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(BQ) Part 1 book Introduction to computational chemistry has contents: Introduction, force field methods, electronic structure methods Independent particle models, electron correlation methods, basis sets, density functional methods.
Introduction to Computational Chemistry Second Edition Frank Jensen Department of Chemistry, University of Southern Denmark, Odense, Denmark Introduction to Computational Chemistry Second Edition Introduction to Computational Chemistry Second Edition Frank Jensen Department of Chemistry, University of Southern Denmark, Odense, Denmark Copyright © 2007 John Wiley & Sons Ltd The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the 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Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wily & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, Canada, L5R 4J3 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Library of Congress Cataloging-in-Publication Data Jensen, Frank Introduction to computational chemistry / Frank Jensen – 2nd ed p cm Includes bibliographical references and index ISBN-13: 978-0-470-01186-7 (cloth : alk paper) ISBN-10: 0-470-01186-6 (cloth : alk paper) ISBN-13: 978-0-470-01187-4 (pbk : alk paper) ISBN-10: 0-470-01187-4 (pbk : alk paper) Chemistry, Physical and theoretical – Data processing Chemistry, Physical and theoretical – Mathematics I Title QD455.3.E4J46 2006 541.0285 – dc22 2006023998 A catalogue record for this book is available from the British Library ISBN-13 978-0-470-01186-7 (HB) ISBN-13 978-0-470-01187-4 (PB) ISBN-10 0-470-01186-6 (PB) ISBN-10 0-470-01187-4 (PB) Typeset in 10/12 Times by SNP Best-set Typesetter Ltd., Hong Kong Printed and bound in Great Britain by Antony Rowe This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production Contents Preface to the First Edition Preface to the Second Edition Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Fundamental Issues Describing the System Fundamental Forces The Dynamical Equation Solving the Dynamical Equation Separation of Variables 1.6.1 Separating space and time variables 1.6.2 Separating nuclear and electronic variables 1.6.3 Separating variables in general Classical Mechanics 1.7.1 The Sun–Earth system 1.7.2 The solar system Quantum Mechanics 1.8.1 A hydrogen-like atom 1.8.2 The helium atom Chemistry References Force Field Methods 2.1 2.2 Introduction The Force Field Energy 2.2.1 The stretch energy 2.2.2 The bending energy 2.2.3 The out-of-plane bending energy 2.2.4 The torsional energy 2.2.5 The van der Waals energy 2.2.6 The electrostatic energy: charges and dipoles 2.2.7 The electrostatic energy: multipoles and polarizabilities xv xix 8 10 10 11 12 12 13 14 14 17 19 21 22 22 24 25 27 30 30 34 40 43 vi 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 CONTENTS 2.2.8 Cross terms 2.2.9 Small rings and conjugated systems 2.2.10 Comparing energies of structurally different molecules Force Field Parameterization 2.3.1 Parameter reductions in force fields 2.3.2 Force fields for metal coordination compounds 2.3.3 Universal force fields Differences in Force Fields Computational Considerations Validation of Force Fields Practical Considerations Advantages and Limitations of Force Field Methods Transition Structure Modelling 2.9.1 Modelling the TS as a minimum energy structure 2.9.2 Modelling the TS as a minimum energy structure on the reactant/ product energy seam 2.9.3 Modelling the reactive energy surface by interacting force field functions or by geometry-dependent parameters Hybrid Force Field Electronic Structure Methods References Electronic Structure Methods: Independent-Particle Models 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 The Adiabatic and Born–Oppenheimer Approximations Self-Consistent Field Theory The Energy of a Slater Determinant Koopmans’ Theorem The Basis Set Approximation An Alternative Formulation of the Variational Problem Restricted and Unrestricted Hartree–Fock SCF Techniques 3.8.1 SCF convergence 3.8.2 Use of symmetry 3.8.3 Ensuring that the HF energy is a minimum, and the correct minimum 3.8.4 Initial guess orbitals 3.8.5 Direct SCF 3.8.6 Reduced scaling techniques Periodic Systems Semi-Empirical Methods 3.10.1 Neglect of Diatomic Differential Overlap Approximation (NDDO) 3.10.2 Intermediate Neglect of Differential Overlap Approximation (INDO) 3.10.3 Complete Neglect of Differential Overlap Approximation (CNDO) Parameterization 3.11.1 Modified Intermediate Neglect of Differential Overlap (MINDO) 3.11.2 Modified NDDO models 3.11.3 Modified Neglect of Diatomic Overlap (MNDO) 3.11.4 Austin Model (AM1) 3.11.5 Modified Neglect of Diatomic Overlap, Parametric Method Number (PM3) 3.11.6 Parametric Method number (PM5) and PDDG/PM3 methods 47 48 50 51 57 58 62 62 65 67 69 69 70 70 71 73 74 77 80 82 86 87 92 93 98 99 100 101 104 105 107 108 110 113 115 116 117 117 118 119 119 121 121 122 123 CONTENTS 3.12 3.13 3.14 3.11.7 The MNDO/d and AM1/d methods 3.11.8 Semi Ab initio Method Performance of Semi-Empirical Methods Hückel Theory 3.13.1 Extended Hückel theory 3.13.2 Simple Hückel theory Limitations and Advantages of Semi-Empirical Methods References Electron Correlation Methods 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 Excited Slater Determinants Configuration Interaction 4.2.1 CI Matrix elements 4.2.2 Size of the CI matrix 4.2.3 Truncated CI methods 4.2.4 Direct CI methods Illustrating how CI Accounts for Electron Correlation, and the RHF Dissociation Problem The UHF Dissociation, and the Spin Contamination Problem Size Consistency and Size Extensivity Multi-Configuration Self-Consistent Field Multi-Reference Configuration Interaction Many-Body Perturbation Theory 4.8.1 Møller–Plesset perturbation theory 4.8.2 Unrestricted and projected Møller–Plesset methods Coupled Cluster 4.9.1 Truncated coupled cluster methods Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 4.10.1 Illustrating correlation methods for the beryllium atom Methods Involving the Interelectronic Distance Direct Methods Localized Orbital Methods Summary of Electron Correlation Methods Excited States Quantum Monte Carlo Methods References Basis Sets 5.1 5.2 5.3 5.4 Slater and Gaussian Type Orbitals Classification of Basis Sets Even- and Well-Tempered Basis Sets Contracted Basis Sets 5.4.1 Pople style basis sets 5.4.2 Dunning–Huzinaga basis sets 5.4.3 MINI, MIDI and MAXI basis sets 5.4.4 Ahlrichs type basis sets 5.4.5 Atomic natural orbital basis sets 5.4.6 Correlation consistent basis sets vii 124 124 125 127 127 128 129 131 133 135 137 138 141 143 144 145 148 153 153 158 159 162 168 169 172 174 177 178 181 182 183 186 187 189 192 192 194 198 200 202 204 205 205 205 206 viii 5.5 5.6 5.7 5.8 5.9 5.10 5.11 CONTENTS 5.4.7 Polarization consistent basis sets 5.4.8 Basis set extrapolation Plane Wave Basis Functions Recent Developments and Computational Issues Composite Extrapolation Procedures Isogyric and Isodesmic Reactions Effective Core Potentials Basis Set Superposition Errors Pseudospectral Methods References Density Functional Methods 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Orbital-Free Density Functional Theory Kohn–Sham Theory Reduced Density Matrix Methods Exchange and Correlation Holes Exchange–Correlation Functionals 6.5.1 Local Density Approximation 6.5.2 Gradient-corrected methods 6.5.3 Higher order gradient or meta-GGA methods 6.5.4 Hybrid or hyper-GGA methods 6.5.5 Generalized random phase methods 6.5.6 Functionals overview Performance and Properties of Density Functional Methods DFT Problems Computational Considerations Final Considerations References Valence Bond Methods 7.1 7.2 7.3 Classical Valence Bond Theory Spin-Coupled Valence Bond Theory Generalized Valence Bond Theory References Relativistic Methods 8.1 8.2 8.3 8.4 8.5 The Dirac Equation Connections Between the Dirac and Schrödinger Equations 8.2.1 Including electric potentials 8.2.2 Including both electric and magnetic potentials Many-Particle Systems Four-Component Calculations Relativistic Effects References Wave Function Analysis 9.1 9.2 Population Analysis Based on Basis Functions Population Analysis Based on the Electrostatic Potential 207 208 211 212 213 221 222 225 227 229 232 233 235 236 240 243 246 248 250 252 253 254 255 258 260 263 264 268 269 270 275 276 277 278 280 280 282 284 287 289 292 293 293 296 6.5 EXCHANGE–CORRELATION FUNCTIONALS 253 of exact exchange to give the PBE0 functional (also denoted PBE1PBE in the literature),62 where the mixing coefficient for the exact exchange is argued to have a value of 0.25 from perturbation arguments.63 Similarly, the third-rung TPSS functional has been augmented with ~10% exact exchange to give the TPSSh method.64 Inclusion of exact HF exchange is often found to improve the calculated results, although the optimum fraction to include depends on the specific property of interest The improvement of new functionals by inclusion of a suitable fraction of exact exchange is now a standard feature At least part of the improvement may arise from reducing the self-interaction error, since HF theory is completely selfinteraction-free 6.5.5 Generalized random phase methods At the fifth level of the Jacob’s ladder classification, the full information of the KS orbitals is employed, i.e not only the occupied but also the virtual orbitals are included The formalism here becomes similar to those used in the random phase approximation (Section 10.9), but very little work has appeared on such methods Inclusion of the virtual orbitals is expected to significantly improve on, for example, dispersion (such as van der Waals) interactions, which is a significant problem for almost all current functionals One approach that can be considered as falling into this category is the class of Optimized Effective Potential (OEP) methods.65 The central idea is that the energy as a functional of the density is unknown (or at least the exchange–correlation part is), but the energy as a function of the orbitals is well known from wave function theory to a given order in the correlation, as defined for example by a perturbation expansion Since the density is given by the sum of the square of the orbitals, this implicitly defines the energy as a function of the density By requiring that the density derived from a Kohn–Sham calculation using a single-determinant wave function exactly matches the density derived from a (correlated) wave function, this implicitly defines the exchange–correlation potential The reference wave functions have so far been based on an MBPT type expansion (Section 4.8) The OEP1 method is defined by terminating the reference density at first order in the perturbation series Since correlation only enters the perturbation expansion at order two, this yields the exchange-only potential Terminating the expansion at second order defines the OEP2 method and corresponds to constructing a KS determinant that yields the (generalized) MP2 density From the condition that the MP2like density matrix matches that from the KS determinant, one may derive a set of coupled equations at the orbital level that provides the exchange–correlation potential correct to second order in the correlation The OEP2 method is computationally equivalent to an iterative MP2 calculation, i.e such calculations are computationally more expensive than standard DFT methods Furthermore the OEP2 method has basis set requirements similar to other correlated wave function methods and thus cannot benefit from the faster basis set convergence of other DFT methods Not surprisingly, OEP2 provides results of roughly MP2 quality although, in favourable cases, the performance may approach those from coupled cluster calculations It does have the desirable feature that it can describe for example dispersion interactions, which are problematic with almost all traditional functionals 254 DENSITY FUNCTIONAL METHODS Whether one should consider the OEP method as a density or wave functional theory is an open question, as it clearly tries to combine the best of both worlds It has the advantage of being able to systematically improve the results towards the exact limit, but inherits also the wave function disadvantages of a slow convergence with respect to basis set size 6.5.6 Functionals overview The introduction of GGA and hybrid functionals during the early 1990s yielded a major improvement in terms of accuracy for chemical applications, and resulted in the Nobel prize being awarded to W Kohn and J A Pople in 1998 Progress since this initial exciting developments has been slower, and the (in)famous B3LYP functional59 proposed in 1993 still represents one of the most successful in terms of overall performance Unfortunately, neither the addition of more fitting parameters, the addition of more variables in the functionals, nor imposing more fundamental restrictions for the functional form have (yet) provided models with a significantly better overall performance.66 Although the performance for a given property can be improved by tailoring the functional form or parameters, such measures often result in the deterioration of the results for other properties It should be noted that the implicit cancellation of the long-range part of the exchange and correlation energies implies that the two functional parts should be at the same level of the ladder, and preferably developed in an integrated fashion A popular topic in the literature is to search for a magic combination of exchange and correlation functionals, perhaps with a few adjustable scaling parameters and a choice of basis set, in order to reproduce a selected set of experimental data This is not a theoretically justified procedure and should be considered merely as data fitting without much physical relevance Nevertheless, such a procedure can of course be taken as an “experimental” fitting function that can be useful for predicting specific properties for a series of compounds Table 6.1 shows an overview of commonly used functionals given by their acronym, and placed in the Jacob’s ladder classification One may furthermore differentiate the functionals based on their use (or lack) of experimental data for assigning values to the parameters in the functional forms The non-empirical ones such as the PW86, Table 6.1 Perdew classification of exchange–correlation functionals Level Name Variables Examples Local density GGA r r, ∇r Meta-GGA Hyper-GGA Generalized RPA r, ∇r, ∇2r or t r, ∇r, ∇2r or t HF exchange r, ∇r, ∇2r or t HF exchange Virtual orbitals LDA, LSDA, Xa BLYP, OPTX, OLYP, PW86, PW91, PBE, HCTH BR, B95, VSXC, PKZB, TPSS, t-HCTH H+H, ACM, B3LYP, B3PW91, O3LYP, PBE0, TPSSh, t-HCTH-hybrid OEP2 6.6 PERFORMANCE AND PROPERTIES OF DENSITY FUNCTIONAL METHODS 255 PW91, PBE and TPSS functionals use the free parameters to fulfil as many of the requirements in Section 6.5 as possible at each level Empirical ones such as the BLYP, B3LYP, HTCT and VSXC, on the other hand, attempt to improve the performance by fitting the free parameters to give good agreement with experimental data This means that these functionals often perform (slightly) better than the non-empirical ones for systems that resemble those in the parameterization set Since the parameterization data are usually molecular systems, this means that they are often preferred for chemical purposes, but often give inferior performance for, for example, periodic systems such as metals Note also that most common functionals belong to levels and 4, as inclusion of HF exchange historically has preceded the development of functionals using derivatives beyond first order As one moves along the rungs of the ladder, it is expected (or hoped) that the accuracy will improve, but there is no guarantee that this is the case 6.6 Performance and Properties of Density Functional Methods An evaluation of the performance of the plethora of different functionals for a variety of properties is a major undertaking.4 We will here just quote two sets of results: (1) Root Mean Square (RMS) errors of atomization energies, ionization potentials, electron and proton affinities over the data set of 407 compounds selected from the G3 data set against experimental data.67 In addition the RMS error for the residual gradient at the experimental equilibrium geometry is taken as a measure of the accuracy of the functionals for predicting equilibrium geometries It should be noted that the evaluation data are the same data used for optimizing the parameters in the HTCT functional, and this functional will therefore naturally display good performance The results are obtained by using a TZP type basis set (2) Mean Absolute Deviation (MAD) of atomization energies over the 223 molecules in the G3 data set against experimental data.64 The results were obtained using the 6-311++G(3df,3pd) basis set While results with the above basis sets are not converged to the basis set limit, the residual basis set errors are presumably well below the inherent errors in the functionals, and the performance thus reflects the quality of the exchange–correlation functionals (Table 6.2) Note that the performance ordering of the functionals is not the same for the two sets of results Only a minor part of this discrepancy can be attributed to the difference in basis sets, the remaining discrepancy is due to differences in the data sets The LSDA method performs somewhat better than Hartree–Fock, but all the gradient-corrected methods are clearly far superior The PW91 and PBE functionals are somewhat poorer than the other GGA functionals, reflecting the fact that these not contain parameters that have been fitted to give a good performance for these systems Hybrid methods including exact exchange tend to perform (slightly) better than the corresponding pure functionals (e.g BLYP/B3LYP and PBE/PBE0), but several of the more recent “pure” functionals such as OLYP and VSXC are comparable to for example B3LYP Since the inclusion of HF exchange is computationally expensive for implementations relying on plane waves for expanding the orbitals, or for programs taking advantage of various density fitting schemes, this represent a computational 256 DENSITY FUNCTIONAL METHODS Table 6.2 Comparison of the performance of DFT methods Functional HF LSDA PW91 PBE PKBZ BLYP PBE0 OLYP B3LYP VSXC HTCT t-HCTH t-HCTH-hybrid TPSS TPSSh RMS (gradient) RMS (kJ/mol) MAD (kJ/mol) 35 16 15 16 21 19 11 14 11 11 11 11 10 649 439 80 87 75 41 50 40 40 39 33 31 26 885 510 99 93 29 40 28 25 21 14 30 24 16 advantage more than a fundamental theoretical improvement In general, it is found that DFT methods often give geometries and vibrational frequencies for stable molecules of the same or better quality than MP2, at a computational cost similar to HF For systems containing multi-reference character, where MP2 usually fails badly, DFT methods are often found to generate results of a quality comparable to those obtained with coupled cluster methods68 (see also Section 11.7.3) Handy and Cohen have argued that the BLYP and B3LYP forms are probably close to the optimum with respect to performance for a functional depending only on the gradients of the density.69 A significant advantage is that DFT methods based on unrestricted determinants (analogous to UHF, Section 3.7) for open-shell systems are not very prone to “spin contamination”, i.e 〈S2〉 is normally close to Sz(Sz + 1) (see also Sections 4.4 and 11.5.3) This is a consequence of electron correlation being included in the singledeterminantal wave function (by means of Exc) Actually, it has been argued that “spin contamination” is not well defined in DFT methods, and that 〈S2〉 should not be equal to Sz(Sz + 1).70 The argument is that real systems display “spin polarization”, i.e there are point in space where is larger than rb (assuming that the number of a _electron is larger than the number of b electrons) This effect cannot be achieved by a restricted open-shell type determinant (analogous to ROHF), only by an unrestricted treatment that allows the a _and b orbitals to be different It is somewhat unclear whether this argument hold for cases with 〈S2〉 values very different from Sz(Sz + 1), as in for example systems with multiple open-shell fragments.71 Another consequence of the presence of Exc is that restricted type determinants are much more stable toward symmetry breaking to an unrestricted determinant (Section 3.8.3) than Hartree–Fock wave functions For ozone (Section 4.4), for example, it is not possible to find a lower energy solution corresponding to UHF for “pure” DFT methods (such as LSDA or BLYP), although 6.6 PERFORMANCE AND PROPERTIES OF DENSITY FUNCTIONAL METHODS 257 those including exact exchange (such as B3LYP) display a triplet instability This “inverse” symmetry breaking is in some cases problematic In radical cations, for example, DFT methods usually refuse to localize the spin and charge, and thereby create unrealistic energy surfaces The Lagrange multipliers arising in Hartree–Fock theory from the orthogonality constraints of the orbitals are molecular orbital energies, and the occupied orbital energies correspond to ionization potentials in a frozen orbital approximation via Koopmans’ theorem The corresponding Lagrange multipliers in DFT not have the same formal relationship, since Koopmans’ theorem does not hold unless the exact exchange–correlation functional is employed For approximate XC functionals, the Lagrange multipliers can be interpreted as the derivative of the total energy with respect to the occupation number of the orbital, often called the Janak theorem72 but discussed first by Slater,73 and this is of course also closely related to experimentally measured ionization potential ∂E = ei ∂ni (6.53) The Lagrange multipliers may also be considered as approximations to ionization potentials using relaxed orbitals, and in practice give quite accurate results for the valence orbitals.74 In earlier work the orbital energies resulting from Kohn–Sham calculations were not considered to have any physical relevance, since they often showed poor agreement with ionization potentials, and orbital energy differences correlated poorly with excitation energies It is now clear that part of the poor agreement was due to the self-interaction error embedded in LDA and GGA methods, while more modern functionals yield much improved results Another difference is that the unoccupied orbital energies in Hartree–Fock theory are determined in the field of N electrons and therefore correspond to adding an electron, i.e the electron affinity The virtual orbitals in density functional theory, on the other hand, are determined in the field of N − electrons and therefore correspond to exciting an electron, i.e unoccupied orbitals in DFT tend to be significantly lower in energy than the corresponding HF ones, and the highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) gaps are therefore much smaller with DFT methods than for HF This also means that orbital energy differences in DFT are reasonable estimates of excitation energies, in contrast to HF methods where excitation energies involve additional Coulomb and exchange integrals The LSDA method usually underestimates the HOMO–LUMO gap, leading to the incorrect prediction of metallic behaviour for certain semiconducting materials Although it is clear that there are many similarities between wave mechanics HF theory and DFT, there is an important difference If the exact Exc[r] was known, DFT would provide the exact total energy, including electron correlation DFT methods therefore have the potential of including the computationally difficult part in wave mechanics, the correlation energy, at a computational effort similar to that for determining the uncorrelated HF energy Although this certainly is the case for approximations to Exc[r] (as illustrated above), this is not necessarily true for the exact Exc[r] It may well be that the exact Exc[r] functional is so complicated that the computational effort for solving the KS equations will be similar to that required for solving the 258 DENSITY FUNCTIONAL METHODS Schrödinger equation (exactly) with a wave mechanics approach Indeed, unless one believes that the Schrödinger equation contains superfluous information, this is likely to be the case Since exact solutions are generally not available in either approach, the important question is instead what the computational cost is for generating a solution of a given accuracy In this respect, DFT methods have very favourable characteristics 6.7 DFT Problems Despite the many successes of DFT, there are some areas where the current functionals are known to perform poorly • Weak interactions due to dispersion forces (part of van der Waals type interactions) arise from electron correlation in wave function methods, but this is poorly described by current DFT methods.75 Rare gas atoms should show a slight attraction, but most functionals display a purely repulsive energy curve, and those that predict an attraction underestimate the effect and the variation between systems.76 Furthermore, none have the correct R−6 limiting behaviour in the long distance limit, although very recent developments appear to provide quite accurate results with only a single parameter.77 In some approaches, an empirical attraction term is added that improves the performance,78 but this is clearly an ad hoc repair Owing to the general overestimation of bond strengths, LSDA does predict an attraction between rare gas atoms, but significantly overestimates the magnitude Hydrogen bonding, however, is mainly electrostatic and is reasonably well accounted for by many DFT functionals • Loosely bound electrons, such as anions arising from systems with relatively low electron affinities, represent a problem for exchange–correlation functionals that not include self-interaction corrections or correct for the incorrect long-range behaviour of the exchange–correlation potential Since loosely bound electrons by definition have most of the associated density far from the nuclei, this may cause the self-interaction error to be larger than the actual binding energy, and thus lead erroneously to an unbound electron In actual calculations using a limited basis set, this may not be obvious, since the outer electron is confined by the most diffuse basis function A positive HOMO energy, however, is a clear warning sign, and extending the basis set with many diffuse functions in such cases may cause the outer electron to drift away from the atom This means that only systems with high electron affinities have a well-defined basis set limiting value Nevertheless, a medium-sized basis set with a single set of diffuse functions will in many cases give a reasonable estimate of the experimental electron affinity.79 The basis set confines the outer electron to be in the correct physical space, and the exchange–correlation functional gives a reasonable estimate of the energy of this density It should be noted that the relatively good performance is in essence due to a correct physical description, rather than a correct theoretical methodology • For chemically bonded systems, analysis80 similar to the H2 system in Section 6.4 suggest that bonds involving: ° two-centre two-electrons (e.g normal covalent bonds), ° two-centre four-electrons (e.g steric repulsion between closed shell systems), and 6.7 DFT PROBLEMS 259 ° three-centre three-electrons (e.g radical abstraction) should be reasonably • • • • described by gradient-corrected methods Systems involving: ° two-centre one-electron (e.g radical cations), ° two-centre three-electrons (e.g radical anions), and ° three-centre four electrons (e.g atom transfer transition structures) are, however, predicted to be too stable The dissociation of charged odd-electron systems is a problem for most DFT methods, with the dissociation energy profile displaying an artificial barrier and an incorrect dissociation energy, often in error by as much as 100 kJ/mol Transition structures are similarly predicted to be too stable (barriers are underestimated) by functionals that not included exact exchange Since Hartree–Fock overestimates activation barriers, hybrid methods involving exact exchange, however, often give reasonable barriers The absence of a wave function makes a direct description of excited states with the same symmetry as the ground state problematic Excited states must be orthogonal to the ground state, which is easy to enforce if the spatial or spin symmetry differ, but difficult to ensure for excited states having the same spatial and spin symmetry Excited state properties, however, can be calculated by time-dependent DFT (linear response) methods, since the excited state is never needed explicitly Such calculations can give for example excitation energies and transition moments, as well as gradients of the excited surface, which allows excited states to be optimized The accuracy of excitation energies is typically ~0.5 eV for valence states, but Rydberg states, where the electron is excited into a diffuse orbital, can be in error by several eV This problem has the same physical reason as the anion problem above, and can be solved by using corrections for the asymptotic behaviour of the exchange–correlation potential.81,82 Such Asymptotic Corrected (AC) functionals display much improved predictions for response properties The exchange–correlation functional is inherently local, depending only on the density and possibly its derivatives at a given point, and this causes DFT methods to be inherently unsuitable for describing charge transfer systems, where an electron is transferred over a large distance Such systems are predicted to have excitation energies that are too low by several eV.83 Relative energies of states with different spin multiplicity are often poorly described In HF theory, the energy difference between a singlet and triplet state with the same orbital occupancy is given by an exchange integral In DFT, this must be described by the exchange–correlation functional, which only depends on the electron density If the two spin states arise from the same electron configuration the two electron densities are very similar, and this makes the results sensitive to the details of the exchange–correlation functional These problems are especially problematic for transition metal systems, where several low-energy spin states are often possible, and many of these cannot be described by a single determinant Pure DFT methods favour low spin states while HF favours high spin states, and hybrid methods with a suitable parameterized amount of exact exchange perform better.84 These problem can perhaps be improved by adding current density terms to the DFT formalism but this is not yet a commonly used procedure since it requires that the orbitals be allowed to become complex 260 DENSITY FUNCTIONAL METHODS • Individual spatial components of a spin multiplet may have different energies, even in the absence of a magnetic field The boron atom, for example, has the electron configuration 1s22s22p1, and the single p-electron can be in either a p−1, p0 or p+1 orbital These should all have the same energy, but since the density associated with the p0 orbital is different from that of a p±1 orbital, their energies as a result differ by ~25 kJ/mol This is clearly non-physical, but can be significantly improved by introducing current density terms.85 6.8 Computational Considerations The strength of DFT is that only the total density needs to be considered In order to calculate the kinetic energy with sufficient accuracy, however, orbitals have to be reintroduced Nevertheless, Kohn–Sham DFT displays a computational cost similar to HF theory, with the possibility of providing more accurate (exact, in principle) results Once an exchange–correlation functional has been selected, the computational problem is very similar to that encountered in wave mechanics HF theory: determine a set of orthogonal orbitals that minimizes the energy Since the J[r] (and Exc[r]) functional depends on the total density, a determination of the orbitals involves an iterative sequence The orbital orthogonality constraint may be enforced by the Lagrange method (Section 12.5), again in complete analogy with wave mechanics HF methods (eq (3.34)) L[ r ] = EDFT[ r ] − N orb ∑l (f f ij i i − d ij ) (6.54) ij Requiring the variation of L to vanish provides a set of equations involving an effective one-electron operator (hKS), similar to the Fock operator in wave mechanics (eq (3.36)) h KSf i = N orb ∑l f ij j j h KS = 12 ∇ + Veff Veff (r ) = Vne(r ) + ∫ (6.55) r (r ′ ) dr ′ + Vxc (r ) r − r′ The effective potential contains the nuclear contribution, the electronic Coulomb repulsion and the exchange–correlation potential, which is given as the derivative of the energy (eq (6.29)) with respect to the density Vxc (r ) = dExc[ r ] d e (r ′ ) = e xc[ r(r )] + ∫ r(r ′) xc dr ′ dr(r ) dr(r ) (6.56) A unitary transformation that makes the matrix of the Lagrange multiplier diagonal may again be chosen, producing a set of canonical KS orbitals The resulting pseudoeigenvalue equations are known as the Kohn–Sham equations h KSf i = e i f i (6.57) The KS orbitals can be determined completely by a numerical procedure, analogously to numerical HF methods In practice, such procedures are limited to small systems, 6.8 COMPUTATIONAL CONSIDERATIONS 261 and essentially all calculations employ an expansion of the KS orbitals in an atomic basis set fi = M basis ∑c ca (6.58) a The basis functions are often the same as used in wave mechanics for expanding the HF orbitals, although basis functions specifically optimized for DFT have recently been proposed (see Section 5.4.7 for details) The variational procedure again leads to a matrix equation in the atomic orbital basis that can be written in the following form (compare to eq (3.51)) h KSC = SCe hab = c a h KS c b (6.59) Sab = c a c b The hKS matrix is analogous to the Fock matrix in wave mechanics, and the one-electron and Coulomb parts are identical to the corresponding Fock matrix elements The exchange–correlation part, however, is given in terms of the electron density, and possibly also involves derivatives of the density or orbitals ∫c a (r )Vxc[ r(r ), ∇r(r )]c b (r )dr (6.60) Since the Vxc functional depends on the integration variables implicitly via the electron density, these integrals cannot be evaluated analytically but must be generated by a numerical integration G ∫ c a (r)Vxc[r(r), ∇r(r)]c b (r)dr ≈ ∑ Vxc[r(rk ), ∇r(rk )]c a (rk )c b (rk )∆v k (6.61) k As the number of grid points G goes to infinity, the approximation becomes exact In practice, the number of points is selected based on the desired accuracy of the final results, i.e if the energy is only required with an accuracy of 10−3, the number of integration points can be smaller than if the energy is required with an accuracy of 10−5.86 There are also some technical skills involved in selecting the optimum distribution of a given number of points to yield the best accuracy, i.e the points should be dense where the function Vxc varies most The grid is usually selected as being spherical around each nucleus, making it dense in the radial direction near the nucleus, and dense in the angular part in the valence space For typical applications, 1000–10 000 points are used for each atom.87 It should be noted that only the larger of such grids approach saturation, i.e in general the energy will depend on the number (and location) of grid points In order to compare energies for different systems, the same grid must therefore be used The grid plays the same role for Exc as the basis set for the other terms Just as it is improper to compare energies calculated with different basis sets, it is not justified to compare DFT energies calculated with different grid sizes Furthermore, an incomplete grid may lead to “grid superposition errors” analogous to basis set superposition errors (Section 5.10) With an expansion of the orbitals in basis functions, the number of integrals neces4 sary for solving the KS equations rises as M basis , owing to the Coulomb integrals in the 262 DENSITY FUNCTIONAL METHODS J functional (and possibly also “exact” exchange in the hybrid methods) The number of grid points for the numerical Exc integration (eq (6.61)) increases linearly with the system size, and the computational effort for the exchange–correlation term rises as GM 2basis, i.e a cubic dependence of the system size When the Coulomb (and possibly “exact” exchange) term is evaluated directly from integrals over basis functions, DFT methods scale formally as M basis However, as discussed in Section 3.8.6, the Coulomb (and exchange) part can be calculated with an effort that scales only as M 1basis for large systems with for example fast multipole methods The numerical integration required for the exchange and correlation parts may also be reduced to a computation cost that scales linearly with system size, i.e with modern techniques DFT methods have true linear scaling.88 This opens up the possibility of performing accurate calculations on systems containing thousands of atoms, which is likely to have impacts on many areas outside traditional computational chemistry Nevertheless, the formal M basis scaling has spawned approaches that reduce the dependence to M basis This may be achieved by fitting the electron density to a linear combination of functions, and using the fitted density in evaluating the J integrals in the Coulomb term M fit r ≈ ∑ aa′ c a′ (6.62) a The density fitting functions may be the same as those used in expanding the orbitals, but more often an auxiliary basis that is optimized for density fitting is used The fitting constants a′a are often chosen such that the Coulomb energy arising from the difference between the exact and fitted densities is minimized, subject to the constraint of charge conservation.89 The J integrals then become eq (6.63), which only involves three basis functions, thereby reducing the computational effort to M 3basis ∫c a (1) c b (1) c g′ ( 2)dr1dr2 r1 − r2 (6.63) Alternative versions where the Coulomb part of the Kohn–Sham matrix is assembled using plane waves as the auxiliary basis have also been proposed and, properly implemented, these achieve linear scaling even for small systems and for large basis sets.90 The use of grid-based techniques for the numerical integration of the exchange–correlation contribution has some disadvantages when derivatives of the energy are desired For this reason, there is also interest in developing grid-free DFT methods where the exchange–correlation potential is expressed completely in terms of analytical integrals.91 The computational cost of a DFT calculation depends strongly on the implementation strategy The use of DFT in the chemical community has to a large extent been introduced by modifying existing programs designed for wave function methods, and in these cases the numerical integration of the exchange–correlation energy adds a small overhead relative to an HF calculation Programs designed for DFT from the outset, on the other hand, can exploit the reductions arising from density fitting, and can consequently run significantly faster than a wave function HF calculation.92 Furthermore, the use of grid-based methods for evaluating the Coulomb and 6.9 FINAL CONSIDERATIONS 263 exchange–correlation contributions means that almost any kind of basis functions can be used, including Slater type orbitals Finally, DFT methods are one-dimensional just like HF methods, and increasing the size of the basis set allows a better and better description of the KS orbitals Since the DFT energy depends directly on the electron density, it has an exponential convergence with respect to basis set size, analogously to HF methods, and a polarized triple zeta type basis usually gives results close to the basis set limit 6.9 Final Considerations Should DFT methods be considered ab initio or semi-empirical? If ab initio is taken to mean the absence of fitting parameters, LSDA methods are ab initio but gradientcorrected methods may or may not be The LSDA exchange energy contains no parameters and the correlation functional is known accurately as a tabulated function of the density The use of a parameterized interpolation formula in practical calculations does not represent fitting in order to improve the performance for atomic and molecular systems Some gradient-corrected methods (e.g the B88 exchange and the LYP correlation), however, contain parameters that are fitted to give the best agreement with experimental atomic data, but the number of parameters is significantly smaller than for semi-empirical methods The semi-empirical PM3 method (Section 3.11.5), for example, has 18 parameters for each atom, while the B88 exchange functional only has one fitting constant, valid for the whole periodic table Functionals such as VSXC contains a moderate number of parameters (21), while other functionals such as PBE are derived entirely from theory and can consequently be considered “pure” ab initio If ab initio is taken to mean that the method is based on theory, which in principle is able to produce the exact results, DFT methods are ab initio The only caveat is that current methods cannot yield the exact results, even in the limit of a complete basis set, since the functional form of the exact exchange–correlation energy is not known At present it is easier to systematically improve on a wave function description than adding corrections to the energy functional in DFT Methods using reduced density matrices are still in their infancy, but promising results have been obtained in recent years It is perhaps a little disturbing that seemingly very different functionals give similarquality results.93 Levy and Perdew94 and others95 have shown how wave functions of near exact quality (such as CCSD(T)) can be “inverted” by a “constrained search” method to give near exact KS orbitals and corresponding exchange–correlation potentials Comparisons of such “exact” Vxc potentials with those discussed in the previous subsections have revealed large deviations and erroneous functional behaviour.96 Since many of these functionals perform well in practical applications, it is clear that the performance is not particularly sensitive to details in the functional, and that the good performance to some extent is due to error cancellations Although gradient-corrected DFT methods have been shown to give impressive results, even for theoretically difficult problems, the lack of a systematic way of extending a series of calculations to approach the exact result is a major drawback of DFT The results converge toward a certain value as the basis set is increased, but theory does not allow an evaluation of the errors inherent in this limit (such as the systematic overestimation of vibrational frequencies with wave mechanics HF methods) Fur- 264 DENSITY FUNCTIONAL METHODS thermore, although a progression of methods such as LSDA, BLYP and B3LYP has provided successively lower errors for a suitable set of reference data (such as that used for calibrating the Gaussian-2 model), there is no guarantee that the same progression will provide better and better results for a specific property of a given system Indeed, LSDA methods may in some cases provide better results, even in the limit of a large basis set, than either of the more “complete” gradient-corrected models The quality of a given result can therefore only be determined by comparing the performance for similar systems where experimental or high-quality wave mechanics results are available In this respect, DFT resembles semi-empirical methods Nevertheless, DFT methods, especially those involving gradient corrections and hybrid methods, are significantly more accurate (and the errors are much more uniform) than those of for example the MNDO family, and DFT is consequently a valuable tool for systems where a (very) high accuracy is not needed References P Hohenberg, W Kohn, Phys Rev., 136 (1964), B864 P.-O Löwdin, Int J Quant Chem., S19 (1986), 19 R G Parr, W Yang, Density Functional Theory, Oxford University Press, 1989; 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O V Gritsenko, R van Leeuwen, E J Baerends, J Chem Phys., 104 (1996), 8535; D J Tozer, V E Ingamells, N C Handy, J Chem Phys., 105 (1996), 9200; J B Lucks, A J Cohen, N C Handy, Phys Chem Chem Phys., (2002), 4612 96 C J Umrigar, X Gonze, Phys Rev A, 50 (1994), 3827; S Hirata, S, Ivanov, I Grabowski, R J Bartlett, K Burke, J D Talman, J Chem Phys., 115 (2001), 1635 ... 80 82 86 87 92 93 98 99 10 0 10 1 10 4 10 5 10 7 10 8 11 0 11 3 11 5 11 6 11 7 11 7 11 8 11 9 11 9 12 1 12 1 12 2 12 3 CONTENTS 3 .12 3 .13 3 .14 3 .11 .7 The MNDO/d and AM1/d methods 3 .11 .8 Semi Ab initio Method Performance... Atomic natural orbital basis sets 5.4.6 Correlation consistent basis sets vii 12 4 12 4 12 5 12 7 12 7 12 8 12 9 13 1 13 3 13 5 13 7 13 8 14 1 14 3 14 4 14 5 14 8 15 3 15 3 15 8 15 9 16 2 16 8 16 9 17 2 17 4 17 7 17 8 18 1... Concepts 11 .1 11. 2 11 .3 11 .4 Geometry Convergence 11 .1. 1 Ab Initio methods 11 .1. 2 Density functional methods Total Energy Convergence Dipole Moment Convergence 11 .3 .1 Ab Initio methods 11 .3.2 Density