the molecules contain only first and secondrow atoms and no transitionmetal atoms, which does limit the generality of the results. Later, they compared the performance of different densityfunctional schemes.38They found that the average absolute deviation could be reduced from about 91 kcal mol' for a specific LDA to 720 kcal mol' for different types of GGA, and further to about 3 kcal mol' for the best hybrid methods. This should not be taken as a surprise since the parameters of the hybrid functionals (k.,the relative weighting of HartreeFock and densityfunctional exchange) are often determined by optimizing this performance. Petersson et ~ 1 have . later ~ ~ argued that the error for the hybrid methods in fact can be reduced even further to about 1.5 kcal mol . These studies tell that in particular with the hybrid methods very accurate binding energies and consequently also relative energies for systems containing covalent bonds between lighter atoms are expected to be accurate when using densityfunctional methods. On the other hand, localdensity calculations are not reliable in that respect as is also the case for HartreeFock calculations (which typically underestimate bond strengths), and it is not guaranteed that the same accuracy is reached for other types of atoms or bonds.
'
5.2 Comparing Isomers.  As mentioned above, Hrovat et al.31studied the Cope rearrangement of 1,2,6heptatriene. This system involves only lighter atoms as well as covalent bonds and we will therefore expect that the densityfunctional calculations give accurate information. The experimental totalenergy difference between product and educt (cf. Figure 4) is 14.5 or 15.7 kcal mol', whereas CI calculations gave 13.8 kcal mol' in very good agreement with the experimental values, and hybrid calculations gave a slightly worse agreement, 11.5 kcal mol Nevertheless, this single example supports the general consensus that densityfunctional methods are applicable for such problems. When turning to transitionmetalcontaining compounds the situation changes somewhat. Yoshizawa et a1.40 studied the benzene hydroxylation by ironoxo species. As an initial study to this, they considered the dissociation energy of FeO+. Whereas the experimental value lies about 8 1 kcal molvarious CIlike methods yielded values between 25 and 75 kcal mol', and a hybrid method gave 75 kcal mol'. Thus, the error is larger, but the densityfunctional calculations gave some of the most accurate results. Therefore,
'.
',
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33 1
although their subsequent results for the reaction may include some uncertainties, it is likely that their overall reaction scheme as calculated with the densityfunctional method is reliable.
6 Chemical Reactions
Having established that the densityfunctional calculations can produce information on energetics that is in good agreement with available experimental information, we can extend the analysis and use the calculations to also obtain information that can only with difficulty, if at all, be obtained from experiments. To these belong, e.g., energy barriers for chemical reactions that will be analysed in the first subsection and the concepts of hardness and softness, etc., that will be the subject of the second subsection. 6.1 Transition States.  The three (meta)stable structures for the Cope rearrangement of 1,2,6heptatriene of Figure 4 are  as is the case for all other (meta)stable structures for any material  specified through the requirement that the forces on the nuclei vanish. This is actually also the case for the transition states along the reaction coordinate connecting the different structures, but the difference is that (at least) one of the eigenvalues of the Hessian (i.e.,the matrix containing the secondorder derivatives of the total energy with respect to the nuclear coordinates) is negative. The two transition states for the reaction of Figure 4 were identified by Hrovat et al.3' and, relative to the total energy of the educt, they found the results of Table 7. This table contains also experimental information as well as results from CIlike calculations. We see that the variations in the total energy along the reaction coordinate are well reproduced by the densityfunctional and CI calculations. Torrent et al.41used densityfunctional calculations in exploring the Dotz reaction of Figure 5 for which three different routes, as shown in the Figure, Table7 Calculated total energies in kcal mol'for the Cope rearrangement of Figure 4 . E, I , and P denote the educt (left in the figure), intermediate (middle), and the product (right), whereas TS(EI) and TS(IP) are the transition states for the two reaction parts. Finally, CI and hybrid specify the method of calculation, and the energies are given relative to that of the educt. All results are from ref. 31 Structure
Exp.
Hybrid
CI
E TS(EI) I TS(1P) P
0 27.5 16.6 24.8  14.5
0 31.2 14.9 29.5 11.5
0 25.9 14.3 22.6  13.8
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(COhCr=c'
OH

C2H3
1
co
HCsCH
1
2
,OH
\
H
3
OH
ti 7 OH I
4
OH
6 Figure 5 Schematic representation of the three diflerent routes for the Dotz reaction (Reproduced by permission of the American Chemical Society from ref. 41)
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6: DensityFunctional Theory
were proposed. They calculated the variations in the total energy along the three different routes and obtained the results shown in Figure 6 . From these they concluded that route B in Figure 5 was highly unlikely, whereas the two other routes (of which route C was proposed by the authors) both could occur. Thus, this example shows how the calculations can give information that extends the experimentally obtained information. As the final example we show in Figure 7 results by Yoshizawa et al.40on the benzene hydroxylation by FeO+. They found that the energy barriers depend significantly on the spin state of the system, as shown in Figure 7. 6.2 Hardness, Softness, and Other Descriptors. When two molecules, A and B, interact, the way they interact is to a large extent often determined by properties of the isolated A and B molecules. In order to quantify these properties, the hardness and softness of Pearson4*can be useful. These quantities were put on a rigorous formal foundation by Parr et al.43within densityfunctional theory and have since then be the subject of some theoretical studies. Here, we shall briefly review their definitions and subsequently consider the various quantities for some few systems. We analyse how the chemical potential changes when the system is modified, whereby the modifications can be due either to changes in the number of . changes can occur electrons or to changes in the (external) potential V e ~These when the system of interest starts interacting with some other system. Consider

ROU~CA
..... Route B
. Route C
Figure 6
Variation of the total energy for the three diflerent routes of the Gotz reaction of Figure 5 (Reproduced by permission of the American Chemical Society from ref. 41)
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Fa+(‘A)
+ CgHg
Fe*(4F) + C&OH
Fe+(‘DO, + C&OH
n
Reactant complex
Figure 7
Hydroxy intermediate
Product complex
Variation of the total energy (in kcal molI) f o r the direct benzene hydroxylation of FeO+ in the sextet and quartet spin states. As shown, the reaction proceeds in two steps (Reproduced by permission of the American Chemical Society from ref. 40)
ing only infinitessimal changes we find [cf. Eq. (31)]
The first quantity on the righthand side is related to the chemical hardness,
where the second identity has been obtained through a finitedifference approximation to the derivative and where I and A is the ionization potential and electron affinity, respectively. The second quantity on the righthand side of Eq. (47) is related to the socalled Fukui function4
Actually, p(3) may be a nonanalytic function of the number of electrons N
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(since electrons are added to the LUMO but removed from the HOMO), and one may therefore introduce two different Fukui functions,
that are useful for studying nucleophilic and electrophilic attacks, respectively. The last identi ties have been obtained through finitedifference approximations. Finally, the global softness is defined via the reverse of the global hardness,
All quantities that have been defined involve the changes in some quantity when the total number of electrons N is changed. We may now analyse these changes in even more detail by noticing that when N is changed the electron density p ( 3 is also changed, in principle at every single point. Therefore, we can introduce positionresolved quantities that describe how the properties of interest change when the electron density at one single point is modified. This information can be very useful when studying the initial states of chemical reactions. One can imagine that two molecules interact and that the interaction involves some electron transfer between the two. Then, it is of advantage for the interaction that the two molecules are so placed relative to each other that the electronaccepting part easily can accommodate this extra electron density and that, equivalently, the electrondonating part easily can donate this density. Accordingly, from the global softness S one defines a local one,
Equivalently, a local hardness can be defined as
s and 21 are each
other's reverse in the sense that
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Finally, also the Fukui function can be resolved in position space,
As mentioned above, these quantities can be useful in analysing how different molecules interact. The global quantities have led to the socalled HSAB principle, i.e., that hard Lewis acids prefer to interact with hard Lewis bases whereas soft acids will interact with soft bases.42But through the introduction of the positionspaceresolved quantities much more insight into the interactions can be obtained. As two examples we show in Figure 8 the local softness s+(a (once again, due to the nonanalyticity of the electron density on the number of electrons we have to distinguish between adding and removing electrons, i.e., to have two different local functions) for acetic acid, and in Figure 9 the Fukui function for three different molecules in comparison with the total electron density. We shall not discuss these results further here. We add that the field is young and many further studies are needed before these routinely can be applied in a predictive manner.
\
4.02
/
O
0
H
0.005
Figure 8 Contour plot of the local softness s+(q for acetic acid. The thick curve represents the contour value 0 (Reproduced by permission of the American Chemical Society from ref. 45)
.
.
..
. .
. ._
.
....
.
..
. ..
..
__ .. ..
... ..
..
Figure 9 Contour plots of the Fukui functionsf+ (second row) a n d f (last tow) together with the electron density of the LUMO (first row) and of the HOMO (third row) for CO (left column), HCN (middle column), and H2CO (right column) (Reproduced by permission of the American Chemical Society from ref. 46)
.. .. ..
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7 Weak Bonds Very often, when studying the formation of bonds between different atoms theoretically, it is assumed that only the orbitals of the valence electrons change so that one can keep the orbitals of the core electrons as for the isolated atom. Then, only a smaller part of the total energy, i.e., the socalled valence energy, has to be calculated. Nevertheless, also this quantity is often quite large, i.e., it is often several 100 eV or even more per atom. Therefore, it is a far from trivial pursuit to calculate the changes in the energy due to the formation of covalent bonds, since these typically are only about at most some few eV per bond. It is even more difficult when calculating the bond energies of weaker bonds such as van der Waals or hydrogen bonds. In this section we shall explore how densityfunctional calculations perform for those. 7.1 Van der Waals Bonds.  PkrezJorda and B e ~ k studied e ~ ~ the six raregas diatomics He2, Ne2, ArZ, HeNe, HeAr, and NeAr using different density functionals. According to experiment, the bond lengths of these range from 5.6 to 7.1 a.u., and the bond energies from 0.9 to 12.3 meV, which clearly shows that these bonds are very weak. Their localdensity calculations gave bond lengths that systematically were underestimated by roughly 1 a.u. and in agreement with this, the bond energies were overestimated by a factor of 210. On the other hand, GGA calculations were not capable of predicting stable structures. This was also the case for the hybrid calculations. As an alternative the authors suggested another functional, which led to accurate results for these systems. But the results taken as a whole show that when attempting to describe these weak bonds with densityfunctional calculations one has to be very careful. 7.2 Hydrogen Bonds.  Hydrogen bonds are not as weak as the bonds between the raregas atoms, and are, moreover, very important for biological systems. The energies of these bonds are up to about 0.5 eV per bond. The hydrogen bonds in infinite chains of hydrogen fluoride are among the strongest ones. Such chains are found, e.g., in crystalline H F and each chain contains a zigzag arrangement of the F atoms with the H atoms placed asymmetric between the F atoms, such that they form a covalent bond to one of the two F neighbours and a hydrogen bond to the other. Using densityfunctional calculations with a localdensity approximation we found48 FF and FH interatomic distances of 4.72 and 1.85 a.u., respectively, and FFF bond angles of 125", which compares very well with experimental values of 4.714.73 a.u., 1.801.83 a.u., and 116120'. A part of this success might be due to the abovementioned fact that these hydrogen bonds are unusually strong. Therefore, when performing similar calculations on linear chains of hydrogenbonded HCN, which has a weaker hydrogen bond, we found49 hydrogenbond lengths that were 1020% too small.
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Similar results were also found by Sim et al.” who studied various finite molecules containing intramolecular hydrogen bonds. They used more different density functionals and found overall that the localdensity approximation underestimated the length of the hydrogen bonds considerably (in some case, the H atoms became placed almost at the midpoint between their two nearest neighbours), whereas GGA calculations led to realistic results. The problems related to the LDA might be due to the fact that the LDA tends to overestimate the energy of any bond between two atoms, and accordingly to prefer as many bonds as possible. Thereby, also a too short hydrogen bond with some covalent character becomes energetically favourable. Also Hamann” found in his study on the Ih crystal structure of ice that the LDA calculations led to too short hydrogen bonds, whereas the GGA calculations led to realistic structures. Surprisingly, the hybrid calculations led to a much too expanded structure. Furthermore, the energies of the hydrogen bonds were overestimated by the LDA calculations, reasonably accurate (but no more) in the GGA calculations, and underestimated in the hybrid calculations. The hydrogen bonds can also take an active part in chemical reactions, e.g., of the type AB.C+A..BC. Also these have been studied with densitywho functional calculations. These studies include that of Johnson et considered one of the simplest possible reactions of this type, H2 H + H H2, using a number of different LDA and GGA functions. They found that the results depended very strongly on the applied functional (which is a highly unsatisfactory situation) and that in some cases it was found that the transition state (corresponds to H3) was stabler than the educts and products. Also Mijoule et al.53 found in their study on the reaction H 3 0 + H20 + H20 + H3O that the energy barrier was considerably underestimated with those GGA functionals they considered. Finally, also Porezag and P e d e r ~ o nstudied ~ ~ such hydrogen exchange and abstraction reactions and found that the results from LDA calculations were highly unreliable, and those of GGA calculations were only in some cases realistic. Before closing this section we mention one completely different aspect. The nuclei of the hydrogen atom is the absolutely lightest one and as such may be the one for which deviations from the BornOppenheimer approximation first can be expected. This means that one might have situations where the quantum nature of the protons manifests itself. For a hydrogen atom that is bonded covalently to one neighbour and via a hydrogen bond to another, the potential felt by it is weak and anhannonic. Thus, if the potential becomes very shallow, the proton may be more delocalized than is usually assumed. Whether this could be the case has been studied by Tuckerman et a/.” for H s O ~and H30,. They used densityfunctional calculations with a GGA but also treated the protons as quantum particles using Feynman’s pathintegral method. It did turn out that in some cases a nonclassical behaviour of the protons was found.
+
+
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8 The Total Electron Density
By construction, the present densityfunctional methods are expected to yield not only accurate total energies but also accurate total electron densities. In this section we shall study these. The total electron density is often a fairly unstructured object with dominating maxima at the sites of the nuclei. In order to obtain information about the bonds between various atoms when forming a compound from the individual atoms, the difference density is much more useful. This density is the difference between the density of the compound and that constructed from the superposed atomic densities placed at the positions of the atoms in the compound. This density might also be much more sensitive to inaccuracies. Recently, Lee et aLS6 studied this density for bis(diiminosuccinonitrilo)nickel, Ni(C4N4H&, using both experimental and different theoretical methods. In Figure 10 we show the structure as well as the experimental and theoretical difference densities, where the theoretical densities are from either HartreeFock or LDA calculations. It is clear that there are differences between the three sets of results, although the overall features are identical. The authors estimated also the net atomic charges for the various atoms of the molecule using the different approaches. These results indicate that the LDA calculations yield results that are in slightly better agreement with the experiment than the HartreeFock calculations. Moreover, it was found that the HartreeFock calculations led to too large electron transfers, whereas the LDA calculations underestimated those. This is in fact a general observation. As a further example of this we consider the density of the crystalline, ionic compound aMnS that was studied by Tapper0 and Li~hanot.’~ According to their HartreeFock calculations, the material can be considered as consisting of Mn2+ and S2 ions, whereas the densityfunctional calculations predict the effective charges to be closer to f1.6.
9 The Orbitals As discussed in Section 2, the KohnSham orbitals are in the rigorous sense not electronic orbitals, although it may be a good approximation to neglect this formal problem when taking the similarity between the HartreeFock and the KohnSham equations into account, in particular when applying the Xa approximation in the HartreeFock equations. Furthermore, also the singleparticle eigenvalues of the KohnSham equations can only with difficulty be given a physical interpretation, except for the highest one of the occupied orbitals. But also here it may be reasonable to study them as they were electronic singleparticle energies. In this section we shall address these issues further. Stowasser and Hoffmann studied recently the singleparticle orbitals for some few smaller systems using different theoretical method^.'^ In Figure 11 we show their results for the water molecule as obtained with a GGA method and with HartreeFock calculations. The orbitals are indeed very similar, supporting the idea that one may analyse the KohnSham orbitals as electronic orbitals.
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34 1
a
b
C r
a
I
Figure 10 (a) Schematic structure of the molecule Ni(C4NdH2)2 and (b)(d) diflerence electron densities obtained by subtracting superposed atomic densities from the molecular density. (b), (c), and (d) were obtained from experiment, from HartreeFock calculations, andfrom LDA calculations, respectively (Reproduced by permission of the American Chemical Society from ref. 56)
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Figure 11 Contour plots of the four occupied orbitals for the water molecule from (left) densityfunctional and (right) HartreeFock calculations in a plane perpendicular to that of the molecule andpassing through the oxygen atom (Reproduced by permission of the American Chemical Society from ref. 58)
These authors found also that the singleparticle energies from the KohnSham equations were shifted upwards in energy compared with experiment, whereas those of the HartreeFock equations were shifted slightly downwards, and in addition spanned a slightly broader energy interval than the experimental values. Only the energy of the highest occupied KohnSham orbital is supposed to have a physical meaning, and therefore we first study this. In Table 8 we show the calculated energies of the highest occupied orbitals for a set of isolated atoms together with experimental ionization potentials. The table confirms the conclusion above, i.e., that the KohnSham energies are too high. Furthermore, these tend to have a too weak dependence on the system, i.e., when passing from the left to the right in the periodic table, the experimental first ionization potential increases more rapidly than predicted from the densityfunctional calculations. This finding might actually explain why the densityfunctional calculations that were discussed in the previous section led to too small effective atomic charges. Politzer and AbuAwwad61 studied the calculated singleparticle energies for 12 smaller molecules and compared them with experimental ionization potentials. They found that the KohnSham orbital energies were at least 2 eV too high
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Table 8 The energy of the highest occupied orbital from LDA densityfunctional (first number) and HartreeFock (second number) calculations for isolated atoms compared with experimental first ionization potentials (third number) asfunctions of atomic number. The results are takenfrom refs. 59 and 60 and are given in eV ~~
H 7.56 13.60 13.60
He 15.59 24.98 24.58
Li 3.39 5.34 5.39
Be 5.69 8.42 9.32
B 4.25 8.43 8.30
C 6.33 11.79 11.26
N 6.50 15.44 14.54
0 7.43 17.19 13.61
F 10.48 19.86 17.42
Ne 13.34 23.14 21.56
Na 3.30 4.95 5.14
Mg 4.87 6.89 7.64
A1
Si 4.80 8.08 8.15
P 6.48 10.66 10.55
S 6.23 11.90 10.36
c1
3.19 5.71 5.98
8.34 13.78 13.01
Ar 10.60 16.08 15.76
K 2.82 4.01 4.34
Ca 3.94 5.32 6.11
Ga 3.04 5.67 6.00
Ge 4.54 7.82 7.88
As 6.07 10.05 9.81
Se 5.83 10.96 9.75
Br 7.62 12.44 11.84
Kr 9.52 14.26 14.00
sc 3.90 5.72 6.56
Ti 2.79 6.01 6.83
V 3.47 6.27 6.74
Cr 4.19 6.04 6.67
Mn 4.65 6.74 7.43
Fe 4.42 7.02 7.90
Co 3.75 7.28 7.86
Ni 4.36 7.52 7.63
Cu 5.20 6.49 7.72
Zn 6.15 7.96 9.39
(i.e., less negative) and often more. The best agreement was found with the hybrid methods, which might be due to the fact that the KohnSham eigenvalues usually underestimate and the HartreeFock eigenvalues usually overestimate the ionization potential whereby some kind of average becomes reasonable. Finally, we show in Figure 12 the experimental band structures together with the calculated ones for polyethylene.62These densityfunctional results (with a localdensity approximation) demonstrate a very good agreement between theory and experiment, but it should be stressed that the experimental data have been shifted rigidly about 2 eV upwards in energy, which is consistent with the results above. We add that band structures from HartreeFock calculations were in general too wide,63 which is a common deficiency of HartreeFock calculations.
10 Excitations
Any experimental study of the properties of a given system involves perturbing the system somehow and by measuring the response of the system trying to
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5.0
10.0 n
2
C
>r
E)
15.0
a
S
L!J
20.0
I
25.0
0.0
0.5
1.0
k Figure 12 Experimental (points) and theoretical (solid curves) band structures for polyethylene (Reproduced from ref. 62)
extract its underlying chemical and physical properties. In order to make a comparison between theory and experiment more direct it is therefore desirable also that the theoretical studies are capable of describing these responses. Among those, excitation energies are of fundamental importance and these are the topic of this section. Electronic excitations involve changing the populations of the various orbitals. A special case is that of ionization for which one electron is removed from an occupied orbital and, in principle, brought infinitely far away from the molecule. In the reverse case, one adds an electron to an unoccupied orbital. The excitation of an electron from an occupied to an unoccupied orbital is, in principle, the combination of the two former processes. As discussed in Section 2, the densityfunctional methods are, in principle, capable of describing only the properties of the energetically lowest state for
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each symmetry representation, which puts clear limitations on those excitations that can be studied. In order to calculate the ionization energies, electron affinities, and transition energies for the processes above one may accordingly do different calculations for the different configurations and from their totalenergy differences extract the energies of interest. This is the socalled ASCF approach. An alternative, approximate, approach is to use Slater’s transitionstate method? Then the energies of interest are calculated directly by performing calculations for configurations in which not one but only half an electron has been removed, added, or excited. Perdew et al.65calculated the ionization potentials and electron affinities for a number of atoms using the first approach and for different computational methods. They found that the ionization potentials were on the average 1.04, 0.32, and 0.25 eV wrong when using HartreeFock, LDA, and GGA methods, respectively, whereas the. same numbers were 1.51, 0.38, and 0.28 eV for the electron affinities. Moreover, the ionization potentials and electron affinities from the HartreeFock calculations were in general too small, whereas the densityfunctional calculations showed a less clear trend. Stener et a1.66 studied carbon 1s + n* and oxygen 1s + n* as well as ionization potentials in carbonylcontaining molecules using the transitionstate method. First, they considered the isolated CO molecule, and subsequently they studied the shifts when the carbonyl group was a part of a larger molecule. Furthermore, they used both densityfunctional and CI calculations, and they calculated the oscillator strengths, too. As a representative example we show in Table 9 their results for some of these systems. It is immediately seen both that the densityfunctional calculations perform at least as well as the CI calculations and also that the approximate transitionstate method of Slater is accurate. It
Table 9 Experimental and theoretical ionization potential ( I P ) and excitation energy ( E E ) in eV as well as oscillator strengths (f)for 1s n* transitions for carbonylcontaining molecules. The energies are given as shifts relative to the valuesfor the isolated CO molecule, and DFT and CI denote results from densityfunctional and CI calculations, respectively. From ref. 66 +
Molecule Cls H2CO HFCO F2CO 01s H2CO HFCO F2CO
IP DFT
IP
2.18 0.16 2.63 3.22 2.66 2.16
IP Exp.
EE DFT
EE CI
EE Exp.
2.18 0.93 3.73
1.73
1.33 0.94 3.27
0.94 1.72 4.35
3.32 2.30 1.38
3.02
3.14 2.00 0.96
3.03 1.89 0.41
CI
3.44
1.83
f
f CI
f
1.40 5.3 0.80 6.2 3.50 8.1
4.6 5.8 7.2
6.6 10.4 13.2
3.40 2.10 1.50
2.82 2.85 2.83
7.2 4.5 2.6
DFT
3.4 3.0 2.8
Exp.
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has to be stressed that the calculation of the differences, as, e.g., shown in the Table, is, in principle, more difficult than the calculation of the quantities themselves, although, on the other hand, one may benefit from fortuitous cancellations of errors. The fact that the transitionstate method of Slater is accurate was also found by Andrej kovics and who calculated excitation energies for four smaller molecules using different densityfunctionalbased methods. They used both the ensemble theory of Gross, Oliviera, and Kohn68969as well as the densityfunctionaltheory formulation of GOrling7' whereby one can obtain exact formulas for the excitation energies (the interested reader is referred to the original works for further details), so that the transitionstate method is  in principle  the less accurate one of those studied. It turned out, however, that the other approaches did not yield more accurate results than the transitionstate method. 11 Spin Properties The spin of the particles (Le., nuclei and electrons) may act as local probes for the properties of a given molecule and a number of powerful experimental techniques have been developed for exploiting this. Also methods for calculating these properties have been developed, and a few examples of those will be reviewed here. 11.1 NMR Chemical Shifts.  Nuclear magnetic resonance (NMR) experiments explore the electronic density in the closest vicinity of nuclei whose spin is nonzero. Although this density is dominated by the core electrons, small effects due to differences in the chemical surroundings of the atom of interest cause small shifts in the resonance energies that, accordingly, give information on the chemical bonds. Salahub and coworkers have employed densityfunctional methods in calculating these chemical shifts for different systems. They used a perturbationtheory method for calculating the responses of the system to the external magnetic field. In one work71 they studied the shifts for 13Cnuclei in transitionmetal complexes and compared these with available experimental information. In Table 10 we reproduce some of their results. They explored also whether the presence of the core electrons could be replaced by a simpler effective potential and, as the results of the table show, both the methodology as such and the effective core potential lead to accurate results. Since transitionmetalcontaining systems are difficult to treat with wavefunctionbased methods due to the large number of d orbitals closest to the Fermi level, their findings are very encouraging. In a subsequent they studied the ''N chemical shifts for a biologically relevant system, the gramicidin channel. In particular they also explored how these change when the conformation of the system is changing, whereby they used moleculardynamics simulation to follow the structural changes. Due to
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Table 10 Experimental and theoretical I3C chemical shifts (in ppm) for some transitionmetal carbonyl complexes. ECP denotes densityfunctional calculations with the application of egectve core potentials, whereas DFT marks allelectron densityfunctional calculations. From ref. 71 Molecule
Exp.
ECP
LIFT
V(C0)6 Cr(C0)6 Mn(CO)SH, cis Mn(CO)SH, trans Ni(C0)4
225.7 21 1.2214.6 211.4 210.8 191.6193.0
216.8 205.8 209.6 207.5 197.6
215.7 204.0 206.2 203.3 192.9
the size and complexity of the system, this work represents an impressive application of densityfunctional methods. Lee et ~ 1 has. presented ~ ~ a densityfunctional method that directly includes the presence of the magnetic field and then used the formalism in calculating the nuclear shielding tensor elements. These are the secondorder derivatives of the total energy with respect to the external magnetic field and the nuclear magnetic moment and describe how the electrons screen an external magnetic field at the site of the nuclei. That is, with 2 e x t being the applied magnetic field, the field at a certain position is
with the field induced by the electrons being
CT is the shielding tensor. 
In Table 11 we present their calculated isotropic shielding constants for different nuclei of various systems using different computational schemes. It is obvious from the Table that the agreement between the calculated and the measured values is far from perfect, independent of the calculational method. It shall, however, be stressed that, being the secondorder derivatives of the total energy, these quantities are very sensitive to any numerical inaccuracies.
11.2 Electron Spin.  With electron spin resonance (ESR) experiments one studies the spin properties of the electrons. The hyperfine coupling constants describe the interactions between the total spin of the electrons and those of the nuclei and they are only then nonvanishing when none of the two spins vanishes. This means that most often such systems have an odd number of electrons, although deviations from this exist. These coupling constants depend essentially on the difference in the spinup and spindown electron densities at
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Table 11 Isotropic nuclear shielding constants for diferent nuclei of various systems using HartreeFock (HF), LDA, or GGA calculations in comparison with experimental values. From ref. 73 System
Method
Nucleus
Value
HF
HF LDA GGA Exp. HF LDA GGA Exp. HF LDA GGA Exp. HF LDA GGA Exp. HF LDA GGA Exp. HF LDA GGA Exp.
'H
28.03 29.09 29.82 28.5 & 0.2 410.40 411.12 405.05 410 k 6 23.67  20.82  15.35 3.0 & 0.9  84.25  87.39  77.14 42.3 f 17.2  110.01 91.63  84.82 61.6 & 0.2  167.31  283.49 271.70 232.8
co
I9F
l3C
* 70 4N
'
I9F
the sites of the nuclei, and are for two reasons difficult to calculate: the potential diverges at those positions whereby the wavefunctions may become nonanalytic at these, and the densityfunctional (and other electronicstructure) methods have been developed for optimizing the total electron density and the total energy and not for calculating the density in a single point. The problems related to the behaviour of the wavefunctions at the sites of the nuclei were studied by Ishii and S h i m i z ~ ,who ~ ~ calculated the isotropic hyperfine coupling constants for some small systems using different types of basis functions. In particular, Gaussians have per construction a vanishing slope at their origin, whereas Slatertype orbitals (which might be closer to the exact KohnSham or HartreeFock orbitals) of s type have a nonvanishing slope. As their results show (Table 12), this difference can have significant effects on the accuracy of the results, but the table shows also that accurate results can be obtained. Also Cohen and C h ~ n gexamined ~ ~ the basisset dependence of these hyperfine coupling parameters by studying a larger set of smaller firstrow molecules. They found that larger basis sets do not necessarily lead to more accurate coupling constants, which might be understood from the discussion
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Table 12 Isotropic hyperfine coupling constants (in G ) for various small systems as obtained with LDA calculations and diferent types of basis functions (GTO or STO for Gaussians or Slatertype orbitals, respectively) together with experimental values. In some cases more values are reported corresponding to diferent sizes of the basis sets. From ref. 74 System Method ' H
H B C N 0
F CH NH
OH
STO Exp. GTO STO Exp. GTO STO Exp. GTO STO Exp. GTO STO Exp. GTO STO Exp. GTO STO Exp. GTO STO Exp. GTO STO Exp.
Ill3
l3C
I4N
I7O
I9F
518 508 5.9,  10.7 0.7,0.5 4.1 2.2, 8.5 6.6, 7.0 7.0 0.1, 1.7 4.9,4.2 3.7 0.8, 0.4  16.4,  11.6  12.3 2.7 109 108
 16 19.1, 19.7 20.6  18 21.1 23.6 21 23.1, 23.3 26.1
9 16.7, 17.1 16.7 3 7.7, 7.3 6.9 6 20.0, 16.8  18.3
above. Furthermore, they found that hybrid methods gave an overall better agreement with experiment than LDA or GGA methods. Barone et al.76 studied the transitionmetal complex CUCZHZ using different densityfunctional methods. They found that the experimental hyperfine coupling constants were well reproduced by GGA calculations with fairly large basis sets. But all studies indicate that these constants can be calculated accurately only with great care.
11.3 Electronic SpinSpin Couplings.  For compounds with valence electrons that are well localized to the individual atoms and for which these electronic shells not are filled, one may ascribe a total electronic spin to the individual atoms. Moreover, these spins of different atoms may arrange themselves in an energetically favourable way which, e.g., for solids may lead to magnetic behaviour. But also for finite, molecular systems such couplings between the spins may exist and have important consequences for the overall properties of the system.
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The transitionmetal systems M2Xi with M = Cr, Mo, W and X = C1, Br are one class of such systems. Here, the structure may be considered consisting of three parallel triangles of halogen atoms separated by the transitionmetal atoms. Requiring that the spinsoof the two M atoms were identical led to MM distances up to almost 1 A too short, whereas breaking the symmetry and allowing the two spins to be different gave excellent agreement with experiment .77 An apparently very simple system is the Cr2 molecule. Here, Edgecombe and B e ~ k showed e ~ ~ that only by treating the spins very carefully could one obtain a good agreement with experiment regarding bond length and, in particular, the variation of the total energy as a function of bond length. In both cases, as well as in others, one may model the spin behaviour of the system of interest using a Heisenberg Hamiltonian,
s~
where is the spin of the Mth atom, and the Js are coupling constants that can be determined experimentally or theoretically. We shall here, however, not discuss this point further.
11.4 Nuclear SpinSpin Couplings.  Also the spins of the nuclei may couple, whereby the coupling is mediated by the electrons. The coupling can be quantified through the Fermi contact terms, and these were, e.g., recently . ~ calculated for various carboranes and boron hydrides by Onak et ~ 1 They calculated over 100 such coupling constants and found an overall agreement with experimental values for the couplings between atoms that were connected via chemical bonds of about 5%. As for the hyperfine coupling constants these quantities depend mainly on the electron densities at the sites of the nuclei and are accordingly sensitive to numerical inaccuracies.
12 Electrostatic Fields The electronic and structural properties of a molecule may change when it is being exposed to external electrostatic fields. In some sense, the problem of calculating these responses is complementary to those discussed above for the spindependent properties. The DC field leads to an extra term in the KohnSham equations
where 2is the field vector of the DC field. Thus, whereas the potential due to the nuclei has a rl dependence, and some of the spindependent properties depend almost exclusively on the behaviour of the wavefunctions closest to the nuclei,
~
35 1
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this operator has its dominating parts far away from the nuclei. Therefore, here the longrange behaviour of the wavefunctions is important. One may quantify the response of the system to the external field through the dipole moment, M k= 1
n
J
which may have both linear and nonlinear dependences on the DC field,
with p(O) being the static dipole moment, a the polarizability, and B, y , . . . hyperpolarizabilities. The (hyper)polarizabilities can be calculated either directly by including the field in the calculations or via perturbation theory. For highly symmetric systems and properly chosen coordinate systems, the matrix containing the polarizabilities become diagonal with only two different values, all = azzand a1 = a, = ayy.Then the average polarizability is
and the polarizability anisotropy is
McDowell et a1.*' calculated p(O) = lfi(o)l and Aa for a set of smaller molecules using different theoretical methods. These results are reproduced in Table 13. They show that the dipole moment in general is underestimated by the HartreeFock calculations and overestimated by the LDA and GGA calculations. This is somewhat surprising, since  as we have seen above  densityfunctional calculations tend to underestimate charge transfers whereas these are overestimated by HartreeFock calculations. The Table shows also that the polarizabilities, which essentially are the first derivatives of the dipole moment with respect to field strengths, show some more scatter. More recently, Cohen and Tantirungrotechai8' explored the performance of several densityfunctional methods in calculating dipole and quadrupole moments as well as polarizabilities of some smaller molecules. Besides confirming the conclusions above, they found less clear trends for the quadrupole moments as well as the fact that the hybrid methods yield slightly better agreement with experiment both for the static dipole moment and for the polarizabili ties. On the other hand, Champagne et al.82found that the polarizability and hyperpolarizability of various finite oligomers of polyacetylene were inaccurate
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Table 13 Experimental and theoretical dipole moments p(O) and dipole polarizability Aa anisotropy for some small molecules, The theoretical results have been obtained using HartreeFock (HF), LDA, and GGA methods, and all quantities are given in atomic units. From ref. 80 System
p)
p)
/&O)
p)
HF
LDA
GGA
Exp.
4.89 16.43 8.50 29.89 16.13 29.97 12.94 29.93 8.51 23.65 23.75 15.85 28.33
6.17 18.43 8.82 31.70 18.01 34.28 15.57 32.52 10.54 26.13 26.41 17.80 29.10
6.26 18.54 8.96 3 1.97 17.82 33.14 15.62 32.14 10.63 25.94 26.75 17.97 29.3 1
5.60 17.39 8.38 30.35 17.27 3 1.90 14.56 32.03 9.64 24.71 25.61 17.51 27.70
Au HF
Au LDA
Au GGA
1.29 1.15 9.01 18.29
1.10 1.30 5.49 16.11
1.16 1.33 5.78 16.46
0.51 0.85 1.15 0.39 12.53 12.10 12.05
2.67 2.38 0.24 1.24 13.16 13.96 11.17
2.68 2.34 0.30 1.66 13.43 14.03 11.15
Au Exp. 1.33 1.45 17.53 1.94 0.66 0.65 13.0 13.3 11.4
when calculated with densityfunctional methods, independent of whether LDA, GGA, or hybrid methods were used. 13 Solvation
So far we have considered systems that were supposed to be isolated and not interacting with any other systems. Accordingly, these calculations correspond to experiments in the gas phase. However, many experiments are carried through in other media, including solutions, and therefore it is desirable to be able to determine the effects on the molecule of interest (the solute) due to the surrounding medium (the solvent). Methods for such studies have been developed, and as a first approximation they can be classified as belonging to one out of two different classes. Here, we shall describe these two approaches and consider a single example of their application for each type of method. 13.1 Dielectric Continuum.  In the one approach one assumes that the interactions between the solvent and the solute are mainly of an electrostatic nature and, therefore, that there are no bonds between the solute and the solvent molecules (including hydrogen bonds which could be important, e.g., when the solvent is water). Then the solvent is modelled as a dielectric (polarizable) continuum with a cavity occupied by the solute. The continuum responds to the charge distribution of the solute by becoming polarized which in turn may lead to a redistribution of the charge distribution of the solute, and so on. Therefore, the electron density of a dissolved molecule may be different from that of the molecule in the gas phase, which ultimately may even lead to changes in its
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structure. This approach suffers from the problem that the size and shape of the cavity are not uniquely defined. Therefore, the calculated quantities may depend more or less strongly on these two (nonphysical) quantities. This approach has been used by Andzelm et al.83 who studied various molecules in the gas phase and dissolved in a continuum that was supposed to model water. Here, we shall just mention a few of their results. As one example they studied relative basicities for different molecules. These are defined as the negative enthalpy of the protonation process A + H+ , AH+. They studied the set of molecules NH3x(CH3)x with 0 5 x 5 3. Relative to the values for NH3 the basicities drop from 1020 kcal mol' in the gas phase to 12 kcal mol' in the solvent according to experiment. This change is well reproduced by the calculations. Secondly, they studied two isomers of acetic acid CH3COOH and found that upon solvation the structure changed slightly. But the changes in the bond lengths never exceeded 0.02 Furthermore, the dipole moment changed from about 1.7 D to 2.6 D for the synform and from about 4.25 D to about 6.15 D for the antiform upon solvation. Finally, they showed that for glycine a different isomer is the stabler one in the solution than in the gas phase.
A.
13.2 Point Charges.  The other approach for including the solvent corresponds to replacing the complete, roughly infinite set of solvent molecules by a finite set whose effects on the solute moreover are reduced to those of an electrostatic potential. Thereby the electrostatic potential of each solvent molecule is supposed to be that of a set of point charges. For water this could correspond to placing charges +q on each hydrogen atom and 2q on the oxygen atoms. The position and orientation of the solvent molecules are chosen randomly under certain constraints (e.g., two solvent molecules should not be placed too close to each other or to the solute, and the density of the water molecules might be fixed). Then, the properties of the system of interest are calculated including this extra potential. In order to reduce effects due to the precise positions and orientations of the solvent molecules, more calculations with different positions and orientations of these molecules are carried through and the averages of the properties determined. The main disadvantage of this approach is that there is no selfconsistent interaction between the solvent and the solute, i.e., the charge distribution of those solvent molecules that are closest to the solute does not change due to the presence of the solute. in their study This approach was applied, e.g., by KnappMohammady et of the structure of LalanylLalanine in aqueous solutions. They found that the structures depended very strongly on the presence of the water molecules. 14 Solids
With solutions we have entered the field of extended systems. Of these, crystalline solids are those materials that have been treated most extensively with
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densityfunctional methods. Actually, for a long period (i.e.,from the end of the 1960s to the middle of the 1980s) densityfunctional methods were applied almost exclusively to such systems. Intuitively, one might expect the electron density inside these materials to be more uniform than it is for a molecular system and, accordingly, that the localdensity approximation is better for solids than for molecules. Due to this long tradition for applying densityfunctional methods for crystalline solids, there is an overwhelming amount of papers devoted to such studies and it is absolutely impossible to give a fair review of all those. Therefore, we shall here restrict ourselves to an absolute minimum realizing that this means not giving the proper reference to all the other works on various systems and properties that they actually deserve. 14.1 Band Structures.  Conceptually the simplest approximation amounts to assuming that the crystalline solid is infinite and periodic in all three dimensions. Then, when expanding the solutions to the KohnSham (or HartreeFock) singleparticle equations in a set of atomcentered basis functions [cf. Eq. (13)], the translational symmetry can be utilized in constructing symmetryadapted basis functions from the identical ones of different unit cells, i.e.,
Here, nlabels different unit cells whose total number is denoted Al.Furthermore, k describes the symmetry representatiqn and is simultaneously closely related to a momentum. Only those values of k that lie within the first Brillouin zone need to be considered. With this, the KohnSham equations become dependent,
and depicting E j as a function of k' gives the band structures. The main differences between treating finite molecules and infinite, periodic solids lies then in the fact that for the solids one has to perform a number of (approximately) infinite summations over the entire lattice+and that Eq. (65) in principle has to be solved for each value of a continuous k in the first Brillouin zone. In practice only a finite subset of ;values is considered and the lattice summations are truncated after so many terms that they can be considered converged. 14.2 Applications.  In Table 14 we have collected some representative examples for calculated properties of some simple crystalline solids as obtained with densityfunctional methods. It is seen that both LDA and GGA calculations yield accurate lattice constants, whereas the cohesive energy is somewhat overestimated with the LDA calculations but accurate with the GGA calcu
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Table 14 Calculated structural properties of some simple crystals using either a LDA or a GGA density functional and compared with experimental values. Thefirst row for each material contains the lattice constant a in the second the bulk modulus B in GPa, and the third the cohesive energy Ecoh in eV. The results are taken from refs. 85 and 87
A,
System
Property
LDA
GGA
Exp .
A1
a B
Si
a B
3.96, 3.93 87,88 4.05,4.14 5.37, 5.38 98,97 5.37, 5.38 3.27 172 8.15 3.94 178 3.91 5.57 75 4.53 5.51 77 8.58
4.04,4.03 79 3.39, 3.45 5.46, 5.45 89,84 4.59,4.40 3.32 166 6.64 4.04 1 24 2.78 5.73 58 3.41 5.76 66 6.00
4.05 72 3.39 5.43 99 4.63 3.30 170 7.57 3.89 181 3.89 5.66 77 3.85 5.65 76 6.52
&oh
&oh
Nb
Pd
a B Ecoh a B &oh
Ge
a B &oh
GaAs
a B &oh
lations. This is once more a manifestation of the LDA tendency to overbind. Also the bulk modulus, being the second derivative of the total energy with respect to volume at the totalenergy minimum, is accurate. In a more recent study, Baraille et a1.88compared HartreeFock and densityfunctional calculations on crystalline Mg. They found results similar to those above as well as a tendency for the HartreeFock calculations to yield a too small cohesive energy; i.e., also for the solids the HartreeFock approximation underbinds. Equivalently with the tendency of HartreeFock calculations to produce too large vibrational frequencies for finite molecules, the elastic constants for this material were overestimated by the HartreeFock calculations, whereas there was some scatter for the densityfunctional results. As seen in Table 15, the HartreeFock calculations lead to band structures that are too broad, whereas they are accurate (or at most slightly too narrow) for the densityfunctional calculations. The former can be understood as being due to the lack of screening, i.e., distant electrons in the solid interact too strongly within the HartreeFock approximation. Also in their recent work on crystalline ZnO, Massidda et al.89found that the band structures were too wide with HartreeFock calculations, but here the LDA bands were somewhat too narrow. ZnO may actually be a material for which correlation effects are quite important (as also is the case for the highT, superconductors, other transitionmetal oxides, some heavyfermion systems,
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Table 15 LDA, HartreeFock (HF), and experimental (Exp.) valuesfor various electronic properties for four semiconductors. The width is the total valenceband width, the gap is the smallest gap between valence and conduction bands, and Ecoh is the cohesive eneyy. _The last two entries are bandenergy diflerences at the r point (k = 0 ) . All energies are given in e V (the cohesive energy in e Vlatom), and the results are from ref. 86 ~~~
Property ~
Method
C
Si
Ge
LDA HF Exp. LDA HF Exp. LDA HF Exp . LDA HF Exp. LDA HF Exp.
21.3 29.9 24.2 4.1 12.1 5.5 5.7 14.6 7.3 13.4 23.7
12.0 18.0 12.5 0.5 5.6 1.2 2.7 8.7 3.4 3.0 9.3 4.2 5.5 2.9 4.7
12.7 18.9 12.6 0.0 4.2 0.7 2.7 7.9 3.2  0.2 4.3 1 .o 4.1 1.2 3.9
aSn
~~
Width
Gap
crls  Er;, Er;
 Er;,
&oh
9.2 5.3 7.6
10.6 16.0 0.0 2.6 0.0 2.3 6.6 2.6 0.9 2.6 0.1 2.8 0.1 3.1
and other materials), so that the approximations inherent in the LDA are too crude. Finally, we mention that densityfunctional methods have also been used in studying much more complex systems than simple crystals. A recent example is the study of Galli et al.” who studied the microfracture in an aSic. Thereby, they modelled the system as consisting of periodically repeated units of 128 atoms. After optimizing the structure of this, they increased one of the unitcell lengths and monitored the positions of the atoms as a function of this elongation until a fracture occurred. Despite this simple description it should be stressed that such calculations are very complicated and represent some of the most sophisticated ones that at present can be carried through with densityfunctional methods.
15 Liquids
The crystalline solids possess both short and longrange order whereby, through the construction of the symmetryadapted Bloch waves of Eq. (64), the calculations become feasible. This is not the case for liquids that may possess shortrange but not longrange order. In order to circumvent this problem one has to consider a simplified model system, whereby one most often considers a
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Table 16 Number of atoms of type B around atoms of type A for liquid Lio.blNa0.39 at diflerent temperatures. From ref. 91 A
B
500 K
600 K
900 K
Li Na Li
Li Na Na
4.9 3.7 1.9
4.5 3.4 2.1
3 .O 2.9 2.2
large, periodically repeated, cell with several atoms. Through this construction, a longrange order is introduced but an intermediaterange order may be lacking. One recent example of such studies is that of Senda et aZ.91who studied liquid LiNa alloys. They considered a cell containing 61 Li atoms and 39 Na atoms, thus modelling a Lio.61Nao.39 alloy. Subsequently, they applied a molecular dynamics method based on the densityfunctional theory in calculating the structure of the system for different temperatures. In Table 15 we reproduce some of their key results. They analysed the number of Li nearest neighbours about the Li and Na atoms as well as the number of Na nearest neighbours about the Li and Na atoms. When the material is homogenous, each atom, independent of being Li or Na, should have 0.61/0.39 = 1.56 as many Li as Na nearestneighbours. The table shows, however, that for lower temperatures, Li atoms tend to cluster about Li atoms and Na atoms about Na atoms whereas the number of LiNa pairs is reduced, i.e., a phase separation occurs. There are many other similar studies on other systems. In particular, one may calculate paircorrelation functions that can be compared directly with experimental information. We shall, however, not discuss these further here. 16 Surfaces as Catalysts
In Section 14 we considered crystalline solids and approximated their structures as if the materials were infinite and periodic in all three dimensions. This may be a good approximation when studying the bulk properties of the material, but for some properties the fact that the material is finite is of ultimate importance. Among these are those properties that are related to the existence of surfaces. The fact that the atoms at the surfaces have a lower coordination than those in the interior and accordingly have dangling bonds leads to the existence of surfacespecific properties. These include surface reconstructions for which the surface atoms adopt a structure different from that of the interior, as well as a higher reactivity of the surface atoms. The latter is the topic of the present section. In studying the surfaces one may take one out of three different approaches. The introduction of a single surface in a threedimensional crystalline solid
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breaks the translational symmetry in one direction but may still keep it (neglecting symmetrylowering surface reconstructions) in the other two. One can develop computational schemes for treating this situation, but when furthermore introducing extra molecules that shall interact with the surface, these have to be introduced periodically along the surface in order to keep the twodimensional periodicity. Alternatively, one may reintroduce the translational symmetry in the third direction by constructing a fictitious system consisting of repeated slabs separated by regions of vacuum. The slabs should be so thick that the central region resembles the inner part of the infinite crystal. Moreover, also here extra molecules will have to be introduced periodically. Finally, one may consider a finite cluster as a model system for the surface without or with adsorbants. In this case there may be dangling bonds in all directions, although one may be interested in only those of one of the surfaces. The other dangling bonds may be saturated by, e.g., hydrogen atoms. The last approach was used by Neyman et aZ.92 in their study of the adsorption of CO on the MgO(001) surface. However, they augmented the cluster with extra point charges to model the remaining parts of the crystal. They found that the bond between the surface and the CO molecule leads to a small change in the C0 vibrational frequency, although the bond length was only slightly modified. The adsorption energy was small, about 0.1 eV as found in their GGA calculations. Ultimately an adsorbed molecule may have modified electronic properties compared with the molecule in the gas phase. Thereby it may react more easily with another molecule, either coadsorbed on the surface or in the gas phase, so that the surface has acted as a catalyst. There has been some work devoted to such studies, but these will not be described further here. Instead we consider a special case of a surface, i.e., the inner surfaces of a zeolite. The channels and cavities of zeolites are often so large that smaller molecules can be accommodated there and, in some cases, interact with the zeolite host. An example of this was studied by Broclawik et aZ.93They studied the dissociation of methane on a gallium site in the socalled ZSM5 zeolite. They considered a finite cluster as a model system, and by performing densityfunctional calculations for that system they were able to identify the reaction mechanism, including educts, products, and transition states. In principle, the calculations are very similar to those discussed above in Section 6, except for differences in the systems.
17 Intermediatesized Systems Intermediatesized systems are systems that are larger than the molecules we have considered sofar but not so large that they can be considered as approximately infinite. This class of materials covers a broad spectrum of different types of systems, each with its special chemical and physical properties as well as problems when treating them theoretically.
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One group of such systems is formed by macromolecules. In some cases these may be considered as being approximately infinite in one dimension and finite in the other two. When it furthermore can be assumed that they are periodic in the one dimension one may use this periodicity in forming symmetryadapted Bloch waves equivalent to those of Eq. (64) for the threedimensional periodicity. This has been described further in ref. 94, where also various applications are presented, but for the present purpose it suffices to mention that, except for the complexity in dealing with the combination of finite and infinite extensions, the calculations yield results that are compatible with those we have reported above. In other cases it is not possible to apply the abovementioned periodicity. Then, for large systems one may approximate the system by a smaller one containing that part of the larger one that is relevant for the problem at hand (this could, e.g., by an active site plus its nearest surroundings for chemical reactions), whereas the remainder is either completely omitted (eventually saturating dangling bonds with hydrogen atoms or other small groups) or treated with a simpler molecularmechanics scheme. Another group of such systems is formed by clusters and colloids. These are fairly closedpacked molecules containing 1010 000 atoms of often only one or two kinds. In some cases, the dangling bonds on the surface are passivated by (often organic) radicals. Due to their after all finite size, quantumconfinement effects lead to properties (e.g., optical) that are different from those of the crystalline analogues and which may be tuned through variation of the size of the system. For metal clusters, also the occurrence of magic numbers (i.e., special number of atoms for which the cluster is particularly stable) has attracted atten tion. One example of such systems was recently treated by Hakkinen et aZ.95with densityfunctional methods. They studied a Au38 cluster passivated by 24 SCH3 groups on the surface yielding Au38(SCH3)24. They examined both structural and electronic properties of this quite large system, with special emphasis on how extra charge is distributed inside the cluster. There are many other related systems. For all of them, the number N of electrons is large, but not so large that it can be considered approximately infinite. Moreover, they do not possess a high (e.g., translational or rotational) symmetry that can be used in making the calculations tractable. Finally, chemically equivalent atoms may become inequivalent through differences in their surroundings (e.g., differences in the types and number of neighbours and in the distance from the center). These systems represent a challenge to theory for which there currently are being developed the socalled orderN methods, i.e., methods that scale linearly with the number of electrons (see, e.g., ref. 96). 18 Conclusions
In this contribution we have concentrated on first describing the fundamentals behind densityfunctional methods as well as the approximations that are
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introduced in practical applications of this theory. In particular, we have showed how the present methods conceptually are similar to the HartreeFock methods. In both cases one arrives at a set of singleparticle equations although the form of the singleparticle operator differs slightly. Therefore, it should not be surprising that in many cases similar results are found, and that the interpretations that are applicable within one approach can also to a good approximation be applied within the other. There is, however, a fundamental difference between the two methods for parameterfree electronicstructure calculations. The HartreeFock approximation represents a first approximation to the exact solution to the exact manybody Schrodinger equation, so that one may in principle systematically improve the solution. On the other hand, the KohnSham equations are currently approximated and it is not obvious whether more accurate solutions yield better agreement with reality. However, for all but the smallest systems, one can solve the KohnSham equations more accurately than one can solve the Schrodinger equations. Taking everything together this suggests that both approaches have advantages and disadvantages and that the best way of developing the understanding of the properties of materials is to continue to apply both types of method and not to abandon any of them. We discussed a number of different applications of densityfunctional methods on various types of systems and properties and, often, compared the results with experimental information or with results of HartreeFock or CIlike calculations. We saw that, in most cases, densityfunctional calculations give accurate information but also that there are exceptions. We shall here not go through the examples again but instead close in stressing that the systems, properties, works, and methods that were considered were chosen very subjectively and are far from being complete. Instead, it is hoped that they give a flavour of the variety of systems and properties that are sought understood using densityfunctional methods and that the interested reader will find it worthwhile to study his or her systems with such methods. On the other hand, it is hopefully also demonstrated that, despite the impression sometimes given by various commercial companies that distribute densityfunctional program packages, densityfunctional electronicstructure calculations can currently not be considered simple blackbox packages that routinely can provide (nearly) exact results on any system at hand, but, instead, each system has to be studied with care and the results always checked for credibility by comparison with any other available information.
Acknowledgments
The author is grateful to Fonds der Chemischen Industrie for very generous support.
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References 1. P. Hohenberg and W. Kohn, Phys. Rev. B, 1964,136,864. 2. W . Kohn and L. J. Sham, Phys. Rev. A , 1965,140,1133. 3. DensityFunctional Methods in Chemistry and Materials Science, ed. M. Springborg, Wiley, Chichester, 1997. 4. T. Koopmans, Physica, 1933,1, 104. 5. P. Gombas, Die statistische Theorie des Atoms und ihre Anwendungen, Springer, Wien, 1949. 6. J. C. Slater, Phys. Rev., 1951,81,385. 7. R. Gaspar, Acta Phys. Acad. Hung., 1954,3,263. 8. Proceedings of Applications of the Density Functional Theory to the Chemical and Physical Properties of Inorganic Systems, Arles, France 1988, J. Chim. Phys., 86, (1989). 9. R. G. Parr and W. Yang, DensityFunctional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. 10. M. Levy, Phys. Rev. A , 1982,26, 1200. 11. E. H. Lieb, Int. J. Quant. Chem., 1983,24,243. 12. S. Huzinaga and M. Klobukowski, Chem. Phys. Lett., 1993,212,260. 13. 0.Gunnarsson and B. I. Lundqvist, Phys. Rev. B, 1976,13,4274. 14. C.0. Almbladh and U. von Barth, Phys. Rev. B, 1985,31,3231. 15. J. P. Perdew and M. Levy, Phys. Rev. B, 1997,56,16 021. 16. J. F. Janak, Phys. Rev. B, 1978,18,7165. 17. E. J. Baerends, D. E. Ellis, and P. Ros, Chem. Phys., 1973,2,41. 18 0.Gunnarsson, J. Harris, and R. 0. Jones, J. Phys. C, 1976,9,2739. 19 J. Harris, Phys. Rev. A , 1984, 29, 1648. 20. .A. D. Becke, J. Chem. Phys., 1993,98,5648. 21 P. Pulay, Mol. Phys., 1969,17, 197. 22 W . H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, Cambridge University Press, Cambridge, 1992. 23 R. Car and M. Parrinello, Phys. Rev. Lett., 1985,55,2471. 24 M. Nendel, K. N. Houk, L. M. Tolbert, E. Vogel, H. Jiao, and P. von R. Schleyer, J. Phys. Chem. A , 1998,102,7191. 25 Handbook of Conducting Polymers, ed. T. A. Skotheim, R. L. Elsenbaumer, and J. R. Reynolds, Marcel Dekker, New York, 1998. 26 J. Ashkenazi, W. E. Pickett, H. Krakauer, C. S. Wang, B. M. Klein, and S. B. Chubb, Phys. Rev. Lett., 1989,62,2016. 27. M. Springborg, S. L. Drechsler, and J. Malek, Phys. Rev. B, 1990,41, 11 954. 28. M. Springborg and A. Pohl, J. Phys. Condens. Matt., 1999,11,7243. 29. C. S. Yannoni and T. C. Clarke, Phys. Rev. Lett., 1983,51, 1191. 30. C. R. Fincher, Jr., C.E. Chen, A. J. Heeger, A. G. MacDiarmid, and J. B. Hastings, Phys. Rev. Lett., 1982,48, 100. 31. D. A. Hrovat, J. A. Duncan, and W. T. Borden, J. Am. Chem. SOC., 1999,121, 169. 1998,120, 11 456. 32. J. Li and B. E. Bursten, J. Am. Chem. SOC., 33. P. J. Hay and R. L. Martin, J. Chem. Phys., 1998,109,3875. 34. R. 0.Jones and 0. Gunnarsson, Rev. Mod. Phys., 1989,61,689. 35. N. Aquino, G. Campoy, and H. YeeMadeira, Chem. Phys. Lett., 1998,296, 111. 36. M. D. Halls and H. B. Schlegel, J. Chem. Phys., 1998,109,lO 587. 37. L. A. Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople, J. Chem. Phys., 1991,94,7221. .
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38. L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys., 1997,106,1063. 39. G. A. Peterson, D. K. Malick, W. G. Wilson, J. W. Ochterski, J. A. Montgomery, Jr., and M. J. Frisch, J. Chem. Phys., 1998,109, 10 570. 40. K. Yoshizawa, Y. Shiota, and T. Yamabe, J. Am. Chem. SOC.,1999,121,147. 41. M. Torrent, M. Duran, and M. Sola, J. Am. Chem. SOC.,1999,121, 1309. 42. R. G. Pearson, J. Am. Chem. SOC.,1963,85,3533. 43. R. G. Parr, R. A. Donnelly, M. Levy, and W. E. Palke, J. Chem. Phys., 1978, 68, 3801. 44. R. G. Parr and W. Yang, J. Am. Chem. SOC.,1984,106,4049. 45. F. De Proft, S. Amira, K. Chobo, and P. Geerlings, J. Phys. Chem., 1994,98,5227. 46. A. Michalak, F. De Proft, P. Geerlings, and R. F. Nalewajski, J. Phys. Chem. A, 1999,103,762. 47. J. M. PCrezJorda and A. D. Becke, Chem. Phys. Lett., 1995,233, 134. 48. M. Springborg, Phys. Rev. B, 1988,38, 1483. 49. M. Springborg, Ber. Bunsenges. Phys. Chem., 1991,95, 1238. 50. F. Sim, A. St.Amant, I. Papai, and D. R. Salahub, J. Am. Chern. SOC., 1992,114, 4391. 51. D. R. Hamann, Phys. Rev. B, 1997,55,10 157. 52. B. G. Johnson, C. A. Gonzales, P. M. W. Gill, and J. A. Pople, Chern. Phys. Lett., 1994,221,100. 53. C. Mijoule, Z. Latajka, and D. Borgis, Chem. Phys. Lett., 1993,208,364. 54. D. Porezag and M. R. Pederson, J. Chem. Phys., 1995,102,9345. 55. M. E. Tuckerman, D. Marx, M. L. Klein, and M. Parrinello, Science, 1997, 275, 817. 56 C.4. Lee, T.S. Hwang, Y. Wang, S.M. Peng, and C.S. Hwang, J. Phys. Chem., 1996,100,2934. 57 R. Tapper0 and A. Lichanot, Chem. Phys., 1998,236,97. 58 R. Stowasser and R. Hoffmann, J. Am. Chem. SOC.,1999,121,3414. 59 S . B. Trickey, Phys. Rev.Lett., 1986,56,881. 60 S . Huzinaga and M. Klobukowski, Chem. Phys. Lett., 1993,212,260. 61 P. Politzer and F. AbuAwwad, Theor. Chem. Acc., 1998,99,83. 62 M. Springborg and M. Lev, Phys. Rev. B, 1989,40,3333. 63 K. Seki, N. Ueno, U. 0. Karlsson, R. Engelhardt, and E.E. Koch, Chem. Phys., 1986,105,247. 64 J. C. Slater, Adv. Quant. Chem., 1972,6, 1 . 65 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B, 1992,46, 6671. 66 M. Stener, A. Lisini, and P. Decleva, Chem. Phys., 1995,191, 141. 67 I. Andrejkovics and A. Nagy, Chern. Phys. Lett., 1998,296,489. 68 E. K. U. Gross, L.N. Oliveira, and W. Kohn, Phys. Rev. A, 1988,37,2805. 69 E. K. U. Gross, L. N. Oliveira, and W. Kohn, Phys. Rev. A, 1988,37,2809. 70 A. Gorling, Phys. Rev. A, 1996,54,3912. 71. M. Kaupp, V. G. Malkin, 0. L. Malkina, and D. R. Salahub, Chem. Phys. Lett., 1995,235,382. 72. T. B. Woolf, V. G. Malkin, 0. L. Malkina, D. R. Salahub, and B. Roux, Chem. Phys. Lett., 1995,239, 186. 73. A. M. Lee, N. C. Handy, and S. M. Colwell, J. Chem. Phys., 1995,103,lO 095. 74. N. Ishii and T. Shimizu, Chem. Phys. Lett., 1995,235,614. 75. M. J. Cohen and D. P. Chong, Chem. Phys. Lett., 1995,234,405.
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76. V. Barone, R. Fournier, F. Mele, N. Russo, and C. Adamo, Chem. Phys. Lett., 1995, 237, 189. 77. T. Lovell, J. E. McGrady, R. Stranger, and S. A. MacGregor, Inorg. Chem., 1996, 35,3079. 78. K. E. Edgecombe and A. D. Becke, Chem. Phys. Lett., 1995,244,427. 79. T. Onak, J. Jaballas, and M. Barfield, J. Am. Chem. SOC.,1999,121,2850. 80. S . A. C. McDowell, R. D. Amos, and N. C. Handy, Chem. Phys. Lett., 1995,235,l. 81. A. J. Cohen and Y. Tantirungrotechai, Chem. Phys. Lett., 1999, 299,465. 82. B. Champagne, E. A. P e M , S. J. A. van Gisbergen, E.J. Baerends, J. G. Snijders, C. SoubraGhaoui, K. A. Robins, and B. Kirtman, J. Chem. Phys., 1998,109, 10489. 83. J. Andzelm, C. Kolmel, and A. Klamt, J. Chem. Phys., 1995,103,9312. 84 M. KnappMohammady, K. J. Jalkanen, F. Nardi, R. C. Wade, and S. Suhai, Chem. Phys., 1999,240,63. 85 Y.M. Juan, E. Kaxiras, and R. G. Gordon, Phys. Rev. B, 1995,51,9521. 86 A. Svane, Phys. Rev. B, 1987,35,5496. 87 A. Garcia, C. Elsasser, J. Zhu, S. G. Louie, and M. L. Cohen, Phys. Rev. B, 1992,46, 9829. 88 I. Baraille, C. Pouchan, M. Causa, and F. Marinelli, J. Phys. Condens. Matt., 1998, 10,10969. 89 S . Massidda, R. Resta, M. Posternak, and A. Baldereschi, Phys. Rev. B, 1995, 52, 16 997. 90 G. Galli, F. Gygi, and A. Catellani, Phys. Rev. Lett., 1999,82, 3476. 91 Y . Senda, F. Shimojo, and K. Hoshino, J. Phys. SOC.Japan, 1998,67,2753. 92 K. M. Neyman, S. Ph. Ruzankin, and N. Rosch, Chem. Phys. Lett., 1995,246,546. 93 E. Broclawik, H. Himei, M. Yamada, M. Kubo, A. Miyamoto, and R. Vetrivel, J. Chem. Phys., 1995,103,2102. 94 M. Springborg, in DensityFunctional Method in Chemistry and Materials Science, ed. M. Springborg, Wiley, Chichester, 1997, p. 207. 95 H. Hakkinen, R. N. Barnett, and U. Landman, Phys. Rev. Lett., 1999,82,3264. 96 S . Goedecker, Rev. Mod. Phys., 1999,71, 1085.
7 Manybody Perturbation Theory and Its Application t o the Molecular Electronic Structure Problem BY S. WILSON
1 Introduction During the 1960s, Kelly'7 pioneered the application of what is today the most widely used approach to the description of correlation effects in atomic and molecular systems; namely, the manybody perturbation theory.*' Using numerical solutions to the HartreeFock equations, he applied the manybody perturbation theory to the correlation problem first in atoms'3 and subsequently in molecule^."^ Kelly's molecular applications were limited to simple hydrides for which a onecentre expansion could be employed, treating the hydrogen atom(s) as an additional perturbation. Correlation energy calculations based on numerical solutions of the HartreeFock equations involve an integration over the continuum which requires some care in its implementation. The range of applicability of the manybody perturbation theory as a method for describing correlation effects in molecules changes radically in the 1970s when a number of author~'~''demonstrated how the introduction of the algebraic appro~imation,'~'*i.e. the use of finite basis sets, could facilitate calculations for arbitrary polyatomic molecular systems. By employing basis sets consisting of subsets centred on each of the component atoms in the system under study (and perhaps on other centres, such as bond midpoints), applications to any molecular system are, in principle, possible. In the algebraic approximation, the integration over the continuum which arose in Kelly's approach based on numerical solutions of the HartreeFock equations becomes a summation over the virtual states obtained by solution of the matrix HartreeFock equations. Essentially, the integration over the continuum becomes a quadrature. During the late 1970s and early 1980s, it became evident that the assumption made at the time of the birth of quantum chemistry that the effects of relativity were of "no importance in the consideration of atomic and molecular structure,
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and ordinary chemical reactions, in which it is, indeed, usually suficiently accurate if one neglects relativistic variation of mass with velocity and assumes only Coulomb forces between the various electrons and atomic nuclei”20 is not correct.2128Relativity has to be considered for molecules containing heavy atoms but its effects can also be significant in the study of some properties of lighter molecules. There has been a great deal of progress over the past twenty years on the treatment of correlation effects within the framework of relativistic quantum mechanic^.^"^* Manybody perturbation theory and the underpinning theory of quantum electrodynamics has played a central role in these developments. The implementation of the algebraic approximation in calculations based on the fourcomponent Dirac equation was not only the key to successful practical applications to molecular systems, but also provided a representation of the negative energy branch of the Dirac spectrum which is not usually available from numerical DiracHartreeFock calculations for atoms. Early nonrelativistic manybody perturbation theory studies of correlation energies in molecules established that the error associated with truncation of the finite basis set is most often much more significant than that resulting from truncation of the perturbation e ~ p a n s i o n .The ’ ~ chosen basis set is required to support not only an accurate description of the occupied HartreeFock orbitals but also a representation of the virtual spectrum. Over the past twenty years significant progress has been reported on the systematic design of basis sets for electron correlation studies in general and manybody perturbation theory calculations in particular. l 8 Over the same period, radical improvement in the performance of algorithms and computer program for carrying out manybody perturbation theory studies have resulted from the relentlessly increasing power of computers and from evolving computer architectures. The introduction of “direct” algorithms, first in selfconsistent field calculation^^^ and then in loworder perturbation treatments of correlation has removed the problem of storing large numbers of twoelectron integrals. Vector processing computers434 and, in more recent years, parallel processing4546 machines have significantly extended the horizons of electron correlation energy calculations enabling more accurate studies of small molecular species and ab initio treatment of quite large systems. The fact that the theoretical properties of a method for describing correlation effects in molecules influence its properties when implemented in a parallel computing environment cannot be overestimated. The manybody perturbation theory leads to algorithms which are not iterative and capable of being separated into parts which can be performed concurrently. The resulting concurrent computation ManyBody Perturbation Theory (ccMBPT) offers an efficient and accurate approach to the electron correlation This report describes progress made on the application of manybody perturbation theory to the problem of molecular structure in recent years. However, since it is almost two decades since the last report” on this subject in Specialist Periodical Reports, the opportunity will be taken to provide some historical perspective on the developments during the past two decades. My 1981 report5’ provided a description of the theoretical background to the
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manybody perturbation theory approach to the electron correlation problem in molecules. The emphasis in that report was on the establishment of an appropriate theoretical apparatus and associated computational technique for a practical scheme of calculation. At that time, there were few published computer program^'^'^ and only a handful of applications were available to demonstrate the power and utility of the manybody perturbation theory approach to the molecular structure problem. Although it should be noted that programs for evaluating the triple excitation component of the correlation energy, which remains the most demanding part of many contemporary calculations, were already extant in 1981.56 Today, the methods presented in the 1981 report are firmly established as the most widely used ab initio quantum chemical technique for molecular electronic structure studies. The theoretical background of the methods are now described in both undergraduate and graduate text books, such as those by P.W. Atkins and R.S. Friedman (Molecular quantum mechanicss7) and by R. McWeeny (Methods of molecular quantum mechanics5‘). Perturbation theory based calculations are increasingly being exploited in a problem solving r6le in a wide range of research areas in the molecular sciences. This is due, to a large extent, to the availability of efficient computer software, most notably the GAUSSIAN series of program^,'^ across a range of computer platforms. Such programs have made quantum chemical calculations, in general, and perturbation theory treatments of correlation effects, in particular, available to a broad user base. Calculations, especially when accessed via a graphical user interface, such as the UNICHEM program6’ (see also ref. 61), have become almost routine and readily accessible to the nonexperts. Particularly popular have been loworder studies, which, because of their computational efficiency, have facilitated applications to systems of a size not accessible to more computational demanding methods. Now, the simplest form of manybody perturbation theory is that taken through a secondorder with respect to the socalled MollerPlesset reference H a m i l t ~ n i a n . ~This ~ “ ~is often designated “MP2”, particularly by users of the GAUSSIAN quantum chemical programs. This abbreviation, which appears unique to the field of quantum chemistry, was employed in a literature search over the past decade. The results are displayed in Figure 1. In 1989, the abbreviation “MP2” was used in the title or key words of just three publications, but in two years this number had risen to 173, by 1995 it stood at 603 and in 1998 grew to 854. A similar analysis was performed for density functional theory using the abbreviation “DFT”, an abbreviation which is also used to a very limited extent in cardiovascular research. In 1989, the number of occurrences of “DFT” in titles and key words stood at 7, increasing to 65 within two years and to 279 by 1995. In 1998 the incidence of “DFT”in titles and key words was 733. In Figure 1, we also display any number of occurrences of the abbreviation “CCSD” (coupled cluster with single and double excitations) which is sometimes interpreted as an “infiniteorder”, manybody perturbation In 1976, Pople, Binkley and Seeger63 listed the properties that ideally one would like to ascribe to a “theoretical model chemistry”. Almost a quarter of a
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looo
i4umbor of publkrtiono
1989 1990 1991 1992 1993 1994 1996 1996 1997 1998
Year of publication ~
mMP2
=DFT
CCSD
Figure 1 Incidence of the acronyms “MP2”, “DFT” and “CCSD” in the title or keywords of publications in each year over the past ten years
century later, it is interesting to reexamine their list in the light of more recent developments. They suggested that a useful theoretical method should (i) provide well defined results for the energies of the electronic states for any arrangement of fixed nuclei, leading to a set of continuous potential energy surfaces; (ii) require a level of computation which does not increase too rapidly with the size of the system; (iii) be sizeconsistent; (iv) yield upper bounds to the exact solution. Each of these points requires some modification in the light of more recent work. The first point is taken to include arbitrary electronic states and arbitrary dispositions of the nuclei, including dissociative processes. The advent of parallel computing machines has necessitated the modification of the second point somewhat in that, provided a method can be effectively implemented in a parallel processing environment exhibiting near linear speedup factors, an increased amount of computation with system size can be accommodated provided the elapsed time does not increase markedly. The term “sizeconsistent”63 (see also ref. 201) and the more recently introduced term “sizeare used interchangeably by many authors causing some confusion. It is preferable to state that a theoretical method should support energy and other expectation values which scale linearly with the number of electrons in the system. The last point, point (iv), has been largely discarded. A good approximation to the exact solution is preferable to a poor approximation which happens to provide an upper bound. Furthermore, to be really useful an
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upper bound should be accompanied by a lower bound of comparable quality. The determination of accurate lower bounds remains a challenge for molecular structure theorists. To the properties of a theoretical model chemistry, listed by Pople, Binkley and Seeger63in 1976, the present author would add the following requirements: (i) be amenable to systematic refinement. If a particular level of theory is not adequate for a given application then one would like to be able to refine the approach until a satisfactory description is obtained; (ii) 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; (iii) provide a rigorous account of relativistic and quantum electrodynamic effects within a unified theoretical framework. Hence a revised list of properties that one might wish for a theoretical model chemistry to have reads as follows: (i) 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; (ii) be amenable to systematic refinement; (iii) 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; (iv) support energy and other expectation values which scale linearly with the number of electrons in the system; (v) provide a rigorous account of relativistic and quantum electrodynamic effects within a unified theoretical framework. We reconsider this list at the end of this review and briefly discuss the extent to which the manybody perturbation theory satisfies these requirement in applications to the molecular electronic structure problem. The plan of this report is as follows: in Section 2 an overview of the theoretical apparatus of the manybody perturbation theory is presented together a discussion of practical algorithms. A review of some of the more important applications made during the period covered by this report will be given in Section 3. Finally, in Section 4, future directions for research on the electron correlation problem in general and the manybody perturbation theory expansion in particular will be given. 1.1 A Personal Note.  My interest in the manybody perturbation theory approach to the correlation problem began with some lectures on diagrammatic techniques delivered by P.W. Atkins at the 1972 Oxford Theoretical Chemistry Summer School directed by the late C.A. Coulson. Atkins demonstrated beautifully how diagrammatic techniques could cut through complicated
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algebra and expose the essential physics of the problem in hand. Writing in the Oxford Theoretical Chemistry Progress Report in 1973, Coulson described the outlook as follows: “I ought to mention that my annual Summer School in Theoretical Chemistry is continuing, ... Dr. Peter Atkins, from the Physical Chemistry Laboratory, is giving a course on diagram techniques for the chemist. For some time I have been worrying about what new developments in theoretical chemistry are likely to be important in the next 1015 years. One has to hazard a guess. Twenty years ago we opted for group theory, and turned out to be absolutely right ... Today, something new must be encouraged. Rightly or wrongly we have decided on diagram methods. Our experience to date is highly favourable ... Peter Atkins has prepared some admirable lecture notes ... we shall now have to wait another 15 years to find out whether we have been successful in out forecast.” After the Summer School, I returned to Bristol, where I was working with the late J. Gerratt on an early implementation of the theory of spincoupled wave functions, and attended a course on advanced quantum theory, given by members of the Theoretical Physics Department in the Royal Fort which housed the Physics Laboratories. Through Gerratt, I had established contact with D.M. Silver at The Johns Hopkins University Applied Physics Laboratory. Silver, together with a postdoctoral coworker, R.J. Bartlett, had begun to explore the application of manybody perturbation theory to molecular systems using finite basis set expansions. I gained a postdoctoral position at Johns Hopkins University and arrived in December, 1974, to work with Silver. (Bartlett had departed during the summer.) Silver and I wrote computer program^^^^^ for performing calculations that were complete through third order in the energy and were soon able to demonstrate the accuracy of loworder manybody perturbation theory provided a well chosen reference was employed. l5 2 Theoretical Apparatus and Practical Algorithms 2.1 Quantum Electrodynamics and Manybody Perturbation Theory.  In his derivation of the linked diagram theorem of manybody perturbation theory in 1957, Goldstone7* made use of the formalism of quantum electrodynamics developed in the late 1940s by T ~ m o n a g a , ~S’ ~. ~h~w i n g e rand ~ ~ F. ~e ~~ n m a n . ~ ~ In particular, he used a timedependent formulation, even though the problem involves a timeindependent perturbation. In his work, Goldstone7’ introduced the graphical techniques into manybody physics making use of Feynmanlike diagrams. However, interactions were taken to be instantaneous and the effects of relativity were ignored. In recent years, the growing interest in the treatment of relativistic and quantum electrodynamic effects in atoms and molecules is necessitating the reintroduc
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tion of physics that has been known for over forty years. The appropriate machinery is Furry's bound interaction picture of quantum electrodynamic^^^ published in 1951. (For recent discussions see Quiney, Grant and Wilson3' 734 and Grant and Q ~ i n e y . ~ ~ ) Goldstone used a second quantized particlehole formalism based on an arbitrary choice of vacuum state. The interaction representation, which is intermediate between the Schrodinger and Heisenberg pictures, was employed and the energy was evaluated by the GellMannLow f~rmalism'~with Hamiltonian
where a is a factor which is set to 0 at the end of the discussion. The model state at t = 00 is allowed to evolve to the fully interacting state at t = 0
and Wick's theorem7' is employed to evaluate the timeordered products of creation and annihilation operators arising in the evolution operator, Ua(0, a)A more symmetrical treatment was presented by Sucherso which allows the interacting state to evolve to t = +oo
This S matrix approach leads to the modified GellMannLow formula for the energy shift za
a
AE = limln(Sa) a+O 2 aA
(4)
where A, which is set to 1 at the end of the derivation, is a factor multiplying H I . The S matrix approach'' has the advantage that it can be represented in terms of fully covariant Feynman diagrams. The Furry bound state interaction picture of quantum electrodynamic^^^ relies on an expansion of the secondquantized electron field operator in terms of singleparticle solutions of the Dirac equation for a static external field. This external field may be thought of as some mean atomic or molecular potential, whose singleparticle spectrum can be divided into positive and negativeenergy branches. This can always be done for the usual elements of the Periodic Table, although problems arise for superheavy atomic nuclei. Investigation of the connection between the ManyBody Perturbation Theory (MBPT) approach and the Furry representation of Quantum Electrodynamics (QED) has been shown to allow a precise definition of QED effects.'* Every MBPT diagram has a corresponding Feynman diagram, but there are Feynman diagrams that have no MBPT counterpart.
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Figure 2 Examples of MBPT and QED diagrams
In Figure 2, examples of MBPT and QED diagrams are given. Sapirstein has observeds2 “The interface of these two apparently diflerent, but actually intimately related approaches contains a great deal of beautiful physics.’’
QED provides a framework for describing the role of the negative energy states in the Dirac theory and the divergences which arise in studies of electrodynamic interaction^.^^^' In Section 2.3 it will be shown how the use of the algebraic approximation to generate a discrete representation of the Dirac spectrum has opened the way for the transcription of the rules of QED into practical algorithms for the study of manyelectron systems. We now have a well defined prescription for the calculation of the properties of atoms and molecules within a relativistic formulation. As in the nonrelativistic case, the relativistic manybody perturbation theory becomes increasingly complicated in higher orders and in practice it is possible to take the expansion to about fourth order with the size of basis set that is required for calculations of useful accuracy. Sapirstein82 has recently reiterated the view that “Allorder methods allow for certain subsets of MBPT expressions of higher order to be calculated, but in general other subsets will remain uncalculated.’’
As in nonrelativistic studies, it is often convenient to assume that those terms which are difficult, or indeed impossible, to calculate are negligible; an assumption which may have little justification in reality. 2.I .I The Ndependence of perturbation expansions.  Historically, there appeared two distinct approaches to perturbation theory for quantum systems  that associated with the names of LennardJones,86 Brillouins7 and Wigne88
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and the approach proposed by Rayleigh and S ~ h r o d i n g e r . ~ Both ~ ’ ~theories begin by writing of the total Hamiltonian operator in the form
where HO is the zero order Hamiltonian defined by the model problem, which for manybody systems of usually an independent particle models and, in particular, the HartreeFock approximation. H1 is the perturbation and ll is a parameter by means of which the perturbation is “turned on”. The Schrodinger equation for the perturbed system is written
where the eigenfunctions are normalized (Y I Y) = 1
(7)
For the unperturbed “model” system, the Schrodinger equation may be written
where the model eigenfunctions, @, are normalized
(0I @) = 1 and satisfy the socalled intermediate normalization condition (@ I Y) = 1
The exact wave function can then be written as IT) := 10) + I X )
with @ and x satisfying the orthogonality condition (@
Ix) =0
The Schrodinger equation for the perturbed system may be written
where
AE=&E
(9)
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Rearranging the perturbed Schrodinger equation (13) and using (8), we get (&  HO  AH1)z = (AH1  AE)@
(15)
The reduced resolvent, Go(z),of HOis defined by
Go(z).(z Ho) = Q
(16)
where Q is the projection operator
Q = 1 
[email protected]>(@(
(17)
Applying GO(&)to the left hand side of (1 5)
which after formal inversion of the operator in brackets gives
The perturbation expansion of LennardJones,86 Brillouin8' and Wigner88 is obtained from (19) by making a Taylor expansion in powers of A
The energy correction is obtained from (9,(6) and (14), together with (9), (10) and (12 ), as
or, substituting (1 l), as
so that, on substituting (20) into (22), we get
The RayleighSchrodinger perturbation series is obtaining by applying Go(E) from the left to equation (1 5) after rewriting it in the form
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This results in the following equation [l  Go(E)(LH1 AE)]x = iZGo(E)
[email protected] (24)
which, by formally inverting the operator in brackets [...I, gives
x = L [ l  Go(E)(AHl  AE)]'Go(E)
[email protected] (25)
and expanding in powers of L, gives the following expansion for x:
x
= {LGo(E)Hl+ LGo(E)(LHl  AE)Hl+
LGo(E)(AHl  AE)Go(E)(LHl AE)H1 + ...}@
(26)
Substituting (26) into (2 1) gives an expansion for the energy correction AE
)@I +
AE = L(@IHi A2(@1HiGo(E)
[email protected])+ A2((DlH1Go(E)(AH1 AE)Go(E)
[email protected])+ ...
(27)
The RayleighSchrodinger perturbation expansion is obtained by ordering AE in powers of 1
The resulting energy coefficients are
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This should be compared with the energy coefficients in the LennardJones, Brillouin, Wigner expansion (23) which can be rewritten in the form
where the energy coefficients are
The LennardJones, Brillouin, Wigner perturbation expansion is not a simple power series in 1 since each depends on the exact energy, 1. Each energy coefficient in the RayleighSchrodinger perturbation expansion consists of a principal term of the form
ak)
which is similar in form to the coefficients in the LennardJones, Brillouin, Wigner series but depends on Go(E) rather than Go(€), together with (in all orders beyond the second) a “renormalization” term such as
Let the eigenstates and eigenvalues of the zero order Hamiltonian, Ho, be OF and Ep, ( p = 0, 1,2, ...,oo),respectively, which we assume to be discrete. Note that = @o. The “spectral representation” of the reduced resolvent then takes the form
Introducing (40) into the expressions (2932) for the RayleighSchrodinger energy coefficients yields the sumoverstates form. For the second order energy we have
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Id2)= (@]Hi Go(E)
[email protected]) p= 1
and for the third order energy we have
p= 1
The fourth order energy component takes the form
and the sumoverstates form of the fourth order RayleighSchrodinger energy component takes the form
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2.I .2 The Linked Diagram theorem.  Methods for describing electron correlation effects in atoms and molecules leading to expressions for the correlation energy and other expectation values which scale linearly with the number of electrons in the system are based, directly or indirectly, on the linked diagram theorem. This theorem has its origins in the work of Br~eckner,~' Goldstone'' and others92393 in the mid 1950s. It can be easily shown that the LennardJones, Brillouin, Wigner perturbation expansion the energy coefficients are not proportional to the number of electrons, N , in the system. This is apparent even in second order since the exact energy for the system occurs in the denominator of the sumoverstates expression. In contrast, the second order RayleighSchrodinger energy coefficient is proportional to N . Brueckner investigated the Ndependence of some higher order terms in the RayleighSchrodinger expansion. He demonstrated that in third and fourth order there are terms which have a nonlinear dependence on N, but that, in each order, these terms mutually cancelled leaving only terms having a linear dependence on the number of electrons in the system. The third order term takes the form
The contributions to the first term which have a nonlinear dependence on N
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can be shown to be exactly cancelled by the second term. In sumoverstates form these terms are
p,v= 1
When the single determinant manyelectron functions @ are constructed from canonical HartreeFock orbitals, the excited functions, and QV, are doubly excited with respect to the reference function 0.The second term in the third order energy expression cancels diagonal components for which p = v in the first term. The principal term in the fourth order energy expression has the form
p,v=l
Both QP and @A can involve double replacements with respect to the single determinantal reference function 0 constructed from canonical HartreeFock orbitals. The intermediate state (11, can involve single, double, triple and quadruple replacements with respect to the reference function. The renormalization terms in (44)involve only double replacement. Brueckner” showed that diagonal terms in the principal term which have a non linear dependence of N are completely cancelled by the renormalization terms. This cancellation is incomplete if the level of replacement employed in generating the intermediate states is restricted. Brueckner” did not obtain a general proof of the cancellation of unphysical terms in each order of the RayleighSchrodinger perturbation series because the explicit demonstration of this cancellation becomes more tedious in higher orders. Goldstone,” Hubbardg2 and H ~ g e n h o l t zfound ~ ~ general proofs. The most elegant of these was that published by Goldstone using the methods of quantum electrodynamics. This proof has often been regarded as somewhat esoteric. K ~ t z e l n i g gdescribes ~~ the proofs of G~ldstone,~’ Hubbard205 and Hugenholtzg3 as “unexpectedly farfetched and not easy to understand”. In fact, quantum electrodynamics (QED) provides a theory for electron interacting via a quantized radiation field. With the growing recognition of the importance of relativistic and QED effects, the Goldstone treatment might appear the most natural. For example, the timedependent formalism used by Goldstone which appeared somewhat unnecessary in the treatment of timeindependent, nonrelativistic problems is entirely appropriate providing a well defined prescription for the calculation of the properties of atoms and molecules within a relativistic/QED formulation.
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In Goldstone's treatment, the model state at t = 00 is allowed to evolve to the fully interacting state at t = 0. The timedependent Schrodinger equation
.a q t )
H @ ( t ) = 1
at
with
H
= No I311
(49)
where the precise form of 310 and 311 need not be specified at this stage. The interaction representation, which is intermediate between the Schrodinger and Heisenberg pictures, is employed, so that
and
which gives
.a
I$
at
=
This can be integrated to yield
The time evolution operator, Ua(t , to) itself satisfies the timedependent Schrodinger equation
a
iat U(t,to) = H ( t ) U ( t ,t o ) with the initial condition U ( t 0 , t o )= 1
(55)
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which is equivalent to the integral equation
This integral equation can be formally solved by iteration
Introducing the chronological ordering operator,
with
allows the introduction of an .common upper limit in the Dyson form U ( t ,to) = 1
+
xn= 1
dt,
(i)n n!
l'
...
n=l
The energy can be evaluated by the GellMannLow formalism7* with Hamiltonian H
= 310
= 310
+ e%,
+
7 i 1 (t)
where a is a factor which allows the perturbation to be switched on adiabatically and which is eventually set to 0.
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38 1
The wave function for the fully interacting state is written
$(o) = u++O lim uu(O, oo)$(oo)
(64)
Writing the total energy as a sum of the zeroorder energy and a level shift, i.e.
gives
Remembering that the time evolution operator satisfies the timedependent Schrodinger equation (55) allows (66) to be written as
or
Thus, the determination of AE requires the evaluation of the evolution operator with respect to the unperturbed wave function. To make further progress, the zeroorder Hamiltonian and the perturbation must be written in second quantized form. Recall that the annihilation operator, ai, and the creation operator, a:, satisfy the following anticommutation relations
"1 ai,aj
+
+
= 6,
where [x, y]+= xy yx and 6u is the Kronecker delta. The relations (69) ensure that the system obeys FermiDirac statistics. The normal product of the operators xj, xj2...xj,, where x = a or at, is defined by
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where p is the parity of the permutation operator (not uniquely defined)
The vacuum expectation value of an operator in normal form vanishes. A contraction (or pairing) of two creation or annihilation operators is defined by
A normal product with contractions is defined by
wherep is the parity of the permutation 1 2 il jl Wick's
... 2r 2r + 1 ... ... j , kl ...
k,
(74)
states that
where the summation runs over normal products with all possible contractions. The vacuum mean field of a product of creation and annihilation operators can be reduced to a sum of all possible fully contracted terms since terms which are not fully contracted vanish:
In second quantized form the zeroorder Hamiltonian may be written in terms of creation and annihilation field operators in the form
or in terms of creation and annihilation operators as
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The perturbing operator may be written in terms of creation and annihilation field operators in the form
or in terms of creation and annihilation operators as
In describing manybody systems in their ground or low lying excited states it is convenient to redefine the vacuum state to contain the single particle states occupied in the ground state, 0. This is usually termed the Fermi vacuum. A set of creation and annihilation operators can be defined with respect to the Fermi vacuum as follows not occupied in
Q,
occupied in 0 not occupied in occupied in Q,
Q,
These operator satisfy the anticommutation relations
[bf,bj] = 0
+
[bj,bj] + = 6~ In this particlehole formalism, the normal product is written
and Wick's theorem takes the form
all contractionr
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2.2 Manybody Perturbation Theory.  The zero order Hamiltonian operator is the sum of oneelectron terms N
+
7 f o = C(h(i)V N ( i ) ) i= 1
where h(i) is the sum of the kinetic energy term and the term arising from the interactions between the electrons and the nuclei.
The VN(i)operator is the Fock operator and describes the averaged interactions of the electrons in the system. The perturbation operator is sometimes termed the fluctuation potential. It consists of a sum of terms describing the electronelectron interactions together with a term compensating for the Fock operator in the zero order Hamiltonian. N
N
i>j= 1
i= 1
The HartreeFock energy is the sum of the zero order and the first order energy coefficients
The correlation energy is given by the sum of the energy coefficients in second and higher order
In practice, for a well chosen reference, the second order energy component accounts for the vast majority of the correlation energy. The truncation perturbation expansion may be written
The zero order energy is just the sum of the orbital energies of the occupied spin orbitals
i=occupied
The first and higher order terms can be written in terms of the orbital energies of
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both the occupied and the unoccupied spin orbitals together with the matrix elements
and
which describe one and twoelectron interaction, respectively. Specifically, each term in the perturbation series consists of a sum over occupied and unoccupied spin orbitals of a quotient, the numerator of which is a product and one and/or twoelectron matrix elements and the denominator being a product of factors consisting of sums of orbital energies. Diagrams provide a more physical representation of the terms which arise in the manybody perturbation series than the algebraic expressions. Moreover, the diagrams can be related to the corresponding algebraic expression via a precise set of rules. A number of different diagrammatic conventions are in common use. Here we adopted one of the most widely used that first introduced by bran do^.^' The basic elements of the Brandow diagrams are shown in Figure 3. These are a oneelectron matrix element, a twoelectron matrix element, a particle line represented by a line with an upward directed arrow and a hole line represented by a line with a downward directed arrow. Particle lines represent the particle lines created above the Fermi level when an electron is excited whilst hole lines represent the hole which are simultaneously created below the Fermi level. A timedependent physical interpretation of the diagram may be given. An example of such an interpretation is given in Figure 4. The algebraic expression corresponding to a given diagram of the Brandow type may be obtained by following the following simple set of rules: ( a ) Downward directed lines are labelled by a unique “hole” index: i,j , k, I, ... (6) Upward directed lines are labelled by a unique “particle” index: a, b, c, d, ...
Figure 3 Basic elements of the diagrams introduced by Brandow: (a) particle line; (b) hole line; (c) oneelectron interaction; (d) twoelectron interaction
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to
tl
t2
t3
Figure 4 Interpretation of diagrams in terms of the particlehole formalism
Figure 5 Types of oneelectron interactions which can arise in diagrams
(c) There is a summation over each unique hole and each unique particle index, covering all possible values of these indices. ( d ) The numerator of the summand consists of the product of one and/or twoelectron matrix elements. The possible types of oneelectron integral which can arise are shown in Figure 5. The twoelectron matrix elements include the permutation operator which interchanges the coordinates of the two electrons. The possible types of twoelectron matrix element are summarized in Figure 6 . The indices p , q, r, and s are read from the diagram in the order leftout, rightout, leftin, rightin, respectively. ( e ) The denominator of the summand consists of a product of factors of the form
There is a summation of the type for each part of the diagram lying between adjacent interaction lines. The first summation extends over all hole lines
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Figure 6 Types of twoelectron interactions which can arise in diagrams
between two adjacent interactions whilst the second extends over the corresponding particle lines. (f) Equivalent lines are defined as those which both start at a given interaction line and end at another with both arrows pointing in the same direction. There is a multiplicative factor of
where n is the number of equivalent pairs of lines in a diagram. ( g ) There is a further multiplicative factor of (
where h is the number of unique hole lines and I is the number of Fermion loops, i.e. the number of continuous loops in the given diagram.
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Figure 7 An example of the translation of a diagram into the corresponding algebraic expression
(4 A
(b) A
Figure 8 Second order correlation energy diagrams for a closedshell system described in zero order by a single determinantal HartreeFock wave function constructed from canonical orbitals. The one Brandow diagram of this type is shown in (a). The exchange diagrams are shown in Goldstoneform in (b)
An example of the translation of a diagram into the corresponding algebraic expression is given in Figure 7. 2.2.1 Closedshell molecules.  Secondorder energy components. The secondorder component of the correlation energy for a closedshell system described in zeroorder by a single determinantal wave function constructed from canonical HartreeFock orbitals is described by the diagram shown in Figure 8a. It describes the excitation of a pair of electrons at some time to creating a two holes below the Fermi level and two particles above the Fermi level followed at some later time tl by the return to the Fermi vacuum. This term consists of a direct part associated with the Goldstone diagram shown in Figure 8a and an exchange part associated with the Goldstone diagram shown in Figure 8b. The corresponding algebraic expression takes the form
This is the expression given in 1934 by Mnrller and Plesset.62It usually accounts for the vast majority of the correlation energy of a system which is well described by a closedshell single determinantal reference function. The remaining second order energy components for a closedshell system described by a single determinantal reference function are associated with the diagrams collected in Figure 9a. These diagrams involve intermediate states
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Figure 9 Second order correlation energy diagrams for a closedshell system described in zero order by a single determinantal wave function. The Brandow diagrams of this type are shown in (a). The exchange diagrams are shown in Goldstone form in (b)
which are singly excited with respect to the single determinant reference function. The first of the diagrams involves only oneelectron interactions, the second and third diagrams involve both oneelectron and twoelectron selfenergy or “bubble” interactions, and the fourth diagram involves two “bubble” interactions. The second, third and fourth diagrams also give rise to exchange terms and the Goldstone diagrammatic representation of these is given in Figure 9b. The corresponding algebraic expressions may be written
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Third order energy components. There are three third order energy components for a closedshell system described in zero order by a single determinantal wave function constructed from canonical HartreeFock orbitals. The diagrams are displayed in Figure 1Oa. All three diagrams only involve intermediate states which are doubly excited with respect to the reference. Each diagram describes a correlation process which begins with the an excitation at time to leading to the creation of two holes and two particles. The third order diagrams shown in Figure 10a are distinguished by the nature of the central interaction line “particleparticle” (pp) in the case of A , “holeparticle” (hp) in the case of B, and “holehole” (hh) in the case of C. Thus in case A at time tl the two particles that were created at time to are annihilated and two further particles created. In case B at time tl one of the holes and one of the particles created at time to are
Figure 10 Third order correlation energy diagrams for a closedshell system described in zero order by a single determinantal HartreeFock wave function constructed from canonical orbitals. The three Brandow diagrams of this type are shown in (a). The exchange diagrams are shown in Goldstoneform in (b)
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39 1
annihilated and a further holeparticle pair is created. Finally, for case C two holes are destroyed at time tl and two new holes created. The exchange diagrams in Goldstone form corresponding to the Brandow diagrams shown in Figure 10a are given in Figure lob. The corresponding expressions take the form
There are two and threebody components of the hp term and two, three and four body components of the hh term, whereas thepp term has only a twobody component. These components are shown is Figure 11. There are a total of 43 third order Brandow diagrams for the correlation energy when the expansion is developed with respect to an arbitrary single
Figure 11 Twobody, threebody and fourbody correlation energy diagrams which can arise through third order for a closedshell system described in zero order by a single determinantal HartreeFock wave function constructed from canonical orbitals
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Figure 12 Third order correlation energy diagrams involving singly excited intermediate states and three oneelectron interactionsfor a closedshell system described in zero order by a single determinantal wave function
determinant. The remaining 40 terms may be divided in to those which involve only singly excited intermediate states, those which contain both single and double replacements and those which contain only double replacements. There are 20 terms which only involve single replacements and they can be further subdivided into: (i) those which contain three oneelectron matrix elements in the numerator. There are two of these. The diagrams are shown in Figure 12 and the algebraic expressions may be written
(ii) those containing two oneelectron matrix elements in the numerator. There are seven of these. The diagrams are collected in Figure 13a and the expressions have the form
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E (a6) K
w
G Figure 13 Third order correlation energy diagrams involving singly excited intermediate states and two oneelectron interactionsfor a closedshell system described in zero order by a single determinantal wave function. The seven Brandow diagrams of this type are shown in (a). The exchange diagrams are shown in Goldrtone form in (b)
The exchange terms which correspond to the above Brandow diagrams are given in Goldstone form in Figure 13b. (iii) those containing only one
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oneelectron matrix element in the numerator of which there are eight. The Brandow diagrams are shown in Figure 14a. The energy expressions are as follows:
The exchange terms which correspond to the above Brandow diagrams are given in Goldstone form in Figure 14b. (iv) finally, there are three terms which involve only twoelectron matrix elements and only singly excited intermediate states. The Brandow diagrams of this type are shown in Figure 15a. Examples of the exchange terms which correspond to these Brandow diagrams are given in Goldstone form in Figure 15b. The corresponding expressions are
This completes our discussion of the third order terms which involve only single replacements in intermediate states. There are 16 third order terms which involve both single and double replacements in intermediate states. They may be further subdivided into (i) those which involve two oneelectron matrix elements in the numerator of which there are two. The relevant diagrams are shown in Figure 16a, examples of the
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E EG E (b10) P
( b l l )P
(bl2) P
E (b16) R
Figure 14 Third order correlation energy diagrams involving singly excited intermediate states and one oneelectron interac'tionsf o r a closedshell system described in zero order by a single determinantal wave function. The eight Brandow diagrams of this type are shown in (a). The exchange diagrams are shown in Goldstoneform in (b) (continued overleaf)
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(b17) R
9 (b21) S
Figure 14 continued
G
(b19) S
(b18) R
(622) T

(b23) T
(b24) T
n
Figure 15 Third order correlation energy diagrams involving singly excited intermediate states and only twoelectron interactions f o r a closedshell system described in zero order by a single determinantal wave function. The three Brandow diagrams of this type are shown in (a). Examples of the exchange diagrams are shown in Goldstone form in (b)
corresponding exchange terms are given in Goldstone form in Figure 16b. and the algebraic expressions may be written:
(ii) those which involve one oneelectronmatrix element in the numerator. The eight diagrams of this type are shown in Figure 17a with examples of the
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Figure 16 Third order correlation energy diagrams involving both singly and doubly excited intermediate states and two oneelectron interactionsfor a closedshell system described in zero order by a single determinantal wavefunction. The two Brandow diagrams of this type are shown in (a). The exchange diagrams are shown in Goldstone form in (b)
Figure 17 Third order correlation energy diagrams involving both singly and doubly excited intermediate states and one oneelectron interactions f o r a closedshell system described in zero order by a single determinantal wave function. The eight Brandow diagrams of this type are shown in (a). Examples of the exchange diagrams are shown in Goldstone form in (b)
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exchange diagram given in Goldstone form in Figure 17b. The associated algebraic expressions may be written as follows:
(iii) those which involve only twoelectron matrix elements. There are six diagrams of this type and they are shown in Figure 18a. Examples of the
(4j
(a2) i
(a6) m
(a4)k
u
Figure 18 Third order correlation energy diagrams involving both singly and doubly excited intermediate states and only twoelectron interactions for a closedshell system described in zero order by a single determinuntal wuve function. The six Brandow diagrams of this type are shown in (a). Examples of the exchange diagrams are shown in Goldstone form in (b)
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corresponding exchange diagrams are given in Goldstone form in Figure 18b. The corresponding expressions are
This completes our discussion of the third order terms which involve both single and double replacements in the intermediate states. There are seven third order terms which involve only double replacements. They may be subdivided into (i) those that contain one oneelectron matrix element in the numerator. This set of diagrams is collected in Figure 19a with examples of the associated exchange diagrams given in Goldstone form in Figure 19b. The corresponding expressions are
(ii) those which involve only twoelectron matrix elements in the numerator. There are a set of five diagrams are this type. Three of these have already been
Figure 19 Third order correlation energy diagrams involving doubly excited intermediate states and one oneelectron interaction for a closedshell system described in zero order by a single determinantal wavefunction. The two Brandow diagrams of this type are shown in (a). The exchange diagrams are shown in Goldstone form in (b)
400
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Third order correlation energy diagrams involving doubly excited intermediate states and only twoelectron interactions for a closedshell system described in zero order by a single determinantal wave function. There a r e j v e Brandow diagrams of this type, three aregiven in Figure 9 and the others are shown in (a). The corresponding exchange diagrams are shown in Goldstoneform in (b)
considered in Figure 10a. The other two diagrams of this type are displayed in Figure 20a with examples of the exchange diagrams displayed in Figure 20b in Goldstone form. The corresponding terms are
This completes our discussion of the third order diagrams and associated algebraic expressions for a single determinant reference function. 2.2.2 Openshell molecules.  The choice of zero order Hamiltonian is crucial to the success of perturbative approaches to the correlation problem and this choice is not as straightforward for openshell systems as it is in the case of closedshell species. A number of openshell version of the manybody perturbation theory have been developed over the years beginning with the work of HubaE and t a r ~ k in y ~1980, ~ which was extended through fourth order in the energy by the present author.97 Much of the work98103on openshell systems has concentrated on states of high spin. The description of arbitary openshell systems requires a multireference perturbation theory which is discussed in Section 2.6. 2.3 Relativistic Manybody Perturbation Theory.  Since the early 1980s, we have witnessed a growing interest in the effects of relativity on the electronic structure of atoms and molecules. Over the past decade the theoretical and computational machinery has been put in place for a relativistic manybody perturbation theory of atomic and molecular electronic s t r u ~ t u r e . * ~  ~ ~  ~ ~ * ~ ~ Pyykko'04 has surveyed the influence of relativity on periodic trends and provided a concise summary of previous reviews of relativistic electronic structure theory. This development can be seen, on the one hand, as a result of a growing awareness of the importance of relativity in accounting for the properties of heavy atoms and molecules containing them. The inadequacy of
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physical models which either neglect relativity or which treat it as a small perturbation has fueled this development. For heavy atoms, the mean speed of electrons close to the nucleus is a substantial fraction of the speed of light, so that a fully relativistic electronic structure theory is required, both at the independent particle level and beyond. The treatment of relativity and the introduction of quantum electrodynamic effects is essential since these may be more important than electron correlation in heavy elements, and there is no evidence that these effects are simply additive. On the other hand, the implementation of relativistic electronic structure theories is dependent on technological developments, which have resulted in computing machines powerful enough to make calculations on heavy atoms, and molecules containing them, meaningful. Vector processing and, more recently, parallel processing techniques are playing a vital role in rendering the algorithms which arise in relativistic electronic structure studies tractable. In a report presented by the author at the 1988 Symposium on Manybody , ~ ~suggested that methods in quantum chemistry held at Tel Aviv U n i ~ e r s i t ywe
“rfone were to hazard a guess as to where the major new developments in quantum chemistry will be in the next Jifteen years, the proper treatment of relativity and the introduction of quantum electrodynamic efects seem to us to be likely candidates” and “...firmfoundations for this project have already been laid down”.
An overview of the salient features of the relativistic manybody perturbation theory is given here concentrating on those features which differ from the familiar nonrelativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic manybody perturbation theory are considered in more detail below: the representation of the Dirac spectrum in the algebraic approximation is discussed; the nonadditivity of relativistic and electron correlation effects is considered and the use of the DiracHartreeFockCoulombBreit reference Hamiltonian demonstrated; effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. The theoretical description of any many body system is usually approached in two distinct stages. First, the solution of some independent particle model yielding a set of quasiparticles, or dressed particles, which are then used to formulate a systematic scheme for describing the corrections to the model. Perturbation theory, when developed with respect to a suitable reference model, affords the most systematic approach to the correlation problem which today, because it is noniterative and, therefore, computationally very efficient, forms the basis of the most widely used approaches in contemporary electronic structure calculations, particularly when developed with respect to a MarllerPlesset zero order Hamiltonian.
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discrete levels
n
negative energy branch
Schriidinger spectrum
Dirac spectrum
Figure 21 Schematic representation of the nonrelativistic and relativistic spectra for an electron in a typical mean field potential
In Figure 21, the nonrelativistic and relativistic spectra for an electron in a typical mean field potential are shown schematically. The most significant distinguishing feature is the presence of the negative energy branch for c < 2c2 in the latter. According to Dirac’’’ the levels in this lower continuum are always filled, subject to the Pauli principle, in the true vacuum thereby preventing the radiative decay of bound, positiveenergy states. The negative energy “sea” is not observable as an entity since only differences between states can be detected. The negative energy continuum and the occupied positive energy states are taken to be below the Fermi level within the particlehole formalism in which the events which occur in a relativistic manybody system may be described. In contrast to the nonrelativistic particlehole formalism, the relativistic extension involves no restriction on particle number. Electrons can be excited from the negative energy continuum into an unoccupied level in the positive energy branch thereby producing a vacancy in the lower continuum which can be related to a positron state. Only the total charge is conserved and not the individual numbers of electrons and positrons. Even hydrogenic systems are implicitly infinitely manybodied theories, making second quantized formulations mandatory, and, although more elegant formulations’06 have followed Dirac’s original theory, the physical content of these theories is essentially unchanged. For low energy processes electronpositron pairs will appear as intermediate states in perturbation summations. At higher energies ( 1 MeV) real electronpositron pairs may be produced. Dirac comments’07 ‘Oandfl>l The use of such a series is based on the observation that independent optimization of the exponents with respect to the energy in selfconsistent field calculations yields an almost linear plot of In [ k against k. The use of orbital exponents which form a geometric series was originally advocated by McWeeny"' and later by Reeves and Harrison'387139 and the idea was revived and extensively employed by Ruedenberg and his collaborators. The eventempered bais set concept was further developed by a number of authors. 156189 A number of significant advantages accrue to the use of eventempered basis sets: ( i ) They have only two parameters, a and D, which have to be determined for each group of atomic functions belonging to the same symmetry species as opposed to one optimizable orbital exponent per basis functions. The determination of orbital exponents by energy minimization is a nonlinear optimization problem and there is little possibility of performing a full optimization particularly for polyatomic molecules if all orbital exponents are taken to be independent. (ii) The further restriction of using the same exponents for all values o f t , so that there are only two nonlinear parameters, a and P, per atom, does not produce a large difference in calculated energies. (iii) The question of the proper mixing of basis functions, in terms of principal quantum number, is superfluous since no mixing is employed. ( i v ) It is evident that an eventempered basis set approaches a complete set in the limit a + 0, f l + 1, f l N + 00 as N + 00. ( v ) An eventempered basis set cannot become linearly dependent if fl > 1. (vi) The parameter fl provides control over practical linear dependence in the basis set. As the size of the basis set is increased, the determinant of the overlap matrix decreases and a point is eventually reached where reliable calculation is impossible with a given numerical wordlength. (vii) Eventempered basis sets have a unique "spacecovering" property. The overlap matrix for eventempered basis sets of exponentialtype functions or Gaussiantype functions has the band structure which is illustrated in Table 4. (viii) Restriction of the basis functions to fewer analytical forms leads to simpler and thus more efficient integral evaluation procedure. 2.4.3 Systematic sequences of basis sets.  Atomic and molecular electronic structure calculations are often performed using a single basis set which is constructed in an ad hoc fashion using experience gained in previous studies of similar systems and little effort is made to examine the dependence of calculated properties on the quality of the basis set. Ruedenberg and his c o  ~ o r k e r s ' ~ ~ ~ ' ~ ~ and the present a ~ t h o r ' ~ have ~  ' ~reiterated ~ the view first put forward by
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Table 4 Structure of the overlap matrix for an eventempered basis set on one centre
Schwartz thattthe convergence of calculations with respect to the size of basis set is a very important problem. In 1963, Schwartz wrote:’89a
The first essential in talking about convergence rates is to have an orderly plan of procedure. That is, one must choose a set of basis functions to be used and then gradually add more and more of thee terms to the variational calculation in some systematic manner. The old habit of picking the “best” (chosen by art) choice of ajixed number of terms is to be discarded fi one wants to see how the problem converges. Once one embarks on any very large scale program, such an orderly plan of attack would be natural just from bookkeeping considerations; this is, however, essential for any mathematical analysis of the convergence rates. Ruedenberg and his c o  w ~ r k e r s58’ ~devised ~ ~ ~ schemes for systematically extending eventempered basis sets. They noted that in order for an eventempered basis set to tend to a complete set as the number of functions, N , tends to infinity, a (a > 0) and p (p > 1) must be functions of N such that a = a(N)+0 asN
+
00
(153)
On the basis of atomic matrix HartreeFock calculations using eventempered sets of Gaussiantype functions of various sizes in which a and /3 were optimized with respect to the energy, Ruedenberg and his coworkers proposed empirical functional forms for the dependence of cc and p on N ; in particular, Schmidt and R ~ e d e n b e r g ”put ~ forward the forms In In p = b In N+b‘
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with
l are given in phartree. 62(l>is the diference between E2(l>and the corresponding.finite element energy
30s30p 40s40p 50s50p 60s60p 70s70p 80s80p 90s90p 100s1oop 56s36p
128.547 128.5147 128.5847 128.547 128.547 128.547 128.547 128.547 128.547
097 947 098 107 098 109 098 109 098 109 098 109 098 109 098 109 098 025
 192 05 1.070 192 067.484  192 073.232 192 075.800 192 077.133 192 077.899 192 078.373 192 078.684 192 078.671
28.4 12.0 6.3 3.7 2.4 1.6 1.1 0.8 0.8
detailed comparision of the methods. A comparsion of the correlation energy of the ground state of the neon atom at the sp limit is made in Table 17. The results suggest that finite basis set expansions are able to support a level of accuracy approaching that of finite element calculations for atoms.
4 Future Directions The manybody 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 function 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 molecular 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 incorplorated by recognizing that QED provides a covering theory for MBPT. In conclusion, it is worth recalling the list of properties that one might wish for a theoretical model chemistry to have. It reads as follows: 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; (ii) be amenable to systematic refinement; (iii) be amenable to efficient parallel computation so that the elapsed time
0)
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required for calculations does not increase too rapidly with the size of the system; (iv) support energy and other expectation values which scale linearly with the number of electrons in the system; (v) provide a rigorous account of relativistic and quantum electrodynamic effects within a unified theoretical framework. While the manybody perturbation theory satisfies most of these requirements, the development of a robust formalism for dissociative processes and for arbitrary excited states remains an obstacle to routine applications in these areas.
Acknowledgments
The support of the Engineering & Physical Sciences Research Council under Grants GR/L65567 and GR/M74627 is acknowledged.
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J. Kobus, L. Laaksonen and D. Sundholm, Comput. Phys. Commun., 1996,98,346. D. Heinemann, A. Rosem and B. Fricke, Physica Scripta, 1990,42, 692. T.H. Dunning, Jr., J. Chem. Phys., 1989,W 1007. R.A. Kendall, T.H. Dunning, Jr. and R.J. Harrison, J. Chem. Phys., 1992,96,6769. W. Klopper, J. Chem. E’hys., 1995,102,6169. D. Moncrieff and S. Willson, J. Phys. B: At. Mol. Phys., 1996,29,2428. D. Sundholm and J. Olsen, J. Chem. Phys., 1991,94, 5051 and references therein. D. Sundholm, J. Olsen, P. Mamqvist and B.O. Roos, in Numerical Determination of the Electronic Structure of Atoms and Molecules, ed. M. Defranceschi and J. Delhalle, Kluwer, Dordrecht, 1988. 245. D. Heinemann, B. Frick.e and D. Kolb, Phys. Rev. A, 1988,38,4994. 246. J.R. Flores and D. Kolb, J. Phys. B: At. Mol. Opt. Phys., 1999,32,779. 247. D. Moncrieff and S. Wilson, J. Phys. B: At. Mol. Opt. Phys., 1999,32, 5379.
237. 238. 239. 240. 241. 242. 243. 244.
8 N e w Developments on the Quantum Theory of Large Molecules and Polymers ~~~~~
BY JANOS J. LADIK
1 Introduction
In the last ten years, since the publishing of a book on the quantum theory of polymers,' the fast development of the field has continued. Another book treating mostly the applications of the theory was published in 1991.* (In this Report we shall consider as polymers only onedimensional covalently bound chains, I D stacks and 1D hydrogen bounded systems. For 2 and 3D polymer networks one has to refer to the literature3.) Since that time a large number of ab initio HartreeFock (HF) and correlation corrected band structure calculations have been published even on periodic polymers with larger unit cells. Of course we cannot discuss all these works here, but besides the description of the methods mainly used we shall show some illustrative examples. The large number of semiempirical calculations (including the valence electron Hamiltonian approach) could not be included in this Report. Before treating these band structure calculations in Section 3 it will be shown how methods mostly used in 1D aperiodic soldis can also be applied for very large molecules like proteins or DNA stacks (Section 2). There is a number of density functional (DFT) calculations on 1D polymers and their properties. Here again one has to restrict oneself to describing the most frequently used methods and presenting some of the applications (see Section 4). There has been an explosionlike development in the treatment of nonlinear optical properties of 1D polymers and clusters. In Section 5 besides referring to the latest book in this field: we have again restricted ourselves to the main methods applied and to some illustrative examples. In Section 6 soliton dynamics in 1D polymers and stacks will be discussed together with the presentation of the basics of the necessary formalism. After a short look to developments in the near future, ample references will close this chapter.
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2 The Treatment of Large Molecules Using Solid State Physical Methods Developed for Aperiodic Chains The ab initio H F treatment with correlation of large molecules is by no means a simple problem. On the other hand, without taking into account correlation effects only the ground state properties of a molecule in its equilibrium geometry can be calculated in a more or less reliable way. Further the standard method for the treatment of Correlation, the configuration interaction (Cl) method, cannot be used well even for medium size systems, because it is not size consistent. Therefore, one has to apply either some form of many body perturbation theory (MBPT) or the coupled cluster (CC) approach, both in a certain approximation (both methods are size consistent). The latter methods can be used even for medium size molecules without using too much CPU time, if one can find a reasonable way' to subdivide the molecule into different regions, uses orbitals localized into these regions and takes into account virtual excitations within these regions and between neighboring regions. We have tried this out using MoellePlesset MBPT in the second order (MP2)6 and the coupled cluster formalisms7 in the f 2 f 2 approximation for the four nucleotide bases' using a valence split (431 G ) basis set.' For the localization of the orbitals the Boys'' procedure was applied.' The Boys procedure has provided rather good localized orbitals even for the virtual n ones with a few exceptions. In those cases in addition to the Boys localization the EdminstonRuedenberg'2 localization procedure was used. " We have applied this method first to formaldehyde and to loxy3!zabutadiene (see Figure 1). Both calculations have shown, in the MP2 and T2f2 approximation, that the method provides about 90% of the correlation energy, if one takes into account only excitations to the first neighbors as compared with the case where the orbitals are not localized and one takes into account all excitation^.'^'^ Since in the case of the larger nucleotide bases we could not perform a full $& calculation, we could use the above described
'
I I
I I
I
i c, "\ C
/
11
F I
N
L c I

01
I I

I
I I
Figure 1 (a) Formaldehyde with its three subregions; (b) IOxyEazabutadiene with its three subregions
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H
H
H Figure 2
The parent molecule of the psoralene molecule series
approximation only with localized orbitals. Since the results seem to be reasonable ( 30 eV correlation energy in the case of cytosine which has 48 valence electrons) one can conclude that this approach gives tolerably good correlation energies. Of course if one _coyld havc used a much better basis and go at least until MP4 or include the T2T2 and terms in the coupled cluster method, one could most probably obtain in this way 6575% of the true valence correlation energy. It is obvious that the method sketched above  even if perhaps it could be used for somewhat larger molecules, like the psoralenes (see Figure 2)  cannot be applied to really large molecules like proteins or a larger segment of aperiodic DNA, because the number of first neighbor excitations between neighboring regions would quickly become prohibitively large. (One should mention that in our calculations we have neglected the correction which is necessary if one applies localized orbitals for correlation calculations. l4 However, most probably these corrections are not very significant in the case of larger molecules.) Therefore one is forced to look for methods used in the theory of disordered condensed matter (solids or liquids). The simplest of these systems is a disordered 1D chain for which fast methods are available to obtain the density of states (DOS) (in other words their level distribution) and the localized wave functions belonging to it. These methods together with some attempts to define an effective total energy per unit cell will be described subsequently.
2.1 The Negative Factor Counting Methods with Correlation and Methods to Calculate Effective Total Energy per Unit Cell of Disordered Chains.  2.1.1 The Matrix Block Negative Factor Counting Method.  Let us assume we have a linear chain of N units and take into account in the simple tightbinding (Huckel) approximation only one orbital per unit cell. (In this way we obtain the DOS belonging, for instance, to the valence bands of a binary or multicomponent system if the positions of these bands are not very different). The corresponding Huckel determinant will be tridiagonal if only firstneighbors’ interactions are considered,
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Here ui and are the usual Hiickel parameters and 1 denotes the unknown root (energy eigenvalue) of the determinant. In a disordered chain the values of Ei and are different from each other. The determinant IH(1)lcan be easily transformed into a didiagonal form with the help of successive Gaussian eliminations. To achieve this we must substract from the second row of the determinant the first row multiplied by P2/(ct1 1). This will eliminate the element P 2 . Continuing this procedure in a similar way we can obtain zeros for all the elements of the lower diagonal of the originally tridiagonal determinant. Therefore the determinant
i= 1
can be rewritten in the form where the diagonal elements of the determinant are given by the recursion relation.
which is the consequence of the Gaussian elimination procedure. Comparison of equations (2) and (3) clearly shows that for a given value of 1,the number of :must equal the number of negative factors E j ( 1 ) eigenvalues less than 1 (1i 1) in equation (3).'' [The calculation of the eigenvalues A1 for a long chain ( N = lo4 or lo3) is difficult but the computation of the factors ~~(1) with the help of equations (4a) and (4b) is very rapid.] By giving 1 different values throughout the spectrum and taking the difference of the number of negative quantities ~i(1) belonging to consecutive values of A, one can obtain a histogram for the distribution of eigerivalues (density of states, DOS) of H to any desired accuracy. This procedure was applied first by Dean'' for the calculation of the DOS of the vibrational spectra of disordered systems. For the electronic structure of long disordered polypeptide chains this method was first used by Seel.l6 In actual calculations, one has to compute the band structure for each component of the disordered chain assuming that it is repeated periodically.
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Then the values a1 (diagonal elements of H) can be determined from the positions of the bands of the components (the middle point or weighted middle points of the bands) and the offdiagonal elements f l i from the widths of the bands. The method is very fast, but from obvious mathematical reasons ( 5 diagonal or 7 diagonal matrices even in the first neighbors' interactions case) it cannot be applied for 2D or 3D aperiodic systems. The negative factor counting (NFC) method described above, using Dean's negative eigenvalue theorem,' can be easily generalized to disordered quasionedimensional systems with an arbitrary number of orbitals per site in an ab initio form.l7 In this case one has the secular determinant instead of a tridiagonal in a triblockdiagonal form,
=o
where Aiand Bi 1 are, respectively, the diagonal and offdiagonal blocks of the Fock matrix, and Si and Q i + l are the corresponding blocks of the overlap matrix. Since the chain is disordered generally Ai # A,, Bi+ 1 # B,+ 1, Si # S,, and finally Qj+ 1 # Q, (i # j ) . One can show17that ( 5 ) can be rewritten in the form det M(A) = det S'/2det(F  11) det S'j2 = det S'/2 det S*/2det(F  11) = det S det(F  21)
F = S1/2F S1/2 The original determinant IM(A)I can be easily brought to a diblockdiagonal form again with the help of successive Gaussian elimination. In this way one obtains for its diagonal blocks, in completely analogy to equation (4a),
This means that the original expression for the value of det M(1), if expressed in the form given by the last expression in equation (6) can be written as
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Here the quantities si are the eigenvalues of S, and A. are the roots of the generalized eigenvalue equation
Fcj = AjSCj
(9)
Further, uik(A)denotes the kth eigenvalue of the matrix block U,(A) defined by (7), and li is the dimension of the ith diagonal block. The matrices UXA) can be easily diagonalized for a given value of A. The number of negative uik(A)s must again equal to the number of eigenvalues A.that are less than the chosen Avalue. On changing A, the whole spectrum can be scanned again and the tota.1 DOS curve of the disordered system (taking into account all the bands) can be obtained in this way to any desired accuracy. In actual calculations one uses the socalled overlapping dimers approximation. That means that in the simplest case of a binary disordered chain ABAABA . . . one performs an ab initio AB, BA, AA, AB, BA etc. calculation. For the first dimer one obtains in this way for the Fock matrix
Al
=
FAA,B2 = FAB,A2 = FBB
and a similar expression for the overlap matrix of the dimer. For the second dimer BA one can write correspondingly
One should observe that the: matrix block FBB occurs in both expressions (1Oa) and (lob), respectively. Therefore we have used for it in the construction of the Fock matrix of the whole disordered chain the mean value FBB = 1/2(FBB(')+ FBB(')).Further one should realize that overlapping dimer approximation works only if one uscs for all dimers the same coordinate system. To find out the error cawed by the dimer approximation we have solved the problem of a disordered chain of 30 units both directly and with dimers.'' It has turned out that the DOSs obtained with the help of both methods hardly differ from each other. One should further mention that the above described socalled matrix block NFC method was extended also for the case of bridges between different segments of the disordered1 chainIg (extended matrix block NFC method) which was necessary to take into account the disulfur bridges in proteins. This rather complicated formalism we don't present here, but refer to the original paper.lg
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After determining the DOS histograms of a disordered chain, one can also compute the wave functions (eigenvectors) belonging to the different energy levels (which are defined by the step length in the grid used for the calculation of the DOS curves). For this the standard technique of the inverse iteration method2' can be used. The results obtained in the cases of disordered polypeptides and nucleotide base stacks show around the gap a strong Anderson localization to one or two amino acid residues, or base pairs, respectively (see below). 2.1.2 The Inclusion of Correlation in the Calculation of Density of States of Disordered Chains.  We can easily substitute instead of the matrix blocks Fjj and Fg into the determinant ( 5 ) such matrix blocks of dimers which take into account also the major part of the valence correlation. For this we can write for a dimer the more complicated expression given by Liegener21and by Liegener et a1.21Modifying their expression we can write
have the same definitions as before, the unitary matrix U Here Fdimer and is formed from the solutions ui of the generalized eigenvalue equation
and Z is the selfenergy matrix. We can apply for the diagonal elements of the selfenergy matrix, X(wj) in the MoellerPlesset (MP) many body perturbation theory (MBPT) in the second order (MP2) approxima tion
tIf we multiply equation (11) from the left by U + and from the right by U, take into account the normalization condition U+SU = , l , the relation U+FdimerU= E ~ and L apply ~ ~ the diagonal approximation, we recover the inverse Dyson equation in its diagonal approximation oi
=
Ej
+ (Z)i,i
(15)
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r= 1
In (13) the summation _runsover all the g basis functions of the dimer. Having constructed Fdinler according to equation (l1), (where [C(~i)]j,i has to be calculated in an iterative way) one can substitute Fdimer instead of Fdjmer into equation (5) using again the overlapping dimers approximation. Applying the matrix bloizk NFC method defined by equations (6)(8) with the matrices Fdimer, one can calculate the total DOS with correlation in the same way as before. 2.1.3 The Calculation of Efective Total Energy per Unit Cell.  The NFC method does not provide, however, an effective total energy per unit cell of a disordered chain. For this purpose one can apply either a cluster method22 or the socalled elongation method.23 In the cluster method one starts for instance with the first ten units (from the left) of a disordered chain clf say 100 units. In this way we obtain the total energy We can then shift the cluster stepwise to the right by one unit until we reach the cluster of units 91100, eliminating at the same time the furthest left unit, respectively. The cluster length remains in this way constant (10 units in our example). Adding all the cluster energies one can write
Since in this way the energy of each unit is counted several times, one has to substract from (16) the sum of energies of the previous clusters reduced by one unit: 91
i=3
The difference of equations (16) and (17) gives the total energy of the complete finite disordered chain
Dividing (18) by the number of units N one obtains an effective energy per unit cell for the disordered chain
E &luster =N
N
Ered
N
Seemingly this method requires very large CPU times, because many clusters have to be computed (in our example 91). However, in reality after calculating
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the first cluster one has to calculate for the next one only integrals between basis functions of the first one (without its furthest left unit) and the newly added monomer. The same set of integrals can be applied also for the clusters reduced by one unit (equation (18)). The elongation method, which is a generalization of the cluster approach described above, has been described in detail in a number of papers.23Therefore we sketch here only its main ideas. One chooses again a starting cluster and elongates it by adding new units stepwise at its right hand side. In this method at the second step one doesn’t eliminate the furthest left unit, but “freezes” it. This means that with the help of a localization procedure one localizes all the MOs of the new (with one unit elongated) cluster. In this way one can distinguish between the localized molecular orbitals (LMOs) of the “active” space and the “frozen” space (at the second step only one unit). The interactions between the LMOs belonging to the active and frozen space, respectively, are negligible also after an arbitrary number of elongations. The method finally calculates the energy of the total system using a density matrix approach (for details see ref. 23). The method until now has only been applied in different semiempirical formalisms and in an ab initio form to disordered (atactic) polypropene.22 2.2 Application to Proteins and Nucleotide Base Stacks.  The NFC method described above (in its matrix block f ~ r m ’ ~ *has ’ ~ been ) applied for different proteins and nucleotide base stacks. In these calculations the DOSs of active and inactive pig insulin2472’were first computed. The two forms of pig insulin have the same sequence, but somewhat different conformations. The calculations were extended also for the determination of the hopping conductivities of these proteins. The method of these calculations and the results are described in detail in refs. 2426 and in a review paper.27 Therefore the results obtained are mentioned only very briefly in this Review. As further steps the DOS calculations were applied also to active and inactive egg white l y ~ o z y m e , ~to~ hog ’ ~ ~ fish insulin,29 to subtilisin28 and to acidic phospholipase A2 (PLA2).30These proteins contain 901 25 amino acid residues and their structure has been determined by Xray diffraction. The calculations have used these structures. Their DOS histograms are similar, but their hopping conductivities are at high frequencies (o> 1Olo sl) in the range of lo’lo’ 0 cm’. Generally their conductivity is strongly dependent on their conformation, especially that of the active site of an enzyme.29 Computations were performed also for the nucleotide base stacks and base pair In the case of the cytosine ( C ) stack we have investigated the DOS and hopping conductivity as a function of the basis set and correlation corrections (inverse Dyson equation with MP2 self energy; see above). In Figure 3 we show how the DOS changes if one uses instead of an STO3G a 63 1G basis and applies correlation corrections. The hopping conductivity of the stack increases in the same time by 1 order of magnitude with the better basis and by further 2.5 orders using correlation corrected level schemes for the d i m e r ~ . ~ l In ’ ~this way one obtains at o = 10’’ s’ a value of 1 0’ cm’ for the hopping conductivity for a C stack of 100 units. (The same result was
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400
200
0 14
12
10
8
14
12
10
8
2
4
6
400
200
0
Figure 3
1
3
5
The DOS of a single aperiodic nucleotide base stack using (a) a STO3G basis, (b) a 631G basis arid (c) 631G + MP2
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obtained previously for a base pair stack of 100 units.34) Therefore all the conductivity values obtained for the proteins and base (pair) stacks with the help of the minimal, STO3G basis can be considered only as lower bounds. 2.3 Possible Application of the Negative Factor Counting Method to Large Molecules.  Finally it should be mentioned that if one combines the matrix block NFC method of a quasi ID system containing cross links (extended negative factor counting method; ENFC3’ method already applied for proteins with SS bridges2430)with the of the MOs, there is a hope that the NFC technique with correlation corrections21can be applied not only to aperiodic chains but also to large molecules. The only condition is that it should be possible to partition the molecule after localization of its MOs into consecutive subspaces. If this is possible one can apply the overlapping dimers approximation for the perimeter of the molecule and the remaining cross bonds can be treated like the SS bridges in proteins in the ENFC procedure. The idea is shown in the case of a naphthalene molecule (see Figure 4). In this procedure the original delocalized HF MOs would serve as starting point. After a of the MOs one can define “units” (which have two or a few centers) and “dimers” of two consecutive units. One can then define new Fock and overlap matrices for these “dimers”. After that one can proceed in the same way as in the case of dimers of an aperiodic chain: (1) solution of the generalized eigenvalue equation of the new Fock operators belonging to the
H
H
H
Figure 4 Application of the combination of the extended matrix block negative factor counting method with partitioning a molecule into subspaces with the help of localized orbitals in the case of a naphthalene molecule
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dimers, (2) application of equations (1 1)( 15) to correct the physically interesting energy levels for corre:lation, (3) application of the ENFC method to the whole molecule (in this case the matrix F in equation ( 5 ) would be constructed from the correlation corrected Fdimer matrices). The proposed method has not been applied yet, but we plan after coding to perform in this way correlation corrected calculations for the lowest unfilled and lowest filled levels of large molecules. One should be aware that most probably the method works the better, the more heteroatomic bonds are in the investigated molecules.
3 Correlation Corrected Eiiergy Band Structures of Different Periodic Polymers 3.1 Methods.  3.1.1 Inver'se Dyson Equation with MP2 Self Energy.  In the previous section this method was already formulated for molecules [dimers: see equations (1 1)(15)]. To apply it to periodic polymers one has only to substitute the level indices i, j , a , b in these equations by capital letters I, J , A , B. These capital letters stand for clombined indices: I = i, ki; J =j , kj; A = a, ka; and B = b, kb. Here i etc. stands for the ith band and ki etc. for a particular quasi momentum in the band. Of course the quasi momenta have to fulfil the conservation law
In actual calculations, especially in the case of broad bands, it can easily happen that the real part of equation (14) goes to zero. In such a case one has either to use the principal value theorem36or can shift the denominator of (14) by a small but finite quantity.37 Another problem is that if one solves the nonlinear system of equations (1 1)(15) i t e r a t i ~ e l y(the ~ ~ graphical solution is much slower), one of course obtains numerous solutions. In the cases of molecules and of insulators or semiconductor polymers with a not very small gap one can always find out the physically relevant solution on the basis of the condition for the pole strength,
which is always fulfilled for these systems. However, for metals (21) is not valid and therefore one cannot apply the inverse Dyson equation in its diagonal approximation (there are many PI values of order 0.14.2). For metallic polymers one has to return to the original electronic polaron in its ab initio formulationm (for details see also refs. 1 and 34). Closing Section 3.1.1, we should like to derive the inverse Dyson equation in the diagonal approximation (equation 15), because this is not known by all chemists. One can start from the oneparticle Green's matrix equation (Dyson
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equation in matrix form) with the Green’s matrix (Go) of the unperturbed uncorrelated system and that of the perturbed system (G)
respectively, defined as (in an already Fourier transformed form)
Here S is the selfenergy matrix, and the diagonal matrices o and E ~ respectively, contain the quasi particle (correlation corrected) and the H F oneelectron energies. Multiplying equation (22) from the left by G;’ and from the right by G’ one obtains G;’ = G’ Z. Applying for this equation the diagonal approximation (which in the case of the calculation of the excited or ionized states of molecules did not cause a large error41), one can write (G;l)I,I= (Gl)I,I (Z)[,, since (Gl)I,Iat a pole is 0, (01  &YF  XI,I = o), one obtains finally
+
+
which is the inverse Dyson equation (15 ) in a more detailed form and applied for a polymer. 3.1.2 Formulation of the Coupled Cluster Method for Quasi I D Polymers.  At the beginning of Section 2 the application of the coupled cluster method7 to larger molecules*3using localized MOs was briefly discussed. This method was formulated and applied (see Section 3.2.1) also for periodic polymers.4245Since the application of this formalism (which is perhaps the most reliable sizeconsistent method which can be used for correlation calculations in large systems) seemed to require very large CPU we had to localize in an optimal way43the delocalized crystal orbita!s (CO) to Wannier functions.46 In the case of periodic polymers in the T2T2 approximation of the coupled cluster method we can write7
(26) ij
a,b
is the ground state Slater determinant constructed from the localized Wannier functions Wi.1
~
,
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n being the number of basis functions per unit cell and N the number of cells in the chain, respectively. As it is well known Wj,lf is the Wannier function localized at site It determined by the COs of the band through the Fourier transform
(a is the elementary trans1at;ion).
Here qj(k,?) is the kth CO of the band j . Finally in equation (26) @$' is a doubly excited Slater determinant in which @O was excited from the filled . Wannier functions wi,/,, wj,!Jf to the originally unfilled ones Wa,lff , w~,~,,, Substituting the expression (25) into the Schrodinger equation H(Y) = EIY) one obtains
This provides the correlation energy projected on the HF ground state as
or
Projection of equation (29) on the space of doubly excited Slater determinants (expressed by Wannier functions) gives
Here the unprimed or simply primed indices refer to Wannier (W) functions occupied in the ground state, while the doubly and triply primed indices are in the ground state unoccupied W functions. Since the W functions, like COs, are orthonormal, expression (3 1) is exact; the higher powers of (f';) woulcl not give a nonzero contribution. We can rewrite equation (31) in the form
where Qoexc is a shorthand notation for the function on the left hand side of equation (31). Evaluation of Eq. (31) provides the coupled cluster doubles (CCD) system of nonlinear equations which can be written in a complex form
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Here
T2 can be expressed in a similar way but one has to write instead of l/rl2 the operator f2. In equation (33) the indices a, p and p' run over all double excitations and the matrices A and B are composed from two electron integrals and Fock matrix element^.^ As mentioned above a crucial step in solving the system of equations (33) is to localize the W functions as well as possible. As is well known, one obtains from the HF CO calculations crystal orbitals with an arbitrary phase factor e''i(k) of unit absolute value. For the best possible localization of the W functions one can require that the variation of the integral
with
should disappear
~J~[A = ,o; I
(37)
From equation (37) one obtains for the determination A,; the
A,k = arg [ / d ? ~ c x p [   i k , !,]At(? &) I
Here V, is the volume of the unit cell and the function AT contains the summation over the basis functions and their coefficients belonging to band j . Since in equation (38) :A occurs, it can be solved only iteratively, but from experience ~ b t a i n e dit~converges ~ . ~ ~ rapidly in a few steps.In equation (36) 6 stood originally for the unit operator 1,47but according to the experiences of Knab et al.,43 depending on the band index j the optimal choice for _the best localization of the different W functions can be also 0 = :($'+ HN)(kNbeing the oneelectron part of the Fock operator) or 6 = (Z2)", n = 1 or 2 (2 is the direction of the polymer axis). (For further details see ref. 43).
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3.1.3 Analytic Energy Grizdients.  For the calculation of vibrational and mechanical properties of 1a.rge molecules or polymers one needs the analytical second derivatives of the total energy (per unit cell). This formalism is much more advantageous, especially for large systems, than performing the corresponding numerical differentiations (both from the point of view of accuracy and of the CPU time required). In the case of a periodic polymer (assuming that all nuclei move in phase with each other in the direction of x), Hirata and Iwata4* have given analytical expressions for the first and second derivatives according to the nuclear coordinates in the H F level. Writing for the total energy per unit cell
In Eq. (39) p, v, A, o are basis function indices and their summation runs from I to g (number of basis functions x in the unit cell), q, 41, q 2 are cell indices and they have to be summed up from 1 to N (number of unit cells in the chain). Further the notation 11 in the twoelectron integrals stands for (1/r12)(1  4Fle2) (the operator Plc*2 exchanges electrons 1 and 2),
HNbeing the oneelectron part of the total Hamiltonian H. Finally the generalized chargeborder matrix elements are defined as
and ENRis the nuclear repulision energy per unit cell. With the help of the orthonormality condition of the COs the expression
was derived.42Here the quantity (the energy weighted density)
was introduced.
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Assuming again that all nuclei move in phase in direction y one obtains in a similar way
To be able to evaluate equation (43) we need the derivatives (ap$ay) and (aJ+$?/8y), respectively. For this purpose we have to know the derivatives of the coefficients occurring in the COs, (aC,,,(k)/ay). They can be obtained from the solution of the coupled HF(CPHF) equation^.^' Following the notation of Pople e t one can write
Solving equations (44) for the quantities uai, one can use the relation
to obtain the derivatives (ac,Jay). In equation (44) qa etc. are MOs. However, equations (44) can be easily rewritten for periodic polymers, if one introduces everywhere instead of the MO indices a, b, i, j the composite CO indices A(a, ku), B(b, k b ) , I(i, ki),J ( j , kj). The matrices H and J are the one and twoelectron parts, respectively, of the skeleton (core) first derivative Fock matrix in the CO basis. The definitions of these matrices are obtained from their molecular counterparts" by substituting instead of the MO coefficents Cm,n) the corresponding CO coefficients
(for more details see refs. 50 and 5 1).
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Having solved equations (44)for the vectors u$ one obtains in a straightforward way the expressions
and a more complicated, but similar expression for (a w',i),/ay) [see equations (41) and (42) of ref. 501. In a subsequent paper52 it was shown how the in the H F level developed formalism for the energy gradient of periodic polymers can be extended to the MP2 level. Assuming again in phase displacements of all the atoms of the polymers one obtains with a generalization of the expression given by Pople et al. for this case5'
with
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The quantities ~ l , a ( k 2 can ) be obtained again from the solution of the CPHF equations and the matrix elements”
with
and finally
It should be noticed that the authors did not provide an analytical expression for the second derivative of the total energy per unit cell in the MP2 case. Sun and Bartlett53did not constrain the motion of the nuclei to be in the same phase to derive the first and second analytic derivatives of the H F COs and their energie~.’~Since in this way they lose the periodic symmetry, they have introduced a new socalled semiorthogonal basis set which made the derivations possible. Using their results they have derived also expressions for the analytic derivatives of E / N of a polymer.53 Since, however, their formalism is quite complicated, and will not be repeated here, it is advised that the interested reader should consult the original paper.53
3.2 Examples of CorrelationCorrected Band Structuresof Quasi 1D Polymers. The first correlation corrected (quasi particle band structure) was calculated by S ~ h a for i ~ alternating ~ trans polyacetylene (PA) using a G3 1G** basis (polarization functions both on the C and H atoms) and an MP2 self energy (without relaxation). Performing a geometry optimization also at the correlated level, he has obtained 2.98 eV for the fundamental gap. In this calculation ten neighbors’ interactions were taken into account using for the different types of
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integrals his cutoff crite~ia,’~ but not applying a multiple expansion for the longrange Coulomb integrals. In the case of the basis applied one can estimate the valenceshell correlation, which is in its calculation about 7075%. [The total energy per unit cell at the MP2 level is 77.168 a.u., at the HF level 76.893 and their difference is 7.5 eV.” On the other hand the valence shellcorrelation energy of an acetylene unit was estimated to be 10 eVs6]. Extrapolation to 100% correlation has given a gap of 2.5 eV.54 In a later calculation on PA Suhais7using Dunning’s basis set58has obtained 2.75 eV for the gap. In subsequent calculations he has used a much better basis [5s3p3d2f for the C atoms; and 4s2pld for the Hs] and has also added a MP4 correction to the MP2 0nc5’ In this way he has obtained 1.8 eV lower total energy per unit cell. Since, however, the usually positive MP3 correction was not taken into account in this calculation, the result obtained does not seem to be very reliable. In this calculation no attempt was made to determine the quasi particle band structure and the fundamental gap. In a still more recent calculation6’ again the G31G** basis was applied for the selfenergy, the correct expression with relaxation6’ was used and the Coulomb integrals were slimmed up to infinity using a multiple expansion.62 They have 21 different kpoints in the first Brillouin zone and they extended the calculation to 21 neighbors’ interactions (they have reached a saturated band structure in their MP2 calculation only in this case). On the other hand they have not used different cutoff radii for the different explicitly computed integrals. They have obtained for the gap in this way (using not an optimized but an estimated geometry based on experiment), and not solving in an iterative way the inverse Dyson icquation (instead applying a formalisms4 which corresponds to the inverse Dyson equation without iterations) a value of 3.22 eV. It is very difficult to settle the dispute between Suhais4and Sun and Bartlett.53 At certain points one calculation is superior, at other points the other one. To come to an etalon about the quasi particle band structure of alternating trans PA one has to apply (1) a very good basis, (2) use for the self energy at least a MP2 + MP3 + MP4 calculation and (3) solve with it in an iterative way the inverse Dyson equation. Further (4) the calculated band structure has to be saturated according to the number of neighbors (better the saturation of the band structure with respect of the different cut off radii of the different types of integrals) and (5) according to the number of kpoints, (6) application of a multiple expansion for the longrange part of the Coulomb integrals, (7) geometry optimization at the highest level of correlation and (8) finally there should be at least an estimalion for the effect of the phonon polaron on the gap. Of course, the fulfillment o f all these requirements requires a rather large CPU time, but at least in the case of PA it would be worthwhile to do it. Using diagrammatic techniques Sun and Bartlett have given general criteria for the convergence of different orders of MBPT, of CC methods and of Green’s functions methods with lattice summation^.^^*^ For the details of these rather complicated investigations we refer to the original papers.

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From the results of the same authors using Koopmans’ theorem one obtains a theoretical (QP) ionization potential of 5.30 eVS3for PA which is in a rather good agreement with experiment (4.60 eV6’). Unfortunately, there is no experimental electron affinity. For polyparaphenylene (PPP) there exists also a H F + MP2 band structure73 calculation. Clementi’s double [ basis was used with a set of d functions added on the C atoms.68 The geometry was taken from Xray diffraction data.74The calculation used second neighbors’ interactions using 25 kpoints in the H F and seven in the QP(MP2) level. The gap obtained is 4.9 eV and the theoretical ionization potential estimated on the basis of Koopmans’ theorem is 7.10 eV. They are both too large compared with the corresponding experimental values (2.7 eV7’ and 5.76 eV,76 respectively). It should be mentioned, however, that besides the shortcomings of the theoretical calculations the discrepancy of 1.4 eV in the ionization potential is in the usual error range if one uses Koopmans’ theorem for the determination of the ionization potential. The difference between the theoretical and experimental values of the gap is at least partially due to the procedure used to estimate the experimental value of the gap on the basis of the onset of the UV absorption peak (which gives a substantially smaller value for the gap if there is an exciton band within the gap) instead of measuring the inverse photoelectron spectrum. There are also QP (quasi particle; correlated band structure) calculations for polyethylene (PE)66967 and polytetrafluorethylene ( t e f l ~ n )In . ~the ~ PE case a G31G** and Clementi’s double [ basis,68 respectively, was applied. In both calculations66967a full geometry optimization was performed. With the G31G** basis a gap of 10.3 eV was obtained, it increased, however, with the poorer double [ basis of Clementi to 11.6 eV, E ” , ~ ~ ~ (= ( Q )Zp) lies at  8.2 eV while the experimental values of the ionization potential are at 7.68.8 eV.69On the other hand the gap value estimated on the basis of experiment is at 8.8 eV,69 i.e. the theoretical value owing to the not good enough approximations used, is too large. It should be mentioned that Sun and Bartlett (second part of ref. 66) also studied PE at the MP2 level. They have found a quite good agreement with the experimental Xray (XPS)70and UV photoelectron (UPS) ~pectra.~’ In the case of Teflon the QP band structure67is similar to that of PE (broad valence and conduction bands), but it is strongly shifted downwards due to the effect of the four negative F atoms in the unit cell. The lower limit of the conduction band lies at 5.5 eV,67 which indicates the possibility of donor doping which could cause nconduction in it. For a cytosine (C) stack (with the experimental geometry of DNA B, again using second neighbors’ interactions and 25 kpoints at the HF and seven at the QP level and Clementi’s double basis) from the inverse Dyson equation with an MP2 self energy a gap of 8.66 eV was obtained.77This gap is still rather large, though according to our investigations the band structure was saturated for the number of neighbors and kpoints, respectively. Therefore we have introduced additionally two sets of pfunctions centered on the nuclei of “phantom” molecules situated at the middle of the stacking distance. These phantom molecules have no extra electrons and nuclear charges; their nuclei serve only

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as points on which the extra pfunctions are centered (for details see ref. 76). Using this trick the gap of the C stack went down to 6.60 eV. This value is already not far from the gap value of 5.5 eV which could be estimated on the basis of the UV spectrum of a C stack.79 Further, it should be mentioned that the coupled cluster (CC) theory in the ?2 f'2 approximation was successfully applied to calculate E / N of trans for its bond a l t e r n a t i ~ nand ~ ~ finally also for the correction of its band structure.45 In the latter case a gap value of 2.5 eV was obtained which agrees with Suhai's estimate54for 100% correlation. Though the calculations at present require too much time, one can expect that with still better localization techniques (localization also inside the unit cell, WannierBoys orbitals) and with ever more powerful computers they will be feasible in the future also for systems with larger unit cells. This would be highly desirable, because the CCT seems to be the only reliable size extensive theory for extended systems (the MBPT theory has convergence difficulties at higher orders, but in some cases even at low order). Finally, turning to the analytic energy gradients, Hirata and Iwata have applied their theory in the H F + MP2 to calculate the harmonic vibrational frequencies of' alternating trans PA. In the case of a G31G basis they have obtained vibrational frequencies with an error of 4 1 2 % of the experimental ones" both for (C2H2)x and ( C Z D ~ while ) ~ , the H F value had an error of 1133%. These results prove the superiority of the methods using correlation corrections and the procedure applying analytic energy gradients.


4 Application of First Principles Density Functional Theory (DFT) to Polymers
The DFT has been successfully applied in different levels of approximation to the ground state properties of 3D solids and of molecules. Though there are a few DFT calculations for polymers in the literature,80981 these calculations have not included a systematic investigation of the dependence of their total energy per unit cell on their conformation and of their valence electron structure (photoelectron spectra).
4.1 Methods.  The program used in our calculations was developed by Mintmire for polymeric c I h a i n ~ ~and ~  for ~ ~ 2D periodic systemsS5based on LDA molecular programs using a linear combination of G a u ~ s i a n s . ~ ~ ~ ~ ~ The chain program was developed to be able to take into account simple translation as well as helical symmetry. In the case of a helical polymer we can define a screw operator $(a, 4)88with a translation along the zaxis (the long axis of the polymer) and with a righthanded rotation 4 about the zaxis. Applying this operator to the position vector i! one obtains
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Since the symmetry generated by the screw operator S is isomorphic with the onedimensional translational group, Bloch's theorem is valid also in this case. Therefore, the oneelectron wave functions will transform under S as
that is K is in the 1D first Brillouin zone. In the case 4 = 0 (pure translation) K = ka. If there is more than one orbital in the unit cell, the $;s have to be constructed from a linear combination of Bloch function qj,
(52)
with
(53) m= 1
(g is the number of Gaussians in the unit cell and N is the number of cells in the chain). The electronic density is then given by the diagonal part of the first order density matrix
Here nXK) stands for the occupation number of the oneelectron states, and p;: are the coefficients the expansion of the density matrix (in direct space),
py, =
$JJ 271
dlcn;(K )
(
j l K ) Cj,j ( K )
eiKm
n
The total energy per unit cell of the polymer is given (in atomic units) as
(55)
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Here Z , and i: denote the nuclear charges 5nd coprc!inates within a single cell, the nuclear coordinates in the mth cell (Rr= Y R : ) . The fourth (Coulomb) term in Eq. (56) can be written as
&’
Since this is a longrange term, the code uses a multiple expansion61 to calculate those Coulomb integrals which were explicitly not taken into account (the program automatically cuts off all the integrals below certain threshold values in any system). The exchangecorrelation potentials E,,[P(?)] are treated separately in Mintmire’s code
These potentials as well the density in the Coulombterm were expressed as linear combinations of Gaussians. For E, the GasparKohnSham (GKS)89390 functional and the KohnSham functionalg0 were used. For the correlation functional the electrongas correlation results of Ceperley and Alder’’ were analytically fitted by Perdew and Zunger (PZ).92 The Gaussian basis anti an auxiliary basis used were optimized for LDF calculations by Goudbut tit al.93In Huzinaga’s notationg4 the H atoms had a (41/1*) and the C atoms a (71 11/41l/l*) contraction pattern, respectively. Auxiliary fitting functions (or four sets of s, p and d functions) (4,4; 4,4) were used for the C atoms and an auxiliary basis of (3,l; 3,l) for the H atoms, respectively, to fit the dens]ty and the exchange potential. 4.2 Examples of LDA calculations on Polymers.  The above described LDA method and program has been applied first of all to helical polyethylene (PE), calculating its total energy per unit ce!l and its band stru~ture.~’ The theoretically found CC bond lengths of 1.53 A (in the GKS case) and 1.52 (in the PZ case) are in good agreement with the experimental value of 1.53 obtained by Xray d i f f r a ~ t i o n The . ~ ~ calculated carbon backbone angle of 1 13” obtained with both GKS and PZ agrees also well with the experimentally found range of 1081 The calculated torsional angles at the minima of E / N are in both
A
A
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the GKS and PZ cases at 5560" and 180", respectively. The rotational barrier lies at a dihedral angle of 120" (eclipsed conformation) with a height of 3.63.7 kcalmol'. The band structures are quite similar in the GKS and PZ cases. The calculated gaps (7.7 eV for GKS and 8.0 eV for PZ) are in a rather good agreement with the value of 8.8 eV estimated on the basis of e ~ p e r i m e n tThis .~~ result is somewhat surprising, because LDF calculations usually underestimate the gap in semiconductors by 2040%. As a next step the LDA method was applied to helical polytetrafluorethylene (Teflon).99 In this case the E / N versus dihedral angle curve again shows a shallow minimum at 60" and two deeper ones at 90" and 64" (gauche forms) in both the GKS and PZ cases. They show also maxima at torsional angles of 70" and 125". Their heights are (measured from the nearest local minima) 11 and 20 kcal mol respectively. The reason for the higher rotational barriers in teflon is obviously the larger repulsion between the F atoms in teflon than between the H atoms in PE. The calculated gaps of the band structures are 5.5 and 5.8 eV at the most stable conformation. These values are somewhat smaller than the one obtained by Kasowski et aL8' using a linear combination of muffin tin orbitals in their LDA calculation. It should be mentioned that these values are still essentially smaller than the gap of 7.2 eV which we have obtained for our quasi particle band structure at the MP2 One should observe also that the lower limit of the conduction band is essentially higher ( 1.52.0 eV), than in the HF + MP2 calculation ( 5.7 eV67).Being, however, negative in both approximations (GKS and PZ, respectively) it still indicates a possibility of ndoping. Finally an LDA calculation was also performed for a polyparaphenylene (PPP) helix."' This conducting polymer (if doped) is quite interesting for its applications in light emitting diodes'" (especially because it emits blue lightlo2), in fieldeffect transistor^"^ and in nonlinear optical devices. lo4 A geometry optimization has provided an intraring CC distance (no distinction has been made between the two possible CC interring distances) I1 = 1.39 an interring CC distance of 1.47 A, a CH distance of 1.10 and a valence angle of 121". These values"' are within the range of the different experimental data. lo' The total energy per unit cell (E/N) has its theoretical minimum at the torsional angle 34.8" (the angle between the planes of two consecutive phenyl rings), while another LDA calculation" has provided 27.4" and the experimental value^'^' are between 20" and 27". It seems so that our torsional angle is too large by at least 7" (though still far better than the H F value of 53"Io6,*). The band structures of PPP were calculatedIm for the torsional angles of O", 34.8" and 90", respectively. In the most stable conformation (which does not differ substantially from the planar one; see Figure 3 in ref. 98) the fundamental gap is 2.54 eV, which is substantially smaller than the H F + MP2 value of 4.88 eV73and is close to the probably underestimated experimental value (on

A,
*No HF

',
+ MP2 torsional angle is available for PPP.
A
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the basis of the UV spectrum, see remarks above in Secion 3.2) of 2.7 eV. It should be mentioned that the LDA calculation using linearized augmented plane waves (LAPW) has given for the planar conformation a gap of 3.40 eV. It can be estimated that at the conformation of 35" torsional angle (a) this gap value would increase to 3.9 eV. One should notice that the lower limit of the conduction band has at a = 34.8" the value of 2.2 eV, indicating again the possibility of ndoping. Further the PZ LDA ionization potential of PPP is in its most stable conforma1:ion 4.9 eV, which lies closer to the experimental photoelectron spectrum76 value of 5.7 eV than the H F + MP2 value of 7.10 eV.73

5 Nonlinear Optical Properties of Polymers Nonlinear optics (NO) hias a great practical importance in electrooptics, in optical switches and modulators. '07'09 The discovery of lasers has provided the ideal tool to study nonlinear optical phenomena in molecules and polymers experimentally. 5.1 Theory of Nonlinear Optical Properties of Quasi 1D Periodic Polymers. 5.1.1 Solid State Physical Methods.  The theoretical treatment of a molecule or a polymer in the presence of an electric field or more generally of a laser beam presents a formidable problem. Here we shall remain first within the framework of the BornOppenheimer approximation and shall not consider the change of the phonons in the presencc: of an electric field because we shall work in a fixed nuclear (framework). Further, first we shall not take into account the effect of the interaction between the linear polymers on their polarizabilities and hyperpolarizabilities either although both effects are nonneglible. lo' l 2 They will be treated subsequently. There are some rather successful calculations for static and dynamic polarizabilities and hyperpolarizabilities of smaller molecules. 13' l 5 It is questionable, however, how well the perturbational method used by the authors would work for larger molecules interacting with laser light. With polyTer_s there is the additional problem that the potential of an electric field E, G , is unbounded and this destroys the translational symmetry of a periodic polymer. Because of this difficulty in a large number of calculations various author:$have applied different extrapolation methods for the (hyper)polarizabilities starting from oligomers with increasing number of units. Only in a few cases have attempts been made to treat infinite polymers at the tight binding and ab inifio HartreeFock level. The latter calculations use, however, a formalism which. is so complicated that its application to polymers with larger unit cells seems 1.0 be prohibitive (for a review see the Introduction of ref. 116). The purpose of the present paper is to present a full theory for static and dynamic (hyper)polarizabilities of periodic quasi 1 D polymers at an ab initio HartreeFock + correlation level. The theory will be developed at two different
'
'
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levels: (1) interaction of an electric field with a periodic polymer and (2) interaction of a laser pulse (taking into account both electric and magnetic field strengths) with a quasi 1D periodic polymer. To be able to formulate_the thzory one has to treat first of all the problem of the unbounded operator E?. If E is homogeneous (which is fulfilled in a good approximation within a laser beam) we can apply, following Mott and Jones' l 9 and Kittel12' (see also ref. 116), the nablaoperator V to a Bloch function
(here, as is wellknown u f l ( i ,3 is lattice periodic):*
Multiplying both sides of equation (60) by  i e i one obtains after reordering the terms e&q,(;,
F) = ie,!?eiiiV,
(eikiq,(K,  F) + ie&p,(E',
F))
(61)
If we qultiply (61) from the left by a Bloch function belonging to band rn with a value k', we find for the matrix element of the first term of (61) on the r.h.s.,
+
+
vanishes unless k' = k. In the latter case the remaining integrand and with it the integral is lattice periodic. Since it allows interband mixing+(generally rn # n), it describes the polarization of the system in the presence of E. On the other hand, the second matrix element originating from the r.h.s. of (61)
is not lattice periodic, Since, h_oweyer,this term allows the change of the quasi momentum vector, K # k , Ak = k'  k, this term corresponds to a current. Therefore, it need not be taken into account if we want to treat theoretically the nonlinear optical properties of a periodic system 791* which depend on the (hyper)polarizabilities (charge redistribution) in a molecule or a chain in the presence of an electric field, but not on the movement of charges (current).
''
*One could argue that by writing down the Bloch function (59), oqe had assumed a priori that our quasi 1 D chain remains periodic in the presence of an electric field E. This is certainly true, but if the band n is not very narrow the derivative V,will not be large and likewise the matrix element (62).
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Interaction of Static and Timedependent Electric Fields with Quasi ID Polymers ( a ) Derivation of the Coupled HaJtreeFock Equations. Let us assume that we have a homogenous electric field E, +
E = E,, + Em= ESt+ 
L
*
M
+ eimwt)= zst+
M
2 0 (eimmf m 1
go2 cos(mwt)
(64)
m= 1
zst
Here is the static field and in the case of the timedependent field J!?~ we have taken into account alsothe overtones. At tbe sal;ne+time we+assume that in this case no magnetic field H i s present, that is H = O(H = curlA). In this case the total Hamiltonian of an nelectron system can be written as
where HOis the unperLurbed Hamiltonian of the nelectron system. The fielddependent part of H, H' can be expressed, if we take into account the first term on the r.h.s. of equation (61) as well (64)
2(TI , . . . ,Tn;Jest, Eo, 0,t) =
One can substitute (65) and (66) into Frenkel's variational principle' l9 which provides the condition for the existence of a stationary state.
We apply for the field and timedependent nelectron wave function the Ansatz
Here Wois the total energy of the nelectron system in the fieldfree case. 2 is the antisymmetrizer and the oneelectron orbitals in the presence of the electric field are
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481
One should mention that the effect of the static field ESt one can calculate with the help of a modified Fock operator (& is the Fock operator belonging to I&,)
with the help of simple firstorder perturbation theory. In this way one obtains instead of
gSt)
with respect to cp,(?). a shift As:' with respect to E?) and the correction cp:'(F, To determine the effect of the timedependentfield Ew one has to substitute equation (68) with (69) into (67) and perform the variation of J with respect to the functions A(P;~and Aq,,, repectively. One obtains in this way the coupled HartreeFock (RPA) equations for a closed shell system.
( i = 1 , 2,...,n; m = 1,2,...,M) Here
and
Further in the case of a quasi 1D periodic polymer
where N is the number of unit cells in the polymer, N , the nupber of nuclei in the unit cell, 2, the charge of the ath nucleus and, finally I?; is the position vector of the ath nucleus in the Ith cell.
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In the system of equation (73) one should observe that because of the occurrence of the unknown functions A(pifm in the integrals (last two terms of the 1.h.s. of (73) the coupled HartreeFock (HF) equations (as the simple H F ones) are nonlinear and, therefore, have to be solved in an iterative way.
( b ) LCAO Approximation for the oneelectron wave functions. Introducing a basis set (x!4)(?)] for the whole polymer chain, where x$q) is the rth basis function in the qth cell, one can write
q = l r=l
and
q=l r=l
(6is the number of basisJunctions per unit cell). The coefficients Cf2tir(E,,) can be obtained by solving the generalized matrix equation
i,n the same way as one does in the case of periodic systems in the absence of Est.12G122 [Since Fo and F,, are, if we use periodic boundary conditions, both cyclic hypermatrices, they can be blockdiagonalized and the problem of a long finite or infinite chain can be reduced to the problem of 6 x 6 mat rice^.'^^] Substituting the expansion (76b) into the coupled H F equations (73) one arrives at the hypermatrix equation.
Here the matrices A and B have the elements
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and
Equation (78) can be rewritten as
B : occurring in equations (82) are, in the case of Since all the matrices AiFs and m a linear chain with periodic boundary conditions, cyclic hypermatrices, they can be blockdiagonalized with the help of the unitary matrix U [the p,qth block of u is = I / (N") exp [i2npq11.l2'
Using the notations obtain
= U+AlmU etc.,
qm = U'c;fm
and Gj = UDj we
This system of equations can be reduced easily in the usual way to such matrix equations in which each matrix has only the order riixrii For N + 00 one can introduce the continuous variable k = (2np)/(aN)) ( n/u 5 k 5 n/a) and can rewrite equations (84) as
Putting the two equations in the hypermatrix form one has
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(86)
whi_chcan be solved at each point k and value E, to obtain the vectors C$!$ (klEo). For this we need the back transformation
o+,
[Actually U and U' are those blocks of 3 and respectively, that are necessary for the transformations of these vectors.] Furthermore from the unitary transformations performed in equations (82) it follows.'2~122 N
eikquA&(g)
= q= 1
N
Btm( k ) =
eikquB:m(4) q= 1
N
Gi ( k ) =
C eikqaGi( 4 ) q=l
Finally_the +CAO crystal orbitals can be written in tlle presence of the electric fields E,, + E,
(c) MoellerPlesset Perturbation Theory. After having computed the quasi oneelectron orbitals and quasi one electron energies Ei one can apply (following Rice and Handy' 13,' 14) the MP/2 expression of the+second +order correlation correction of the quasi total energy for given k, u,t, E,, and Eo. In this case the MoellerPlesset perturbation will be
ei
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(m = i N ) A
where the quasi Fock operators f a r e defined as
j= 1 A
A
The quasi Coulomb operators J;. and quasi exchange operators K, are defined in the usual way but not with the help of the unperturbed crystal orbitals qj but with the perturbed ones Gj. Furthermore,
( H is the oneelectron operator of the unperturbed polymer). With the definition (90), one can derive in the standzrd way_123the secondorder correction to the quasi total energy for given t,m,Est and EO
Here @,(?I
&4p2
= &,p2(?,
W,
&, &),
because the perturbed crystal orbitals
. . .) and quasi oneelectron energies
depend on the same variables (this is meant in the argument of the qzetc. by the three points). The i, k and rndependence is expressed by the combined indices I = i,ki,mi, etc. Therefore, the fourfold summation includes the threefold integration over k (because of the conservation of momenta (ki + k, = k, + kb); for the details, how one calculates the matrix elements (@rijJIIGAqB)occurring in (931, again see.123 After having calculated EMP~, we can write for the total energy with MP2correlation corrections
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Here i H F ( k ,t) is the quasi HartreeFock total energy calculated in the presence of and at a certain value of k and at time t.
zSt zu
( d ) Calculation of the Polarizabilities and Hyperpolarizabilities. Having the total %nergy (95), we can formally express it in the presence of the electric field E = E,, + Eo as
Ef&
where and E~p2are the HartreeFock and MP2 energies of the polymer per unit cell in the absence of the field and
is the part of the induced dipole moment due to the static field. It can be easily calculated with the aid of the expression
x [qi(c, k)
1 ) dk + corr. correc.
+ mi,&, Zst,k )
where jii = lelri is the dipole moment operator and corr. correc. stands for terms coming from the correlation corrections in the wave function.'I4 Using the expansion (96) of i one obtains for the components of the static and dynamic (hyper)polarizabilities, respectively,
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(100a) 1 51/3yoo;0; 0)+ fly 20; 0,0)cos(20t) = 2
(100b) (10Oc)
zst
In these equatiorfs &t,p and i w , 2 , respectively, are the p t h component of and the Ath one of E,, respectively. The series expansion indicated above provides also the symmetry relation.' l 4
There are, of course, more combinations of 0 in /3 possible, but they are not measurable at the present time.' l4 Formally one can write also for the elements of y similar expressions as fourth derivatives of I?. For instance, it can be proven' l 4
For the numerical implementation of these formulae one can either use a perturbation expansion for the functions Aqf,,, '13,' l4 or derive coupled HartreeFock (CHF) equations for each order.'24 This procedure seems to be more advantageous for numerical calculations, but the resulting expressions are quite complicated. For this reason we do not reproduce them here, but refer to the original paper.'24 It seems to be an acceptable compromise to use the second order CHF equations for the dynamic a s and ps, but use simple perturbation theory for the dynamic ys using as unperturbed wave functions the results of the solutions of the first and second order CHF equations. One should point out, however, that in our calculations '24 the second numerical derivatives
were not calculated with a satisfactory accuracy. Therefore, the best numerical way to obtain the tensor elements of the dynamic hyperpolarizabilities cannot be settled yet.
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( e ) Interaction of a QuasiID Polymer with a Laser Pulse This problem has already been treated, in a previous paper,12' but the paper contains s_everal inconsistencies. Further, the lattice periodic part of the operator was not introduced. Since in the previous section [starting from equation (64)] a rather general formalism was described, it is easy to extend it to the case of a laser pulse. Let us assume to have a laser pulse of Gaussian shape,
that i,s in @meT/2 the value of i ( t ) decreases to i / e . Further, in this case and A # 0. Further we can write
A(t,W ) =
M
M
m=l
m= 1
zst= 6
C A m ( t , w ) = C pieimwf+ &eimwr1
Using the expression
(Coulomb gauge12') one obtains for V (taking into account equation (61)
If one introduces th_egeneralized momentym @  SA(t, w ) to take into account the magnetic field H(t, w ) one obtains for H' instead of equation (66)
The nelectron wave function will be in this case
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as before, but the definition of the oneelectron orbitals in the presence of the electromagnetic field changes to
m= 1
x @ ( t E t')
Here WOis again the eigenvalue of
@(t E t') =
+ AqTm(C,&,20)
[email protected](tE t') and
T 1 if 1005 2 0 otherwise
T 2
t < 100
+
Substituting again H = H H f with the definition (106) of H f and (107) with the definition (108) of the Gis into equation (67) (Frenkel's variational principle) one obtains instead of (73) the generalized coupled H F equations. 125
Here
& is the field free Fock operator of the polymer and the new operators
& I , Q2, Q 3
and
fi*
are defined as follows (llla)
(lllc) (llld)
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4A
and has the same definition only in the first and third terms on the r.h.s. 20 and A; are exchanged. If we use again an LCAO expansion for the A$:,, we obtain again the on the r.h.s. (t means transposed) hypermatrix equation (78) but with (Dr0;)' with the somewhat changed definitions
The matrix elements of Bo* %re th_e same as in equation (80) but instead ofcIps',,s(&t) etc. (because now& = 0) we have the cg's. Finally, the components of the vectors D* are now defined as
s= 1
(The coefficien? c$? are th+eeigenvector components of the Fock operator of the unperturbed (Est = e, = 0) polymer). The new matrices A&, are also cyclic hypermatrices and can be blockdiagonalized in the same way as before. Therefore, one obtains finally again equation (86) but on the r.h.s. we now have
Equation (89), which defines th_e oneelectron orbitals Gi, is nearly the same as before but instead Of+ci,st;s(k,Est) we have c?,(k) and the coefficients depend now also on Ao. The e_xpressionsin paragraphs (b) a_"d ( c ) remain the same except that instead of Est the quantities depend on Ao. To be able to calculate the (hyper)polarizabilities [paragraphs ( d )of the previou_ssection] one has to introduce again in all equations of this section a fictitious Est.
(f)The Application of the Polarization Propagator to Infinite Chains. Another way to compute the longitudinal polarizability of a closedshell infinite system is the polarization propagator technique. 126 In the framework of this method one can write in the RPA level for the polarization p r ~ p a g a t o r ' ~ ~
Here 42 is the matrix of the dipole transition strength which is spin and number conserving. 127Therefore, the poles of (1 14) provide the singletsinglet excitation energies and the corresponding residues are the transition moments. The static (oindependent) and dynamic (adependent) electric dipole polarizabilities per unit cell are then given by
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The terms Xb,i(k,o)and Yb,j(k,o)can be obtained from solving the system of linear equations (A + 0 l ) X + 4BY =
(1 16a)
B*X + (A*  01)
(116b)
=
a*
a
If N +00 the matrices A, B, X, Y and are of infinite dimension. Taking into account the periodic symmetry of the polymer the infinite sums over k (crystal momentum) can be transformed into an integration over k in the first Brillouin zone giving
Here
If one neglects the matrix B in equations (1 16) and the second and third terms in equation (1 18a), every coupling between the COs 4i(k)etc. disappears and one obtains the sum of states (SOS) expression for the polarizability per unit cell.128 The described polarization procagator method circumvents the problem of the unbounded Hamiltonian term I?, but with its aid only dipole polarizabilities (a)and not hyperpolarizabilities (p, y etc.) can be calculated. ( g ) Methodfor the Calculation of the Vibration Contribution to the Polarizability of InJinite Polymers. As is well known, the electric field changes the geometry of
492
Chemical Modelling: Applications and Theory, Volume I
a polymer and gives rise to a vibrational force field which leads to vibrational hyperpolarizabilities. The vibrational contribution to the (hyper)polarizabilities to an infinite chain can be quite large, reaching the magnitude of the electronic contribution, as K i r t m a x ~ ' has ~ ~ found by extrapolating his results from oligomers. One can implement the finite field treatment13' in the framework of the ab initio H F CO method.13' For this purp$se one calculates the chang? of P (p, cc, p, y etc.) in the presence of the field E a t the equilibrium geometry Ro with respect to the field free case:
and the difference o,f the quantity P if one uses in the presence of the field the adjusted geometry RE
If E is not very large one can write (Apu)% = a,p(elect)Ep
+ iflupy(elect)EpEy + 
(121a)
In a similar way
with amp= cc,p(elect)
+ cc,p(displ)
( 123a)
The total Hamiltonian of the system will be the same as in equation (65) and for its periodic part one can use the procedure given in equations (59)(62). As before one can derive the system of equations (78) with the definitions (79)(8 1) for the LCAO case. Finally taking into account that for a periodic polymer all matrices occurring in equation (78) are cyclic hypermatrices, one arrives aJter blockdiagonalization again to equations (86) (in this case without the field Eu). To calculate the static linear polarizability one has to compute first the tot_al dipole moment (by fixed geometry) at different trial electric field strecgths ET (six were applied). The dipole moments as a function of the different ETSwere fitted to a polynomial expansion in the components of ET. From the coefficients of this expansions it is easy to obtain the static polarizabilities on the basis of the expression132
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In actual polymer calculations one has to optiyize the geometry in the field freecase and in the presence of an external field E v p In the latter case two values were applied and to each one the trial field ET values were added. For the geometry optimization the SIMPLEX method, developed for large molecules, was adapted for polymers. Having the geometries Ro and & as well the quantities cr,p(elect) one can calculate from equation (121a) with the help of euation (123a) a,g(displ), the vibrational contribution to the static polarizability of a periodic linear polymer. ~
5.1.2 Large Clusters and Extrapolated Oligorners.  There is a larger number of papers in which their authors have calculated static and dynamic (hyper)polarizabilities at the ab initio HF, H F + MP2 level or using density functional methods of oligomers and have extrapolated their results for long/infinite chains. In the earlier papers they have used the sum of states methods (which will not be discussed here), while in the latter ones they have applied the coupled H F (RPA) equations or the density matrix renormalization group. These calculations have been extended also to take into account the vibrational contributions and the effect of interactions between chains, or the environment on a linear chain, respectively. Mosley et ai. have applied different density functional methods for a molecular hydrogen chain. 134 The results obtained with the aid of Romberg's extrapolation procedure' 35 overestimate the static longitudinal polarizability as compared to the CHF results. Several papers deal with the electronic contribution of the (hyper)polarizabilities using the CHF equations and an extrapolation procedure from a series of oligomers to polymers. Frequencydependent (hyper)polarizabilities were calculated taking into account also correlation at the MP2 level for polyacetylene'36 (the results will be discussed in the next section). Further static azz and pZzvalues were computed for polymethineimine chains with different donor or acceptor groups. 37 The static longitudinal second hyperpolarizability of a H2 molecular chain was determined again with Romberg's extrapolation' 35 introducing correlation at the MP2 + MP3 + MP4 In other works'39 the static longitudinal polarizability and second order hyperpolarizabilities at the H F level for polyyne using an improved extrapolation procedure (the socalled KTRH method14') were computed. For polymethineimine chains the static a,, and pzz values were calculated in different conformations and with have different donor/acceptor end group^.'^' In a recent paper Dalskov et aZ.142 computed the static azz and a,,(o,o) for the polyyne series (C2nH2) with n = 50 in the RPA (CHF) and n = 13 in the SOPPA (correlated secondorder polarization propagator approximation) and have investigated the stability of different extrapolation procedures. They have found that the purely mathematically based nonlinear sequence transformation method 143 gives the most satisfactory results. In two recent a symmetrized Density Matrix Ren~rmalization'~~
'
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Group method was applied for dynamic aL and y L using ID Hubbard and UI/ models and a HubbardPeierls model Hamiltonian in the case of linear polyenes. Wortmann and Bishop'47 have successfully extended Onsager's 14' reaction field model to nonlinear optical properties in condensed media,'47 which is applicable in any order of dynamic hyperpolarizabilities. Turning to the vibrational contributions to the nonlinear optical properties of linear chains, in a paper of Kirtman et it was shown that low frequency collective modes contribute to the vibrational (hyper)p~larizities'~~ of polyacetylene. The calculation was performed at the double harmonic level of approximation and has applied the sum of states method. In a number of papers just the vibrational contributions to the longitudinal hyperpolarizabilites were calculated for polyacetylene' 50 or polyyline,15' respectively (using again the double harmonic approximation). In some other papers the electronic and vibrational parts of the longitudinal static polarizability aL and second hyperpolarizability y L were computed simultaneously using the CHF equation and the double harmonic approximation, respectively. For poly(dimethylsi1ane) EL, lS2 for polyacetylene and polybutatriene yL lS3 and for alltranspolysilane again y L lS4 was computed. The results show that the vibrational contributions are of the same magnitude as the electronic ones. It should be emphasized that it is impossible to mention all the large number of papers which have appeared in this field in the last few years. Therefore, we have tried to select the most interesting looking ones which have either introduced new theoretical techniques or attacked new physical problems. 5.2 Results of Calculationsof NLO Properties and Their Discussion.  5.2.1 Solid State Physical Calculations.  The calculation of static linear polarizabilities a z z / N = aL/N for the polymers poly(H2), and the Hbonded systems poly(H20) and poly(LiH) using the above described solid state physical formalism has given much larger values' l 6 than those obtained by extrapolating oligomer values with the aid of differengrocedures. Further computations for the static E L / N values of poly(/
\),
poly(/'\)
CH2
and poly(/
\)
have
provided similar results. Subsequently using' 24 separate firstorder and secondorder CHF equations for (125a) and
 ei20r
'pi(2) ( k ,
7
(125b)
1,
in the series expansion Acpi(k,