Theoretical Chemistry (SPR Theoretical Chemistry (RSC)) (v. 3)

A Specialist Periodical Report

Theoretical Chemistry Volume 3 A Review of the Literature Published up to the End of 1976

Senior Reporters

R. N. Dixon, Department of Theoretical Chemistry, Universify of Bristol C. Thomson, Department of Chemistry, University of St. Andrews Reporters

D. G. Bounds, Universify of Manchester lnsfifufe of Science and Technology A. Hinchliffe, Universify of Manchesfer lnsfifute of Science and Technology 1. L. Robertson, University of York A. J. Stone, University of Cambridge

The Chemical Society Burlington House, London, WIV OBN

British Library Cataloguing in Publication Data Theoretical Chemistry Vol. 3qChemical Society. Specialist periodical reports). 1. Chemistry, Physical and theoretical I. Dixon, Richard Newland IT. Thomson, Colin 111. Series 541’.2 QD453.2 73-9291 1 ISBN 0-85 186-774-X ISSN 0305-9995

Copyright 0 1978 The Chemical Society All Rights Reserved No part of this book may be reproduced or transmitted in any form or by any means -graphic electronic, including photocopying, recording, taping or information storage and retrieval systems - without written permission from The Chemical Society

Printed in Great Britain by Adlard and Son Limited Bartholomew Press, Dorking

Foreword

This third volume of the Specialist Periodical Reports on Theoretical Chemistry continues the coverage of the subject outlined in the foreword to the first volume, and includes topics which have not previously been reviewed in this series. The survey of calculations on molecules containing five or six atoms bridges the gap between the subject matter covered by Thomson and by Duke in Volume 2. The theory of chemical reaction rates is still at a much more qualitative level of development than theories of molecular properties, but recent advances have made a considerable impact on organic chemistry. Stone gives a very comprehensive account of theories of organic reactions, with particular emphasis on models which can lead to predictions of relative rates of reactions. Hinchliffe and Bounds review in detail the calculation of the electric and magnetic properties of molecules. Finally, the use of pseudopotentials in molecular calculations is extending the range of ab initio calculation to molecules containing heavy atoms, and Dixon and Robertson survey this rapidly growing field. As in previous volumes the Reporters have not attempted to restrict themselves to SI units, but conversion factors to SI units are given on page ix. R. N. DIXON C. THOMSON

Contents Chapter 1 Ab lnifio Calculations on Molecules containing Five or Six Atoms By C. Thomson 1 Introduction

1

i

2 Molecules containing Five Atoms A. H5, H5+,and H5B. AH4 CHI and CH4+ BH4, NH4, and OH4 SiH4, PH4, and SH4 Miscellaneous AH4 C. AB4 Molecules AX4 BF, and CF4 ClF4 and SF4 Transition-metal halides A04nD. HAB, E. H2N-X F. H2C=NX G. Nitrenes H. Diazomethane Cyanogen Azide I. H 2 C - a J. Carbonium Ions K. Miscellaneous Five-atom Molecules

3 3 3 3 7 8 9 9 9 9 10 11 11 12 13 14 16 16 17 17 20 21

3 Molecules containing Six Atoms A. AtB4 CzH4 N2H&P2H4, NzF4, and P2F4 Boron Tetrahalides, B2X4 Nitrogen Tetroxide, N 2 0 4 B. AH,

23 23 23 28 30 30 31 33 33 34 36 37

c. A x ,

D. E. F. G.

Formamide, HCONHa MeXY Glyoxal, (CHO), Miscellaneous Papers

Contents

vi

Chapter 2 Theories of Organic Reactions By A. J. Stone

39

1 Introduction

39

2 Topological Methods

39

3 Validity of the Woodward-Hoffmann Rules

51

4 Single Perturbation Methods

58

5 Rigorous Perturbation Theories

68

Chapter 3 The Quantum Mechanical Calculation of Electric and Magnetic Properties By A. Hinchliffe and D.G. Bounds

70

1 Introduction

70

2 Electromagnetic Properties Multipole Moments Induced Moments

74 74

3 First-order Properties A Consequence of the Instability in First-order Properties Ways to Improve Expectation Values Configuration Interaction The MCSCF Method UHF Methods Many-body Perturbation Theory Constrained Variations Local Energy Methods Bond Properties

76 81 82

4 Second-order Properties Polarizability Calculations of Higher Polarizabilities and Semi-empirical Calculations Magnetic Susceptibility Gauge Invariance Magnetic Shielding

88 89

75

82 83 a4 84 85

86 88

93 94 96 98

Chapter 4 The Use of Pseudopotentials in Molecular Calculations 100 By R. N. Dixon and I . L. Robertson 1 Introduction

100

Contents

vii 2 Core-Valence Separability and the Formal Derivation of Pseudohamiltonians Molecules with Several Atomic Cores Non-uniqueness of the Pseudopotential Open-shell Pseudohamiltonians

101 105 107 109

3 The Parameterization of Pseudopotentials The Method of Explicit Core-Valence Orthogonality

111 115

4 Discussion and Comparison of Particular Cases LiH N,

1 20 122 123 123 126 128 129 130 131

F2

ciz Br, and HBr Metal Carbonyls: Ni(CO)*, Pd(C0)4, Pt(CO), HgH HZ0 5 Conclusions

Author I ndex

131

135

Abbreviations AE A0 CDHF CEPA CGTO CHFEP CI CNDO DZ DZ+P DIM EHF FFT GIAO GVB HF HOMO IEPA IP LCAO LUMO MBS MCSCF MIND0 MNDDO MO MSXa ODC

ovc

PE PMO PNO SCF

sos

STO STO-nG UHF VB VE VNDDO VSIP

WH

All electron Atomic orbital Coupled Hartree-Fock method Coupled electron-pair approximation Contracted Gaussian-type orbital Coreless Hartree-Fock effective potential Configuration interaction Complete neglect of differential overlap Double-zeta basis Double-zeta basis and polarization functions Diatomics in molecules Extended Hartree-Fock Finite-field technique Gauge-invariant atomic orbital Generalized valence bond; variants are denoted G1, G2, and G F Hartree-Fock Highest occupied molecular orbital Independent electron-pair approximation Ionization potential Linear combination of atomic orbitals Lowest unoccupied molecular orbital Minimal basis set Multiconfigurational self-consistent field Modified intermediate neglect of differential overlap Modified neglect of diatomic differential overlap Molecular orbital Multiple-scattering Xa method Optimized double configuration Optimized valence configuration Potential energy Perturbational molecular orbital Pseudo-natural orbital Self-consistent field Sum over states Slater-type orbital Slater-type orbital expanded in n Gaussian-type orbitals Unrestricted Hart ree-Fock Valence bond Valence electron Valence electron with neglect of diatomic differential overlap Valence-shell ionization potential Woodward-Hoffmann

Units

A number of different sets of units are used throughout this volume. Conversions to SI units are as follows: Energy: 1 a.u. (hartree) = 4.359 828 aJ 3 2625.47 kJ mol-1 1 eV = 0.160 210 aJ = 96.4868 kJ mol-l 1 cm-1 = 1.986 31 x J = 11.9626 J mol-l Length: 1 a.u. (bohr) = 0.529 177 x 10-lo m I A (ingstrom) = 10-10 m Cm Dipole moment: 1 D (debye) = 3.335 64 x Magnetic moment: 1 ,UB (Bohr magneton) = 9.2732 x J T-l

1 Ab initio Calculations on Molecules containing Five or Six Atoms BY C. THOMSON

1 Introduction The present Report attempts to survey calculations carried out during the past few years by ab initio methods which were not covered in Volume 2 of the present series. In the latter, the Report by Thomsonl dealt with molecules containing up to four atoms, and the article by Dukea dealt with large molecules. Since the completion of Volume 2, the literature on ab initio calculations has continued to grow rapidly and it was soon clear that space limitations would preclude any comprehensive coverage of the literature dealing with medium-sized molecules. Therefore this Report is restricted to calculations on molecules containing five or six atoms, and even within this group it is not possible to refer to all such calculations which have been published. Those studies which seem of particular interest to the Reporter have therefore been surveyed so that the selection is somewhat subjective. During the past five years there have been no spectacular advancesin fundamental theory; rather there has been a consolidation of earlier experience in ab initio methodology3and a more widespread use of existing methods in tackling problems of interest to more chemists in general, i.e. there have been many more applications to medium sized polyatomic molecules, usually employingminimal basis sets (MBS). There have also been many more calculations in which geometry optimizations are carried out, and in which the basis sets in SCF calculations have been extended to DZ or DZ+P quality, and more recently one sees an increasing use of methods which include at least some electron correlation, especially via configuration interaction (CI). Examples of the latter calculation were until recently restricted to molecules containing up to three atoms, but the recent development of efficient CI programmes had enabled these calculations to be carried out without too great expense on a variety of larger molecules, and this work is referred to later on in this Report. In order that this Report be useful to non-specialists interested in earlier ab initio work in this area, it is useful to cite several reviews and books relevant to the subject matter of this chapter. The bibliography by Richards and co-workers4 has been 1 2

3 4

C. Thomson, in ‘Theoretical Chemistry’, ed. R. N. Dixon and C. Thomson, (Specialist Periodical Reports), The Chemical Society, London, 1975, Vol. 2, p. 83. B. J. Duke, in ‘Theoretical Chemistry’, ed. R. N. Dixon and C. Thomson, (Specialist Periodical Reports), The Chemical Society, London, 1975, Vol. 2, p. 159. H. F. Schaefer, ‘Electronic Structure of Atoms and Molecules’, Addison Wesley, Boston, 1972. W. G. Richards, T. H. Walker, and R. Hinkley, ‘Bibliography of ab-initio Molecular Wave Functions’, Oxford University Press, London, 1971.

1

2

Theoretical Chemistry

updated to 1973,6 and contains a list of all the earlier ab initiu calculations. The proceedings of the First International Congress on Quantum Chemistry,O and that of a conference on ‘Quantum Chemistry: The State of the Art’,’ contain many review papers and survey many of the currently interesting areas in quantum chemistry. A volume devoted to theoretical chemistry* has appeared in Series Two of the MTP International Review of Science, and an excellent survey of recent developments in molecular electronic structure theory by Schaefer has recently a ~ p e a r e dThis . ~ review gives a more comprehensive list of books and reviews than is possible here. We should, however, mention that a comprehensive series of eight volumesl0 on ‘Modern Theoretical Chemistry’ is starting to appear and this series in particular should give an up to date and comprehensive survey of ab initio calculations. We have also not attempted in this Report to survey the individual molecules containing five and six atoms which are studied usually together with the larger molecules in the series of papers from Pople’s group. Recent reviews of this work have appeared and the reader is referred to these11-13 for further details and references. The general availability of the Gaussian 70 programme developed by Pople and co-workers (via the Quantum Chemistry Program Exchange14)has encouraged many non-specialists to venture into this field and to extend their investigations to larger molecules. However, it is important that such packages are not used in an uncritical way, and the limitations of the SCF procedure, and of minimal basis set calculations in certain instances, should be borne in mind. A recent book by Csizmadia is useful in this light, dealing with applications to organic molecules.1s As in the previous Report,’ developmentsin theoretical and computational methods as such will not be dealt with. The results of calculations will usually be quoted in atomic units* (distances/Bohr,energiesrnartree) but occasionally electron volts (ev) or kilojoules (kJ) for energies are used. A list of commonly used abbreviations is given at the beginning of this volume. The calculations described are organized into sections defined by the general formulae of the species. This is to some extent an arbitrary division but serves to group together those molecules of similar geometrical structure. As mentioned above, discussion will be restricted usually to work carried out during the period 1973-6.

*

1 Bohr=0.528 18 A; 1 Hartree=27.21 eV=2625.46 kJ.

W. G. Richards, T. E. H. Walker, L. Farrell, and P. R. Scott, ‘Bibliography of ab-initio Molecular Wave Functions. Supplement for 1970-73” Oxford University Press, London, 1974. 6 R. Daudel and B. Pullman, T h e World of Quantum Chemistry’, Dordrecht, Reidel, 1974. 7 V. R. Saunders and J. Brown, ‘Quantum Chemistry: The State of the Art’, Science Research Council, London, 1975. II ‘Theoretical Chemistry’, ed. A. D. Buckingham, MTP International Review of Science. Physical Chemistry, Series Two, 1975, Vol. 1. * H. F. Schaefer, Ann. Rev. Phys. Chem., 1976,27,261. 10 ‘Modern Theoretical Chemistry’, Plenum Press, New York, 1976-77, Vols. 1-8. 11 W. A. Lathan, L. A. Curtiss, W. J. Hehre, J. B. Lisle, and J. A. Pople, Progr. Phys. Org. Chem., 1974, 11, 175. 1%W. A. Lathan, L. Radom, P. C. Hariharan, W. J. Hehre, and J. R. Pople, Topics Current Chem., 1973,40,1. 1 3 W. J. Hehre, Accounts Chem. Res., 1975, 8, 369. l4 Quantum Chemistry Programme Exchange, Indiana University, Bloomington, Indiana, U.S.A. l5 I. G. Csizmadia, ‘Theory and Practice of MO Calculations on Organic Molecules’, Elsevier, New York, 1976. 6

Ab initio Calculations on Molecules containing Five or Six Atoms

3

2 Molecules containing Five Atoms

These are divided into the following classes, where in a particular class we also consider the relevant charged species: H5, AH4,AB4,HAB3, H2NX,H2CNX, Nitrenes, Diazomethane, H2CXY, Carbonium ions, Miscellaneous penta-atomic molecules. A. H5, H5+, and H,-.-The simplest penta-atomic molecule is H,+ and it has been the subject of several recent studies. The mass spectrum is well known and earlier work on the stability of this molecule is referred to in a paper by Huang et al.ls These authors investigated several geometrical structures by either carrying out a VB calculation with CI, or by obtaining SCF wave functions using a flexible basis set. The VB-CI calculations showed no stability for H5+ in a D z d configuration (in ~ 7 the authors concluded that contrast to previous predictions by Poshusta et ~ 1 . and the method is unreliable for this type of ionic system. However, the SCF calculations predict a binding energy of 0.007 Hartree (17.8 kJ mol-l) with an overall C,,symmetry. A more recent VB-CI study by Salmon and Poshustale used a more flexible basis set and gave similar results to the SCF calculations, and it is clear that polarization of the basis orbitals is very important in improving the VB results. Other calculations on H5+and Ha+ (n c 15) have been reported,19but the most extensive work to date is that of Ahlrichs,20who used the PNO-CI and CEPA methods. Reviews of these methods have been given elsewhere,lS but essentially they go beyond the SCFtype wave function and include electron correlation. In the PNO-CI m e t h ~ d , ~all l-~~ doubly excited configurations in addition to the HF function are included, and the CEPA21s23method also accounts for the effects of higher than doubly substituted configurations in an approximate way. For H5+the two methods give very similar results. In Ahlrichs' work,2oa large CGTO basis of lobe functions was used and the orbital exponents were carefully optimized so that the various different kinds of interaction such as ion-dipole, ion-quadrupole, dispersion, etc. were all accounted for, It was claimed that the relative energy errors should be no larger than Hartree. Various geometrical configurations were investigated and, for the most important of these, geometry optimizations were carried out. The minimum energy structure was found to be of Dzd symmetry whereas the HF geometry is of Czvsymmetry, as found in other work. The potential surface near the D2d structure is, however, extremely shallow. The author concludes that at room temperature the structure is mainly H2H3+. The computed value of De is 0.012 Hartree (30.9 kJ mol-l) and & e a o o ~ 1.364 kJ mo1-l. B. AH4.--CH, and CH,+. The number of calculations on CH4listed in Richards' bibliography4v5is 75 (up to 1973), and most of the current methods in use in quantum chemistry have been tested on this molecule. Several calculations on CHI have been concerned with the calculation of innershell or outer-shell ionization energies. '1 17

10

a1

83

J.-T. J. Huang, M. E. Schwartz, and G . V. Pfeiffer, J. Chem. Phys., 1972,56,755. R . D. Poshusta, J. A. Haugen, and D. F. Zetik, J. Chem. Phys., 1969, 51, 3343. W. I. Salmon and R.D. Poshusta, J. Chem. Phys., 1973,59,4867. S . W. Harrison, L. J. Massa, and P. Solomon, Nature, 1973, 245, 31. R. Ahlrichs, Theor. Chim. Acta, 1975, 39, 149. R. AhIrichs, H. Lishka, V. Staemmler, and W. Kutzelnigg, J. Chem. Phys., 1975, 62, 1225. R. Ahlrichs and F. Driessler, Theor. Chim. A d a , 1975, 36, 275. W. Meyer, J . Chem. Phys., 1973,58, 1017.

4

Theoretical Chemistry

The most reliable method has proved to be the ASCF method, in which the core binding energies are obtained by subtracting the SCF energies of the ground state from the SCF energy of the system with one of the core electrons Several papers have dealt with the calculation of the K-shell ionization energies of CHI, and also the Auger spectrum. Bagus and co-workers have reported two such studies. In the first paper,25the authors computed a ‘Al ground-state energy of -40.207 34 Hartree, using a (10,6/6,1)-+[7,5/5,1]* basis set supplemented by Rydberg functions on both C and H. Calculations of the Rydberg states of the lal-l hole state were carried out and the results were in good agreement with experiment. In the second paperz6a more extensive basis set (12,7,2/6,2)-+[8,5,2/5,2] gave an SCF energy of - 40.214 178 Hartree for R(CH) = 2.066 Bohr. The computed Auger energies were in excellent agreement with experiment and the authors conclude that SCF wave functions provide a qualitative basis for the analysis of molecular Auger spectra. It is emphasized that multiplet splittings of the final states are important, and play a key role in determining the spectra. Deutsch and CurtissZ7have investigated in more detail how the core ionization energies of CHI, HzO, NH3, and HF depend on the size and completeness of the basis set. Calculations were carried out both with the RHF and UHF procedures and eight different basis sets from a minimal to essentially a DZ + P basis set. The authors conclude that a large and flexible basis set is needed to obtain good agreement with experiment, with polarization functions being less important for highly symmetrical molecules like CH,. A related topic is the computation of valence-shell ionization potentials (VSIP). The calculation of vertical ionization potentials via Koopmans’ theorem28leads in many cases to serious errors, and a version of the ASCF method has been used to compute VSIP for several small molecules, including CH4.z9All the valence hole states of the molecule were computed. Agreement with experiment was substantially better than in the calculations using Koopmans’ theorem. Other authors have used less accurate SCF wave functions to investigate various other properties of CH, such as the force constants, previously studied by Meyer and Pulay,3O using Gaussian lobe functions. Schlegel et ~ 1 1 have . ~ ~ used the popular STO-3G and STO 4-3 1G basis sets to compute force constants in a variety of first- and second-row hydrides, including CH, and SiH,. The 4-31G basis set gives reliable values for the harmonic constants but the STO-3G basis does not. The higher force constants were also investigated. A more recent paper has used the same programme and a 431G basis for studies on several hydrocarbon^.^^ Similar results were The notation (A,B,C/D,E) refers to a primitive basis set of A s-type GTO,Bp-type GTO,and C d-type GTO on atoms other than H ; for H, D s-type and E p-type GTO.Such a basis set is usually contracted and the notation for the CGTO is [A,B,C/D,E]. z4 P. S. Bagus, Phys. Reo., 1965, 139A, 619. P. S. Bagus, M. Krauss, and P. E. LaVilla, Chern. Phys. Letters, 1973, 23, 13. 26 I. B. Ortenburger and P. S. Bagus, Phys. Rev., 1975, 11A, 1501. 25

27 28

29

30 31 32

P. W. Deutsch and L. A. Curtiss, Chem. Phys. Letters, 1976, 39, 558. T. Koopmans, Physica, 1933, 1, 104. M.F. Guest and V. R. Saunders, Mol. Phys., 1975, 29, 873. W. Meyer and P. Pulay, J . Chem. Phys., 1972, 56, 2109. H. B. Schlegel, S. Wolfe, and F. Bernardi, J. Chem. Phys., 1975, 63,3632. C.E. Blom, P. J. Slingerland, and C. Altona, Mol. Phys., 1976, 5, 1359.

Ab initio Calculations on Molecules containing Five or Six Atoms

5

obtained, but the bond lengths and angles were more extensively optimized in order to compute the equilibrium geometries. The one-electron properties such as the deuteron quadrupole coupling constant have been less studied until recently, but Dixon et aZ.33have recently reported the results for this property and also values of the diamagnetic shielding and susceptibility computed with a medium-size basis set. Moderate agreement with experiment was obtained. Several papers have dealt with the evaluation of wave functions including correlation in various ways. Birnstock3*has calculated the 13Cshielding constants in CHI and several other small molecules using an approximate form of uncoupled HartreeFock theory and the minimal basis set wave functions of Palke and L i p s ~ o m bThe .~~ results were similar to those obtained earlier by Ditchfield et aZ.3s CI calculations have also been reported of the states involved in Auger transitions in CH4.37Using a (9,5/5)+[5,3/3] basis set and valence-shell CI, good agreement was obtained with experiment, and also with the earlier SCF 26 The authors also used the same method for the Auger spectra of HF, HzO, and CO and it is clear that a modest CI using a medium size basis is capable of describing these spectra. There has been renewed interest in recent years in the calculation of Compton profiles and momentum expectation values. Much of the earlier work involved the use of SCF wave functions, but the results obtained using large-scale CI wave functions have recently been published.38It was concluded, however, that it is not necessary to go beyond the near-HF wave function in order to compute reasonable profiles, providing large, well balanced basis sets are used, a conclusion also reached by Tanner and E p ~ t e i n . ~ ~ The problems involved in the calculation of nuclear spin-spin coupling constants are well known, and Roos and co-workers earlier used a perturbation procedure to calculate JHHfor H, with encouraging results using ab initio wave function^.^^ The method involves correlating the zeroth-order wave function by a large CI calculation and treating the coupling between protons by perturbation theory involving the excited triplet configurations. Since this approach should be applicable also to polyatomic molecules, the authors41have studied CH4, H20,and NH3,using GTO basis functions. Several basis sets of different sizes were used. It is clear that correlation effects play an important role in the case of indirect coupling between nuclear spins separated by two bonds, and the basis set is also very important. Except for HzO, however, the agreement with experiment was not very good, and it was suggested that vibrational effects might be significant. It is crucial to include the doubly excited triplets in the calculations. The calculation of a substantial fraction of the correlation energy, particularly the 34 35 36 37

38 39 40

41

M. Dixon, T. A. Claxton, and R. E. Overill, J. Magn. Resonance, 1973,12, 193. F. Birnstock, Mol. Phys., 1973, 26, 343. W. E. Palke and W. N. Lipscomb, J. Amer. Chem. SOC.,1966, 88, 2384. R. Ditchfield, D. P. Miller, and J. A. Pople, J. Chem. Phys., 1971, 54, 4186. J. H. Hillier and J. Kendrick, Mol. Phys., 1976, 31, 849. T. Ahlenius and P. Linder, Chem. Phys. Letters, 1975, 34, 123. A. C. Tanner and J. R. Epstein, J. Chem. Phys., 1974,61,4251. 3. Kowalewski, B. ROOS, P. Siegbahn, and R.Vestin, Chem. Phys., 1974, 3, 70. J. Kowalewski, B. ROOS,P. Siegbahn, and R. Vestin, Chern. Phys., 1975, 9, 29.

6

neoretical Chemistry

valence-shell correlation, and the detailed analysis of the contributions to the correlation energy now seems to be feasible for small polyatomic molecules and Ahlrichs and co-workers4*have reported detailed results on Be&, BH, BH,, CH3-, CHI, NH,, HzO, and OH,+. Discussion here is confined to the results for CHI. The calculations were by the PNO-CI and CEPA-NO methods, details of which were given in ref. 21. The basis sets used were ca. DZ+P or a larger basis with both d and ffunctions, but two sets of p functions on H. These basis sets were bigger than in previous calculations by this method, and the authors claim to have obtained ca. 8 5 % of the correlation energy in each case. For CH,, the best energy obtained was - 40.425 22 Hartree. Ahlri~hs*~ has also described in more detail the method used in this work. Werner and M e ~ e have r ~ ~continued work with their version of the PNO-CI and CEPA methods,23computing the static dipole polarizabilities of CH,. Various basis sets were used in this study, mostly with better results than in earlier work. Meyer also studied the energy surface of CH,+ in earlier work with this method.23 MC-SCF calculations on polyatomic molecules are still rather rare, although there have been many such calculations on triatomic and diatomic molecules. Levy4s has described the results of such calculations using a minimal STO basis set for CH,, C2H4,and CZ&. A quadraticallyconvergent method was described and the results of localizing the orbitals were investigated. The GVB method developed by Goddard and c o - ~ o r k e r s has ~ * been ~ ~ ~applied to alkanes, ethylene, and a~etylene,,~ and CH, was among the molecules studied. The advantages of this type of wave function were discussed in Volume 2, and in the current work minimal, DZ, and DZ + P basis sets were used. One interesting observation is that heats of reaction for reactions involving the breaking of single bonds are quite reasonably described, i.e. for the reaction (1) ~ S C F 363.6 = kJ mol-1 for a CH4+CH3

+

H

(1)

D Z + P basis, whereas the GVB value is 409.2 kl mol-l, the experimental value being 430.9 kJ mol-l, One feature of interest for CHI is that the hybridization is spILal, rather than sps as in the usual VB description. Potential energy (PE) surface calculations are now becoming feasible, though expensive, for larger systems and several authors have described work on CHI in this connection. Eaker and Parr40 have used the diatomics in molecules method (DIM) to obtain the potential energy surfaces for CHn and obtained a heat of atomization which was ca. 84 kJ smaller than experiment. Wiberg and co-workers60have studied the energy changes for four angular deformation modes of CH, using STO-3G, STO 4-31G, and DZ + P basis sets. The 4-31G basis set appears to be capable of 42

R. Ahlrichs, F. Driessler, H. Lischka, V. Staemmler, and W. Kutzelnigg, J. Chem. Phys., 1975, 62, 1235.

43 44

45 46 47

R. Ahlrichs and F. Driessler, Theor. Chirn. Acta, 1975, 36, 275. H.-J. Werner and W. Meyer, Mol. Phys., 1976, 31, 855. B. Levy, Chem. Phys. Letters, 1973, 18, 59. W. A. Goddard and R. C. Ladner, J . Amer. Chem. SOC.,1971,93, 6750. W. A. Goddard, T. H. Dunning, jun., W. J. Hunt, and P. J. Hay, Accounfs Chem. Res., 1973,6, 368.

41 49

50

P. J. Hay, W. J. Hunt, and W. A. Goddard, J. Amer. Chem. SOC.,1972, 94, 8293. C.W. Eaker and C. A. Pam, J. Chem. Phys., 1976,64, 1322. K. R. Wiberg, G . B. Allison, and J. J. Wendoloski, J. Amer. Chem. SOC.,1976, 98, 1212.

Ab initio Calculations on Molecules containing Five or Six Atoms

7

giving a reliable description of the bending modes and these were related to those involved in the formation of several types of small-ring compound. There have been three other studies of the lowest singlet PE surface of CHI. Two of these were SCF studies using a minimum basis ~ e t . Both ~ ~ predict s ~ ~ a barrier height for the reaction (2) of > 209 kJ mol-l. This is in agreement with orbital symCH2(.41)

+ H2+CH4

(2)

metry arguments which predict this process to be forbidden, and therefore one expects a large barrier height. More recently, Bauschlicher et aZ.53have shown that a singIe-configurationwave function cannot describe the least-motion surface continuously, and have reported CI calculations using a CGTO basis of DZ quality. All singly and doubly excited valence-shell configurations were included (1192). The saddle point was located and lies 11 1.7 kJ above the separated reactants. This work is a good example of what can currently be accomplished for larger molecules concerning the reaction pathway. The electronic structure at the transition state was compared with that of the reactants and products, and it resembles CHI much more than the products. Electron correlation is far more important at the saddle point than at either of the two end points and reduces the barrier substantially. Although the least-motion approach has a high barrier, if the approach is by a pathway which avoids the singlet CH, lone pair, no barrier at all is found for the reaction via this non-least-motion a p p r o a ~ h . ~ BH4, NH4, and OH4. Very little work has been reported on these species. Calculations on BH4- were reported some years ago by several group^,^^-^^ and calculations using STO-3G and STO 4-31G basis sets are briefly mentioned in a paper dealing with the role of BH5 in the hydrolysis of BH4-.58 Pople and co-workers57have investigated the interaction of H, with several simple Lewis acids, including BH2+,and have calculated the energies for several different symmetries in the case of BH4+.All bound states were found to be of C,,symmetry, although the molecule has also been predicted to be square planar. The authors conclude that the latter symmetry should dissociate into BH2++Hz,and that BH4+ should be observable only at low temperatures. In this series of calculations, basis sets of up to 6-31G* have been used; the asterisk refers to the fact that &orbitals, and in some cases p-orbitals, on H are included. NH4+ has received rather little attention recently. Pople and co-workers studied the geometry with the 4-31G basis and Hopkinson and Csizmadia5*reported calculationswith a (8,3/3)+[2,1/1] basis. In a rather differently motivated study, basis sets of ca. DZ quality were used by Claxton et aZ.59who varied R(NH) and the proton exponent in calculations of the deuteron quadrupole coupling constants. These 51

52

53

54 55 56

57

51 59

J. N. Murrell, J. B. Pedley, and S. Durmaz, J.C.S. Faraday IZ, 1973, 69, 1370. P. Cremaschi and M. Simonetta, J.C.S. Faraduy 11, 1974, 70,1801. C . W. Bauschlicher, jun., H. F. Schaefer, and C. F. Bender, J. Amer. Chem. Soc., 1976,98,1653. P. Pulay, Mol. Phys., 1971, 21, 329. J. H. Hall, D. S. Marynick, and W. N. Lipscomb, Inorg. Chern., 1972, 11, 3127. I. M. Pepperberg, T. A. Halgren, and W. N. Lipscomb, J. Amer. Chem. Soc., 1976, 98,3442. J. B. Collins, P. von R. Schleyer, J. S. Binkley, J. A. Pople, and L. Radom, J. Amer. Chem. Soc., 1976,98, 3436. A. C. Hopkinson and I. G. Csizmadia, Theor. Chim. Acta, 1973, 31, 83. T . A. Claxton, M. Dixon, and J. A. S. Smith, J.C.S. Faraday ZI, 1972, 68, 186.

8

Theoretical Chemistry

turned out to be rather larger than found experimentally. Robb and co-workersG0 have examined the concept of the size of an electron pair and reported calculations on NH,+ in which the stereochemistry of the lone pairs was computed. NH, itself has been less thoroughly studied, but Pople et al. showed that the tetrahedral structure is not a stable minimum but a saddle point.57 Only a very loose complex is predicted for this system, stable by only ca. 3.4 kJ mol-I. SiH,, PH4, and SH,. There have been several papers recently concerned with these second-row hydrides. Some earlier work on these molecules is also mentioned briefly. Silane, SiH,, has been the most extensively studied. Palke61used both an MBS of STO and also a second basis set with 3d orbitals on Si, and optimized the geometry. The best total energy (- 290.6047 Hartree) was much higher than that obtained by Rothenberg et aL6, in an earlier SCF calculation with a large CGTO basis (- 291.24 Hartree). The force constants and dipole moment derivatives were not in good agreement with experiment except for the symmetric stretching constant. It was noted that d-orbitals on Si had a negligible effect on the calculated properties. SiH, 63 was also studied by the MsXa method.64Ionization potentials and transition energies were in fair agreement with experiment, but the binding energy was poor, although the bond length was ca. 3 % off the experimental values owing to the restrictions of the present muffin tin form adopted for the potential. Chong and co-workers 6s have also investigated the ionization potential, using a perturbation theory method which gives corrections to Koopmans’ theorem.66An STO basis set intermediate in quality between an MBS and a DZ basis set was used (each 1s function represented by a single STO; 2s and 2p orbitals by two STO’s) and various hydrides including SiH, were studied. The agreement with experiment was good. Reference was made above to Ahlrich’s studies21 with the IEPA and CEPA methods. In a later paper in this MgH,, AIH,, SiH,, PH,, H,S, and HCl were investigated. This work represents one of the few investigations concerning a series of molecules containing second-row atoms including electron correlation. The basis sets on the heavy atoms were (1 1,7)+[7,4] plus polarization functions; in the most extensive basis set these included &functions in addition to d-functions. As in the earlier work,21it was found that the polarization functions contribute much more to the correlation energy than to the SCF energy. A detailed analysis of the valenceshell correlation energies was given. If one uses the localized representation the inter-orbital pair correlation energies and the IEPA error are smaller in absolute value than those of the corresponding first-row hydrides. Schlegel and c o - w ~ r k e r salso ~ ~ calculated the force constants for SiH, as in the work on CH, referred to above, but comparisons with experiment showed somewhat similar agreement to that found earlier by Palke.sl 8o

61

G2

63 64 65 66 67

M.A. Robb, W. J. FIaines, and I. G. Ciszmadia, J . Anzer. Cliem. Soc., 1973, 95, 42. W. E. Palke, Chem. Phys. Letters, 1971, 12, 150. S. Rothenberg, R. H. Young, and H. F. Schaefer, J. Ainer. Chem. SOC.,1970, 92, 3243. M. L. Sink and G ,E. Jurim, Chem. Phys. Letters, 1973, 20,474. K. H. Johnson, Ado. Quantum Client., 1973, 7, 143. D. P. Chong, F. G . Herring, and D. McWilliams, J. Clrem. Phys., 1974, 61, 3567. D, P. Chong, F. G . Herring, and D. McWilliams, J . Chern. Phys., 1974, 61, 78. B. Ahlrichs, F. Keil, H. Lishka, W. Kutzelnigg, and V. Staemmler, J . Chem. Phys., 1975, 63, 455.

Ab initio Calculations on Molecules containing Five or Six Atoms

9

Ionization potentials calculated by the ASCF24method were studied by Guest and Saunder~,2~ who found that this combined SCF procedure successfully accounts for the relaxation energies computed by independent ASCF calculations. An interesting and detailed paper on SH4and SH6has appeared.68SF, and SF, are both well known and are the simplest examples of hypervalent sulphur comp o u n d ~However, .~~ the compounds SH4 and SH6have so far not been prepared and Schwenzer and Schaefer 68 have computed SCF wave functions for these molecules using a large contracted basis set (for S , (12,9)+-[7,5]; for H, (5)+[3]} which was augmented with polarization functions. Geometry optimizations were carried out, although it was pointed out that there is no guarantee that an absolute minimum has been reached. The results predict the structure of SH4to be analogous to that of SF4. It was found that SH, lies energetically above SH2+ Hz. Populations analyses and molecular one-electron properties were computed. Miscellaneous AH4. Calculations on the hydrides of several molecules containing an argon core, i.e. third-row hydrides, have been rather few in view of the expense and necessity of including d-orbitals. Several of these species, including TiH4, were recently investigated using the FSGO method, with complete geometry optimization.?*The argon core was approximated in a well defined way and the calculations show that the results should be of qualitative significance and in many cases may be useful guides if any of these molecules are detected experimentally. C. AB4Molecules.-It is convenient to divide these molecules into tetrahalides AX4, where X = F, C1, Br, or I, and AY4, where Y = 0 or S . In view of the successes of minimal basis set SCF calculations in the prediction of qualitatively correct molecular g e ~ m e t r i e s , ~we . ~ ~should - ~ ~ first refer to an important paper by Ungemach and Schaefer.74These authors also point out that usually DZ basis sets provide geometry predictions approaching quantitative accuracy, and the addition of polarization functions to the basis set had very little effect on the geometrical predictions. The authors have shown, however, that for AB4 molecules these general conclusions may not be valid. We refer to specific examples below, but it is clear that for AB, species the predicted geometries are very sensitive to the choice of basis set. It is particularly difficult to distinguish between square-pyramidal (CdV)and ‘detached octahedral’ (C2V)geometries. Examples are given in the following sections. AX4. BF4 and CF4. The only calculation75found on BF4 involved a rather small basis set SCF study. Although several papers have dealt with CF, at a minimal basis set level, the more extended basis set results of Brundle et aZ.,76Clementi et aZ.,77and Adams and Clark78have not been improved upon. 68 69 70

71 72

73 74 75 76

77 78

G. M. Schwenzer and H. F. Schaefer, J. Amer. Chem. SOC.,1975, 97, 1393. J. I. Musher, Angew. Chem. Internat. Edn., 1969, 8, 54. E. R. Talaty, A. J. Fearey, and G . Simons, Theor. Chim. Acta, 1976, 41, 133. J. A. Pople, in ‘ComputationalMethods for Large Molecules and Localized States in Solids’, ed. F. Herman, A. D. McLean, and R. K. Nesbet, Plenum Press, New York, 1973, p. 11. H. F. Schaefer, in ‘Critical Evaluation of Chemical and Physical Structural Information’, ed. D. R. Lide, National Academy of Sciences, Washington, 1974, p. 591. J. A. Pople, in reference 10. S. R. Ungemach and H. F. Schaefer, Chem. Phys. Letters, 1976, 38, 407. R. M. Archibald, D. R. Armstrong, and P. G. Perkins, J.C.S. Faraday II, 1973, 69, 1793. C. R. Brundle, M. B. Robin, and H. Basch, J . Chem. Phys., 1970, 53, 2196. E. Clementi and A. Routh, Internat. J. Quantum Chem., 1972, 6, 525. D. B. Adams and D. T. Clark, Theor. Chim. Acta, 1973, 31, 171.

Theoretical Chemistry

10

Adams and Clark 7 8 used a large basis set of better than DZ quality and calculated core binding energies and shifts for several fluoro- and chloro-methanes,including CF4. These were obtained using Koopmans' theorem, hole state calculations, and equivalent cores ~ a l ~ ~ l a t ithe o nlatter ~ , ~ giving ~ the best results for minimal basis sets, but there was little difference between the three methods for the more extended basis sets. NF,+ was also studied in this paper. ClF, and SF,. The interhalogen compounds CIF4, ClF4+, and ClF,- have been studied recently in view of conflictingexperimental and theoretical predictions of their geometries. Guest et aLS0used an essentially DZ P basis and found that ClF4- is square planar, in agreement with experiment, but that the cation ClF4+has equal FClF angles of 154", whereas the n.m.r. spectrum is consistent with C,, symmetry. The ab initio calculations, however, predict a very flexible molecule. Walsh-type diagrams were presented. The need for calculations with a more extensive basis set and possibly CI was clear from this study. Ungemach and Schaefer 81 attempted to clarify the situation by carrying out calculations on ClF4+with a larger DZ basis set. Included in this paper were also calculations on CIF2 and its ions, and also on ClF4 and ClF4-. The radical ClF4 has been observed recently by e.s.r. The minimum basis set results for ClF4+agree with those of ref. 80 in predicting a square-pyramidal structure, and the more reliable DZ results also predict a similar ClUgeometry. The ground and excited state of ClF4 and the ground state of CIF4are predicted to be planar by both sets of calculations, with relatively long Cl-F bond lengths. The two highest MO's in the radical were relatively close together. These results, however, have been modified by the inclusion of 3d functions on Cl.74 For CIF4+,the total energy is lowered substantially, and the geometry predicted now changes from C,, to CZu.The latter result is in accordance with Walsh's rules and agrees with the known structure of SF, (see below). The d-functions also reduce the CI-F bond length by 0.15 A. The prediction for ClF4 is also changed, from square planar to square pyramidal. The results are summarized in Table 1.

+

Table 1 Theoretical results for CIF4+arid ClF4 Clbasis

a e

E

raz

'

rcq

-854.9 -851.19 -856.27 -856.432 -856.436

1.66 1.78 1.76 1.61 1.63

1.66 1.78 1.76 1.61 1.57

154 148.2 143.5 144.2 169.6

MBS DZa DZ+Pa

-851.44 -856.75

1.82 1.83 1.69

1.82 1.83 1.69

Planar

-

&,

Ref.

154 148.2 143.5 144.2 109.7

80 81 81 74 74

-

81 81 74

00,

MBS MBS DZa DZ+Pa DZ+Pa

Planar 163

F atom basis was (95)+[42]; CI (12,9)+[64] -DZ; (12,9,1)-+[6,4,l]= DZ+P. Values in degrees.

163

* Values in A.

L.J. Aarons, M.F. Guest, and I. H.Hillier, J.C.S. Faraday II, 1972, 68, 1866. m M. F. Guest, M. B. Hall, and I. H. Hillier, J.C.S. Faraday II, 1973, 69, 1829. S. R. Ungcmach and H. F. Schaefer, J. Amer. Chem. SOC.,1976,98, 1658. I t J . R. Morton and K. F. Preston, J. Chem. Phys., 1973,58, 3112.

w

Ab initio Calculations on Molecules containing Five or Six Atoms

11

Since CIF,- and SF4 are isoelectronic, the question of the electronic structure of SF, was investigated by Radom and S ~ h a e f e rUsing . ~ ~ an MBS, they obtained an erroneous prediction (C4v)but the DZ basis set gave the qualitatively correct C,, geometry. d-Functions were not employed in this study so the computed SF bond lengths were too long, but it seems likely that their inclusion will not change the predicted geometry although this needs confirming. Ungemach and S~haefer,'~ however, emphasize the dangers of predictions for these molecules with inadequate basis sets and it is clearly desirable in cases like these to obtain as accurate wave functions as possible. Transition-metalhalides. There have been a few other calculations on halides containing a metal atom, in particular the tetrachloro-derivatives. Earlier work on species like NiFs4-,84NiF4 , 86 and CuFd2-8s used small basis sets and the first results with a larger basis set were from a careful study of CUC~,~in both %and D,~configurations by Veillard and c o - w ~ r k e r s .88~A ~ ,(12,8,5/10,6) +[5,4,2/3,3] basis gave the most accurate results for the ground state. Doubly occupied orbitals which are mainly metal 3d orbitals were found to be at lower energy than ligand 3p orbitals. The lowest IP are associated with mainly chlorine 3p orbitals using Koopmans' theorem, but relaxation effects can be taken into account and the 3d and 3p orbitals then have comparable IP. The Td structure is computed to be more stable by ca. 76 W than the D4h structure but it is energetically more favourable to distort the geometry to a flattened D,a structure. Veillard8ghas recently surveyed the work of his group in this field and presented results on several other complexes. Hillier and co-workers have also studied many compounds of this type,g0n81 especially with respect to their photoelectron spectra, and a recent paper on CoCI,*is typical.eaThe g-value was calculated and compared with e.s.r. data. In addition, CI calculations were used to study the low-lying excited states. The calculated transition energies were sensitive to the configurations included, but agreement with experiment was quite good. Rather more calculations using the M s X a techniques4have appeared but space restrictions do not permit us to review these in this article. A04n-. Almost all calculations reported in this section concern the ions A04nwhere n = 1 - 4 . Early work on these molecules will not be discussed, but during the past three years there have been a variety of calculations on these ions. One of the most interesting of these is permanganate, MnO,-, which has an interesting spectrum but is still not fully understood. Early a6 initio studies by Hillier

,-

13 84 85 86

87 88 *9 90

91

92

L. Radom and H. F. Schaefer, Austral. J. Chem., 1975, 28, 2069. A. J. Wachters and W. C. Nieuwpoort, Internat. J. Quantum Chem., 1971, 5 , 391. H. Basch, C. Hollister, and J. W. Moskowitz, in 'Sigma Molecular Orbital Theory', ed. 0. Sinaniiglu and W. Kenneth, Yale University Press, New Haven, 1970, p. 449. J. A. Tossel and W. N. Lipscomb, J . Amer. Chem. SOC.,1972, 94, 1505. J. Demuynck and A. Veillard, Chem. Phys. Letters, 1970, 6, 204. J. Demuynck, A. Veillard, and U. Wahlgren, J. Amer. Chem. SOC.,1973, 95, 5563. A. Veillard, in reference 7, p. 211. I. R. Hillier and V. R. Saunders, Mol. Phys., 1971, 22, 1025. I. R. Hillier and V. R. Saunders, Mol. Phys., 1972, 23, 449. I. H. Hillier, J. Kendrick, F. E. Mabbs, and C . D. Garner, J . Amer. Chern. SOC.,1976,98, 395.

Theoretical Chemistry

12

and S a u n d e r ~and , ~ ~Dacre and Elder 94 used minimal basis sets. Hillier and Saundeng6subsequently used a DZ basis set, but more recent extensive studies have been carried out by Mortola et ~ 1 1and . ~ ~by Johan~en.~' The earlier investigations are discussed in more detail by WoodB8who used two very large extended basis sets in SCF calculations and who carried out the first CI calculations on this ion. Additional diffuse functions were added to the primitive basis sets in the hope that these would improve the description of the excited states. However, despite these refinements, agreement with experiment was only fair. Very recently, Pitzer's groupsB have made use of their SCF programme which makes efficient use of symmetry,looand have studied the ground and excited states with large contracted basis sets, obtaining lower energies than in previous work. The authors also carried out open-shell SCF calculations on several excited states, and also studied the MnO,Z- ion. Agreement with experiment was again not very good. It seems clear that very expensive calculations will be needed before this problem can be said to have been solved. The isoelectronic ions C104-, SOP-, and Pod3- have been studied by Johansen.lol A medium size basis set was used, and regular tetrahedral geometry assumed. The energy-level order was in agreement with experiment. Other work on these molecules particularly concerned with the photoelectron spectrum has been reported.lo2 D. HAB3.-Although several molecules of this general formulae have been studied earlier, only two papers which deal with the Lewis adducts of BH, and BF, are referred to here. Runtz and Baderlo3have introduced a virial partitioning of the molecular energies in polyatomic systems, and have discussed this partitioning method and its application to the Lewis adducts BH3H-, BH,F-, BF3H-, and BH3C0.104This method involves a spatial partitioning of the molecule into atom-like fragments, and partitions its properties into a sum of fragment contributions in a manner determined by the topographical properties of the charge distribution. Further details are to be found in references 103 and 104. It was concluded that the relative stabilities of BH, and BF3 to act as Lewis acids is determined more by the properties of the terminal H and F fragments than by the net charge and energy of the B fragment. The reaction of BF:, HF has been looked at from a different viewpoint by Silla et a1.lo6SCF calculations using both minimal and 4-31G basis sets were carried out and the molecular geometry optimized. For BF,- the results were similar for the MBS case to those found earlier by Fitzpatrick.loBThe geometry of the adduct HBF,- was determined. The 4-31G basis set results gave longer bond lengths. Differences in the

+

I. H. Hillier and V. R. Saunders, Proc. Roy. SOC.,1970, A320, 161. P. D. Dacre and M. Elder, Chern. Phys. Letters, 1971, 11, 377. ~ 3 5 I. H. Hillier and V. R. Saunders, Chem. Phys. Letters, 1971, 9, 219, 96 A. P. Mortola, H. Basch, and J. W. Moskowitz, Internat. J . Qirnntum Chem., 1973, 7 , 725. 9 7 H. Johansen, Chem. Phys. Letters. 1972, 17, 569. 98 M. H. Wood, Theor. Chim. Acta, 1975, 36, 309. 99 H.-L. Hsu, C. Petersen, and R. M. Pitzer, J . Chem. Phys., 1976, 64, 791. loo R. M. Pitzer, J . Chern. Phys., 1973, 58, 3111. 101 H. Johansen, Chem. Phys. Letters, 1971, 11, 466. 102 J. A. Connor, I. H. Hillier, V. R. Saunders, and M. Barber, Mol. Phys., 1972, 23, 81. 103 R. F. W. Bader and G. R. Runtz, Mol. Phys., 1975, 30, 117. 104 G. R. Runtz and R. F. W. Bader, Mol. Phys., 1975, 30, 129. lo5 E. Silla, E. Scrocco, and J. Tomasi, Theor. Chim. A d a , 1975, 40,343. 106 M. J. Fitzpatrick, Inorg. Nuclear Chem. Letters, 1973, 9 , 965. 93 94

Ab initio Culcirfations oil Molecules coiitainiiig Fice or Six Atoms

13

electrophilicity and proton donating capability of HF due to the formation of the adduct with BF, were evidenced and discussed. A DZ-quality calculation on HNO, has been reported and used i n the interpretation of the photoelectron

of this type have been studied by several different E. N,N-X.-Molecules workers. Hopkinson and Csizmadialo8reported a detailed study of the reaction (3). NHzX

+

H++&H;jX

(3)

with X = H , CH,, F, OH, CN, CHO, or NO2. A rather small (8,3/3)+[21/1] basis set was used. The geometry of the aminiuni ions was optimized, and experimental + geometries were used for the amines. The rotational barriers in NH,X were all found to be small. The Is orbital energies were correlated with the charges and compared with ESCA data. NH2CN has been studied more recently with a larger basis set, together with NF,CN and PF,CN.lo9 The optimum RCN angles were calculated to be 178.9", 176.6", and 175", assuming fixed bond lengths. A rationalization of the observed angles in terms of nuclear-nuclear and electron-electron repulsions was given, It should, however, be pointed out that the large uncontracted basis set did not include polarization functions, except for d-functions on phosphorus. NH,CI has been carefully studied by microwave spectroscopy and SCF calculations of its molecular properties using an essentially DZ basis have been recently reported.llo The barrier to inversion was computed and analysed in terms of the different contributions to the barrier. The main conclusion was that the barrier is 'attractive' in Allen's terminology.'" The dipole 2nd quadrupole moments were also computed i n this paper. Hydroxylamines have been investigated previously,112y113 but in a more recent paper, Trindle and S h i l l a d ~ have l ~ ~ computed the potential surface for the interconversion of ammonium oxide with hydroxylamine [equation (4)]. NH30

+-+ MzNOH

(4)

A minimal basis set of STO (expanded in Gaussian lobe functions) was used, and the least-energy pathway determined, assuming fixed values for some of the parameters. The complete optimization is not practicable for systems of this size and most surface studies make physically reasonable approximations to reduce the number of variables. The computed pathway is a non-least-motion rearrangement. The implicatiorx of the results with respect to the stability of F,NOF relative to F,NO were also discussed. A related set of calculations by Hart115also studied NH,O and its stability with respect to NH,OH. Gaussian lobe basis functions with a similar quality basis set to that in earlier work were used.116 107

108 lo8 11O

111 112

113

115 116

D. R. Lloyd, P. J. Roberts, and I. H. Hillier, J.C.S. For(idc.v 11, 1975, 71, 496. A. C . HopAi~ihoiiand 1. G. Csizmndia, Tlieor. Chim. Acta, 1974, 34, 93. J. M. Howell, A. R . Roqsi, and R. Bissell, Clzem. Phys. Letters, 1976, 39, 312. G. L. Bcndazalli, D. G. Licter, and P. Palniieri, J.C.S. Fclrndrry 11, 1973, 69, 791 W. H. Fink and L. C. Allcn. J . Clieiiz. P h y ~ . 1967, , 46, 2261. W. H. Fink, D. C. Pan, and I,. C . Allen, J . Chrrn. Phvs., 1967, 47, 985. L. Pedcrson and K. Morukuma, J . Cliem. Pliys., 1967, 46, 3941. C. Trindle and D. D. Shillady, J . Arizcr. Chevn. Sbc., 1973, 95, 703. R . T. Hart, Arrstmt. 3. Chem., 1976, 29, 231. F. R. Burden and B. T. Hart, Artsrrtil. J . Client., 1973, 26, 1381.

14

Theoretical Chemistry

The ground state of N H 3 0 has a long NO bond (1.689 A) and the potential energy curves for the dissociation to NH3(A,) and O ( l 0 ) were obtained. Population analyses were presented, and N H 3 0 was predicted to be ca. 126 kJ mol-1 less stable than NH,OH, a result rather similar to that obtained by Trindle with a smaller basis set. The 3E excited state was investigated and found to be repulsive, which was attributed to the approaching O(3P)causing excitation of an ammonia lone pair into an NO antibonding orbital. A second paper1* computed localized orbitals, one-electron properties, and detailed populations and gave a detailed discussion of the bonding in this species and in hy droxyl amine, The as yet unknown molecule H,NNO was studied several years ago, but a careful study of its isomerization to the diazohydroxide has been carried out recently by Provan and Thomson.lls Using an STO-3G basis set, several pathways were investigated and the minimum energy configuration of the diazohydroxide was established. The relevant transformations are those in equation (5). The calculations

with both STO-3G and STO 4-31 G basis sets predict the traiis,cis (1) and cis,trans (2) configurations of the hydroxide to be of comparable energy (after geometry optimization) and the pathway for the migration of the proton to give these isomers was investigated in detail. The barrier to the isomerization to (1) was calculated to be 381 kJ mol-l whereas that for the out-of-plane rotation which gives (2) was 46 kJ mol-l. Alternative pathways were investigated including one involving the intermediate (3).

Investigation of the corresponding transformations in the monomethylnitrosamine, CH,NHNO, gave rather similar results, and it was concluded that possibly solvation may have an important influence on the barrier. The addition of a water molecule to mimic at least the first stages of solvation resulted in a decrease of the barrier by ca. 60 kJ mol-l, and further investigations of solvation in this reaction and in other reactions involved in this process11Bshould, it is hoped, shed some light on these transformations which are believed to be important in the mechanism of chemical carcinogenesis by N-nitro~amines.~’~ F. H,C=NX.-MethyIeneirnhe, H,C=NH, is the simplest molecule of this type 117 118

119

B. T. Hart, Austral. J . Clrem., 1976, 29, 241. D. Provan, C . Thomson, and S. C . Clark, Ztitertzhf.J . Qrrantrini Chem,, 1977, S4, 205. C . Heidelburger, ,411ti. RPO.Biocliem., 1975, 44, 79.

Ab initio Calculations on Molecules containing Fitye or Six Atoms

15

and has recently been the subject of several papers. Macauley et al.lZ0reported 4-31G geometry optimizations on the ground and low-lying excited states, and compared the orbitals with those of the

\C=O group. The barriers to inversion were computed /

in this work. Kollman and co-workersl*lreported similar calculations and also investigated the protonated form, CH2NH2+,using, however, up to DZ-quality basis sets. In a more recent study, Botschwina122computed the force field with a (73/3) basis set and the method of Pulay and Meyer.12sThe results were in good agreement with the Iimited experimental data. The work on H2C=NH discussed above has been restricted to the parent molecule, but has recently reported a detailed investigation of the seven fluoroderivatives,together with the parent compound. The STO-3G basis was used and the geometries were optimized. For CHF=NF the cis isomer was predicted to be more stable than the trans, and this situation is also found in NF=NF, NH=NH, and CHF=CHF. The previous ab initio 126 computations on the monofluoro-derivative did not include geometry optimization. The above studies are of particular interest in view of the recent synthesis and observation of H2C=PH and H2C=PC1.12* These molecules have also been studied using STO-3G and STO 4-31G basis sets by T h o m ~ o n , lwho ~ ~ optimized the geometries. In CH,=PH, the HOMO is a n-orbital, as in CH2=PCl and CF2=PH, but this is different to the situation in CH2=NH in which the HOMO is a a-orbital. The P atom has a substantial positive charge, and the dipole moments were computed for each molecule. The CPH angle was 95-98' in each case. Table 2 summarizes these results. Table 2 Computed geometry, energies, and dipole moments ( p ) of phosphalkeneswith STO-3G and STO 43-1G basis sets Molecule H&=P H&=PCIa F2C =PH

Basis 3G 43-1G 3G 3G 43-1G

R(CP)/A 1.62 1.67 1.65 1.67 1.74

R(PX)/A 1.39 1.47 2.115 1.39 1.67

CPX/O 97.0 98 -0 99.2 92.6 94.4

p/D 0.357 1.08 3.254 2.60 1.024

E/kJ mol-l

- 375.9883 -379.8149 -830.0383 -570.9110 -570.2519

Calculations with the 4-31G basis set gave convergence problems and were not pursued in view of the expense. 0

120

lel

R. Macauley, L. A. Burnelle, and C. Sandorfy, Theor. Chim. Acta, 1973, 29, 1. P. A. Kollman, W. F. Trager, S. Rothenberg, and J. E. Williams, J. Amer. Chem. Soc., 1973, 95, 458.

P. Botschwina, Chem. Phys. Letters, 1974, 29, 580. P. Pulay and W. Meyer, Theor. Chim. Acta, 1974, 32, 253. lt4 J. M. Howell, J. Amer. Chem. SOC., 1976, 98, 886. lt5 R. Ditchfield, J. E. del Bene, and J. A. Pople, J. Amer. Chem. Soc., 1972, 94, 703. la6M. J. Hopkinson, H. W. Kroto, J. F. Nixon, and N . P. C. Simmons, J.C.S. Chem. Comm., 122 123

lS7

1976,513. C . Thomson, J.C.S. Chem. Comm., 1977, 322.

Theoretical Chemistry

16

G. Nitrenes.-The

importance of nitrenes is well known,12*and the prototype NH has been extensively studied in the past,12gbut until recently no ab irzitio work has appeared on other nitrenes. The simplest alkylnitrene, MeN, has not been studied spectroscopicallybut three recent papers have investigated the ground state and some excited states."* 1309 131 Pople and co-workers,ll using an MBS (STO-3G) computed the ground-state geometry, but a more extensive set of calculations at the SCF level with a DZ basis The 3 A z ground-state and the lE and lAl set has been reported by Yarkony et excited-state geometries were computed and found to be rather similar. The CH bond length and NCH angle are very like those in MeOH, but the CN bond length of 1.47 A seems to indicate a single bond, and this is further substantiated by the population analysis. A variety of molecular properties were also computed with a DZ P basis set. Additional computations on CH2=NH showed that MeN is about 42 kJ less stable than the imine, which may account for the fact that MeN has not yet been detected experimentally. Harrison and Shalhoubl3 have recently investigated the related carbonylnitrenes HCON, FCON, MeCON, and MeOCON, finding a triplet ground state in all cases. Geometry optimizations were carried out using an STO-3G basis set. H. Diazomethane.-This molecule is of considerable importance as a source of methylene, and there have been several detailed studies recently. Following earlier work on the ground state by Snyder and Basch,13, Hart, in 1973, carried out calculations using a Gaussian lobe basis set on CH2N2 and several of its i s o r n e r ~ . l ~ ~ A similar study by Leroy and Sana134in 1974 also employed an essentially minimal basis set (73/3) and also Pople's 4-31G basis. The bonding and charge distribution were discussed and the enthalpy of formation computed. More recently two complementary studies on the dissociation [reaction (6)] have

+

CHzNz-CH2

+

N2

(6)

appeared. Walch and Goddard13j reported a thorough GVB and GVB-CI study of the ground state and several low-lying states with a DZ-quality basis set. The GVB description of the ground state is essentially that of a singlet biradical (cf. 03). The bonding changes on forming 21A1 CH, and XICs+ N, were also examined. Vertical excitation energies were in good agreement with experimental results. A similar study using Gaussian lobe functions (DZ accuracy) and a few calculations with a DZ + P basis were reported by Lievin and Ve~haegen.l~~ Both the F A and 3B1,lB1, and lA,* states were investigated as a function of RCH,RNN,and HCH at several values of RCN. The dissociation energy of CH2N2was predicted to be 0.91 eV and the term energy of the l A l state to be 0.49 eV. Table 3 lists some of the results for the computed geometry and force constants for CH2N2. 'Nitrenes', ed. W. I. Lwowski, Interscience, New York, 1970. See reference 1 for recent work. D. R. Yarkony, H. F. Schaefer, and S. Rothenberg, J . Amer. Chem. Soc., 1974, 96, 5974. J. F. Harrison and G. ShaIhoub, 3. Amer. Chem. SOC.,1975, 97,4172. lS2 L. C. Snyder and H. Basch, J . Amer. Chem. SOC.,1969, 91, 2189. la3 B. T. Hart, Ausfrul. J . Chem., 1973, 26, 461, 477. m G. Leroy and M. Sana, Theor. Chim. Acfa, 1974, 33, 329. 135 S. P. Walch and W. A. Goddard, J. Amer. Chem. SOC.,1975, 97, 5319. 136 J. Lievin and G. Verhaegen, Theor. Chim. Acfa, 1976, 42, 47.

12*

lZ9

l30 l3I

Ab initio Calculations on Molecules containing Fiue or Six Atoms

17

Table 3 Calculated equilibrium geometry and force constants of CH,Nz (lA,) Calculated Property ~ R C H(deg.) RCH (A) RCN(A) RNN(A) RCN R N N(A)

+

ke (erg rad-2) ke-cH (dyn rad-I)

(dyn rad-l) ICCH (dyn cm-1) ~ C H - C N(dyn cm-l) ICCN (dyn cm-1) ~ C N - N N(dyn cm-l) ~ N (dyn N cm-l)

k0-cN

a

Experiment a

Ref. 134

126 1.075 1.300 1.139 2.439 0.631 x 10-11

121.7 1.078 1.282 1.189 2.471 3.6 x 10-11

-0.467 x 10-3 5.411 x 105

-

-

8.34~ 105 1.23 x 105 16.89~ 105

-

-

12.53~105

-

1 7 . 4 8 105 ~

Ref. 136 123 1.080 1.289 1.148 2.436 0.801 x 10-11 0.036 x 10-3 -0.583 x 10-3 6.497 x 105 0.413 x l o 5 10.91 x 105 1.95~ lo5 18.98 x 105

See ref. 134 for references.

Cyanogen Azide. Cyanogen azide (NCN,) has recently been studied experimentally, despite its instability, and a calculation with a (73)-+[2,1] GTO basis has been rep0~ted.l~' The first two IP correlate with the theoretical orbital energies, using Koopmans' theorem, but the lone pair orbital ionization energies are in error by 3 eV. I. H,C-XY.-The most extensive studies on this type of molecule are those on CH,OH and its ions. The radical itself has been discussed by several different authors. A conformational study by Bendazolli et using the counterpoise orbital method13shas focused attention on the barrier to rotation of the OH bond, given fixed values of the other parameters. Essentially an STO-4G basis was used. The counterpoisefunctions are located at different positions in space from the nucleii and balance the difference in quality of the basis between configurations. In CH,OH, the computed barrier (1003 cm-l) is in good agreement with a barrier of 900 cm-1 computed with a fully optimized 4-31G basis. The potential functions obtained were used to predict the temperature variation of the !-proton hyperfine coupling constant. Although this calculation assumed a planar radical, e.s.r. evidence does not support this assumption.140 DZ calculations by Ha141 (RHF) optimizing the geometry resulted in a nonplanar computed conformation and a barrier to inversion of 1.7 kJ mol-l. The barrier to OH rotation for the cis form was ca. 7.6 kJ mol-1 and ca. 15 kJ mo1-1 for the trans. Both these are repulsive barriers. A detailed study of substituent effects on a variety of radicals of the type CH,X has included work on CH,OH and CH,SH.l*, Pople also reported earlier an STO-3G B. Bak, P. Jansen, and H. Stafast, Chem. Plzys. Letters, 1975, 35, 247. G. L. Bendazzoli, P. Palrnieri, and G . F. Pedulli, Chem. Pkys. Letters, 1974, 29, 123. 139 S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, 553. 140 P. J. Krusic, P. Meakin, and J. P. Jesson, J. Phys. Chem., 1971, 75, 3438. 141 T.-K. Ha, Chem. Phys. Letters, 1975, 30, 379. 1,*2 F. Bernardi, N. D. Epiotis, W. Cherry, H. B. Schlegel, M,-H. Whangbo, and S. Wolfe, J. Amer. Chem. SOC., 1976, 98, 469.

137 138

Theoretical Chemistry

18

study," but Bernardi et aZ. fully optimized the geometry with a 4-31G and found a non-planar structure (out-of-plane angle= 27") and inversion barrier ca. 2.1 kJ mol-l. The angles were significantly different from the STO-3G results.ll Conformational preferences and the orbitals were discussed in detail. CH2SH was also found to be very slightly non-planar but with a very low barrier. In both cases, the inversion cross-section is much shallower than that for rotation. CH2F and CH,Cl were also investigated and found to be non-planar. Studies of the anions CH20H- and CH2SH- have been reported, particularly the rotation-inversion behaviour. Plots of the potential energy surface were analysed in terms of possible paths between the different conformation^.^^^ A more recent paper14sdealt in detail with the protonation of these molecules and of their isomers MeX- [reaction (7)]. Several basis sets of different sizes were used which contained CHZXH-

+

H+-+MeXHtH+

+

MeX-

(X = O o r S )

(7)

either sp or sp and dfunctions, and full geometry optimization was performed at fixed CH, SH, and OH bond lengths. There was no evidence for d-orbital effects on any of the properties of these species, and it was proposed that differences in the behaviour of these molecules is due to the longer C-S bond length and greater polarizability of the S atom and not to conjugation effects. Protonated formaldehyde is the cation HOCH2+and this has been the subject of a paper by Tel, Wolfe, and Csizmadia,14*following earlier work by Rosi4' and Lehn.14* The ion is planar and there is a barrier to rotation about the CO bond of 82 kJ mol-l. Imposition of a pyramidal geometry on the carbonium ion centre causes a change in the barrier from 82 kJ mol-1 to 64 kJ mo1-l. The results were compared with calculations on MeCH,+. Although in the above paper MeS- was studied, MeO- was not, but the latter has now been investigated in detail. Lehn148and co-workers computed the PE surface for the simplest nucleophilic addition to a carbonyl group, i.e. reaction (8). A

(73/4)+[32/2] basis set was used. More recently Yarkony et aZ.lsOhave made a detailed study of the geometries of the Z e E and a 2 A 1states of the methoxy radical Me0 and also of the methoxide ion. The three geometries were found to be significantly different (Table 4). The ion is significantly more pyramidal than methanol. A discussion of the possible reasons for this and the long CH bond length in this 143

F. Bernardi, I. G. Csizmadia, H.B. Schlegel, M.Tiecco, M.-H.Whangbo, and

144

Guzzetta., 1974, 104, 1101. S. WoIfe, H. B. SchIegel, I. 2020.

S. Wolfe,

G.Csizmadia, and F. Bernardi, J. Amer. Chem. SOC.,1975, 97,

F. Bernardi, I. G. Csizmadia, A. Mangini, H. B. Schlegel, M.-H. Whangbo, and S. Wolfe, J . Amer. Chem. SOC.,1975, 97, 2209. 148 L. M. Tel, S. Wolfe, and I. G. Csizmadia, Internut. J. Quuntitrn Chem., 1973, 7 , 475. 1 4 7 P. Ros, J. Chem. Phys., 1968, 49, 4902. 14* J. M. Lehn, B. Munsch, and P. Millie, Theor. Chim. Acru, 1970, 16, 351. 149 H. B. Biirgi, J. M. Lehn, and G. Wipff, J. A m w . Chem. SOC.,1974, 96, 1956. 150 D. R. Yarkony, H. F. Schaefer, and S . Rothenberg, J . Amer. Chem. Soc., 1974, 96, 656. 145

Ab inito Calculations on Molecules containing Five or Six Atoms

Table 4 Geometry predictions geometry of MeOH Molecule Me0 Me0 MeOMeOH a

Bond lengths in A.

b

50

19

for Me0 and MeO- compared with the experimental

State

R(CH) a

2E

2Ai

1.08 1.08

lAi lAi

1.12 1.093c

Angles in degrees.

C

R(CO) 1.44 1.65 1.39 1.434

L O(OCH)a 109

102 114

109.5

Not determined experimentally: value is for CHs.

species was given in terms of the Mulliken populations. The Jahn-Teller distortion in the ground state of Me0 is predicted to be small. Turning now to other molecules of this general formula, the most extensively studied has probably been keten, CH,=C=O. Several papers on this molecule, prior to 1973,have appeared, but a recent paper by Dykstra and Schaefer is the most detailed yet.151 The authors compared their calculations with earlier minimal basis set calculations by del Bene,lsZwho also carried out limited CI, and predicted the position of the lowest excited states. Dykstra and Schaefer used DZ, DZ+Rydberg, and DZ+P basis sets. Geometry optimizationswere carried out for the DZ basis and they carried out selected calculations with the larger bases for this geometry. Vertical transitions to 18 different states were computed, and similar results obtained with the different basis sets. It is clear from the results that the inclusion of Rydberg orbitals is important, and electron correlation has a large effect on the AE, emphasizing that CI calculations with a DZ basis set are highly desirable. B a ~ c hinvestigated l~~ the photodecomposition of keten and carried out both SCF and MCSF calculations with an augmented DZ basis set. It was concluded that the first excited triplet state of keten can form 3CH2(3B,)and lCH,(lA relatively rapidly and the first excited singlet state of keten can give 3CH2(3Bz) easily in a near-leastmotion path. However, the formation of lCHz(lA1)from the first excited singlet state of keten by a near-least-motion path seemed to be very improbable. Pendergast and Fink 15*have performed somewhat similar calculations, including CI, on the surface for reaction (9). Partial geometry optimizations were carried out CHz

+ CO+CHzCO

(9)

and reasonable agreement with experiment was obtained. More configurations were used in the CI than in ref. 153 but the basis set used was smaller. Ellinger 155 and co-workers have obtained ab initio electrostatic molecular potential maps 156 for the iminoxy radical H,CNO in investigations of the electrophilic reactivity in nitroxides, using a small basis set. Attack at the nitrogen lone pair is favoured in this molecule. Two recent papers have dealt with the stability of the ‘Criegee intermediate’,

155

C. E. Dykstra and 31. F. Schaefer, J. Amer. Chem. SOC.,1976,98, 2689. J. del Bene, J. Amer. Chem. SOC.,1972, 94, 3713. H. Basch, Theor. Chirn. Acta, 1973, 28, 151. P. Pendergast and W. H. Fink, J . Amer. Chem. SOC.,1976, 98, 648. Y. Ellinger, R. Subra, G. Berthier, and J. Tomasi, J. Phys. Chem., 1975, 79, 2440.

156

E. Scrocco and J. Tomasi, Fortschr. Chem. Forsch., 1973, 42, 95.

151 152

l53 154

Theoretical Chemistry

20

CH202,which has been postulated as an intermediate in the reaction of O 3 with CzH4.15' Ha and c o - w ~ r k e r s 'carried ~~ out Gaussian lobe SCF calculations of approximately DZ quality. Several conformations were considered and the three-membered ring was found to be the most stable, with the CH, and CO, groups perpendicular to one another. A much more detailed study of the same species was by Wadt and GoddardlSS using the GVB method. The ring form was shown to be > 1 eV more stable than the open form, which was also predicted to have a biradical rather than a zwitterionic structure. The total energies in all cases were below those in Ha's work. A detailed qualitative account of the similarity to 0, was given in terms of the GVB orbitals. Unlike many recent papers, an extensive discussion was given of the thermochemistry of the decomposition reaction. GVB-CI calculations gave excitation energies for various states and the calculations shed considerable light on the experimental work on this interesting reaction. J. Carbonium Ions.-Carbonium ions are very important in organic chemistry160 and many carbocations and carbanions have aroused the interest of theoreticians. A detailed report of work in this area has been given by and Hehre,le2and space does not allow for further discussion here. However, we do refer to selected papers of interest which have appeared more recently. One such species which has been thoroughly studied is C2H3+, the vinyl cation. Early STO-3Gls3studies predicted a more stable classical form (4) rather than the symmetrical bridged structure (5). Addition of d-functions was later shown to

(4)

(5)

stabilize the bridged form relative to (4); p-functions on the H atoms also had the same effect. The latter calculations were with a 6-31G basis, and the energy difference was only 23.8 kJ mol-l. Clearly more refined calculations were needed and an IEPA-NO study by Zurowski et al.le' found that the bridged structure was more stable by ca. 29.3 kJ mol-l. The authors discussed the probable accuracy of the resuIts and believe that the qualitative prediction is correct. The same conclusion was found for CzH5+. In an attempt to provide a definite answer to this question, Weber et al. have investigated the optimum path for the rearrangement between the two strucR. Criegee, Rec. Chem. Progr., 1957, 18, 11. T.-K. Ha, H. Kuhne, S. Vaccani, and Hs. H. Gunthard, Clzem. Phys. Letters, 1974, 24, 172. 1 5 9 W. R. Wadt and W. A. Goddard, J . Amer. Chem. SOC.,1975, 97, 3004. 160 See, e.F. 'Carbonium Ions', ed. G. Olah and P. von R. Schleyer, Wiley Interscience, New York, 1968, Vol. I. 161 L. Radom, J. A, Pople, and P. von R. Schleyer, J . Anzcr. Chern. SOC.,1972, 94, 5935. 162 W. S, Hehre, A C C O I ~Client. I I ~ ~ Res., 1975, 8, 369. l G 3P. C. Hariharan, W.A. Lathan, and J . A. Pople, Clzcm. Pliys. Letters, 1972, 14, 385. 164 B. Zurawski, R. Ahlrichs, and W. Kutzelnigg, Cltern. Phys. Letters, 1973, 21, 309.

157 158

9

Ab initio Calculations on Molecules containing Five or Six Atoms

21

+

tures.166J66For each nuclear configuration, a DZ P quality SCF wave function was computed, together with an extensive CI wave function (all single and double excitations). The points on the minimum energy path were in the plane and during the migration of H1, the atom H3 moves trans to H2 with L C1C2H3=188.9' when a = 90.8, and back through the linear position to the final L C1C2H3of 179.1 '. The inclusion of CI alters the path significantly. However, after critically evaluating the errors involved, the authors conclude that the classical and bridged structures have the same energy to within 4-8 kJ mol-1 with (5) probably the more stable, and that the barrier to rearrangement is also low (ca. 4-13 kJ). The importance of geometry optimization was emphasized. Carbanions have been less studied, apart from CH3-,167916* but included in a more recent set of calculations169on several carbanions (CH3-, C2H5-,and ethynyl anion) are calculations on CzH3-. For reliable calculations on this type of molecule, diffuse orbitals must be added to the basis set. Several different basis sets were used, but the geometry of the neutral parent molecule was used in some of the calculations. The main aim of this paper was to investigate the electron density and difference densities, electron affinities, and proton affinities. The inversion barrier in the vinyl anion was ca. 142 kJ mo1-1 which was in good agreement with that found by Lehn et in an earlier calculation. K. Miscellaneous Five-atom Molecules.-We mention in this section calculations on a variety of molecules which are not readily classified. One such molecule is the as yet undetected sulphilimine, H2SNH. A 4-31G basis set computation of the energy surface for rotation-inversion has been presented.170 The optimum HSN angle was much smaller than the observed angle in known compounds R1R2SNH. Hydroxyborane, H,BOH, has been studied171with a (9,5,1)+[4,2,1] basis set. The OH bond length and the BOH angle were optimized. Of particular interest in this study was the rotational barrier about the BO bond. The planar form of the molecule is the most stable, and the computed barrier was 68.6 kJ mol-l. A partial n-bond is superimposed on the cr-bond. An analysis of the energy and population components was carried out. Formic acid, HCOOH, was studied several years ago, but more recently the cation H2COH+(protonated formaldehyde) and the surface for the reaction have HzCOH++HCO+

+

H 2

(10)

been studied. A saddle point was found and the activation energy computed. Acyloxyl cations R - g 0 have been the subject of a detailed study at the MBS \O+ level by Maier and R e e t ~ DZ . ~ ~basis ~ sets were also used. The cyclic dioxiryl 165 166 167 168 16g

170

J. Weber and A. D. McLean, J. Amer. Chem. SOC.,1976, 98, 875. J. Weber, M. Yoshimine, and A. D. McLean, J. Chem. Phys., 1976, 64,4139. F. Driessler, R. Ahlrichs, V. Staemmler, and W. Kutzelnigg, Theor. Chim. Acta, 1973,30, 315. R. Ahlrichs, F. Driessler, H. Lischka, and V. Staemmler, J. Chem. Phys., 1975, 62, 1235. J. E. Williams, jun. and A. Streitweiser, jun., J. Amer. Chem. SOC., 1975, 97, 2634. P. Mezey, A. Kucsman, G. Theodorakopoulos, and I. G. Csizmadia, Theor. Chim. Acta, 1975, 38, 115.

171 173 173

0.Gropen and R. Johansen, J. Mol. Structure, 1975, 25, 161. S. Saebo, Chem. Phys. Letters, 1976, 40, 462.

W. F. Maier and M. T. Reetz, J. Amer. Chem. SOC.,1975, 97, 3687.

lleoretical Chemistry

22

cation (6) was found to be the lowest-energy species. Calculations were reported with R = H , F, Me, or HCsC.

(6)

Goddard and co-workers have reported calculations on another possible intermediate in several interesting reactions, the peroxyformyl radical, HC03.174HF calculations on the 2A’state suggest that it may readily decompose to HO+CO,. This state is predicted to be only 0.4 eV above the ground state, 2A”.It is emphasized that the existence of a low-lying excited state in this case (as in HOz) has an important role in oxidation processes. Hopkinson in 197317s investigated the Wolff rearrangement of diazo-ketones and a-diazo-esters to form ketens. In particular, an MBS was used to investigate the equilibrium [equation (1 I)]” between the parent oxiren (7) and the isomer formyl-

carbene (8). The author concluded that oxiren was more stable by only ca. 1.7 kJ mol-1 and both isomers were less stable than keten by ca. 300 kJ mol-l. Recently a more extensive set of calculations using a DZ basis set with full geometry optimization showed the carbene to be ca. 50 kJ mol-l lower in energy.17* The reaction path for the conversion was also studied, and the authors found Ea for the ring opening of 30.5 kJ mol-l. 17* Carbon suboxide C302has been the subject of several papers. Earlier disagreed as to whether the structure is linear or bent. A (9,5,1)+[4,3,1] basis set calculation 17*concluded that it is essential to include d-orbitals to determine the potential for CCC bending. Although the agreement with experiment was not satisfactory, the molecule is predicted to be non-linear but it was concluded that a more extensive basis set calculation is needed. Calculations including CI are needed here. H,O,- is the hydrated OH- ion, and several papers have dealt with this species. The two most recent studies included correlation via a large C1180,181 following earlier SCF ca1culationsla2with the same basis set. The SCF calculations gave a linear, slightly asymmetric H-bond, and a similar result was obtained by Newton and

N.W. Winter, W. A. Goddard, and C. F. Bender, Chem. Phys. Letters,

1975, 33, 25. A. C. Hopkinson, J.C.S. Perkin II, 1973, 794. IT* 0. P. Strausz, R. K. Gosavi, A. S. Denes, and I. G . Csizmadia, J. Amer. Chem. SOC., 1976,98, 4784. l T 7 J. R. Sabin and H. Kim, J. Chem. Phys., 1972,56,2195. 17* L. I. Weimann and R. E. Christoffersen, J . Amer. Chem. Sac., 1973,95, 2074. 178 H. H. Jensen, E. W. Nilssen, and H. M. Seip, Chem. Phys. Letters, 1974, 27, 338. 180 B. 0.Roos, W. P. Kraemer, and G . F. Diercksen, Theor. Chim. Acra, 1976, 42, 77. ln1A. Stragard, A. Strich, J. Almlof, and B. ROOS, Chem. Phys., 1975, 8, 405. lS2 W. P. Kraemer and G. F. Diercksen, Theor. Chim. Acta, 1972, 23, 398. lT4

175

Ab initio Calculations on Molecules containing Five or Six Atoms

23

E h r e n s ~ nbut , ~ ~the ~ latter authors obtained a smaller barrier for the proton transfer reaction. An earlier CI calculation181with a limited basis gave, however, a symmetric single minimum H-bond, but the most extensive recent work, involving 50280 configurations, and using a basis including polarization functions on each centre give an almost symmetric equilibrium structure, the correlation energy stabilizing the system by 15.1 kJ mol-l. Hillier and Kendrickls4have also carried out CI calculations on the hole states of C,O, in order to interpret the ESCA spectrum; agreement with experiment was quite good. Ab-initiu calculations on H,O, lS5 have also been reported. This molecule is predicted to be stable with a zig-zag chain structure. A more recent study by Blint 18@ and Newton discussed the bonding in detail. S ( C N ) , is an interesting compound which has a non-linear SCN arrangement. The high-resolution photoelectron spectrum has been recorded, and ab initiu calculat i o n have ~ ~ ~ been ~ carried out on the ground state and the 2B1state of S(CN),+ for the ground-state geometry. The calculations predict a lower energy for the non-linear SCN case, although full geometry optimization was not attempted. Electron density maps were presented and the localized MO suggests some episulphide ring contributions to the dominant open-chain structure of S(CN),. Space does not permit the inclusion of any of the papers dealing with hydrogen bonding18*or solvation phenomena,lss and the reader is referred to the above references for more details. It is clear, however, from recent work that it is feasible to include the interaction with several solvent molecules using the ‘supermolecule’ approach, and very interesting and informative results have been obtained in this way. 3 Molecules containing Six Atoms Lack of space prevents a comprehensive survey, and we will deal primarily with the following general classes of molecules: A2B4, AH,, AX6,followed by sectionsdealing with formamide, MeXY, glyoxyl, and a few other molecules. A. A2B4.-C2H4. Several molecules of widespread chemical interest belong to this class but CzH4is by far the most extensively studied. It is not possible to do more than point to some calculations which have furnished interesting detail of the electronic structure, and we confine ourselves to introducing the more recent work in the light of the earlier studies. There have been several calculations on the ground state of the molecule. Earlier SCF calculations are listed in the bibliography,4 and we mention here two more recent studies. 183 184 185

186

187 188 189

2

M. D. Newton and S. Ehrenson, J. Amer. Chem. SOC.,1971, 93, 4271. I. H. Hillier and J. Kendrick, J.C.S. Faraday II, 1975, 71, 1369. B. PlesniEar, S. Kaiser, and A. Aiman, J. Amer. Chem. SOC.,1973, 95, 5476. R. J. Blint and M. D. Newton, J . Chem. Phys., 1973, 59, 6220. P. Rosmos, H. Stafast, and H. Bock, Chem. Phys. Letters, 1975, 35, 275. L. C. Allen, J. Amer. Chem. SOC., 1975, 97, 6921. See, e.g. H. Kistenmacher, H. Popkie, and E. Clementi, J . Chem. Phys., 1974, 61, 799.

24

Theoretical Chemistry

Meza and WahlgrenlgOcarried out SCF calculations with a small contracted basis set on ethylene and fluoroethylene in order to investigate the effect of fluorine substitution on the electronic structure. Detailed population analyses and electron density contour maps were presented, and a variety of one-electron properties computed. The total energy of - 77.9685 Hartree was substantially higher than that obtained by Basch and McKoylgl (- 78.037 32 Hartree), whose work we discuss below. Brundle and c o - w o r k e r ~used ~ ~ ~a larger basis set in a discussion of the photoelectron spectrum of C,H, and B,H,. Agreement with the experimental data was very good. Calculations whose main purpose was the study of the ground and excited states are referred to below. The GVB method has already been mentioned and has been applied to larger molecules during the past few years, and ethylene and its excited states have been the subject of three papers. In the GVB wave functions computed with MBS, DZ, and DZ+P basis sets were described. The orbitals were described in terms of two types of o-bonding pairs, one localized in the C-C region and the second corresponding to the C-H bonds. The o part of the C-C bond involves orbitals with ca. 68 % p-character on their main centre. The n-bond is made up from two n-orbitals, one from each C atom, and this splitting of the pair function increases the n-bond energy by ca. 109kJmol-l compared with the usual doubly occupied n-orbital description. The GVB wave function has an energy of 0.054 Hartree lower than the HF wave function for an MBS calculation. A CI calculation on the ground state using only the four orbitals of the C=C bond lowered the energy by 0.018 Hartree giving a final GVB-GI energy of -77.6978 Hartree. However, the best energy was obtained with the DZ P basis of - 78.1332 Hartree compared with the corresponding SCF energy of - 78.0370 Hartree. The heat of reaction for the process shown in equation (12) was computed to be between 600 and 850 kJ mol-l (depending on the basis) compared with the experimental value of 715 kJ mol-l.

+

CzH4 4 2CHz

(12)

There is a cis-trans barrier of 279 kJ mol-1 for the ground state of CzH4. The n-n*( T ) triplet (3B,,) was computed to have a perpendicular geometry and have a

minimum in energy ca. 7 kJ mol-l lower than the saddle point for the N state. The cis-trans barrier in the Tstate was computed as 131 W mol-l. In a later paper, Wadt and Goddardlg3compared the GVB calculations with various TNDO calculations for various properties of interest, and discussed various problems in the INDO method. Levin et al.lg4have calculated the GVB n-orbitals of C2H4using a fixed m o r e which was obtained from a full HF calculation. The orbitals were spatially projected to obtain the correct spatial and spin symmetry without restricting the nature of the individual orbitals. The agreement with the results of full CI calculations was good for both total energies and excitation energies.

S. Meza and U. Wahlgren, Theor. Chim. Acta, 1971, 21, 323. H. Basch and V. McKoy,J. Chern. Phys., 1970, 53, 1628. lo* C. R. Brundle, M. B. Robin, H. Basch, M. Pinsky, and A. Bond, J . Amer. Chem. Soc., 1970, 92, 3865. lgt W. R. Wadt and W. A. Goddard, J . Amer. Chem. SOC.,1974, 96, 5996. l g 4 G. Levin, W. A. Goddard, and D. L. Huestis, Chern. Phys., 1974, 4, 409. lD1

Ab initio Calculations on Molecules containing Five or Six Atoms

25

The force constant has also been applied to the ground state of C2H, and comparisons have been made with calculations on CzHzand C2H6.1g5 The interesting problem of the excited states is now considered. The lowest (n-n*) states are the 3B1, and lB1,states, usually called the T and V states, and there has been a great deal of controversy concerning the nature of these states. The broad absorption found experimentally at 7.7 eV has been ascribed to the V-N or (n-n*) intra-valencetransition from the ground states.lg6This is believed to be accompanied by a relative twisting motion of the two CH2groups. The n-n* triplet transition lies in the range 4 . 3 4 . 6 eV. Early SCF calculation^^^^^^^^ on the V(lBlu) state predicted that it lies at least 9 eV above the ground state, but Dunning, Hunt, and Goddard,lggin 1969, carried out SCF calculations using additional Rydberg orbitals in the basis set, and found that these had virtually no effect on the predictions for the T state, whose excitation energy is in fair agreement with experiment and which is clearly a valence state. However, for the planar V state, the Rydberg orbitals had a drastic effect, reducing the excitation energy to ca. 8 eV. This state was clearly a Rydberg state since the expectation value <x2> was ca. 50 Bohr2. This calculation provoked considerable discussion. Experimentalists2oo argued that the data indicated a normal valence state for the V state. Basch and McKoy201 carried out various open-shell SCF calculations on the planar and perpendicular states of C2H4.Although a very similar basis set was used to that in ref. 199, the authors found the twisted Vstate to be valence-like, and the planar Ystate to be Rydberg, but they ascribe the latter result to the inadequacy ofthe RHF theory. The twisted state was found to be ca. 0.6 eV below the planar state. Hunt et a1.,201in later work, found a twisting barrier of 1.6 eV. It was clear at this stage that this question could only be resolved by much more extensive calculations including CI. Independently several authors reported CI calculations. Buenker and co-workers2oa carried out both SCF and CI calculations, and obtained nearly quantitative agreement for the V t N singlet transition energy. They also found that the excited state is quite diffuse, but these authors also suggested that the absorption to the Vstate at 7.7 eV is not to the lowest (n-n*)singlet. They also suggest that the fvalue for the transition should be computed to aid in the interpretation of the spectrum. In a later note,203they conclude that non-vertical transitions are responsible for the V-N band, and the broad diffuse nature of the spectrum in this region could be caused by two states in close proximity which have the same symmetry as the twisted ethylene. A more extensive series of calculations including CI and the (9,5) +[4,2] DZ basis set augmented by diffuse functions were carried out by Bender et aL204 195 196 197 198 199

200 201 202

203 204

P. Pulay and W. Meyer, Mol. h y s . , 1974, 27, 473. A. J. Merer and R. S . Mulliken, Chem. Rev., 1969, 69, 639. M. A. Robin, H. Basch, N. A. Kuebler, B. E. Kaplan, and J. Meinwald, J. Chem. Phys., 1969, 48, 5037. 3. M. Schulman, J. W. Moskowitz, and C . Hollister, J. Chem. Phys., 1967, 46, 2759. T. H. Dunning, jun., W. J. Hunt, and W. A. Goddard, Chem. Phys. Letters, 1969, 4, 147. E. Miron, B. Raz, and J. Jortner, Chem. Phys. Letters, 1970, 6, 563. H. Basch and V. McKoy, J. Chem. Phys., 1970, 53, 1628. R. J. Buenker, S . D. Peyerimhoff, and W. E. Kammer, J . Chem. Phys., 1971, 55, 814. R. J. Buenker, S . D. Peyerimhoff, and H. L. Hsu, Chem. Phys. Letters, 1971, 11, 65. C. F. Bender, T. H. Dunning, jun., H. F. Schaefer, W. A. Goddard, and W. J. Hunt, Chem. Plzys. Letters, 1972, 15, 171.

26

Theoretical Chemistry

The HF calculations find the planar V state to be Rydberg-like. n-CI calculations gave results similar to those found by Buenker et aL203 The transition energy AE( V-N)= 8 eV which is slightly lower than the SCF result. However, the Vstate is not found to be valence like, although the CI decreases the size by ca. 30%,it still has a much more diffuse distribution than the triplet state (<x2) ca. 12 Bohr2),but it is not a normal Rydberg state. At about the same time, M o r ~ k u m also a ~ ~found ~ rather similar results, and Ryan and Whitten206carried out two types of CI calculations: the first a n-CI calculation, in which the T state was found to be of valence type. They then included u+o* excitations in the CI, and found that this gave a description of the Vstate which was of essentiallyvalence type. These results, however, were not in accordance with those of Bender and c o - ~ o r k e r s . ~ ~ ~ Levy and Richardzo7have approached this problem in a different way by using an MBS and including electron correlation by second-order perturbation theory. Rydberg orbitals were included and it was shown that the transition energy is nearly independent of the Rydberg character of the n*-orbital. The authors concluded that this result would also hold for larger basis sets, and emphasized that the Rydberg or valence character of the state should not be assessed on purely energy grounds. Basch208has criticized the work of Bender et d 2 0 3 and Ryan et al.,206the first especially because of the lack of 3d orbitals in the basis and that of Ryan20s in omitting possibly important functions from the basis set. Basch carried out MCSCF calculations and used both spatially extended basis functions and polarization functions. However, there was less CI than in ref. 204. The results disagreed with those of Ryan and Whitten,20ebut were in reasonable agreement with those of Bender et aL204The h-n* state becomes increasingly valence-like with improvements in the configuration and orbital bases. However < x 2 ) is still fairly large. Finally, Buenker and Peyerimhoff209 have gone some way towards resolving some of the outstanding questions in this interesting problem. Iwata and Freed2l0have argued previously that inclusion of on -+n*m* configurations would reduce the need for diffuse orbitals in the n* state and to test this and other questions which are raised above they carried O U calculations ~ ~ ~ with ~ (i) inclusion of all single and double excitations with respect to all valence MO's, (ii) inclusion of polarization functions in the bonds, and (iii) the use of approximate natural orbitals in carrying out the CI. The DZ and essentially DZ + P basis sets used (including diffuse orbitals) contained up to 40 contracted functions. The CI involved up to 11OOO configurations and made use of their own CI procedures.211The results were that the N-Vtransition was found to be essentially intravalence with a calculated fvalue of 0.25-0.30, in good agreement with experiment. Diffuse 3d orbitals were quite important in describing the V state, even after extensive CI.The vertical N-V excitation energy was calculated to be 8.1 eV, and the 0-0 transition (90"twisted conformation) was predicted between 6.0-6.2 eV, with very little change in CC bond length on twisting. 205

206 208 209 210

211

K. Morukuma, unpublished work cited in ref. 204. J. A. Ryan and J. L. Whitten, Cltern. Phys. Letters, 1972, 15, 119. B. Levy and J. Ridard, Chem. Phys. Letters, 1972, 15, 49. H. Basch, Chem. Phys. Letters, 1973, 19, 323. R. J. Buenker and S. D. Peyerimhoff, Chem. Phys., 1976, 9, 7 5 . S. Iwata and K. F. Freed, J . Chem. Phys., 1974, 61, 1500. R. J. Buenker and S. D. Peyerimhoff, Theor. Cliim. Acfa, 1974, 35, 33; 1975, 39, 217.

Ab initio Calculations on Molecules containing Five or Six Atoms

27

Thus the inclusion of long-range 3d functions seems to be of crucial importance, as well as CI, in describing the nb state. Table 5 gives the excitation energies for various states of CaH4from this work. Table 5 Calculated vertical excitation energiesfor C,H, (in eV) Method

State GVB(MBS) GVB(DZ)a 0.0 0.0 WAld T(3B~u) 4.70 4.24 V(lB1u) 13.28 9.99 a

Ref. 193.

6

Ref. 209.

C

CIb 0.0 4.34

8.13

Ref. 207 - perturbation theory.

PTc 0

8.8-9.9 Ref. 206.

CId

ExpC

0

0.0 4.4

8.1 C

7.65

Experimental value.

Fischbach et uLzrzhave investigated several different methods of representing the upper orbitals in Rydberg transitions with particular applications to CzH4. Most of the calculations referred to above have assumed a twisted conformation for the triplet state 3(3t,n*),but with flat CH, groups. Possible distortions to a pyramidal geometry have been investigated by Baird and S w e n ~ o nBoth . ~ ~ an ~ MBS and a DZ basis were used and it was found that the pyramidal geometry stabilizes the state by ca. 21 kJ mol-l in the case of the MBS, but only by ca. 5 kJ mol-1 for the DZ basis for the most favourable conformation (cis-flapped). Hence the stabilization effect is rather small and needs studying with larger basis sets and CI. of the barrier to rotaA double-configuration (MCSCF) calculation by tion in the ground state, improves the computed barrier for both CC stretch and CH2 twisting; agreement with experiment was excellent. Ahlrichs et al. have reported the results of IEPA-PNO and CEPA calculations on CzH4 and have analysed the correlation energy contribution^.^^^ Finally, three papers are mentioned which have dealt with the potential energy surfaces for the dimerization of CH, to ethylene, reaction (13), and the transformation of methylcarbene to ethylene, reaction (14). 2CH2 MeCH

__+

__+

CzH4 C2H4

(1 3)

(14)

Basch in 1970 carried out one of the first MCSCF calculations on a polyatomic system216in an investigation of the least-motion, coplanar approach of two methylenes to give CzH4. For two closed shell, singlet state methylenes, the reaction path is purely repulsive, and the reaction occurs for two appropriately oriented bent triplet methylenes. An alternative reaction which gives rise to C2H, is the rearrangement of methylcarbene, MeCH. This reaction can occur in a variety of ways, and Altmann et aL217 carried out SCF calculations with a DZ basis set on two cross-sections of the potential hypersurface. The first cross-section was investigated in an MBS calculation of the variation of E with the CCH angle. 212

213 2l4 216 217

U. Fischbach, R. J. Buenker, and S. D. Peyerimhoff, Chern. Phys., 1974, 5, 265. N. C. Baird and J. R. Swenson, Chem. Phys. Letters, 1973, 22, 183. M. H. Wood, Chem. Phys. Letters, 1974, 24, 239. R. Ahlrichs, H. Lishka, B. Zurawski, and W. Kutzelnigg, J. Chem. Phys., 1975, 63, 4685. H. Basch, J . Chern. Phys., 1970, 55, 1700. J. A. Altmann, I. G. Csizmadia, and K. Yates, J. Amer. Chem. SOC., 1974, 96, 4196.

28

Theoretical Chemistry The ground state of MeCH is a singlet, but there is a low-lying triplet state at ca.

4-5 kJ mol-1 for the most stable staggered conformation. The main conclusions from this study were that the singlet rearrangement is preferred and involves a syn migration with a low barrier (ca. 87 kJ mol-l), a conclusion which is in accordance with the electron density changes found during the migration. In a later paper, the authors218analysed the charge re-distribution in more detail, using electron density contour maps. An energy-level correlation diagram shows no correlations between bonding levels and antibonding levels in the product. The carbene itself, MeCH, has been further studied by StaemmIer,219who also investigated CH2, CHF, and CF, in their singlet and tripIet states. Both singlet and triplet states of MeCH are much higher than that of C2H4,but the triplet is the ground state in these calculations, both at the SCF level and also in the IEPA approximation. This is in contrast to the results of ref. 21 7, but the basis set used was more extensive. Electron correlation has a large effect on the S-Tenergy gap, and the final value is 25 kJ mol-l for MeCH, for the staggered conformation. Although not an A2B4molecule, it is convenient at this point to mention a recent investigation of the silaethylene molecule, in which an Si atom replaces one carbon atom; CH2=SiHz has been detected experimentally and its spectrum is known. In 1975, Schegel et ~ 1 reported . ~ the~ first ~ ab initio results on this molecule. Force constants were computed with the force constants method of P ~ l a yand , ~ ~the wave functions obtained uia STO-3G SCF computations, and later STO-4G. The molecule is planar with Si-C= 1.693 A, Si-H = 1.479 A, CH = 1.074 A, L. HSiC= 122.9", and LHCSi= 122.7".The Si-C bond is highly polarized and the computed dipole moment is 1.1 D. The partial positive charge on Si is believed to be reflected in a relatively high Si-H stretching frequency. A second paper221dealing with the lower electronic states dealt with four geometrically different structures starting from the optimized geometry found in ref. 220. Comparisons were made with full 4-31G geometry optimizations on C,H,. There is a significant difference between the So and Tl state stabilities with respect to C2H4. In CHz=SiHz, the lowest state is a triplet with a singlet only ca. 6 kJ mol-1 higher. Population analyses were also presented for the various states. The low value of A E is in line with the extreme reactivity of the molecule. NzH4, P2H4,N,F,, andP,F,. There have been a few recent calculations on hydrazine, NzH4. Wagner222used a (7,3/3)+[2,1/1] basis set and calculated the barriers to internal rotation. A single rotamer was predicted at 94" for NzH4 with cis and trans barriers of 40 and 15 kJ mol-l, which are in reasonable agreement with experiment. Comparison was made with analogous calculations on N2F4 where there are two stable forms (64"and 180"), the trans configuration being more stable by 6 kJ mol-l. The gauche-trans barrier in this case was 24 kJ. Jarvie et al.223have included bond polarization functions in a flexible sp basis set. The NNH and HNH angles were kept the same, and the barriers to rotation and the 218

J. A. Altmann, I . G . Csizmadia, and K . Yates, J . Anrer. Clzem. SOC., 1975, 97, 5217.

219

V. Staemmler, Theor. Chim. Acta, 1974, 35, 309. H. B . Schegel, S. Wolfe, and K . Mislow, J.C.S. Chern. Comm., 1975, 246. 0. P. Strausz, L. Gammie, G . Theodorakoupoulos, P. G . Mezey, and I. G . Csizmadia, J . Amrr. Chem. S o r . . 1976, 98, 1622. E. L. Wagner, Theor. Chirn. Actn, 1971, 23, 115. J. 0. Jarvie, A. Rauk, and C. Edmiston, Conod. J . Chem., 1974, 52, 2778.

p20

221 222

2 3

Ab initio Calculations on Molecules containing Fiue or Six Atoms

29

equilibrium angles were improved using the basis set. A further paper224examined in more detail the conformational changes involved in describing the potential energy surface. There have been fewer calculations on P2H4, but Wagner, in 1971, investigated both P2H4 and PzF4 226 with essentially a minimal basis set, but with d-functions on phosphorus. In the case of P2H4,two stable rotamers at 75" and 180"of equal energy were predicted, these being separated by a barrier of 2.1 kJmol-l. The cis barrier was computed to be 17 kJ mol-l. The d-functions on P did not appreciably alter the barrier curve, and the barriers are significantly smaller than in the case of N2H4 and NzF, discussed above.222 A 'trans only' form of P2F4 is the theoretical structure, but the curve is quite flat. previously found the gauche structure to be more stable, as found Robert et aZ.22s using a much larger basis above, and very recently a calculation by Albrand et aZ.227 set gave SCF energies ca. 5 Hartree lower than the above results, even though polarization functions were not included. This paper dealt primarily with the ab initio calculation of the spin-spin coupling constant lJ(PP). The computed rotational barriers were 17 and 6.7 kJ mol-l. The method used to calculate the J values was described and the agreement with experiment was quite good. Aminophosphine, H2NPH2,has two different sites for inversion and has been the subject of two recent papers from Csizmadia and co-workers. In the first of these,22D the stereochemistry at the nitrogen was investigated for fixed values of the other parameters, using a DZ-quality basis in SCF calculations. The N atom adopts a trigonal-planar geometry. The calculations show that the PH, group releases electrons to the NH2 group. The addition of polarization functions makes no significant difference to the geometry. The second paper 230 investigated the stereochemistry at phosphorus. The pyramidal nature of the phosphorus group was confirmed, with an inversion barrier of ca. 19 kJ mol-l, slightly greater than in PH3. The barriers for inversion in directlybonded three-co-ordinate atoms is determined principally by the differences in electronegativity of the Aminoborane, H2BNH2,has been the subject of a recent SCF calculation232 with a D Z + P basis set of GTO, but with d-functions only on N. The geometry was optimized and the BN bond lengths found to be 1.378 8, in the planar form and 1.469 A in the orthogonal form; in addition the arrangement about N is pyramidal in the latter. The barrier to internal rotation was 130 kJ mol-l, much smaller than in an earlier as a result of the lowering in energy of the pyramidal form. Similar calculations on vinylborane and H2BSH have been 2 z4 2z5 226

227 228 229 230 23l 232 2S5 234

J. 0. Jarvie and A. Rauk, Canad. J. Chem., 1974,52, 2785. E. L. Wagner, Theor. Chim. Acta, 1971, 23, 127. J.-B. Robert, H. Marsmann, and J. R. Van Wazer, Chem. Comm., 1970, 356. J. P. Albrand, H. Faucher, D . Gagnaire, and J. B. Robert, Chem. Phys. Letters, 1976, 38, 521, C. Barbier and G. Berthier, Theor. Chim. Acta, 1969, 14, 71. I. G. Csizmadia, A. H. Cowley, M. W. Taylor, L. M. Tel, and S. Wolfe, J.C.S. Chem. Comm., 1972, 1147. I. G. Csizmadia, A. H. Cowley, M. W. Taylor, and S. Wolfe, J.C.S. Chem. Comm., 1974,432. S . Wolfe, Accounts Chem. Res., 1972, 5 , 102. 0. Gropen and H. M. Seip, Chem. Phys. Letters, 1974, 25, 206. D . R. Armstrong, B. J. Duke, and P. G. Perkins, J . Chem. SOC.(A), 1969, 2566. H. M. Seip and H. Jensen, Chem. Phys. Letters, 1974, 25, 209.

30

Theoretical Chemistry

Boron Tetrahalides, B2X4.Although these molecules have been extensively investigated by semi-empirical methods, only recently have a6 irtitio calculations been performed. Using minimal basis sets for the core and s-orbitals, but DZ for the valence p-orbitals, SCF calculations on B2C1, and B2F4 (as well as on B,CI, and B4F,) were carried and the results analysed in terms of localized molecular orbitals. The staggered rotamer is predicted to be more stable, in agreement with experiment for B2CI, but not for BzF4,but in the latter case the error in the barrier is only ca. 3 kJ, so it is possible that a more extended basis set would correct this. The wave functions were analysed in detail and used to interpret the photoelectron spectrum of B4CI4. A more recent calculation on B2F4has appeared238using two different basis sets, giving an energy ca. 3-4 Hartree lower than in ref. 235.The rotational barrier was computed to be 1.5 kJ mol-l, favouring the D26structure, as above, although slightly bigger. Larger DZ P calculations are needed here. The interactions leading to this barrier were analysed in detail. Nitrogen Tetroxide, N2 04.The structure and bonding in N 2 0 4have been the subject of much speculation in view of the long and weak N N bond in the molecule. Numerous semiempirical studies are referred to elsewhere, but the past two years have seen two detailed papers dealing with ab initio calculations. The high symmetry reduces the total number of independent twoelectron integrals which have to be computed ~ ' the first but the calculations are still very time consuming, and Griffith et ~ 2 1 . ~in study carried out SCF calculations with an MBS of STO's. Calculations were also reported using two methods of approximating some of the integrals. The lowestenergy state has an energy of - 405.9277 Hartree, and the wave function was N-N antibonding, largely as a result of filling the 661, a*-orbital in preference to the 6a, bonding orbital. The amount of N-N n-bonding was small. The lowest state was significantly stabilized by NNO three-centre interactions. Howell and Van Wazer236also studied N z 0 4with a small basis set, which however, gave an energy of -407.650 Hartree and a barrier of 48.5 kJ mol-l. Griffiths et al. did not investigate the barrier.237The authors do not attribute the weak bond to the occupation of the 6b,, (a*) orbital. Ahlrichs and Kei1238have made a more detailed study including correlation by the IEPA method, but essentially in a form including only a limited amount of correlation, in a manner similar to that used in the OVC method.23eCalculations were performed for various N-N distances and for the skew structure, in addition to the planar structure. The results show that the long N-N bond is due to (i) the delocalization of the bond pair over the whole molecule and (ii) to a rather strong repulsion between the doubly occupied orbitals of the NO, fragments. The coplanarity results from a delicate balance between the repulsive forces favouring the skew structure and the effects of bonding which favour the planar structure. In Table 6 some of these calculations are compared.

+

235

236 237

238

930

M. F. Guest and I. H. Hillier, J.C.S. Furaday If, 1974, 70, 398. J. H . Howell and J. R. Van Wazer, J . Amer. Chem. SOC.,1975, 96, 7902. R. L. Griffiths, R. G. A . R. Maclagan, and L. F. Phillips, Chem. Phys., 1974, 3,451. R. Ahlrichs and F. Keil, J. Amer. Chem. SOC.,1974, 96, 7615. G. Das and A. C. Wahl, J. Chem. Phys., 1966,44, 876.

Ab initio Calculations on Molecules containing Five or Six Atoms

31

Table 6 Computed energies and barriers to rotation for NzO4

a

Basis set

E/kJ mol-l

MBS (STO) (73) (GTO) DZ (GTO)

-405.9277 -407.3650 -407.9072

Experimental value= 12.13 kJ mol-l; see ref. 238.

Rotational barrier a / kJ mol-l

48.5 25.10 I,

Ref. 237 236 238

Correlated wave function.

B. AH,.-A number of recent calculations have investigated the stability of BHS. Detailed calculations with and without correlation by Hoheisel and KutzelniggaPe were reported for five possible structures. SCF calculations indicate that BHs is unstable with respect to BH3 H2, but when correlation is included a binding energy of ca. 9kJmol-l is found for a C8 geometry. A C,, structure is, however, only 38 kJ mol-l higher in energy. Pople and co-workers241 have also investigated several boron-containing compounds, including BH,+ and BHS. With their SCF wave functions, no bound species was found, even with complete geometry optimization, except in the case that correlation was included by unrestricted Moller-Plesset perturbation when a very shallow minimum was found correspondingto a binding energy of ca. 7 kJ rnol-l, rather similar to that found by K ~ t z e l n i g gwhose ,~~~ calculations gave a somewhat lower energy. Similar conclusions on BH, were reached by Pepperberg et al.243 The ions CH5+and CH,- have been investigated for several years but only the most recent studies are considered here. Hariharan et al.244studied CH,+ with an STO 4-31G basis set and in later used their more extended basis and included A slightly lower energy was obtained in an IEPA calculation by Dyczmons et aLa4,who also investigated CHs-. The geometries found by all these authors were in good agreement and the binding energy of 180 kJ mol-1 was in reasonable agreement with that found by Collins et al.241(181 kJ mol-l). The calculations on CH5- dealt only with one structure, that of D3h symmetry. Dedieu and Veillard had previously reported calculations on CH5-, both SCF,246and including CI,**' and showed that the minimum of energy corresponds to a loose complex between CHI and H-. The D3h geometry can be regarded as the transition state for an s N 2 reaction. There have been more detailed studies on the sN2 reaction very recently. In 1975, BaybuttBa8found that for CH4+H- the addition of polarization functions to the basis has very little effect on the barrier for the reaction, but diffuse carbon sfunctions have a more pronounced effect, increasing the barrier by 42-68 kJ mol-1

+

240

241 242

243 244 245

2413 247 248

C. Hoheisel and W. Kutzelnigg, J. Amer. CJiem. SOC.,1975, 97, 6970. J. B. Collins, P. von R. Schleyer, J. S. Binkley, J. A. Pople, and L. Radom, J. Amer. Chem. Soc., 1976,98, 3436. J. S. Binkley and J. A. Pople, Internat. J . Quantum Chem., 1975, 9, 229. I. M. Pepperberg, T. A. Halgren, and W. N. Lipscomb, J. Amer. Chem. SOC.,1976,98, 3442. P. C. Hariharan, W. A. Lathan, and J. A. Pople, Chem. Phys. Letters, 1972, 14, 385. V. Dyczmons and W. Kutzelnigg, Theor. Chim. Acta, 1974, 33, 239. A. Dedieu and A. Veillard, J. Amer. Chem. SOC.,1972, 94, 6730. A. Dedieu, A. Veillard, and B. Roos, 'Proceedings of the 6th Jerusalem Symposium on Quantum Chemistry', 1973, Israel Academy of Sciences, Jerusalem. P. Baybutt, Mol. Phys., 1975, 29, 389.

32

Theoretical Chemistry

in this reaction. This paper gave an interesting analysis of this type of calculation for s N 2 reactions, at the SCF level. Finally, Keil and A h l r i ~ h sreported , ~ ~ ~ PNO-CI and CEPA calculations on several S N reactions, ~ including that giving CH5-. The barrier height computed was 236 kJ mol-l, and the inclusion of correlation decreased the barrier by ca. 30 kJ mol-l. The larger drop found in ref. 245 was probably an artefact of the IEPA method. The errors in these very extensive calculations were believed to be only 10-30 kJ mol-l, which is indicative of the accuracy now attainable. Fukui and co-workers have discussed various factors involved in treating the reaction co-ordinate in surface calculations and applied the method to CH,+ T.a50 There have been several calculations dealing with AH5species where A is a secondrow atom, S, Si, or P. PH, is a hypothetical molecule and has been extensively studied as the prototype quinquevalent phosphorus compound. Earlier calculations found that d-functions had a significant effect on the total energy, by Rauk et al.251 but primarily act to overcome deficiencies in the basis set used {which was (12,9,1/5)+[6,4,1 PD. A similar basis set was used by Walker 252 in an investigation of the E’ vibrational modes of XY5 molecules. It is only recently, however, that studies including correlation have appeared. Keil and K u t ~ e l n i g gusing ~ ~ ~IEPA-PNO, PNO-CI, and CEPAPNO methods used a slightly smaller basis set but of course go beyond the SCF approximation. This paper attempted to clarify the role of the d-AO’s of P and the question of the importance of the electronegativity of the axial ligands, such as F in PH,F, in making the three-centre, four-electron bond stable. It was found that the energy lowering due to d-AO’s was ca. 167 kJ mol-1 in PH,. The molecule is not bound with respect to PH3+ H,. The D3h structure does not represent an absolute minimum on the PE surface, but it is more stable in this conformation with respect to deformation to C3v.There are many interesting points in this paper, concerning the relative stabilities and bonding in PH3F2,PH5, PH4F, PH2FB,PH2F, PH3, and their nitrogen counterparts NH3F2,NH3F+, NH2F, and NH3, but space does not permit more discsussion here. SiH5- has also been the subject of a few recent papers. Wilhite and SpialterZs4 found that the trigonal-bipyramidal form is stable by 71 kJ mol-1 with respect to SiH4+ H- and the tetragonal-pyramidal form stable by 58 kJ mo1-l. The reaction proceeds with H- approaching the face of a silane tetrahedron withEaca. 36 kJ mol-l. A barrier of 12 kJ mol-l to the Berry255pseudorotation was found. Baybutt in the paper referred to above248also studied this reaction, but carried out ab initio geometry optimizations. The barrier computed with the largest basis including diffuse s-functions was 78 kJ mol-l. A recent paper by Payzant et al. 256 has compared recent experimental work with 449 250

25l 252 253 254

255 256

F. Keil and R. Ahlrichs, J . Amer. Chem. SOC.,1976, 98, 4787. K. Fukui, S . Kato, and H. Fujimoto, J. Amer. Chem. SOC., 1975, 97, 1. A. Rauk, L. C. Allen, and K. Mislow, J . Amer. Chem. SOC.,1972, 94, 3035. W. Walker, J . Mol. Spectroscopy, 1972, 43, 41 1 . F. Keil and W. Kutzelnigg, J . Amer. Chem. SOC., 1975, 97, 3623. D. L. Wilhite and L. Spialter, J. Amer. Chem. SOC., 1973, 95, 2100. R. S. Berry, J. Chem. Phys., 1960, 32, 933. J . D. Payzant, K . Tamaka, L. D. Betowski, and D. K. Bohme, J. Amer. Chem. Soc., 1976,98, 894.

Ab initio Calculations on Molecules containing Five or Six Atoms

33

the above predictions, and although the calculations are in accordance with the rate constants the experimental low Ea for H- SiH, (ca. 8 kJ mol-l) is in disagreement with the calculations, possibly because of basis set deficiencies or an incorrect approach geometry. C. AX,.-Turning now to pentahalides, calculations have appeared on the fluorides corresponding to several of the hydrides described above. Wang257computed the lowest-energy structure of CF5- to be the tetrahedral structure with the Fapproaching along one of the CF bonds. PF6 was first investigated at the SCF level by Strich and Veillard,258who found d-functions to play a significant role. Computed energy differencesbetween the DSh, C4v,and Cbstructures support Berry’s 255 pseudorotation mechanism as the mechanism for ligand interchange for PF5. Howell et al.259have also studied this molecule and PF4H, but with a smaller basis set, and have presented electron density contour maps, and more recently Keil and K ~ t z e l n i g g ,have ~ ~ also discussed the relation between the bonding in PF, and in PH4F, PH,F,, and PH,F,. There have not been any calculations on SiF,- but Baybutt 248 investigated F- + SiH,F in his paper on sN2 reactions. D. Formamide, HCONH,.-This molecule is an important model compound for the peptide bond, and several interesting calculations on it have appeared recently. A large SCF calculation in 19702s0using a GTO basis set examined a large number of nuclear configurations. More recently, there have been some papers on this molecule utilizing the GVB method. The orbitals were discussed qualitatively in terms of the valence states of C H 2 0and the ,A1 state of NH,.261The orbitals were calculated using a DZ basis set for the planar and twisted geometries. CI calculations were also carried out. The detailed density maps of the various orbitals show clearly the distinct chemical bonds and lone pairs. The excited states were investigated yielding excitation energies to the 3A”(n*en), lA”(n*~ n ) and , 3A’(n*-n)states of 5.39,5.65, and 6.19 eV. The computed result is in good agreement for the known lA” transition, as is the dipole moment. The results for the rotational barrier are also in excellent agreement with experiment. The many investigations of the dimerization of formamide and its hydration, which have been carried out by the Pullmans,26aare not discussed in this Report, but in a series of papers dealing with interactions of formamide with other molecules SCF calculations were reported by Ottersen et al., using a (7,3/4) +[4,2/2] basis The authors also optimized the geometry, assuming the molecule to be planar, and investigated N2H4. In a second paper264they investigated the keto and en01 tautomers of formarnide, finding the en01 tautomer to be less stable by between 63 and 80 kJ mol-l, depending

+

257 258 260

261 262

S. J. Wang, 2.Naturforsch., 1973, 28a, 1832. A. Strich and A. Veillard, J. Amer. Chem. Sac., 1973, 95, 5574. J. M. Howell, J. R. Van Wazer, and A. R. Rossi, Inorg, Chem., 1974, 13, 1747. D. H. Christensen, R. N. Kortzeborn, B. Bak, and J. J. Led, J. Chem. Phys., 1970, 53, 3912. L. B. Harding and W. A. Goddard, J. Amer. Chem. SOC.,1975, 97, 6300. G. Alagana, A. Pullman, E. Scrocco, and J. Tomasi, Internat. J . Peptide Protein Res., 1973,3, 251.

263 264

T. Ottersen and H. H. Jensen, J. Mol. Structure, 1975, 26, 255. T. Ottersen, J. Mol. Structure, 1975, 26, 365.

34

Theoretical Chemistry

on the basis set. The authors also investigated the hydrogen-bonded HCONH,-H20 c o m p l e x e 286 ~ ~and ~ ~ also ~ the HCONH2-NH, E. MeXY.-A variety of molecules with this general formula will be considered in this section. Yarkony and Schaefer26shave investigated the acetyl cation MeCO+, which is important in mass spectroscopy. The question of the structure of this species is an interesting one since there are several possible geometrical isomers. The authors investigated the four most stable configurations of C2H30+,structures (9)-(12) 0

Ill

7+ 7+

C

H\

,c-c

H"

/

Ed+

\o/

and the results are shown in Tables 7 and 8. In addition, the electronic spectrum of the most stable of these ions (9) was discussed. The excited states were determined and the expressions for the energy in these cases given explicitly. A discussion of the species in terms of molecular fragments showed the usefulness of this procedure. An

Table 7 Ground-state energies and geometries of MeCO+ isomers 26s Isomer H\,1 . o w H-'" 1.460

c-0

Symmetry

E (SCF)

c 3Y

-151.993808

CS

- 151.891656

&409.50

a

Bond lengths in A.

267

T. Ottersen and H. H. Jensen, J . Mol. Structure, 1975, 26, 375. T. Ottersen and H. H. Jensen, J . MoI. Structure, 1975, 28, 223. T. Ottersen, H. H. Jensen, R. Johannsen, and E. Wislsff-Nilssen, J. Mol. Structure, 1976, 30,

268

379. D. R. Yarkony and H. F. Schaefer, J . Chem. Phys., 1975, 63, 4317.

265 266

Ab initio Calculations on Molecules containing Five or Six Atoms

35

Table 8 Charge distribution in the MeCO+ isomers for various states2sa

0e-

Me + 0.37 + 0.40 + 0.45

+ 0.60

+ 0.02

+ 0.51

+ 0.03

4 0.59

-0.04 + 0.55

c

CH2 + 0.26

+ 0.56

CH2

C + o.os'H

O-H

('A)

0.31

,o + 0.72

( ' A , : 7a22e4) ('E: 7a22e33e) (3E:7 a 2 e 4 3 e )

-0.09

+ 0.29 + 0.34

+ 0.34

+ 0.52

,H

'"2,/' -0.20

accurate two-configuration wave function for MeCO+ was also obtained and it was shown that the dissociation products are unambiguously Me+ and CO. Schaefer and c o - w o r k e r 270 ~ ~have ~ ~ ~also studied in some detail the isomerization of MeCN-+MeNC. This process, reaction (15), is one of the simplest examples of a MeCN +MeNC

(15)

unimolecular reaction, and in the first paper the authors report DZ basis set SCF calculations. The predicted exothermicity was 72.8 kJ mol-l, in reasonable agreement with experimental estimates (ca. 62.8 kJ mol-'), and the barrier height was 246 kJ mol-1 (experimental 160.6 kJ mol-l). The latter result was interpreted as indicating that correlation effects would be greatest in the transition state. The calculations also predicted that the Me group would remain pyramidal at the transition point. Detailed population analyses suggest more charges on the methyl carbon in the transition state. The second paper27oinvestigated the barrier to internal rotation of the Me group, also with a DZ basis; 75 points on the potential surface were studied and the barrier was determined. It was also established that this approach (SCF theory) does not predict the existence of a relative minimum in the reaction co-ordinate. MeOH is the simplest alcohol and has been the subject of a number of recent papers. Near-Hartree-Fock wave functions for MeOH and also MeO- and MeOH,+ were obtained by Tel et al. in 1973.271Various geometries were investigated and the proton affinity and barrier to rotation and in-plane inversion were computed. The values obtained were - 832, 60, and 136 kJ mol-l respectively. MeOH,+ showed no barrier to either rotation or inversion. Ha et aZ.272used an essentially DZ basis to calculate the force constants and dipole moment derivatives. 269 270

271

272

D. H. Liskow, C. F. Bender, and H. F. Schaefer, J. Amer. Chem. SOC.,1972,94, 5178. D. H. Liskow, C. F. Bender, and H. F. Schaefer, J . Chem. Phys., 1972, 57, 4509. L. M. Tel, S. Woife, and I. G. Csizmadia, J. Chem. Phys., 1973, 59, 4047. T.-K. Ha, R. Meyer, and Hs. H. Gunthard, Chem. Phys. Letters, 1973, 22, 68.

36

Theoretical Chemistry

Kern and co-workers 273 have applied a previously developed method for analysing internal rotation in terms of localized bond orbital to MeOH, using initially an MBS of STO orbitals for the SCF calculations and then extending these with a DZ basis. The essential result was that the dominant contribution to the barrier is due to overlap (exclusion-principle) interaction between closed-shell, localized bonds. The main effect of the lone-pair electrons on the barrier is partially to shield the oxygen nuclear charge from the bonding electrons. In connection with the analysis of core bonding energies, ad am^^^^ has investigated how electronic relaxation accompanying core ionization varies with the environment of the atom and has shown that this is slight. Calculations on the coreionized species MeOH+ and MeF+ were reported among other molecules studied. Near-Hartree-Fock wave functions for FCH20H have been reported p76 in connection with a study of the gauche effect,277and a more detailed account of this work has appeared.e78The computed PE curve for the ground state shows a single minimum corresponding to the polar CF and OH bonds lying in a gauche conformation. The anticoplanar configuration corresponds to an energy maximum, but in the lowest singlet excited state, the antiplanar conformation is more stable than the gauche. Localized orbitals were obtained in this study. MeNO has been the subject of a series of CI calculations aimed at understanding the The eclipsed equilibrium geometry was used, and the first (n*-n-) and second ( n * t n + ) excited states are located at 2.17 and 7.14 eV, whereas the triplet states at 1.29 and 5.39 eV are of (n*+n-) and (n*-n)nature. F. Glyoxal, (CHO),.-There have been a number of calculations on the ground state of this molecule, and very recently a more detailed study of several excited states. Pincelli et al.zsOinvestigated the trans, perpendicular, and cis configurationswith a DZ basis set, using the experimental bond lengths and angles from the trans form which is experimentally the most stable. Not surprisingly, the trans form was more stable, the energy difference between cis and trans forms being 26.8 kJ mol-l. The barrier height for cis-trans conversion was ca. 33 kJ mol-l. Population analyses, ionization potentials, and the dipole moment were discussed, Hazs1also used a standard geometry for both cis and trans forms and with a near DZ basis he obtained a value of only 13 kJ for the cis-trans separation. This result is probably because the choice of geometry was far from the optimum. Pople and coworkerszs2obtained a lower energy with a 4-31G basis set which should be less than DZ quality. Sundberg and C h e ~ n g used , ~ ~ a somewhat smaller than DZ basis set and per273 274

C. W. Kern, R. M. Pitzer, and 0. Sovers, J. Chem. Phys., 1974, 60, 3583. 0. J. Sovers, C. W. Kern, R. M. Pitzer, and M. Karplus, J. Chem. Phys., 1968, 49, 2592.

D. B. Adams, J.C.S. Faraday II, 1976, 72, 383. S. Wolfe, A. Rauk, L. M. Tel, and I. G . Csizmadia, J. Chem. SOC. ( B ) , 1971, 136. S. Wolfe, Accounts Chem. Res., 1972, 5 , 102. 27a S. Wolfe, L. M. Tel, W. J. Haines, M. A. Robb, and I. G. Csizmadia, J. Amer. Chem. Soc., 1973,954863. 279 T.-K. Ha and U. P. Wild, Chem. Phys., 1974, 4, 300. 2*o U. Pincelli, B. Cadioli, and D. J. David, J . Mol. Strircture, 1971, 9, 173. 2 8 1 T.-K. Ha, J . Mul. Srrucrure, 1972, 12, 171. L. Radom, W. A. Lathan, W. J. Hehre, and J. A. Pople, Austral. J . Chem., 1972, 25, 1601. 2*3 K. R. Sundberg and L. H. Cheung, Chem. Phys. Letters, 1974, 29, 93.

275 276

277

Ab initio Calculations on Molecules containing Five or Six Atoms

37

formed a partial geometry optimization for the cis and trans forms and five intermediate points on the path from cis to trans. The isomerization energy was 20 kJ mol-1 and the energy barrier was 30 kJ mol-l. This study used the even-tempered basis sets introduced by Ruedenberg et aE,284 A more extensive set of geometry optimization calculations was reported by DZ basis gave a lower SCF energy than in Dykstra and S ~ h a e f e r A . ~contracted ~~ previous work and the results for the cis-trans separation are compared with the earlier work in Table 9. All five structural parameters were optimized and shown to give improvements in the results. The nature of the molecular orbitals was analysed in detail. Table 9 summarizes the results for the ground state. Table 9 Glyoxd ground-state energies, cis-trans separation, and rotational barrier

a

Reference

trans Energy a

280 283 28 1 282 285

- 226.4703 - 226.3246

Value in Hartree.

- 226.2477 -226.2428 - 226.5182 b

cis-trans Separationb

Barrier b

26.8 20.1 12.5 25.5 24.7

33.0 30.12 25.9 33 .O 31.4

Value in kJ mol-1.

A second paper286dealt in more detail with the excited states with a similar basis set. The vertical excitation energies were determined for 20 trans and 20 cis excited states, including all singlet and triplet n-n* and n-n* excitations and the lowest n -+o* excitations. Geometry optimization was performed on the three lowest cis and trans states. Two very low-lying and unobserved states trans 3Bu and cis 3B2 are predicted to lie about 15000 cm-1 (1.86 eV) from the corresponding ground states. The experimentally observed states arise from TI +n* excitations, but the lowest triplets, which have very different geometries to the ground state, arise from n-n* excitations and are of biradical-type structure. G. Miscellaneous Papers.-Calculations have been reported on BH3C0,287the In the latter dimerization of BH, and LiH,288and the reaction of H- with POH3.280 study the trigonal bypyramid with the phosphoryl oxygen equatorial turns out to be more stable. The transition state was predicted to be a distorted trigonal bipyramid. Murrell and ~ o - w o r k e r have s~~~ reported MBS and DZ calculations on the nitromethyl anion CH2N0,- in its planar and pyramidal forms. The most stable form is planar, and is expected to protonate on the oxygen, whereas the pyramidal form would protonate on C. This conclusion was not obvious from charges alone, but was evident on examination of potential energy contour maps. 284

285 286

287

288 289 290

K. Ruedenberg, R. C. Raffenetti, and R. D. Bardo, ‘Energy Structure and Reactivity’, Wiley, New York, 1973, p. 164. C. E. Dykstra and H. F. Schaefer, J. Amer. Chem. SOC., 1975, 97, 7210. C. E. Dykstra and H. F. Schaefer, J. Amer. Chem. SOC.,1976, 98, 401. S. Kato, H. Fujimoto, S. Yamabe, and K. Fukui, J. Amer. Chem. SOC., 1974, 96, 2024. R. Ahlrichs, Theor. Chim. Acta, 1974, 35, 59. C. A. Deakyne and L. C. Allen, J. Amer. Chem. Soc., 1976,98, 14. J. N. Murrell, B. Vidal, and M. F. Guest, J.C.S. Furuday 11, 1975, 71, 1577.

38

Theoretical Chemistry

In conclusion, two contrasting papers are mentioned, Radom et aLzB1 have used SCF calculations (STO-3G) to compute the optimized geometries for the C3H3+and C3H+cations, followed by single calculations at the optimized geometry with 4-31G and 6-31G* basis sets. The most stable C3H3+species is the cyclopropenium ion (13), which has a resonance energy of > 251 kJ mol-l. The next most stable form with an energy 142 W mol-l higher is the propargyl cation (14).

-

Although many studies of dimer systems such as (H20)2,NH,. "H3, etc. have appeared, space only permits the mention of two such calculations, a very extensive investigation of the water dimer potential surface based on CI wave functions by Matusoka et a1.2B2 and a similar but more limited study by Diercksen et aLZB3 The computed dimerization energies in ref. 292 corresponding to the potential minima for the linear, cyclic, and bifurcated configurations were - 234, - 20.5, and - 17.6 kJ mol-l, with correlation effects responsible for - 4.6, - 0.8, and - 3.76 kJ mol-1 respectively of the total binding energy for these forms. The error estimates in this work are k 1.7 kJ mol-l which illustrates that quantum chemical calculations of chemical accuracy are certainly now available for molecules containing up to six atoms; however, these remain expensive, and it is important that the choice of the problem for these theoretical investigations is made with care.

*91

a92 a93

L. Radom, P. C. Hariharan, J. A. Pople, and P. von R. Schleyer, J. Amer. Chern. SOC.,1976, 98, 10. 0. Matsuoka, E. Clementi, and M. Yoshimine, J. Chem. Phys., 1976, 64, 1351. G. H. F. Diercksen, W. P. Kraemer, and B. 0. ROOS, Theor. Chim. Acta, 1975, 36, 249.

2 Theories of Organic Reactions BY A. J. STONE

1 Introduction It might seem a truism to define theoretical chemistry as the study of theories about chemistry, but in fact most of the activity of theoretical chemists has been directed not towards the understanding of chemistry, if we understand that term to imply reactions, but towards the understanding of physical properties of static systems. The intractability of the problem has been sufficient reason, but recently progress has been made on a number of fronts, and it is the purpose of this Report to draw attention to some of them. I shall be concerned for the most part with organic chemistry; this implies that the molecules are large by most theoretical chemists’ standards, and that a study of the potential surface must be very limited in scope. Indeed it is rarely possible to do more than estimate the activation energy. It follows that little, if anything, can usefully be said at present about the dynamics of such reactions, in contrast to three-atom and four-atom reactions, where dynamical studies are progressing rapidly.l Many previous reviews have been devoted to particular theories, often in considerable detail. In the present Report I shall attempt to describe, rather briefly, what seem to me to be the important advances in the past ten years or so, and to relate them to one another. In this connection I distinguish between the accurate calculation of a potential-energy surface by established methods, which I regard not as a theory but as a kind of experiment, and the use of some simplifiedmodel or principle which permits useful chemical information - in particular, relative rates of reaction to be derived without such elaborate calculation. Although there is no clear dividing line between these two, we have on trial not the Schrodinger equation or even the Born-Oppenheimer approximation, which we may agree to be valid, or at least to be invalid only in well-understood ways, but the various approximations and assumptions which are necessary to proceed from there to a theory which the ordinary organic chemist can understand and apply. 2 Topological Methods

There is no doubt that the principles usually known as the Woodward-Hoffmann rules have provided a most valuable insight into our understanding of organic reactivity, and in particular of pericyclic reactions. Their applications in organic

R. D. Levine and R. B. Bernstein, ‘Molecular Reaction Dynamics’, Clarendon Press, Oxford,

1974.

39

40

Theoretical Chemistry

chemistry have been extensivelyreviewed 2$ and I shall confine myself to a discussion of their theoretical basis. The idea was first proposed* to explain the stereochemistry of ring opening of cyclobutenes. It was observed that substituted cyclobutenes opened on heating to form the correspondingbutadienes, but of the possible products the only ones found were those which could have been formed by a reaction path (called conrotatory by Woodward and Hoffmann4)in which both methylene groups turned in the same direction (Figure 1). The other possible reaction path, the

Figure 1 Allowed (right) and forbidden (left> products of the ring opening of a substituted cyclobutene disrotatory one, was not apparently possible. Woodward and Hoffman explained this in terms of a postulate relating to the highest occupied molecular orbital and supported by extended Hiickel calculations, and this was rapidly refined into a grouptheoretical argument by Longuet-Higgins and Abraham~on.~ The same type of argument was soon found to account for the failure of two ethylene molecules to combine thermally to form cyclobutane, and for their readiness to do so on photoexcitation, whereas ethylene and butadiene combine readily on heatingbaLet us recall the argument which is used here. We suppose that the two ethylene molecules approach ‘face to face’ so that their n orbitals overlap. This configuration has symmetry DZh,and the molecular orbitals of the individual ethylenes may be combined in the usual way to form symmetry orbitals for the complete system. The sum and differenceof the n-bonding orbitals transform as Alg and while the sum and difference of the antibonding orbitals transform as B,, and Bls (Figure 2). Applying the same procedure to the new bonding and antibonding a-orbitals of the cyclobutane molecule yields the results shown at the right of Figure 2. These are the only orbitals to change substantially during the reaction, and if the D2h symmetry is maintained throughout the reaction, the orbitals on the left must correlate with the orbitals of like symmetry on the right, as shown in 2

R. B. Woodward and R. Hoffmann, Angew. Chem. Internat. Edn., 1969,8,781; ‘The Conservation of Orbital Symmetry’, Academic Press, London, New York, 1970. G. B. Gill, Quarr. Reu., 1968,22, 338; T. L. Gilchrist and R. C. Storr, ‘Organic Reactions and Orbital Symmetry’, Cambridge University Press, London, 1972; R. E. Lehr and A. P. Marchand, ‘Orbital Symmetry’, Academic Press, London, New York, 1972; G. B. Gill and M. R. Willis, ‘Pericyclic Reactions’, Chapman and Hall, New York, 1974. R. B. Woodward and R. Hoffmann, J . Amer. Clzem. SOC.,1965, 87, 395. H. C. Longuet-Higgins and E. W. Abrahamson, J . Amer. Chem. SOC.,1965, 87, 2045. R. Hoffmann and R. B. Woodward, J. Amer. Chem. SOC.,1965, 87, 2046.

41

Theories of Organic Reactions

Figure 2 Orbital correlation diagram for the addition of two ethylene molecules to form cyclobutane

Figure 2. Initially, in a thermal reaction, there are two electrons in each of the ethylene z-orbitals, and it is apparent that if the reaction follows the symmetrical reaction path, the initial state correlates with a highly excited state of the product. Configuration interaction between the two AIg states leads to an avoided crossing, but there is still a considerable activation energy (Figure 3a). The thermal reaction is

A

fY

A"

A"

A'

A'

Figure 3 State correlation diagrams for (a) the addition of two ethylene molecules to form cyclobutane, (b) the Diels-Alder addition of ethylene and butadiene

meoretical Chemistry

42

said to be forbidden. I n contrast, the lowest excited state of the reactants correlates smoothly with the lowest excited state of cyclobutane, without an abnormally large activation energy, so that the photochemical cycloaddition of two double bonds to form a four-membered ring is allowed. It is indeed an important synthetic procedure. There are many conceivable analogues of this reaction. We mention only the famous H, I, reaction, long supposed to proceed via a rectangular transition state, but shown by Sullivan to involve iodine atoms. The rectangular transition state is, in the light of the Woodward-Hoffmann rules, obviously forbidden. If now we look at the corresponding reaction between butadiene and ethylene, the Diels-Alder reaction, the picture is quite different (Figure 4). Classifying these

+

a/

0'

Figure 4 Orbital correlation diagram for the Diels-AIder addition of ethylene arid butadiene to form cyclohexene orbitals under the Cssymmetry of an assumed symmetrical intermediate, we see that the bonding orbitals of the reactants all correlate with bonding orbitals of the product. Consequently in this case there is a smooth correlation between reactant ground state and product ground state (Figure 3b); this reaction is thermally allowed, in accord with its widespread occurrence. Here it is the excited state which correlates with a more highly excited state of the product, so that the photochemical reaction is forbidden. These arguments, like their p r ~ t o t y p eare , ~ based on the symmetry of the system, which is supposed to be maintained throughout the reaction. Pearson8p* has indeed shown that the symmetry cannot change along a reaction co-ordinate, except at a 7

8

9

J. H. Sullivan, J . Chem. Phys., 1967, 46, 73; 47, 1566. R. G. Pearson, Theor. Chim. Acra, 1970, 16, 107; Accounts Chem. Res., 1971, 4, 152; J . Amer. Chem. Soc., 1972, 94, 8287. R. G . Pearson, 'Symmetry Rules for Chemical Reactions', Wiley-Interscience, London, New

York, 1976.

Zheories of Organic Reactions

43

stationary point, though this observation is of doubtful significance since it is not obvious that the lowestenergy reaction path should correspond to any symmetry at all. In any case, most reactions involve molecules, or collections of molecules, with no point-group symmetry, so that the argument cannot depend on the presence or absence of formal symmetry. The power of the Woodward-Hoffmann rules derives from the fact that they apply to a whole class of reactions, irrespective of formal symmetry. The effect of substituents will usually be small, however, except possibly in the neighbourhood of intersections of potential surfaces, and it has recently been shownloSl1 that the effect of a small perturbation is not to remove an intersection but only to shift it to a different place in nuclear configuration space. (A large perturbation may shift an intersection ‘offthe edge’ of nuclear configuration space and so remove it.) Thus substituents may change the potential surfaces quantitatively but usually without changing their topological relationships, and it follows that an argument which uses the symmetry of a special case, but relates to the topological nature of the potential energy surfaces, will remain valid for the unsymmetrical case. Nevertheless it is apparent that the allowed or forbidden character of a reaction is not fundamentally a group-theoretical property, as is, for example, the allowed or forbidden character of a spectral transition, and that the principle of ‘conservation of orbital symmetry’ is badly named. This point may be illustrated by considering the third main type of concerted reaction, the sigmatropic shift. A sigmatropic shift of order [ i , j ] is defined1*as ‘the migration of a bond, flanked by one or more n systems, to a new position whose termini are i - 1 a n d j - 1 atoms removed from the original bonded loci, in an uncatalysed intramolecular process’. Examples are the Cope rearrangement (1) +(2) (a [3,3] shift) and the rearrangement of monosubstituted cyclopentadienes(3) +(4) +(5), a sequence of [1,5] shifts. Both these reactions are allowed, but the [1,3] shift (6) +(7)is not. If we examine the latter case (Figure 5 ) we see that there is no symmetry what-

10 11 12

H. C. Longuet-Higgins, Proc. Roy. SOC.,1975, A344, 147. A. J. Stone, Proc. Roy. Soc., 1976, A351, 141. R. B. Woodward and R. Hoffmann, J. Amer. Chem. Soc., 1965,87, 2511.

Theoretical Chemistry

44

Figure 5 Orbital correlation diagram for the suprafacial [1,3] shift of a proton in propene. The transition-state orbitals are based on the Hiickel orbitals for cyclobutadiene

ever, even of an approximate kind, that we may use to classify the orbitals of the reactant and product. Nevertheless, by considering a configuration in which the hydrogen is loosely bonded to the two terminal carbon atoms, it is possible to construct a correlation diagram, as shown, and to see that the reaction is forbidden. If instead of following this ‘suprafacial’ path, in which the n system is involved at the same side at both ends, the reaction follows the ‘antarafacial’ path in which the II system is attacked from opposite sides (Figure 6), the correlation diagram (Figure 7) ,,

. .. .. . .. . ,

9 .-...

.

,_. .-..

Figure 6 Suprafacial (lef ) and antarafacial (right) [ 1,3] shifts of a proton in propene shows that the reaction becomes allowed. Geometrically it is impossible to achieve the necessary overlap of the hydrogen Is orbital with both terminal carbon atoms without at the same time removing the overlap between the carbon p-orbitals. However, a similar result is predicted for the [1,7] shift which does appear to proceed

Tlteories of Organic Reactions

45

Figure 7 Orbital correlation diagram for the antarafacial [1,3] shift of a proton in propene. The transition-state orbitals are based on the orbitals of Mobius cyclobutadiene (see text) antarafacially,13a but only in open-chain systems. A suprafacial [1,3] shift can occur13b if the hydrogen is replaced by an alkyl group which inverts as the reaction proceeds in the manner of (8)+ (9).

These and similar results have been collected by Woodward and Hoffmann into a few simple rules. They use the notation nns or ,,na to denote a reacting component (a molecule or fragment of a molecule) with a n system containing n electrons and reacting suprafacially (s) or antarafacially (a). A similar notation ,ns or ,ns is used for u systems, where suprafacial reaction is taken to mean that the atoms at the end of the u bond either both retain or both invert their configuration, while antarafacial is taken to mean that one atom retains configuration and the other inverts. According to this notation, the [1,3] suprafacial shift of a hydrogen is a (,,2s+a2,) reaction; and the [1,7] antarafacial shift is (,6,+,2,). The [1,3] the [1,5] shift is (,,4,+,2,), shift of a carbon atom with inversion is a (,,2, + ,28) reaction. Similarly the addition 13

I. Fleming, 'Frontier Orbitals and Organic Chemical Reactions', Wiley-Interscience, London, New York, 1976; (a) p. 99, (b) p. 103.

46

neoretical Chemistry

of two ethylenes is (,,2s+,2,), the Diels-Alder reaction is (,,4s+ ,2,), and the conrotatory and disrotatory ring openings of cyclobutene are (,,2a ,2s) and (n2s c2,J respectively. In this notation the Woodward-Hoffmann rules say simply that for an allowed ground-state reaction the number of (4n + 2)#components plus the number of (4n)&components must be odd; for an allowed excited-state reaction the number must be even. Both the simplicity of these rules and the observation that symmetry is not a prcrequisite for their validity have led several authors to seek a simpler derivation. One of the simplest is based on an idea of Heilbronner’s,lqthen no more than a curiosity: he noticed that a cyclic hydrocarbon with a half-twist in it (a ‘Mobius’ hydrocarbon) has a different pattern of n-orbital energies from the ordinary, untwisted (‘Huckel’) type. The latter’s energy levels are obtained by Frost and Musulin’s16 construction, of inscribing a polygon, with n vertices for CnHn,in a circle of radius 2lSl with one vertex down. The energy levels are then obtained as the projections of the vertices on to the vertical energy axis (Figure 8a). The Mobius case proceeds similarly except

+

+

-Figure 0 Orbital energy levels for (a) Huckel benzene, (b) Mobius benzene

that the polygon is arranged with one side down (Figure 8b). Consequently a Mobius hydrocarbon has a closed shell, and is expected to be aromatic, when it contains 4n n-electrons, in contrast with the 4n+ 2 for the Huckel case. It is a moot point whether such a Mobius hydrocarbon could be realized as a stable molecule, but Zimmerman l o saw that pericyclic reaction intermediates could be viewed as approximations to either the Huckel or the Mobius case. If the intermediate is to be described by a minimum basis set wavefunction, then the basis functions (s or p functions, or whatever hybrids remain after constructing localized bond orbitals for those bonds which persist) form a cyclic array, each overlap between neighbours being tither positive or negative. Changing the sign of one function clearly changes the sign of two overlaps, so that the number of negative overlaps is unambiguously either odd (the Mobius case) or even (Huckel). The number of electron pairs required for stability is then respectively even or odd, so that for an allowedground-state reaction the sum of the number of negative overlaps and the number of electronpairs must be odd. It is clear that the Woodward-Hoffmann rule is just a different statement of this result, since each (4& component introduces one negative overlap while a (4n + Z), 14

l6

E. Heilbronner, Tetrahedron Lerrers, 1964, 1923. A, A. Frost and B. Musulin, J . Chern. Phys., 1953, 21, 572. H. E. Zimmermann, J. Amer. Chem. Soc., 1966,88,1564, 1566; Accoiints Chcm. Res., 1971,4, 272.

Theories of Organic Reactions

47

component introduces an odd number of electron pairs. The number of (4n + 2), and (4n)s components is seen to be irrelevant since neither type affects the parity of the sum. Fukui and Fujimoto17 arrived at a similar result, though not so generally stated, by considering the interaction between the ends of a conjugated polyene, allowing the resonance integral Flnto be either positive or negative. Dewar l8 made essentially the same point, remarking that pericyclic thermal reactions proceed via aromatic transition states. However, Dewar also noted that antiaromatic systems have low-lying excited states, so that the barrier to reaction in the excited state is likely to be low. Indeed, if there is a barrier to the ground-state process, there is likely to be a corresponding well or funnel in an excited state which will positively assist the photochemical reaction.lS Aromatic systems on the other hand have high first excited states (which is part of the reason for their stability) and thus a high barrier to photochemical reaction. Thus the partner to the principle stated in the previous paragraph is that for an allowed photochemical reaction the sum of the number of negative overlaps and the number of electron pairs must be even. The same conclusion was reached using a different argument by Van der Hart, Mulder, and Oosterhoff.20They started from a valence-bond formalism, which in its simplest form fails to account for the antiaromaticity of 4n-membered rings, and indeed predicts them to be aromatic. They showed however that if the valence-bond wavefunction is properly antisymmetrized with respect to all electron permutations, there is a term in the energy, arising from the cyclic permutation of all electrons, which is positive for 4n-membered Hiickel rings (and 4n + 2 Mobius ones) and negative for 4n + 2 Huckel rings (and 4n Mobius ones). Their calculation consequently led to the same result as the molecular orbital approach. Dayz1 has given a careful account of the relationship between the WoodwardHoffmann rules and Mobius/Hiickel aromaticity, and has defined the terms ‘suprafacial’ and ‘antarafacial’in terms of the nodal structure of the atomic basis functions. His approach makes quite explicit the assumption that the transition state involves a cyclic array of basis functions. Thus the interconversion of prismane (10) and benzene, apparently an allowed (,,2s +2,, + ,2,) process, is in fact forbidden because there are additional unfavourable overlaps across the ring.l Alternative approaches have been suggested by Langlet and Malrieu2’ and by T~-indle.~ Langlet ~ - ~ ~ and Malrieu point out that although the correlation diagram method requires the use of symmetry orbitals, which must therefore be delocalized,

17 18

19 20

21 22

23 24 25

K. Fukui and H. Fujimoto, Bull. Chem. SOC.Japan, 1967,40,2018. M. J. S. Dewar, Tetrahedron Suppl., 1966, 8 , 75. W. Th. A. M. van der Lugt and L. J. Oosterhoff, Chem. Comm., 1968, 1235. J. J. C. Mulder and L. J. Oosterhoff, Chem. Comm., 1970,305, 307; W. J. van der Hart, J. J. C. Mulder, and L. J. Oosterhoff, J. Amer. Chem. Sac., 1972, 94, 5724. A. C . Day, J . Amer. Chem. SOC.,1975, 97, 2431. J. Langlet and J.-P. Malrieu, J. Amer. Chem. Soc., 1972, 94, 7254. C. Trindle and 0. Sinanoglu, J. Amer. Chem. SUC.,1969, 91, 4054. C. Trindle, J . Amer. Chem. SOC.,1970, 92, 3251, 3255; Theor. Chim. Acta, 1970, 18, 261. C. Trindle and F. S. Collins, Internat. J. Quantum. Chem. Suppl., 1971, 4, 195.

48

Theoretical Chemistry

an equivalent description of the wavefunction is obtainable in terms of localized orbitals. They point out that the interconversion of cyclo-octatetraene (11) and cubane (12) is apparently allowed if the correlation diagram is constructed using the symmetry without due regard to the spatial relationship of the orbitals. In fact the process involves two (,2,+,2,) ring closures and so is forbidden. They set up a PCILO analysis of the correlation, and are able to determine qualitative, though approximate, orbital correlation diagrams. However, they do not seem to produce any predictions that are not given by the principle of conservation of orbital topology. Trindle and Sinanoglu23 also used localized orbitals, obtained by applying Edmiston and Ruedenberg’s 26 method to CND0/2 wavefunctions, and were able to construct correlation diagrams for interconversion of C4H ions. The principle adopted here is that of constructing a single-determinant wavefunction, and of following its development during the course of the reaction. If the final wavefunction so obtained corresponds to an excited state of the product, the reaction is thermally forbidden. In a further paper 24 Trindle suggests a method for determining whether this is the case. . . of the reactant. As reaction Consider the ground configuration @ = proceeds, this will correlate with some configuration Y/’= Iy1’y2’. . . yn’l of the product. If configuration interaction is ignored, this may or may not be the ground configuration Y = Iy1y2 . . y n l . To find out whether it is, we may determine the overlap between the two configurations:27a

,+

.

S =

=

det (D’) [det (D) det

(1)

where Dil’

= ,

Du =

<WilW>,

D r j = <W’lW’>

(2)

A value of one for S then implies that the ground configurations correlate and that the reaction is allowed,while avalue of zero implies that the initialconfigurationcorrelates with some excited configuration of the product, so that the reaction is forbidden. This is simply a formalization of the basic principle underlying the W-H rules, but Trindle then suggests that Y’may be found approximately by applying a ‘topological identity transformation’ to the initial orbitals of @. By this is meant a distortion of the orbitals which preserves their nodal structure.26Trindle postulates that an adequate approximation may be obtained by using localized orbitals as far as possible,z4 and allowing rehybridization and rotation of atomic orbitals, but not the transfer of amplitude from one atom to another. Trindle gives a procedure for optimizing the transformation subject to these constraints, but that is probably unnecessary, as we may see by considering, as a model example, the suprafacial [1,3] sigmatropicshift in propene. The C-C o-bonds and the non-reacting C-H bonds remain essentially unchanged but the reacting C-H bond and the n bond (Figure 9a) yield, on applying the topological transformation, the rather strange functions shown in Figure 9b. The overlap between the corresponding configuration and the true configuration !P based on the orbitals in Figure 9c is, using (l),

26

27

C. Edmiston and K. Ruedenberg, Rev. Mod. Phys., 1963,35,451; J . Chem. Phys., 1965,43, S97. R. McWeeny and B. T. Sutcliffe, ‘Methods of Molecular Quantum Mechanics’, Academic Press, London, New York, 1969; (a) p. 49, (b) p. 94.

neories of Organic Reactions

49

Figure 9 Suprafacial [1,3] shqt in propene. (a) orbitals for reactant; (b) orbitals for product obtained by a 'topological identity' transformation from (a); (c) actual orbitals for product

where s is the overlap between the hydrogen 1s orbital and the sp3 hybrid. (The determinants are squared because there are two electrons in each orbital.) It is easy to see, by constructing a corresponding diagram for the antarafacial case, that the overlap (Y' I Y ) is then b(l +~ ) ~ / ( 1+s2) - z 0.7 (assuming s z 0.6) corresponding to an allowed process. (We do not expect a value of unity because of the approximations.) Using this method we may now show in simple terms that the conversion from prismane to benzene is forbidden. In this case the three breaking a-bonds of prismane (shown dotted in Figure 10a) are transformed into the three benzene functions in

Figure 10 (a) Prismane; (b), (c), (d) benzene n-orbitals obtained by applying a 'topological identity' transformation to the bonds shown by dashed lines in (a); (e) an antisymmetric n-orbital for benzene

Figure lob, c, and d. Since these all have zero overlap with the antisymmetricbenzene orbital shown in Figure lOe, the overlap determinant vanishes and the interconversion is thermally forbidden. T r i x ~ d l ehas ~ ~given a number of other examples, using his more elaborate procedure, and the method appears to work well. A related method has been proposed by Wilson and Wang2* in terms of the natural orbitals (the eigenfunctions of the exact one-electron density matrix 27b). If, throughout the reaction path, the wavefunction can be represented adequately by a 28

E. B. Wilson and P. S. C. Wang, Chem. Phys. Letters, 1972, 15, 400.

Theoretical Chemistry

50

single Slater determinant, then the occupation numbers of all the principal natural orbitals will remain close to unity and the others close to zero. If however the ground state of the product corresponds to a different configuration, then the graph of occupation numbers against reaction co-ordinate will exhibit a crossing (which may be avoided if the symmetry is low). Wilson and Wang postulate that such a crossing corresponds to an unfavourably high activation energy, and give examples to illustrate the method. . ~ ~studies using the GeneralYet another approach has been used by G ~ d d a r dHis ized Valence Bond (GVB) method, much like the unrestricted Hartree-Fock method except that the orbitals are spin-coupled in pairs as in a valence-bond wavefunction, revealed that the shift of a bond from one side of an atom to the other (as in the reaction H, + D + H + HD) leads to a change in the phase relationships between the three basis orbitals involved. Using this Orbital Phase Continuity Principle Goddard was then able to derive selection rules in agreement with Woodward and Hoffmann. He claimed however to be able to treat open-shell systems more consistently. A similar, but qualitative, procedure has been given by Zimmern-~an.~~ have adopted a diametrically opposite standpoint. If the Halevi and Katriel predictions depend on topology rather than symmetry, then it is legitimate to use the highest possible symmetry, even if it does not correspond to the actual disposition of the molecules in the optimum reaction path. The procedure then consists of classifying the wavefunctions of reactant and product under the highest symmetry common to both, and only then seeking a distortion (reaction co-ordinate) which will permit a correlation between the occupied orbitals of reactant and product. For the isomerization of cyclobutene, for example, the full C,,symmetry of the molecule is used, rather than the more limited C, or C, symmetry of the hypothetical conrotatory or disrotatory intermediates. The 0- and n-bonding orbitals of cyclobutene transform as A l and B, respectively under this group; the two bonding n-orbitals of butadiene transform as B, and A,. The A , and A, orbitals do not have matching symmetry; to make them match it is necessary to lower the symmetry by a distortion transforming according to the direct product A , x A , = A , . The conrotation is just such a distortion, but there are four other A , co-ordinates. Three of these involve the hydrogens, but the fourth is a twist of the carbon skeleton. Halevi’s approach implies that all of these distortions may contribute to the reaction co-ordinate, and Dewar has shownJ3 that the transition state is indeed substantially twisted. The method is illuminating and quite easy to apply, but the restriction to problems with a substantial degree of symmetry is a severe limitation. George and RossJ4set out to derive symmetry rules for chemical reactions as a set of selection rules on elements of the transition matrix. Each element of this matrix describes the probability of transition from a specified state of the reactants to a specified state of the products. One selection rule on such a matrix is the approximate conservation of total electron spin; by making the Born-Oppenheimer approxima31932

2g

30 31 32 33 34

W. A. Goddard, J , Amer. Chent. SOC.,1972, 94, 793. H. E. Zimmermann, Accounts Chem. Res., 1972, 5 , 393. J. Katriel and E. A. Halevi, Theor. Chirn. Acta, 1975, 40, 1. E. A. Halevi, Ilelc. Chirn. Acta, 1975, 58, 2136; Angew. Chem. Zrirernnt. Edn., 1976, 15, 593. M. J. S. Dewar, Clienr. in Britain, 1974, 11, 97. T. F. George and J. Ross, J. Chern. Phys., 1971, 55, 3851.

Theories of Organic Reactions

51

tion, neglecting virtual electronic transitions, and applying a type of Franck-Condon approximation (evaluating electronic integrals at a fixed nuclear configuration) George and Ross were able to derive a symmetry selection rule. By further neglecting all remaining dynamical effects they were able to express this rule in terms of orbital correlation diagrams and the conservation of orbital symmetry. George and Ross discuss at some length the violations of symmetry rules which may occur if the approximations fail, as they not uncommonly do. This work dealt with concerted reactions ; more recently they have discussed symmetry rules for nonconcerted reactions, where one or more stable intermediates are involved.'' Silvers6 adopted a similar approach, and proposed a further category of 'unfeasible' reactions in which neither individual orbital correlation rules nor total symmetry correlationrules (Wigner-Witmer rules) were obeyed. Silver and Karplus s7 have shown how to derive W-H symmetry rules using a simple valence-bond formalism. 3 Validity of the Woodward-Hoffmann Rules There is no shortage of theoretical arguments supporting the W-H rules in one form or another, as there is no lack of experimental verification. Woodward and Hoffmann went so far as to claim2 that there were no exceptions to their rules, since the arguments in their favour were so fundamental and far-reaching. In making this claim, however, they were careful to explain that a forbidden reaction might take place under exceptionallyenergeticconditions, or an allowed one fail in adverseconditions. The rules are to be interpreted simply as stating that a forbidden reaction requires an additional energy of activation that is not required for the corresponding allowed reaction. Other, distinct, effects may favour either mode in particular cases. It is necessary to be careful about terminology here. Following Dewar,S6we regard a concertedreaction as one which takes place in a single kinetic step, with only one maximum along the reaction co-ordinate. A synchronous reaction is a concerted reaction in which the various changes in bonding have occurred to comparable extents in the transition state. A two-stage reaction is concerted but non-synchronous, some of the changes in bonding occur almost entirely between the reactant and the transition state, the rest between the transition state and the product. A two-step reaction is one taking place in two distinct kinetic steps, and involving a stable intermediate. In this language, the implications of the W-H rules are that allowed reactions will proceed in a concerted fashion and that forbidden reactions will not, the argument in the latter case being that some other, non-concerted mechanism is likely to provide a more stable pathway. It is now fairly clear that there are exceptions to both of these generalizations. In this connection the concept of orbital isomerism is helpful. This was introduced by Dewar, Kirschner, and K ~ l l m a rwho , ~ ~postulated that a group of isomeric molecules can be divided into sets in such a way that members of any one set can be interconverted by paths which involve no crossing between the highest 36 37 38 39

H. Metin, J. Ross, and T. F. George, Chem. Phys., 1975, 11, 259. D. M. Silver, J. Amer. Chem. SOC.,1974, 96, 5959. D. M. Silver and M. Karplus, J. Amer. Chem. Soc., 1975, 97, 2645. M. J. S. Dewar, Faraday Discuss. Chem. SOC.,1976, NO.62, p. 197. M. J. S. Dewar, S. Kirschner, and W. W. Kollmar, J. Amer. Chem. SOC.,1974, 96, 5240.

52

Theoretical Chemistry

occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), but members of different sets can be interconverted only via paths which do involve such a crossing. Members of one set are said to be homorners of each other (short for HOMO-HOMOmers) and members of different sets are said to be Zumomers (for HOMO-LUMOmers). Since a HOMO-LUMO crossing implies a relatively high energy barrier, interconversion of two lumomers is a forbidden process, whereas interconversion of two homomers is allowed. This idea does not seem to have been rigorously investigated theoretically; Wilson and Wang’s observation2’ that there is a non-crossing rule for natural orbital occupation numbers suggests that it is strictly speaking invalid, and Komornicki and McIver 40 claim that it is an artefact of restricted Hartree-Fock theory, but Dewar, Kirschner, and Kollmar cite some examples which illustrate its value as a qualitative concept. Furthermoreq1they claim that a forbidden reaction cannot occur more readily by a biradical mechanism, as Woodward and Hoffmann suggest ;2 a biradical must have one electron in each of two nearly degenerate orbitals, so that such a state can only occur near a HOMO-LUMO crossing. However, there is room for some confusion here. If a HOMO-LUMO crossing occurs during the course of the reaction, then it is possible to construct three singlet states, all of which may be important near the transition state. One, HOHO, arises from the initial ground state and has two electrons in the original HOMO. A second, LULU, has two electrons in the original LUMO. The third, ‘HOLU, has one electron in each. In addition, we may construct a triplet state 3HOLU,again with one electron in each. At an arbitrary (unsymmetrical) nuclear configuration near the transition state, these states may all be required for an accurate description of the system. The following points seem to be worth making: (i) A simple picture in which the total energy is thought of as a sum of orbital energies can be highly misleading since it suppresses crucial energy differences between the different states. Note that the HOMO-LUMO crossing is only welldefined for the HOLU states, since the energies of occupied and virtual orbitals (e.g. the HOMO and the LUMO in the state HOHO) are not comparable quantities, one being related to ionization potentials and the other to electron affinities. Moreover the HOMO-LUMO crossing does not lead to a Jahn-Teller effect, as suggested by D e ~ a ragain , ~ ~ because it occurs in a well-defined way only in the HOLU states and there need be no associated degeneracy at this point along the reaction path, or at any other. (ii) Near the transition state, there is no a priuri reason for expecting the states to be ordered in one way rather than another, and there may be significant energy differences between them. Consequently Dewar’s view that the biradical state ‘HOLU will not offer a significantly easier path than the forbidden path represented by HOHO has no validity; neither, on the other hand, has Woodward and Hoffmann’s suggestion that it will. Either case may occur. (iii) The term ‘biradical’ is normally taken to describe a system with two singly occupied orbitals, between which the exchange integral is small, so that the singlet *O 41

A. Komornicki and J. W.McTver, J . Amer. Cliern. Soc., 1976, 98, 4553. M . J. S . Dewar, S. Kirschner, H. W. Kollmar, and L. E. Wade, J . Amtzr. Chenr. SOC.,1974, 96, 5242.

53

Theories of Organic Reactions

and triplet are very close in energy. To use it, or even the term ‘singlet biradical’, to describe the lHOLU state, seems to be an unnecessary invitation to confusion. (iv) The states HQHO, LULU, and lHOLU are merely basis states for the manyelectron problem, and will not usually provide accurate descriptions as they stand. It has been recognized that configuration interaction is needed in the vicinity of a HOMO-LUMO c r o s ~ i n g : ~e.g., ~ - ~an ~ SCF calculation42 gave the difference in energy between the conrotatory and disrotatory paths for the ring opening of cyclobutene as 200kJmol-l, whereas inclusion of CI reduced the difference to about 60 kJ mol-I. (Note that the carbon skeleton was assumed planar in both cases, which is, as we have seen, probably incorrect.) Since inclusion of ordinary CI makes so much difference, it is likely that a satisfactory description calls for a multiconfiguration SCF treatment in which the three states HOHO, LULU, and WOLU have equal (v) It should be noted that the result of such a calculation would be unaffected by any linear transformation of the HOMO and LUMO into two new orbitals. This freedom can be used to eliminate the lHOLU contribution from the ground state, whether or not that state is the singlet component of a biradical. Consequently any success in describing the ground state as a mixture of HOHO and LULU does not exclude the possibility that it is the singlet component of a biradical state. (vi) Since there are four states involved in a forbidden reaction, it is apparent that there are many conceivable energy relationships between them. Some possibilities are illustrated in Figure 11 ; (a) shows an energy profile for a concerted forbidden

I

I

L

I

Figure 11 Hypothetical triplet (dashed) and singlet potential curves for a forbidden reaction: (a) concerted reaction; (b) two-step reaction with possible intersystem crossing

reaction, while (b) shows a profile for a two-step process proceeding via a biradical intermediate, with possible intersystem crossing to a chemiluminescent product. Salem and his co-workers4 7 have classified photochemical reactions on this basis. 42 43 44

45 46

K. Hsu, R. J. Buenker, and S. D. Peyerimhoff, J. Amer. Chem. SOC.,1971, 93, 2117. J. E. Baldwin, A. H. Andrist, and R. K. Pinschmidt, Accounts Chem. Res., 1972, 5 , 402. R. C . Bingham and M. J. S. Dewar, J. Amer. Chem. SOC.,1972, 94, 9107. N. D. Epiotis, R. L. Yates, D . Carlberg, and F. Bernardi, J. Amer. Chem. SOC.,1976,98, 453. R. E. Townshend, G. Ramunni, G . Segal, W. J. Hehre, and L. Salem, J. Amer. Chem. Soc., 1976, 98,2190.

47

W. G . Dauben, L. Salem, and N. J. Turro, Accounts Chem. Res., 1975,8,41; N. J . Turro and A. Devaquet, J. Amer. Chem. SOC.,1975,97, 3859; N. J. Turro, W. E. Fameth, and A. Devaquet, J. Amer. Chem. SOC.,1976, 98, 7425.

54

Theoretical Chemistry

It follows that although the allowed/forbidden distinction is fairly clear-cut, depending apparently on topology rather than on details, predictions of relative rates of allowed and forbidden reactions, and of the course of such forbidden reactions as may occur, are much less reliable. Some of the difficulties may be illustrated by Berson and Salem’s48concept of ‘subjacent orbital control’. They pointed out that in a four-centre reaction such as the [I ,3] sigmatropic shift, the allowed Mobius-type intermediate had the lowest energy (4a+41/28 in Hiickel terms), but that of other possibilities the forbidden Huckel-type intermediate had a lower energy (4a 48) than the biradical allyl+ methyl, at (4a+21/2/3), because it benefited from a lower energy in an orbital ‘subjacent’ to the HOMO. However a subsequent calculation by Hehre, Salem, and W i l ~ o t tshowed ~~ that for the degenerate rearrangement of methylenecyclopropane (1 3) the biradical pathway was preferred not only to the

+

forbidden ones but even to the allowed ones. The idea of subjacent orbital control is a useful one - it has been usedY5Ofor example, to explain the preference for a forbidden reaction path, some 160 kJ mo1-1 below the corresponding allowed path, in the degenerate arrangement of homocyclopropenyl cation - but the quantitative significance of the subjacent orbital effect is difficult to assess without explicit calculation. Schmidt61 has commented that symmetry-forbidden, concerted modes of reaction may be quite accessible if the reaction path is short, the force constants in reactant and product are both small, and the anharmonicities large. He cites a case where a forbidden reaction occurs readily with an activation energy of only 25 kJ mol-l. Epiotis 52 has given a detailed account of stereoselectivity in concerted reactions, and suggested that lack of stereospecificity in [2 + 21 cycloadditions may arise not because the reaction passes through biradical species in which stereochemical information is lost but rather because of competition between several concerted processes leading to stereochemically different products. For a reaction between an electron donor D and an acceptor A, he considered, besides the ground state DAY charge-transfer states D+A- and D-A+ and locally excited states D*A and DA*, and concluded that the W-H stereoselectivitycould be reversed for 4n-electron pericyclic processes [but not for (4n+2)-electron ones] if the reaction were highly polar, i.e. between a strong donor and a strong acceptor. Schilling and SnyderS3obtained a similar result by explicit calculation on the ring closure of a series of heteroacycles isoelectronic with the ally1 anion. Some of their systems obeyed the W-H rules; some 48

49 50 51

52 53

J. A. Berson and L. Salem, J. Amer. Chem. SOC.,1972,94,8917; J. A. Berson, Accounts Chem. Res., 1972, 5 , 406. W. J. Hehre, L. Salem, and M. R. Wilcott, J . Anter. Chem. SOC.,1974, 96, 4328. A. J. P. Devaquet and W. J. Hehre, J. Amer. Chem. SOC.,1974, 96, 3644; W. J. Hehre and A. J. P. Devaquet, J. Amer. Chem. SOC.,1976, 98, 4370. W. Schmidt, Tetrahedron Letters, 1972, 581. N. D. Epiotis, J. Amer. Chem. SOC.,1972,94, 1924, 1935, 1941, 1946; ibid., 1973,95, 1191, 1200, 1206, 1214; Angew. Chem. Znternat. Edn., 1974, 13, 751. B. Schilling and J. P. Snyder, J . Amer. Chem. SOC.,1975, 97, 4422.

55

Theories of Organic Reactions

showed ground configuration correlating with ground configuration for all possible paths, whether formally allowed or forbidden, and some showed ground configuration correlating with excited configuration for all possible paths. Theoretical study of forbidden reactions consequently calls for rather careful calculations. Dewar and his c o - w ~ r k e r s ~54~ have ~ led the field here, mainly because their MIND0/3 ~ r o g r a m ~and ~ -more ~ ~ recently MNDD0/2 38 permit extremely fast computation of reaction energy surfaces with full geometry optimizat i ~ nThese . ~ ~are semiempirical methods, and although they have been very carefully parameterized and checked on many molecules whose structures and energies are known, the results have to be treated with some caution. Dewar claims38an accuracy for MIND0/3 of better than about 45 kJ mol-1 in heats of formation, 0.02 A in bond lengths, and 5" in bond angles, and MNDD0/2 seems to be better. Nevertheless, the reliability of these methods in calculating transition states is disputed strongly both from the experimental5g and the ab initio t h e ~ r e t i c a lstandpoints. ~~ Calculations have also been carried out by ab initio methods, in particular Gaussian 70,c6+60~61 but although these have a more secure theoretical foundation they are much more time-consuming, and full geometry optimization at a high level of accuracy is usually prohibitively expensive. To make matters worse, SCF calculations are usually inadequate for transition states, where bonds are being made and broken, and configuration interaction calculations of sufficientaccuracy are usually out of the question for molecules of the size required. A reasonable compromise might be to locate the transition state using a semi-empirical method, and then to study it more carefully using ab initio methods, but this does not seem to have been tried. The ring opening of cyclobutene has been studied by various workers.42Dewar and Kirschner found 62 that the symmetrical disrotatory path requires an activation energy of 375 kJ mol-l compared with 230 kJ mol-l for the conrotatory path. However, a later calculation 63 using MIND0/3 (whichhas the efficient geometry optimization not available in the earlier MIND0/2) showed that an unsymmetrical disrotatory path, in which one methylene group rotates fully before the other begins to move, requires only 70 kJ mol-1 more activation energy than the conrotatory path. Such a path passes through a non-aromatic configuration where the first methylene is fully rotated, avoiding the antiaromatic configuration of the symmetrical path. The transition state occurs earlier, at a point where the first methylene has rotated through about 45 '. The numerical result is in agreement with an experiment by Brauman and 389

54 55

M. J. S. Dewar, Angew. Chem. Internat. Edn., 1971, 10, 761. R. C. Bingham, M. J. S. Dewar, and D. €I. Lo, J. Amer. Chem. SOC., 1975,97,1285,1294,1302, 1307.

56

57

58

59 60

M. J. S. Dewar, D. H. Lo, and C. A. Ramsden, J. Amer. Chem. SOC.,1975, 97, 1311. M. J. S. Dewar, H. Metiu, P. J. Student, A. Brown, R. C. Bingham, D . H. Lo, C. A. Ramsden, H. Kollmar, P. Weiner, and P. K. Bischof, Quantum Chemistry Program Exchange, Indiana University: Program 279 (CDC version); Programs 308, 309 (IBM versions). J. W. McIver and A. Komornicki, Chem. Phys. Letters, 1971, 10, 303. G. D. Andrews and J. E. Baldwin, J. Amer. Chem. SOC., 1976, 98, 6706. W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, Quantum Chemistry Program Exchange, Indiana University, Program 236; W. J. Hehre, R. F. Stewart, and J. A. Pople, J . Chem. Phys., 1969,51,2657; W. J. Hehre and W. A. Lathan, J. Chem. Phys., 1972,56, 5255.

61 62

63

D. Poppinger, Chem. Phys. Letters, 1975, 35, 550; Chem. Phys., 1976, 12, 131. M. J. S. Dewar and S. Kirschner, J. Amer. Chem. SOC., 1971, 93, 4290. M. J. S. Dewar and S. Kirschner, J. Amer. Chem. SOC.,1974,96, 6809.

3

56

Tjteoretical Chemistry

Archie 64 who pyrolysed cis-3,4-dimethylcyclobutene at 280 "C and found about 0.005 % of the W-H-forbidden product. They inferred that the difference in activation energy was at least 60 kJ mol-l. Earlier Brauman and Golden 65 obtained the same result from a consideration of the isomerization of bicyclo[2,1 ,O]pent-Zene (14)

to cyclopentadiene. This reaction is forced to proceed in a symmetrical disrotatory mode by the geometrical constraints, and provides a formal, if readily understandable, exception to the W-H rules. These results attach a useful energy scale to the W-H rules ; if forbidden reactions generally require an additional activation energy of the order of 60 kJ mol-l, it is easy to see why the reactions proceed so selectively, since such a difference in activation energy leads to a product ratio of better than loa at normal temperatures. The energy difference is likely to vary substantially, of course; the importance of the Woodward-Hoffmann rules lies in the fact that the difference is almost always large enough to give a product ratio close to loo%, and the fact that it may sometimes be smaller or even negative is not very important. Dixon, Stevens, and Herschbach * s have carried out accurate calculations on a number of possible transition states for the H2+ D2$2HD reaction. The most likely candidate for a concerted process is a trimolecular, hexagonal structure, which has an energy of 288 kJ mol-1 above three separated molecules. This is to be compared with 517 kJ mol-1 for the square bimolecular species6'*6 8 and 432 kJ mol-1 for dissociation of H2into atoms. Other species would be allowed intermediatesaccording to the W-H rules, but only H g has an energy lower than is required for the atomic process. A conclusion which emerges from Dewar's work is that transition states in all forbidden reactions and many allowed ones are unsy~nt-netrical.~~, 7 0 If this conclusion is correct, then accurate ab initio calculations of transition states are likely to be difficult to achieve, since no assumptions can be made about geometry, and the energy must be optimized with respect to all nuclear co-ordinates. Efficient ways of carrying out this optimization are not yet available. Furthermore, the potential surface for forbidden reactions seems 71 to be characteristically of the form shown in Figure 12a, rather than that for a typical allowed reaction as in Figure 12b. In the latter case, minimizing the energy with respect to all other geometrical parameters at each value of the reaction co-ordinate yields a well-defined reaction path and transition state (dotted line and point X in Figure 12b), but in the forbidden case the same procedure, starting at either end of the reaction path, leads into a dead end, from s*e

64 65 68

690

J. I. Brauman and W. C. Archie, J. Amer. Cltem. SOC.,1972, 94, 4262. J. I. Brauman and D. M. Golden, J. Amer. Chem. Soc., 1968, 90, 1920. D . A. Dixon, R. M. Stevens, and D. R. Herschbach, Faraduy Discuss. Chem. SOC.,1976,

No. 62, 110.

70

H. Conroy and G. Malli, J . Chem. Phys., 1969,50, 5049. D. M. Silver and R. M. Stevens, J . Chem. Phys., 1973, 59, 3378. J. W. McIver, Accounts Chem. Res., 1974, 7 , 72. T. Minato, S. Yamabe, S. Inagaki, H. Fujimoto, and K. Fukui, Bull. Chem. Soc. Japan, 1974,

71

47, 1619. M. J. S. Dewar and S. Kirschner, J. Amer. Chem. SOC.,1974, 96, 5244.

67 68

6g

Theories of Organic Reactions

57

Figure 12 Typical potential surfaces for (a) forbidden reactions; (b) allowed reactions. Shaded regions are at high energy

which reaction may be impossible (Y in Figure 12a) or may OCCUT only by a catastrophic jump into the exit valley (Z in Figure 12a). Bauschlicher, Schaefer, and Bender7a describe a surface of this kind for the forbidden insertion reaction CHz('A1)+ H:,-+CHI, which they obtained using a contracted Gaussian double-zeta basis set with extensive configuration interaction, but they were obliged to impose C,, symmetry. The true transition state and reaction path may in such cases be very elusive. McIver and KomornickiS8have shown that the transition state may be found as a minimum (actually a zero) of the square of the gradient vector; such a minimum must be characterized as a transition state or saddle point of the potential surface itself, rather than a local minimum or maximum, by showing that the force constant matrix has exactly one negative eigen~alue.'~ The method is easily applicable to semiempirical 74 but less easily to accurate ab initio methods; however, Poppinger has described its implementation in the framework of Gaussian 70, though only at the STO-2G level. Application of the method has led to a confiict between D e ~ a r who , ~ ~ finds the allowed 1,3-dipolar addition of fulminic acid, HCNO, to acetylene to be a two-stage reaction, involving an unsymmetrical transition state, and Poppinger,7s who finds it to be synchronous, involving a relatively symmetrical transition state. Poppinger's transition state has one negative force constant according to STO-2GYbut Dewar, using MNDD0/2, finds a maximum at the same geometry, with two negative force constants. The conflict therefore hinges on the ability of STO-2G and MNDD0/2 to predict force constants accurately for transition-state geometries. The semi-empirical methods are undoubtedly better for force constants at ground-state equilibrium geometries, but whether this holds for transition states is not yet clear. In this case, Dewar has found two alternative, 389

72 73 74

75

C. W. Bauschlicher, H. F. Schaefer, and C. F. Bender, J. Amer. Chem. SOC.,1976,98, 1653. J. N. Murrell and K. J. Laidler, Trans. Faraday SOC.,1968, 64, 371. K. Jug, Theor. Chim. Acta, 1976, 42, 303. D. Poppinger, J. Amer. Chem. Soc., 1975, 97, 7486.

Theoretical chemistry

58

unsymmetrical transition states with much lower energies than the symmetrical configuration, so the reaction does presumably proceed in an unsymmetrical fashion. A similar conflict has occurred over the Diels-Alder addition of ethylene and butadiene, which Townshend et ai.4sfind, on the basis of STO-3G and STO-4-31G calculations with limited CI, to be synchronous. Further investigation of this disagreement is evidently needed. 4 Simple Perturbation Methods The characterization of a reaction as allowed or forbidden, although a valuable first step, does not take us very far. We have seen that it provides no quantitative information, and that even the qualitative information has to be treated cautiously. In many cases there are several allowed reaction paths, and it becomes necessary to distinguish between them, to determine which is most probable, and ideally to estimate yield ratios and reaction rates. A number of perturbative methods have emerged for this purpose. Early work on organic reactivity concentrated on conjugated systems, and various reactivity indices were proposed. The free valence Fr for atom r of a conjugated system is defined by F r = ~ ’ 3 - C pr t (3) t+

r

whereprt is the 7c bond-order of the bond between atoms r and t. The self-polarizability nrr is defined by (4)

nrr = aqr/aar,

the derivative of the n-electron population on atom r with respect to the coulomb integral for that atom. A high value of either index was assumed to imply high rea~tivity.’~a By assuming a model for the transition state, more plausible indices can be derived. The usual procedure invokes a Wheland intermediate (1 5 ) in which both

+ - II

-.+

(15)

the reagent X and the atom being displaced are bonded to the carbon, which is assumed to be roughly tetrahedral. In the localization energy method,76this tetrahedral carbon atom is assumed to be isolated from the remainder of the original n system, and the change in n energy which results, the localization energy Lr, is taken as the reactivity index, on the basis that other contributions to the activation energy are likely to be approximately constant for a given reagent X. Dewar has developed a general perturbational molecular orbital theory (PMO theory) of organic chemistry 76-78 and has shown how perturbation theory can be used to obtain 76 77

78

M. J. S. Dewar, ‘Molecular Orbital Theory of Organic Chemistry’, McGraw-Hill, New York, 1969; (a)p. 362, ( b ) p. 364. M. J. S. Dewar, J. Amer. Chem. SOC.,1952, 74, 3341. M. J. S. Dewar and R. C. Dougherty, ‘The PMO Theory of Organic Chemistry’, Plenum, New York, 1975.

Theories of Organic Reactions

59

very simply an approximate value Nrfor the localization energy, called the reactivity number or Dewar number. The localization energy approach has also been used by Brown, for example in determining relative rates of Diels-Alder reactions.7 9 Fukui and his co-workers observed that electrophilic substitution in aromatic hydrocarbons occurred most readily at the position r for which the frontier electron density&, the square of the HOMO coefficient,was largest.8o This empirical principle was found to work tolerably well for heteroaromatic molecules too,81and was justified in terms of a model in which the Wheland intermediate was stabilized by hyperconjugative interaction between the original n system and a pseudo n-orbital involving the attacking and leaving groups.82The resulting reactivity index is the superdelocalizability S, given by

where vj is the number of electrons in orbital j , v is the number associated with the reagent (0 for an electrophile, 1 for a radical, and 2 for a nucleophile), Crj is the coefficient of orbital j at atom r (the atom at which substitution is taking place), q is the energy of orbital j , and ccn is a coulomb integral for the pseudo n-orbital. For an electrophile the sum reduces to a sum over occupied orbitals only, and if the term with the smallest denominator is taken, we arrive at the approximate superdelocalizability Sr’ =

BC~HO~/(EHO - ah)

(6)

which is proportional to the frontier electron density. Similarly for nucleophiles the appropriate reactivity index turns out to be proportional to the frontier density, calculated this time for the LUMO. Brown s3 proposed a similar index Zr for electrophilic substitution given by between the electron affinity of the electrowhere I Y is the difference (A * -IHO) phile and the ionization potential of the HOMO, and PgZ is the resonance integral between the electrophile and the carbon orbital. Here again only the frontier orbital is considered. The last step in this argument, the ‘frontier orbital approximation’, is evidently of doubtful validity, and it is not surprising that the frontier electron density is not a reliable reactivity index, though at least one example of its failure76b* 84 has been disputeds5@ as attributable to an unsuitable choice of parameters. Sinceneither author gave his parameters, it is difficult to judge the validity of the criticism. We shall, however, return to the question of the frontier orbital approximation later. R. D. Brown, J . Chem. SOC.,1951, 1612. K. Fukui, T. Yonezawa, and H. Shingu, J. Chem. Phys., 1952, 20, 722. 81 K. Fukui, T. Yonezawa, C. Nagata, and H. Shingu, J. Chem. Phys., 1954, 22, 1433. 82 K. Fukui, T. Yonezawa, and C. Nagata, Bull. Chem. SOC.Japan, 1954,27,423. 133 R. D . Brown, J. Chem. SOC.,1959, 2224, 2232. 84 M. J. S. Dewar, Ado. Chem. Phys., 1965, 8 , 110. 85 K. Fukui, ‘Theory of Orientation and Stereoselection’(Reactivity and Structure - Concepts in Organic Chemistry, Vol. 2), Springer-Verlag, Weinheim, 1975 ; (a) ibid., p. 54. 79 80

Theoretical Chemistry

60

Apart from this approximation the superdelocalizability depends on similar assumptions to the localization energy. There has been some vigorous argument*s over their relative merits. Fukui has shown87*88 that if h(A)is the secular determinant in the Huckel approximation and Arr(A) its rr minor, then the reactivity indices Fr, nrr, L,, S,, and Nr can all be expressed in terms of

It is then possible to show that if two such functions Gr and Gs,for reactions involving different molecules or different positions in the same molecule, satisfy Re (Cr(y)>> Re (Cs(y))

for all y

(9)

then all those reactivity indices will agree in finding position r more reactive than position s. However, this condition is a very strong one, and although it is satisfied for different positions in linear polyenes it has not been possible to show that it is generally applicable, although numerical calculations by Koutecky ef d s 9 showed good correlations between the various indices. Moreover, since the approximate indices&, and Zr do not fit into this scheme, their usefulness is in doubt. Whether for these reasons or not, the reactivity indices excited little interest among organic chemists, and it was only after Woodward and Hoffmann’s early papers that perturbational methods were widely invoked again. Fukui and Fujimoto 17, @ O adopted Fukui’s frontier orbital theory to explain the Woodward-Hoffmann rules, and both they 91 and Hoffmann and Woodward 92 were able to explain stereoselective effects [such as the preference in the Cope rearrangement (1) -+(2) for the chair-like intermediate (16) rather than the boat-like one (17), and the tendency for Diels-

Alder reactions to give the endu adduct as in (18)+(19) rather than the exo one as in (18) +(20)] in terms of qualitative perturbation theory. Similar methods were used by Hoffmann and his co-workers in discussion of favourable and unfavourable conand of relative stabilities of isomers.94 formations of conjugated All these treatments are based, implicitly or explicitly, on extended Huckel theory, 86 8’ 88 89 90

9? g3 94

S. S. Sung, 0. Chalvet, and R. Daudel, J. Chem. Phys., 1959,31,553; K. Fukui,T. Yonezawa, and C . Nagata, ibid., p. 550; B. Pullman, ibid., p. 551 ; H. H. Greenwood, ibid., p. 552. K. Fukui, T. Yonezawa, and C. Nagata, J . Chem. Ptiys., 1957, 26, 831. K. Fukui, ‘Molecular Orbitals in Chemistry, Physics and Biology’, ed. P.-0. Lowdin and B. Pullman, Academic Press, London, New York, 1964, p. 513. J. Koutecky, R. Zahradnik, and J. Cizek, Trans. Faraday SOC., 1961, 57, 169. K . Fukui, Tetrahedron Letters, 1965, 2009, 2427; Bull. Cliem. Sac. Japan, 1966, 39, 498. K. Fukui and H. Fujimoto, Tetrahedron Letters, 1966, 251. R. Hoffmann and R. B. Woodward, J . Amer. Chem. Soc., 1965,87,4388,4389. R. Hoffrnann and R. A. Olofson, J . Amer. Cliem. SOC.,1966, 88, 943. R. Hoffrnann, A. Imamura, and G . D. Zeiss, J . Amer. CRern. SOC.,1967, 89, 5215.

Theories of Organic Reactions

61

and as such take no account of the charge distribution or of overlap. Klopman and H ~ d s o nO6, SalemYg7 ~ ~ ~ and Fukui and Fujimoto used a semi-empiricalapproach to develop more elaborate expressions, incorporating these additional effects, for the energy of interaction of two molecules A and B. Their equations take the general form 859

where qs, qb are the electron populations in orbitals a and bybelonging to molecules A and B respectively; Cra, Csb are molecular orbital coefficientsin orbitals r and s of molecules A and B respectively, and Er, cS are the corresponding orbital energies; Qk,Qlare the net charges on atoms k, 2 of molecules A, B respectively, separated by a distance &; p&b and s & b are resonance and overlap integrals; and E is a local dielectric constant. In this equation, often called the Klopman-Salem equation, the first term represents the repulsion energy of filled orbitals, and hence the steric effects; the second term is the ordinary electrostatic interaction between the molecules; and the third term describes a charge-transfer stabilization due to the mixing of filled orbitals on one molecule with empty orbitals on the other. Klopman O6 saw that the second term would dominate (‘charge control’) when the reactants were both highly charged and unpolarizable (‘hard’ in Pearson’s terminology) and that the third term would dominate (‘frontier orbital control’) when both reactants were uncharged and highly polarizable (‘soft’). He was then able to rationalize the observations that hard acids form stable complexes with hard bases, and soft acids with soft bases, whereas the complexes of hard acids with soft bases, and vice versa, are less stable. He was also able to take account of the nature of the reagent in electrophilic or nucleophilic substitution, in a way that the reactivity index approach is unable to do, and to 95 96 O7 98 99

G. Klopman and R. F. Hudson, Theor. Chirn. Acta, 1967, 8, 165. G. Klopman, J. Amer. Chem. SOC.,1968, 90,223. L. Salem, J . Amer. Chem. SOC.,1968, 90, 543, 553. K. Fukui and H. Fujimoto, Bull. Chern. SOC.Japan, 1968,41, 1989. R. G. Pearson, J. Amer. Chem. SOC.,1963,85, 3533; J. Chem. Educ., 1968,45, 581,643.

62

Theoretical Chemistry

explain how the ratio of yields of ortho, meta, and para products in substitution of benzene derivatives depends simply on the competition between charge control and frontier orbital control. The use of the term ‘frontier orbital control’ implies an approximation which is not in fact made in equation (lo), namely the frontier orbital approximation: the neglect of all contributions to the third term except those from the frontier orbitals, the HOMO and LUMO, in each molecule,In many cases a further approximation can be made by taking only one of the two frontier orbital terms, namely that which involves the smaller HOMO-LUMO energy separation, and which is assumed therefore to be the major contributor. If one molecule is an electron donor (high HOMO) and the other an acceptor (low LUMO) the energy separation between donor LUMO and acceptor HOMO will be substantially larger than that between donor HOMO and acceptor LUMO. The frontier orbital approximation has been vigorously advocated by Fukui, in the face of considerable doubts from other theoreticians. The argument against this approximation is plain enough: the contributions to the Klopman-Salem equation from the other orbitals are often considerably larger in number, and are smaller in magnitude only by virtue of a modest increase in the energy denominator, and it is not at all evident that the sum of these other contributions will not outweigh the frontier orbital contribution. Fukui’s answer to this is frankly unconvincing; he bases it on three principles :n5s loo (i) the ‘principle of positional parallelism between Charge-Transfer and Bond-Interchange’, which derives from the relationship between frontier electron density at a given atom and the sum of partial bond orders with neighbouring atoms for the frontier orbital; (ii) the ‘principle of narrowing of the frontier orbital separation’: the HOMO becomes less bonding and the LUMO less antibonding as the nuclear configuration changes in the course of the reaction; (iii) the ‘principle of growing frontier electron density along the reaction path’: the frontier orbitals tend to become concentrated in the region where new bonds are being formed. These principles are unconvincing for various reasons. In the first place, they are generalizations from calculations on the butadiene ethylene Diels-Alder reaction and one or two others,100supported only later by an argument based on Huckel ideas and involving several further approximations. In the second place, the principles themselves, if valid, call for changes in the frontier orbitals which can only come about through mixing with other orbitals, and, if the changes are significant, then either the changed frontier orbitals must be used in the calculation or the contributions of the other orbitals must appear. In the third place, the Klopman-Salem equation must relate to an early point on the reaction path (where the molecules are still distinct, though close enough for a molecular orbital description of the combined system to be valid,lol i.e. at distances of the order of 5-8 bohrlo2)for if it did not, the use of the molecular geometries, wavefunctions, and energies calculated for the isolated molecule would be quite unrealistic. The implicit assumption is that the graphs of energy against reaction co-ordinate for two similar reaction paths are similar in shape (Figure 13a) so that energy differencesat an early point on the reaction path will

+

100 101 102

K. Fukui and H. Fujimoto, Bull. Chem. SOC.Japan, 1969,42, 3399.

A. Devaquet and L. Salem, J . Amer. Chem. SOC.,1969,91,3793. A. Devaquet, MoI. Phys., 1970, 18,233.

Theories of Organic Reactions

63

Figure 13 Possible profiles of energy against reaction co-ordinatefor a pair of similar reactions

reflect the difference in activation energy. The situation shown in Figure 13b is assumed not to occur; this is sometimes confusingly called the ‘non-crossing rule’, but, besides having nothing to do with the familiar non-crossing rule for diatomic molecules, it is not really a rule but only a pious hope. Anyway, if the equation refers to the beginning of the reaction path, Fukui’s principles do not apply, since the postulated changes in the frontier orbitals will not have proceeded far enough to have any significant effect. Nevertheless, frontier orbital theory undoubtedly works, though it is not as universally successful as the W-H rules. The most important example of its success is probably the field of cycloadditi~ns.~~~ The Diels-Alder reaction displays, to a marked degree, (i) stereoselectivity, or the tendency to form endo rather than ex0 adducts. For example (18)-+(19) but (18)#+(20); (ii) regioselectiuity,or preference for one of the possible products obtainable from unsymmetrically substituted reagents: e.g., reaction of (21) and (22) gives a mixture of different amounts of the ortho and meta isomers (23) and (24);

(iii) site selectivity, or preference for one site when several are available: e.g., Diels-Alder reactions of anthracene generally take place across the 9,lO-positions rather than the 1,6positions; (iv) periselectivity. When cyclopentadiene (25) adds to tropone (26), the reaction could, according to the W-H rules, involve a six-electron pericycle (27) or a tenelectron one (28). The latter is preferred, which leads to the product (29). 103

W . C. Herndon, Chem. Rev., 1972,72, 157; K. N.Houk, Accounts Chem. Res., 1975,8, 361.

Theoretical Chemistry

64

I (28)

In 1967, Sauer lo4said, referring primarily to regioselectivity, that 'the relative amounts of structurally isomeric adducts . . . cannot yet be explained'. In fact stereoselectivity was by then understood qualitatively on the basis of frontier orbital theory and site selectivity quantitatively in terms of Brown's paralocalization energy,7Dbut no explanation of regioselectivity and periselectivity was available, Within the year, however, Feuer, Herndon, and Hal1105 were able to explain stereoselectivityand regioselectivity quantitatively in terms of a perturbation method based on Huckel or extended Huckel theory and including only the frontier orbital term, and now all these properties, together with the relative reaction rates, are satisfactorily understoodf3'lo3# l o 7 purely on the basis of frontier orbital theory. Other applications have been described by Fukui,lo8 generally in a qualitative rather than quantitative way, and more recently the method has been extended to allow for the effects of external fieldsl o g110 # and the presence of a third rnolecule,ll*~ ll1 so providing a description of catalysis. If three molecules A, B, and C have HOMOS a, b, c and LUMOs a', b', c', and (for example) sab' is the overlap between the HOMO of A and the LUMO of B, then reaction is facilitated ('orbital catalysis' occurs) when either (i) A and B are donors, C is an acceptor, and SabSac'Sbc' < 0 or (ii) A and B are acceptors, c is a donor, and Sa'b'Sa'cSb'c > 0. It is clear that the frontier orbital term, the third term of equation (lo), cannot be considered in isolation, since charge and steric effects may be important to0,13~107 but nevertheless frontier orbital theory is very powerful, as well as being simple enough to be understood and used by non-mathematical organic chemists. The reader is referred for details to the more specialized reviews.l3985, 112 91p

l o 6 3

J. Sauer, Angew. Chem. Internal. Edn., 1967, 6 , 16. W. C. Herndon and L. H. Hall, Theor. Chim. Acta, 1 9 6 7 , 7 , 4 ; J. Feuer, W. C. Herndon, and L. H. Hall, Tetrahedron, 1968, 24, 2575. lo6R. Sustmann, Tetrahedron Lefters, 1971, 2717, 2721. 107 K. N. Houk, J. Sims, C. R. Watts, and L. J. Luskus, J . Amer. Chem. SOC.,1973, 95, 7301. l08 K. Fukui, Accounts Chem. Res., 1971, 4, 57. H. Fujimoto and R. Hoffmann, J . Phys. Chem., 1974, 78, 1874. 110 A. Imamura and T. Hirano, J. Amer. Chem. SOC.,1975, 97, 4192. l11 K. Fukui and S. Inagaki, J . Amer. Chem. SOC.,1975,97,4445; K. Fukui, Israel J. Chem., 1975,

104 105

14, 1. 112 'Chemical

Reactivity and Reaction Paths', ed. G. Klopman, Wiley, London, New York, 1974.

65

llzeories of Organic Reactions

Why then does frontier orbital theory work? Herndon and Ha111°5calculated the complete charge-transfer term of the Klopman-Salem equation as well as the two frontier orbital contributions, using extended Huckel theory, and found that the larger frontier orbital contribution (HOMO of diene - LUMO of dienophile) was some 6040% of the total, the proportion being rather variable, and found that quantitatively reasonable predictions of relative rates were obtainable either from the complete charge-transfer term or from this major contribution. In their later paperY1O5 they used n-electron Huckel theory and found that the frontier orbital contributions were each about 30% of the total, but that the major frontier orbital contribution again gave the right answer. They were sceptical about the general adequacy of the frontier orbital approximation, in view of the substantial stabilization attributable to the other terms. It has recently become clear 1 1 3 9 11* that simple Hiickel theory, as well as some more elaborate techniques such as MIND0/2, are unreliable for use in conjunction with frontier orbital theory. For example, Huckel molecular orbital coefficientssuggest that acrolein (30) will dimerize to (31), but in fact the product is (32). SCF orbitals @

<*

5 I

0

give the right answer, though the reaction seems to be mainly charge-controlledin any case.lo1*ll6 To understand the source of the difficulty, consider Figure 14, which shows interaction between two molecules, of which A has MO coefficients aka,

a 4

b-p

O+d

b-p

a-d

b+p

Figure 14 Molecular orbital coefficients for two interacting molecules, showing two possible orientations 113 114 115

T. Inukai, A. Sato, and T. Kojima, Bull. Chem. SOC.Japan, 1972, 45, 891. K. N. Houk, J. Amer. Chem. SOC.,1973, 95, 4092. K. N. Houk and R. W. Strozier, J. Amer. Chem. SOC.,1973,95,4094; P. V. Alston and D. D. Shillady, J. Org. Chem., 1974, 39, 3402.

66

Theoretical Chemistry

and B has coefficients b f p ; without loss of generality we can assume that a, b, a, are all positive. Two orientations are possible, as shown in Figures 14a and 14b, and the contribution to the charge-transfer term of equation (10) is, in case (a), proportional to

- [(a+ &)(b+ B) + (a - a)(b - 8)12

and in case (b) to - [(a - a)(b

+ B) + (a + or)(b-

8)]2

+ spy,

(1la)

- orS)2,

(11b)

= - 4(ab = -4(ub

so that the difference between the energies for the two orientations is proportional to

aborp. Thus the greater stabilization is obtained from equation (lla), where the interaction is between the two larger-magnitude coefficients.l16 The conclusion is the same, clearly, if the reaction proceeds unsymmetrically so that only one interaction needs to be considered. The MO coefficients calculated116by CND0/2 are shown in Figure 15; evidently small changes in the calculated coefficients would be enough to

0.58

0.59

oi;

- 0.39

-0-58

0 HOMO LUMO

0.51 LUMO

0 HOMO

Figure 15 Molecular orbital coefficients, calculated by CNDOI2, for the HOMO and LUMO of acrolein

change the sign of a, b, a,or /land hence to change the preferred orientation from the one shown to the incorrect one which leads to (31). Secondary interactions, between atomic orbitals on the two molecules which are close to each other in the transition state but which do not become bonded, can also affect the result,lo71117but like electrostatic and steric effects, they are usually invoked in a qualitative fashion when the primary calculation fails to give the right answer. Further uncertainties arise in the determination of the orbital energies. Houk has noted 114 that Huckel energies, and indeed calculated orbital energies generally, are unreliable, and recommends the use of experimental ionization potentials and electron affinities in place of calculated energies of occupied and virtual orbitals respectively. The ionization potentials are obtained from photoelectron spectra and the electron affinities deduced from n-z* electronic spectra, polarographic reduction potentials, and charge-transfer spectra, and Houk and his co-workers have given representative 116 117

N. D. Epiotis, J. Amer. Chem. SOC.,1973, 95, 5624. P. V. Alston, R. M. Ottenbrite, and D. D. Shillady, J. Org. Chem., 1973,38,4075; S . Inagaki, H. Fujimoto, and K. Fukui, J. Amer. Chem. SOC.,1976, 98, 4054.

Theories of Organic Reactions

67

values for a large number of classes of compound.ll* Houk has gone so far as to suggest114that frontier orbital theory may work because the calculated energy gaps for the non-frontier orbital terms are too small, so that their contributions to the interaction energy are exaggerated. However, he does not cite any evidence for this suggestion. This does not inspire much confidence in the frontier orbital approximation, and there are undoubtedly cases where it fails.l31ll9Further doubts arise if we study Figure 15a again. Here the HOMO coefficientson the two significant atoms are identical, so that interactions involving this orbital do not favour one orientation over the other, Evidently the LUMO-HOMO interaction in Figure 15b must be the more significant one, and this would still be true if the energy denominator for Figure 15b were somewhat larger than in Figure 15a. (In this case of course, the two energy denominators are identical, because the two molecules are identical; moreover, for the sake of illustration, we have ignored the differencebetween the values for the two newlyforming bonds.) In other words, a sufficientlylarge numerator can compensate for a large denominator. In some cases such ‘subjacent orbital effects’ do outweigh the frontier orbital 6o Other failures, in reactions of 1,3-dipoles, have been explained1l8as arising because some such molecules are very flexible and the frontier orbital energies and coefficients are very sensitive to the shape of the molecule. Nevertheless the general success of frontier orbital theory makes it clear that such cases are exceptional. This paradox has not yet been resolved. The work of Epiotis62 offers the following clue. The Klopman-Salem equation involves orbital energies, for which we may substitute ionization potentials and electron affinities, by virtue of Koopmans’ theorem. If we think, as we should, in terms of states rather than orbitals, the excitation energy of a state in which an electron is transferred from a donor D to an acceptor A is the differenceID - A A between the ionization potential and the electron affinityonly if the donor and the acceptor are infinitely far apart. When they are closer, a negative electrostatic term C must be added. The energies of the possible types of state take the form DA: D+A-: D-A+: D*A: DA* :

0 ZD-AA

ZA-AD

+ +

C+C-+

~ Y D

hvA

(D* and A* are locally excited states). Thus the excitation energies of all the electron transfer states D+A- and D-A+ are reduced, and consequently the energy of the frontier orbital D+A- state may be relatively much closer to zero than would appear from an examination of the ionization potentials and electron affinities alone. On the other hand Sustmann and Schubert found that the logarithm of the rate constant for a number of Diels-Alder reactions was inversely proportional to I D - A A , though the points were widely scattered. 118 119 120

K. N. Houk, J. Sims, R. E. Duke, R. W. Strozier, and J. K. George, J. Amer. Chem. SOC.,1973, 95,7287; K. N. Houk and L. L. Munchaussen, J. Amer. Chem. SOC.,1976,98,937. P. Caramella and K. N. Houk, J. Amer. Chem. SOC.,1976, 98, 6397. R. Sustmann and R. Schubert, Angew. Chem. Internat. Edn., 1972, 11, 840.

68

Theoretical Chemistry

Another possibility concerns the resonance integrals p a b which appear in the mopman-Salem equation. In a Hiickel picture, these are independent of the orbital energy, but in a double-zeta or better description we would expect the more tightlybound electrons to have more contracted orbitals, and the higher virtual orbitals to be more diffuse.121It may be that the HOMO and LUMO have the optimum spatial distribution for strong interaction, and that interactions involving more contracted and more diffuse orbitals are weaker.122 Further investigation of this problem is clearly called for, and would be valuable in helping to show where frontier orbital theory can be trusted and where it cannot, and also perhaps in helping to throw some light on other aspects of stereoselectivity which are at present not fully underst00d.l~~ 5 Rigorous Perturbation Theories

Here we look briefly at what might be called ‘proper’ perturbation methods: those which may, at least in principle, be made reasonably rigorous, as distinct from those of the previous section, which generally involve some element of handwaving. One approach of this type is due originally to Bader,12*who used perturbation theory to show that unimolecular reaction is particularly favoured along a normal co-ordinate whose symmetry is the same as that of the transition density between the ground state and a low-lying excited state, because then second-order Jahn-Teller interaction depresses the energy of the ground state. The idea has been taken up by Salem126and Pearson126and used, in conjunction with a semilocalized MO model, to determine the allowed routes in the pyrolysis of cyclobutane and cyc10hexene.l~~ Fukui et aZ.lzehave described a similar method. Rigorous perturbational treatments of the interaction between two molecules belong to the field of intermolecular forces, and I shall not attempt a comprehensive review, since the topic has been reviewed by Stamper129in the previous volume in this series. However, several authors have devised perturbation schemes with a view to their application in problems of reactivity, which is a departure from conventional theory of intermolecular forces, where the possibility of making and breaking of bonds is usually excluded, on the reasonable grounds that the problem is quite hard enough anyway. A major difficulty is the problem of electron indistinguishability. The natural choice of the unperturbed Hamiltonian is the sum of the Hamiltonians for the separated molecules, but this is not symmetric with respect to permutations of electrons on one molecule with electrons on the other. The order of a term in the perturbation expansion then becomes undefined,129and although this difficulty can be the application to large systems is probably not in sight. R. A. Thuraisingham, Ph.D. Thesis, University of Cambridge, 1975. J. N. Murrell, M. Randic, and D. R. Williams, Proc. Roy. SOC.,1965, A284, 566. l a 3 J. E. Baldwin, J.C.S. Chem. Conim., 1976, 734, 738. lZ4 R. F. W. Bader, Canad. J. Chem., 1962, 40, 1164. 1 2 5 L. Salem, Chem. Phys. Letters, 1969, 3, 99. lZ8 R. G. Pearson, J . Amer. Chem. SOC.,1969, 91, 1252, 4947. 127 L. Salem and J. S. Wright, J . Amer. Chem. SOC.,1969, 91, 5947. 128 K. Fukui, S. Kato, and H. Fujimoto, J . Amer. Chem. SOC.,1975, 97, 1. 1 2 9 J. G. Stamper, in ‘Theoretical Chemistry’, ed. R. N. Dixon and C. Thomson, (Specialist Periodical Reports), The Chemical Society, London, 1975, Vol. 2, p. 66. lJO D . M.Chipman, J. D. Bowman, and J. 0. Hirschfelder, J . Chem. Phys., 1973, 59, 2830, 2838.

121

lZ2

Theories of Organic Reactions

69

An alternative approach is to express the Fock operator for the combined system in terms of the Fock operators for the individual molecules, so that the unperturbed wavefunction is a Slater determinant containing the occupied orbitals for the individual molecules. Application of perturbation theory can then in principle yield the exact (correlated) energy of the combined system, but only more modest goals are accessible in practice. Devaquet131has applied this approach to two interacting n systems, and obtained an expression for the interaction energy which is essentially the same as the Klopman-Salem equation, except that the energies and coefficients are SCF energies and coefficients, and the interaction integrals are more precisely defined. Sustmann and B i n ~ c hdescribed l~~ a method which started from the same point, but invoked the zero-differential-overlapapproximation; on the other hand, it was not confined to n-electrons and the perturbation energy was refined iteratively. Using a MIND0 parameterization they then applied the method to Diels-Alder reactions, and were able to account for effects which cannot be explained in terms of simple n-electron theory, such as the preference for endo addition of cyclopropeneto cyclopentadiene. Basilevsky and Berenfeld 133 developed a general SCF perturbation theory, in which they carried out a symmetrical orthogonalization of the molecular orbitah of the individual molecules, and expanded in powers of the overlap. They were able to obtain expressions for Coulomb, exchange, polarization, charge-transfer, exchange repulsion, and dispersion contributions to the interaction energy. Some of these terms are additional to those which appear in the Klopman-Salem equation (though polarization and dispersion energies are unlikely to be important where the chargetransfer term is significant) and they find more general forms for the KlopmanSalem terms. However, the formulae do not seem to have been applied to anything more complicated than the interaction of two H, molecules.134 Gouyet has shown how to carry out a similar procedure without orthogonalizing the orbitals. Instead he uses135biorthogonal orbitals, in terms of which the Hamiltonian matrix is non-Hermitian. He gives a diagrammatic expansion, and shows how to eliminate the ‘intramolecular’ terms, namely those spurious contributions to the intermolecular interaction energy which arise not from the true interaction but from the fact that the basis set for molecule B can improve the description of molecule A. The method can also be used where the unperturbed state is a multiconfigurational function, and where termolecular interactions must be considered,13sbut once again the only application137has been to the interaction of two H, molecules. This is evidently an area where there is ample scope for further work.

A. Devaquet, Mol. Phys., 1970, 18, 233. R. Sustmann and G. Binsch, Mol. Phys., 1971, 20, 1. M. V. Basilevsky and M. M. Berenfeld, Internat. J . Quantum Chem., 1972, 6 , 23, 555. M. V. Basilevsky and M. M. Berenfeld, Internat. J. Quantum Chem., 1974, 8, 467. J . F. Gouyet, J. Chem. Phys., 1973, 59, 4637. 136 J. F. Gouyet, J. Chem. Phys., 1974, 68, 3690. 157 E. Kochanski and J. F. Gouyet, Mol. Phys., 1975, 29, 693. 131 132 133 134 135

3 The Quantum Mechanical Calculation of Electric and Magnetic Properties ~~

~

BY A. HINCHLIFFE AND D. G. BOUNDS

1 Introduction

The aim of this chapter is to review the current status of the quantum-mechanical calculation of electric and magnetic properties of isolated atoms and molecules. In view of the rapid advances made during the past decade in the calculation of ab initio molecular wavefunctions, we will clearly concentrate for the most part on the calculation of such properties using standard ab initio methods such as gaussian orbital LCAO-MO-SCF (linear combination of atomic orbital-molecular orbital-selfconsistent field), configuration interaction (CI), coupled Hartree-Fock, and the like, but will also review similar calculations at the semi-empirical and empirical level where appropriate. For readers unfamiliar with the theory of electric and magnetic properties, the books by Daviesl and by Atkins* review the subject thoroughly, whilst the more technical details of quantum-mechanical calculations on atoms and molecules have been described in many other places.a We should also state what this review is not: we do not intend to give a general catalogue of experimentally measured properties, nor of calculated ones. The latter can be found easily from compilations such as the ‘Bibliographyof ab initio molecular wavefunctions’,’ although experimental results are admittedly often scattered throughout the literature. Also we will not generally be concerned with bulk properties, since molecular quantum mechanics at the moment deals with isolated (gasphase) molecules. Instead we intend to concentrate more closely on methods applicable to the calculation of an individual property, the likely accuracy of the calculation, and possible ways for improving the accuracy. First of all, why are the electromagnetic properties of molecules worth investigation ? There are generally two reasons, the first of which is to enable one to calculate experimentally useful quantities in order to make new predictions : apart from mwltipoles and polarizabilities themselves there are quantities that arise in the theory of the interaction between radiation and matter; it is trite but true to say that the whole of spectroscopy and of light scattering depend on electromagnetic properties. In many 1 2

3

D. W. Davies, ‘The Theory of the Electric and Magnetic Properties of Molecules’, Wiley, London, 1967. P. G . Atkins, ‘Molecular Quantum Mechanics, Part HI’, Oxford University Press, 1970. R. McWeeny and B. T. Sutcliffe, ‘Methods of Molecular Quantum Mechanics’, Academic Press, New York, 1969. W. G. Richards, T. E. H. Walker, and R. K. Hinkley, ‘A Bibliography of Ab Initio Molecular Wavefunctions’, Clarendon Press, Oxford, 1971 ;W. G. Richards, T. E. H. Walker, L. Farnell, and P. R . Scott, ‘Supplement for 1970-1973’, Clarendon Press, Oxford, 1974.

70

The Calculationof Electric and Magnetic Properties

71

cases, calculated quantities can be compared with experiment directly; for example molecular-beam resonance experiments are capable of yielding values of the dipole moments of molecules so accurately that the variation of dipole moment with vibrational quantum number can be clearly detected (Table 1),6 whereas in other

Table 1 a Variation of dipole moment (Debye) with vibrationalquantum number.6The systematic errors are 1 part in 104,relative errors ca. 7 parts in v=o 39K19F

a9K79Br

8.592596 10.62813

v= 1 8.130932 10.67861

v=2

8.8oO904 10.72942

Unless otherwise stated, the atomic system of units is used in this Report: bohr =a0 = atomic unit of length~0.529x 10-lo m; hartree=atomic unit of energy, e2/4neoao-44.359aJ; atomic unit ofelectricdipole=euo 21 8.478 x 10-SOCm; atomic unit of electric quadrupole=eao2cz4.487 x 10-40 C m2 etc. (1

applications it may well be easier to perform a quantum-mechanical calculation and get an approximate result rather than make the necessary experimental measurements (although we should emphasize that such workers ought to have an awareness of the probable accuracy of their calculation). Indeed, theoretical calculations may often offer the only feasible approach: for example the dipole moment of a charged species cannot be measured experimentally, and that of a free radical only with difficulty; quadrupole moments are relatively easy to calculate but difficult to measure; how accurate are the calculated values likely to be?; the moments of a molecule in large crystal fields are unobtainable experimentally; the polarizability of a molecule as a function of internuclear separation is of interest in light scattering and in the theory of intermolecular forces but cannot be measured experimentally, etc. Secondly, comparisons with experimental results where these exist provide criteria for evaluating the accuracy of wavefunctions and the methods of generating them. For very many years the only satisfactory criterion for comparing wavefunctions calculated by the variation method was the total energy and perhaps the dipole moment although, of course, experimental values of other quantities were generally not available. The more recent interest in these other properties gives experimental data for comparison and hence more stringent tests for a molecular wavefunction. Comparison with experiment is necessary to estimate the errors inherent in the methods used and so this is of paramount importance if they are to be applied to the prediction of unknown properties. A systematic treatment of errors in particular properties has yet to be published (with the notable exception of dipole moments6)). Such a treatment would be an invaluable aid to the non-specialists who wish to use molecular wavefunctions for the prediction of electromagnetic properties. Again in the wider view, the forces which hoId molecules together are electromagnetic in origin; the study of electromagnetic properties is therefore of fundamental importance in the theories of bonding and intermolecular forces. At long range the electrostatic interaction between two molecules is expanded in their multipole moments whilst the second-order contribution, the so-called dispersion forces, depends on the polarizabilities. It is significant that the first chapter on inter5

R. van Wachem, F. H. de Leeuw, and A. Dymanus, J. Chem. Phys., 1967,47,2256.

* S. Green, Adv. Chem. Phys., 1974, 25, 179.

72

Theoretical Chemistry

molecular forces in the classic book by Hirschfelder, Curtiss, and Bird is concerned with electromagnetism. Whether a bond is covalent or ionic depends on the relative polarizabilities of the constituents, a high difference favouring ionic bonding. Evidence of the validity of the electric field approach to chemical bonding occurs in the work of Bader et OZ.,~ who compared density difference maps for HF and F, with maps of the density of the molecules in an electric field minus the density of the free molecules as calculated by Stevens and Lips~omb.~ This is illustrated by the Figure, which shows the electron density of LiF in a moderately strong electric field directed along the internuclear axis minus the electron density of the field-free molecule. The Figure should be compared with Figure 3 of ref. 10, a density difference map of LiF minus the overlapped ions Li+ and F-. The two maps show the same characteristicsthe ‘dipole polarization’ of charge density around Li, which is typical of atoms bonding through s-electrons, and the ‘quadrupole polarization’ of charge density around F, typical of atoms that bond mainly through p-electrons.8 Magnetic and electrical properties can be conveniently (but not uniquely) classified into three categories; the first type can be called one-electron or first-order properties, and can be calculated from where Yorepresents the wavefunction for the state in question (often the ground state) and & the operator representing the property. This type includes the electric multipoles and lowfrequency parts of the diamagnetic susceptibility and diamagnetic shielding. The second type can be called response functions (polarizabilities and susceptibilities) which describe the response of the system to an applied field, and are obtained from second and higher orders of perturbation theory, and therefore cannot be calculated directly from !Po.The third type are properties involving spin, such as the hyperfine splitting constants. The first two types can be thought of as being classical in origin, the third type relativistic. A further difference is that the first two types of property usually depend on those regions of the charge distribution far from the nucleus whilst spin properties often involve point densities at nuclei. The calculation of magnetic resonance coupling constants has been extensively reviewed l3 The magnetic shielding constants responsible for chemical shifts are only included briefly since they are calculated using second-order perturbation theory by methods very similar to those used for the diamagnetic susceptibility; indeed, many papers report both simultaneously. Calculations of the frequency dependence of polarizability have also been omitted, not because they are unimportant or the methods of calculation not well developed but because they are calculated by differentmeans and have not yet been extended even to many diatomic molecules. Finally, the many accurate calculations for one-electron systems made possible by the availability of exact solutions of the Schrodinger equation are not considered, since they are clearly inapplicable to larger systems. J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, ‘Molecular Theory of Gases and Liquids’, Wiley, New York, 1964. a R. F. W. Bader, I. Keaveny, and G . Runtz, Canad. 3. Chem., 1969, 47, 2308. R. M. Stevens and W. N. Lipscomb, J. Chem. Phys., 1964,41, 184, 3710. lo A. Hinchliffe and J. C . Dobson, Chem. SOC.Rev., 1976, 5, 79. l1 A. Hinchliffe and D. G. Bounds, unpublished results. l2 C. Thomson, in ‘Electron Spin Resonance’, ed. R. 0.C. Norman, (Specialist Periodical Reports), The Chemical Society, London, 1973, Vol. 1; 1974, Vol. 2; 1976, Vol. 3. l3 W. G. Richards, J. Raftery, and R. K. Hinkley, in ‘Theoretical Chemistry-Quantum Chemistry’, ed. R. N. Dixon, (Specialist Periodical Reports), The Chemical Society, London, 1974, Vol. 1.

The Calculation of Electric and Magnetic Properties

73

I

i I

Figure

SCF-MO density diflerence map for LiF in uniform electric field along internuclear axis minus electron density offield-free rnolecule.11 Contours are drawn corresponding to values of the electron density as follows: A = - 0.800, B= -0.400,C= -0.200,D = -0.080, . . ., J=O, . . . S=0.800electrons

74

Theoretical Chemistry

2 Electromagnetic Properties

We firstly define the electric and magnetic multipoles, then show how the interaction energy with external fields can be expressed in terms of the induced moments and go on to see how these are related to the field-free moments via ‘response functions’. Most of our treatment will concern the electric case, since this is the simpler, and since magnetic effects can be treated by analogy. Multipole Moments.-For any system of charges ei (1 < id n) a set of electric multipoles can be defined. They are tensor quantities, the 2n-th pole being an n-rank tensor. For a set of charges er with position vectors ri relative to some origin, the first few are: n = O

q =

n = l

pa =

n

= 2

$,et a

Q,p

=

C eiria i

E eiriarig i

where we have used the notation that a, /?,y can each be x, y or z. In addition, a repeated Greek subscript implies summation over that subscript - thus ria ria

C rta ria. a

The potential outside the charge distribution and due to it is simply related to the moments, as is the interaction energy when an external field is applied.l* The multipole moments are thus very useful quantities and have been extensively applied in the theory of intermolecular forces, particularly at long range where the electrostatic contribution to the interaction may be expanded in moments. Their values are related to the symmetry of the system: thus, for instance, a plane of symmetry indicates that the component of p perpendicular to it must be zero. Such multipoles are worth calculating in their own right. The energy E(F) of the system in an external field F is treated by expanding in terms of the potential 4i at each rf. The ${ can then be written in terms of the potential and its derivatives at the origin 0.” Substituting the multipole moments and putting

where subscript 0 means ‘evaluated at the origin’, the expression for energy is14 1

E ( F ) = ~ ( 0+ ) q+o- PaFa-Z Q , B G ~

-

..

! ~ , j ~ ~ b ~ .(3)

where as before, a repeated Greek subscript implies summation: thus ,uaFa= puzFz+pyFV pZFzetc. For q and ,u the definitions are unambiguous. For higher multipoles it is often convenient to define the quantities in differentways; the quadrupole moment is often defined as oap = 4 er(3riarib-rr26,p) (4)

+

l4

A. D. Buckingham, Quart. Rev., 1959, 13, 183.

The Calculationof Electric and MagneticProperties

75

An advantage of this redefinition is that 8 vanishes for a spherical charge distribution, and it is often advantageous to work in terms of principal axes, i.e. an axis system related to the molecular axis system by an orthogonal transformation such that the off-diagonal elements of the tensor vanish. In terms of similarly defined higher moments, the interaction energy becomes

This is covered, together with other questions of nomenclature, by Buckinghamf4 and by Bottcher.ls Finally, four properties of multipoles are worth quoting: (i) they are symmetrical in all indices; (ii) only the lowest non-zero moment is origin-independent; (iii) if a system has centre of symmetry, any 22n+1pole is zero if the origin is chosen at the centre of symmetry; and (iv) (a) where there exists an infinite rotation axis, all moments are characterized by a scalar quantity, (b) a generalization of this is: if a molecule has an n-fold axis of symmetry, all 2 m poles can be characterized by a scalar quantity for rn c n. Magnetic multipoles can be defined in a similar manner by expanding the usual vector potential A.' As always in the magnetic case the treatment is more complicated, and the only useful magnetic moment is the magnetic dipole moment p ~which , can be considered as arising from the circulation of an intramolecular current. Thus whilst a single charged particle may have no electric dipole (since this implies a separation of charge), it may have a magnetic dipole moment due to its non-zero angular momentum. A molecule can have one electric dipole moment but three contributions to its magnetic dipole: an orbital contribution due to the circulating electrons, a spin contribution for any unpaired electrons, and a similar term for the nuclei. There are unfortunately various definitions of magnetic multipole moments in the literature - a recent paper by Raabl6discusses the various definitionsand properties, and creates some order from an apparent chaos. Induced Moments.-Applying an external field changes not only the energy but also the shape of the charge density. Thus the multipole moments in equation (5) are those of the charge density in the field. These can be related to the field-free moments by expanding the total moment under consideration as a Taylor series in the field strength (or, in general, all of its non-zero derivatives and adding the various contributions). Thus for a dipole in a uniform field F

(the clash in symbols in the above expression is regrettable but follows normal nomenclat~re'~). Considering only the dipole term in the interaction energy (the weak-field case)

15 26

C. 3. F. Bottcher, 'The Theory of Dielectric Polarisation', Elsevier, New York, 1973, Vol. 1. R. E. Raab, Mol. Phys., 1975, 29, 1323.

Theoretical Chemistry

76 and expanding as a Taylor series in F E(F)=E(O)+

( 3 0 (230 -

F+ f

--

F2

+ ...

we see that, by comparing terms

..

a is the dipole polarizability (usually just called the polarizability), whilst p , y .are the first, second, . . . hyperpolarkabilities respectively. They are tensors, symmetrical in all indices, and their properties have been extensively reviewed by Buckingham When a field with non-zero gradients is present higher multipoles contriand 01-r.'~ bute to the interaction energy and the energy expression involves terms in all the higher derivatives and the cross-termsl4

The magnetic case is treated similarly except that terms in the derivatives of the field are not considered to be important and have not been calculated, the constant H analogue of equation (6) being used. The spin-orbit magnetic moment (&M>

= (&orb

+ F">

= <-Bei-geBes^>

(1 1)

is substituted for I(. and the second-order energy term corresponding to cc is the diamagnetic susceptibility, although this is a more complicated quantity than a. The possibility of a non-zero magnetic moment for the nucleus introduces an extra term; the nuclear moment ,UN may couple with the field H to produce another secondorder energy term of the form &GafipMaHB.This is classed as second order by virtue of perturbation theory, although it is only first order in H;c is the shielding constant responsible for n.m.r. chemical shifts. Thus the response of a system to an external field can be calculated exactly from a complete knowledge of its multipole moments and response functions defined at zero field. Such knowledge is difficult to come by, both experimentally and theoretically.

3 First-order Properties First-order properties can be written as the expectation value of the appropriate operator fi as follows : a =

<~lfil~l>

(12)

and so their calculation needs only the wavefunction Y for the state considered, usually the ground state, and a suitable formula for the integrals involved. They are particularly easy to incorporate into computer programs for calculating Hartree Fock SCF-MO wavefunctions of the gaussian orbital type, and are often routinely calculated.le All one-electron properties fall into this class, which includes the electric field gradient at a nucleus, the multipole moments, and the low-frequency 1'

A. D. Buckingham and B. J. Om, Q w r f . Rev., 1967, 21, 195. See, e.g., D. Neumann, and J. W. Moskowitz, J . Chem. Phys., 1969,50, 2216.

TIte Calculationof Electric and Magnetic Properties

77

parts of the diamagnetic susceptibility and shielding constant. Some authors also report the electric field at each nucleus: this particular property is of interest because the Hellman-Feynman theorem predicts that for a Hartree-Fock wavefunction calculated at the experimental equilibrium geometry the sum of the forces on all the nuclei should be zero (the force being the product of the nuclear charge and the electric field).lsTable 2 shows some typical values of the electric field at the nuclei in Table 2 Electricfield vector E at the nuclei in ethene and ethyne,ll calculated using a large gaussian basis SCF-MO wavefunction (molecules in xy-plane, C - C axis is y-axis). The atomic unit of electricJield is e/4ne0ao2~5.142 x lo1’ V m-l Energy/hartree F/a.u.at H at C

Ethyne - 76.779743 (0, 0.0803,0) (0, -0.0749, 0)

Ethene

- 78.062531 (0,-0.0050, -0.0098) (0, 0.0106,O)

ethene and ethyne calculated using large gaussian basis-set SCF-MO’s, and it is easily seen that the Hellman-Feynman theorem is not generally satisfied. How much of the discrepancy is due to assuming an incorrect molecular geometry and how much due to an incompletely optimized wavefunction is not known, but it may well be that this quantity is worth calculating to give a sensitive indication of the degree of optimization of the wavefunction. The first-order property for which most experimenta120-22 and theoretical results exist is the (electric) dipole moment. The literature on calculations of dipole moment is almost as large as that on wavefunctions. If Y were known exactly, the value of a first-order property calculated from equation (12) would be exact. In practice, only an approximation to Y is known, and it is important to know how the expectation value differs from the exact value. Since errors in calculated dipole moments due to the breakdown of the BornOppenheimer approximation are likely to be small (typically 0.002 am.), and for most molecules relativistic effects can be ignored,‘?there are two separate remaining problems in practice. The first concerns the likely accuracy when the wavefunction is at the Hartree-Fock limit, the second the effect of using a truncated basis set to obtain a wavefunction away from the Hartree-Fock limit. If YHF is a wavefunction at the Hartree-Fock limit for a closed-shell molecule, Y H F can be improved using standard configuration interaction: if represents a configuration where a single electron has been excited from a filled MO to a virtual one, (i.e. a singly excited configuration), Yq a doubly excited configuration. . ., then an improved wavefunction is given by Y/ = CoyHF

19 20

21 22

+

TCiyi t

+

TTCijyij 8 3

+

(1 3)

C . W. Kern and M. Karplus, J. Chem. Phys., 1964, 40, 1374. W. Gordy and R. L. Cook, ‘Microwave Molecular Spectra’, Wiley, New York, 1970. A. L. McClellan, ‘Tables of Experimental Dipole Moments’, Freeman, San Francisco, 1963. R. D. Nelson, D. R. Lide, jun., and A. A. Maryott, ‘Selected Values of Electric Dipole Moments for Molecules in the Gas Phase’, Nat. Bur. Standards (US) NSR DS-NBS10 Govt. Printing Office, Washington D.C., 1967.

78

Tlzeoretical Chemistry

and a first-order property M is given by (-qnsrl!P> =

Co2
+

2co 3 Ci(YHFlA?pf*) a

+ c CtCjC~fIA2IY,>+ 2co cv Cfj

(14)

23

It should be remembered that terms such as (UHFl&fI Yij) will vanish as a consequence of the Slater-Condon-Shortley rules, and if we were to use perturbation theory to calculate the CCetc. we would find Cicc
24 25 26

L. Wharton, L. P. Gold, and W. Klemperer, J . Chem. Phys., 1962,37, 2149. P. E. Cade and W.Huo, J . Chem. Phys., 1967,47, 614. C. F. Bender and E. R. Davidson, J. Chem. Phys., 1968, 49, 4222. A. D. Buckingham, ‘Physical Chemistry, An Advanced Treatise’, Academic Press, New York, 1970, Vol. 4.

The Calculationof Electric and Magnetic Properties

79

The second problem is the much more realistic one of the effect of a limited basis set expansion. This is clearly a more serious problem because only for linear molecules or those with a few first-row atoms can the Hartree-Fock limit be reached at present. For many of the molecules with which theoretical chemists deal, wavefunctions of such accuracy are not available but it may be some comfort to know that even if they were they need not give very good answers It should be mentioned that Brillouin’s theorem applies to any SCF wavefunction, but unless the wavefunction is near the Hartree-Fock limit the electron distribution cannot be expected to be a close representation of the true one. No general treatment of this problem has been given; neither does one seem possible since it would depend on the ways in which the basis set under consideration was weak, and these may be many. For any variational wavefunction which is not near the Hartree-Fock limit the Brillouin theorem is irrelevant, and even for those of Hartree-Fock accuracy lowlying important excited states may invalidate the conclusions drawn from it. The statement that values of one-electron properties are expected to be good ‘because of the Brillouin theorem’ should therefore be regarded with caution. When considering the effects of an approximate wavefunction, !?, it is useful to remember that is usually variationally determined. Effectively the quantity

is minimized. The problem is invariably reduced to matrix form, by expanding the trial function as a linear combination of basis functions of fixed functional form and varying the linear coefficients CZ,i.e. !?= C& until for a change 6ct in any CC, SB=O. This gives the lowest obtainable for this choice of #t and N. However, although the czhave been varied to make approach Etruethere is no such restriction making ~+Yt,,,, and it is quite possible that can be varied in such a way that

xi!l

Table 3 Calculatedfirst-orderproperties for HCN usingfive basis sets which gave the same energy to jive decimal places. Atomic units are used throughout Largest value1a.u. 1 .29068

Smallest valuela.u. 1.29000

- 15.64610

- 15.64602

0.01622 0.47520 - 0.67259 - 27.69539 -26.68683 - 8.89553 -9.56807

0.01619 0.47452 -0.67197 - 27.69373 - 26.68577 - 8.89526 -9.56723

the expectation values of other operators get worse. It is also well known that the stability of other properties with respect to a small variation in ct is much less than that of the energy. The results for some HCN wavefunctions (Table 3) illustrate this. Five different wavefunctions gave the same energy to five decimal places (E= -92.90574 hartree).ll All properties have their usual meanings as defined in the

80

Theoretical Chemistry

POLYATOM2’ programs and elsewhere: the less obvious ones are: c,diamagnetic shielding; F and FG, the electric field and field gradient at the nucleus; x the diamagnetic susceptibility.2 is the internuclear axis. All properties are relative to the carbon atom origin and the values are in the global co-ordinate system and atomic units. From Table 3 it can be seen that as a rough rule, if the energy is determined to n decimal places, first-order properties are stable to 4 2 . There is one other rather interesting result of these calculations not shown in the Table. For all properties other than F and FG the lowest values were those from the fully self-consistent wavefunctions which gave the lowest energy. As upper bound principles do not exist for other operators, this must be a coincidence. Which properties are least well determined by the variational method? The basis functions in the LCAO expansion are either Slater orbitals with an exponential factor e-{r or gaussians, e-ar2; r appears explicitly only as a denominator in the SCF equations: thus matrix elements are of the form (q&lk/rl$t);these have the largest values as r+O. Thus the parts of the wavefunction closest to the nuclei are the best determined,* and the largest errors are in the outer regions. This corresponds to the physical observation that the inner-shell orbitals contribute most to the molecular energy. It is unfortunate in this respect that the bonding properties depend on the outer shells. Several generalizations have been made regarding the calculation of electric 2B (i) dipole moments of polyatomic molecules are in general underestimated by minimal-basis Slater orbital calculations (including calculations using Pople’s STO-nG method2*),although this is not always the case; (ii) dipole moments are usually overestimated when calculated using energy-optimized minimal gaussian basis sets or extended sp basis sets; (iii) the addition of polarization functions may only slightly change the energy but have a considerable effect on the dipole moment and other properties of the outer regions. Dunning’s results29illustrate this point nicely. Contrarily, dipole moments may still be overestimated even near the HartreeFock 31 These three conclusions are illustrated in Table 4.l’ The ST0/4-31G basis set consists of four primitives per basis function for each first-row atom 1s orbital and two valence basis functions of each type, one of the basis functions having three primitives the other one. The large Dunning sp basis set 2 9 is presumably very close to the sp limit, and it was augmented for the second calculation withp-type on H and d-type on C and F. Single polarization functions of each type were used with exponents chosen rather arbitrarily (1 ,O). In view of the above three comments it will be anticipated that second moments, which depend on terms like C x i 2 , will generally

*

Although at the nuclei there is a singularity in the wavefunction not reflected by either STO’s or GTO’s, but this is a different problem. This cusp problem is thc reason for the sometimes poor values of calculated properties near the nucleus.

POLYATOM V/2 System Manual, QCPE Chemistry Department, Indiana University. L. Radom and J. A. Pople, in ‘Theoretical Chemistry’, ed. W. Byers Brown, MTP International Review of Science, Physical Chemistry Series 1 , Vol. 1, Butterworths, London, and University Park Press, Baltimore, 1972. 29 T. H. Dunning, J . Chem. Phys., 1971, 55, 3958. 30 R. S. Mulliken, J. Chem. Phys., 1962, 36,3428. a1 R. K.Nesbet, Adc. Quantum Chem., 1967, 3, 1. 2’

28

The Calculationof Electric and Magnetic Properties

81

Table 4 Dipole moment of CH,F, calculated using several diferent basis sets Elhartree Basis set details -227.611649 0.3194 ST0/2G: minimal basis set with each basis function containing two primitives; 34 primitives, 17 basis functions -236.287916 0.5378 ST0/4G; 68 primitives, 17 basis functions -236.851995 0.5412 STO/GG; 102 primitives, 17 basis functions -237.591927 1.0700 ST0/4-31G; 68 primitives, 31 basis functions -237.821529 1.0700 ST0/6-31G; 74 primitives, 31 basis functions - 237.926359 1.1032 Dunning’s large sp basis set; 29 97 primitives, 57 basis functions Dunning sp + single polarization function of each 0.9509 -237.993483 type; 121 primitives, 81 basis functions Experimental O.77l2l

be less accurately calculated than first moments, and more sensitive to the outer regions of the electron density. Table 5 shows the effect of adding polarization functions to an essentially complete sp basis set, again for CH2FZ.11Multipoles Table 5 Second moments for CH2Fzcalculated using the two Dunning basis sets of Table 4. The H atoms lie in the xz plane, the two F atoms in the yz-plane Basis set sp set sp set plus polarization

<x2>




- 11.1088

-15.2255 - 14.5312

- 13.2967

-11.1143

-12.9074

higher than the first are thus not only difficult to measure, they are also difficult to calculate reliably. A Consequence of the Instability in First-order Properties.-Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places; does an error in the fourth matter? To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability m. In this method a perturbation term - p,(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wavefunction calculated as usual. For small uniform fields,

al=

=

m)-P(O),F,

P a m = P@),

-3a,#,q?

+ %$q?

and the polarizability 01 can be determined from either of these expressions. A small field is required since 01 is defined at F = O and also so that the series expansions of E(F) and p ( F ) in Fcan be truncated to give the equations above, i.e. so that hyperpolarizabilities can be ignored. However, as F+O, E(F)-+E(O), and ,@) +p(0) so any errors in p(0) and p ( F ) assume an additional importance. In any finite field calculation, the choice of F is important because of the conflicting requirements of a small F to make truncation valid and a large F to minimize the rounding errors in the differences. The optimum choice clearly depends on the relative values of p(O), 01, t9,etc. for the molecule concerned. As the errors in p(0) and p ( F ) should be greater than in the SCF energies it is

82

Theoretical Chemistry

attractive at first sight to use the energy expression. In practice this makes but a marginal difference to the results1’ because the difference E ( F )- E(0) is smaller than the difference p ( F ) - p(0) and the improvement is negligible. [This high sensitivity of second-order properties to the wavefunction is quite general: Yoshimine and Hurst 32 minimized the functional corresponding to the second-order energy in a field, an uncoupled approximation to the perturbed Hartree-Fock equations. Their calculations were on atoms and a was obtained from E ( F )= E(0)- +aF2,and they found that good ground-state wavefunctions which gave the same energy to 0.O005 hartree could give polarizabilities different by a factor of 2.1 Ways to Improve Expectation Values.-These may be conveniently divided into (a) methods concerned with introducing correlation and improving the energy beyond the Hartree-Fock limit, and (b) those methods which improve particular expectation values without improving the energy, and possibly at some energy cost. Category (a) includes (i) configuration interaction (CI), (ii) multiconfiguration SCF (MCSCF) and extended Hartree-Fock (EHF) methods, (iii) unrestricted hartreeFock (UHF), and (iv) many-body perturbation theory. Category (b) includes (v) constrained variational methods, and (vi) local energy methods. Methods in category (a) represent an overall improvement in the wavefunction and often require a major computational effort. There is an improvement in the energy and possibly in the expectation values of other operators, but at present the difficulties make them generally unattractive for calculations on medium or large molecules or for studies on smallish molecules at several different geometries. The second category contains methods which do not introduce correlation into the wavefunction : rather they attempt to ‘tailor’ wavefunctions for specific purposes. Configuration Interaction. In the CI method the starting wavefunction is usually a Hartree-Fock function calculated with a reasonably complete basis set, as close as possible to the Hartree-Fock limit. Excited configurations are usually constructed from the virtual SCF orbitals, but since these are not well determined by the SCF method they are not a particularly good choice. Their only virtue is that they are directly available from the SCF calculation. To improve the energy significantly, a large number of configurations must be mixed in, and even restricting the choice of configurations to those thought to be the most important means that CI calculations on diatomics frequently involve hundreds of configurations. At first sight therefore the CI method appears to be even more unattractive for these first-order properties than for energy. Recently, however, Green6*33-36 has shown that it is possible to choose configurations that improve the dipole moment whilst making only a marginal difference to the energy; a dramatic improvement in dipole moment can sometimes be obtained by mixing in only a small number of configurations. The method is based on the following procedure.33All possible doubly excited configurationsare generated from the Hartree-Fock function and their contributions to the second-order Rayleigh-Schrodinger perturbation theory energy computed. Approximately 100 of the most important are used for a CI calculation, all singly M. Yoshimine and R. P. Hurst, Phys. Rev., 1964, A135, 612; s3 S. Green, J. Chem. Phys., 1971, 54, 827. 34 S. Green, J . Chem. Phys., 1971, 54, 3051. 35 S. Green, J . Chem. Phys., 1972, 56, 739.

32

The Calculationof Electric and Magnetic Properties

83

excited configurations are then added to the CI and those with the largest coefficients retained. Finally more doubly excited configurations are added to check the stability of the dipole. If there are too few of the double excitations the single ones are overemphasized; it seems to be necessary to add double excitations down to a level where their coefficientsare similar to those of the single excitations. For open-shell systems higher excitations than double ones must be added.s6The configurations are again chosen using standard perturbation theory and it is necessary to iterate through natural orbitals in order to keep the problem to a reasonable size. The cost of such a calculation is said to be of the same order of magnitude, for closed-shell systems, as the cost of obtaining the starting SCF orbitals. In principle the dipole moment, energy, and any other property should be obtainable to any degree of accuracy by including sufficientconfigurations and using a large basis set. In practice it is only possible to deal with severely truncated sets. Green’s calculations have demonstrated that it is possible to keep the error in the dipole moment for diatomics to about 0.05 D with a suitable choice of configurations. It is interesting that the CI approach may be limited in practical terrns for calculating very accurate dipole moments even for very simple molecules. For LiH an extensive CI calculation has been where 97 % of the correlation energy was recovered but the dipole moment was still in error by 0.015 D, even after proper vibrational averaging. Although this is an error of only 2 parts in a thousand, the experimental error is 1 part in lo4. The MCSCF Method. As noted previously, the primary reason for poor convergence in CI calculations is the use of SCF virtual orbitals in constructing excited configurations; these orbitals are not determined variationally and so are rather poor approximations to the ‘true’ virtual orbitals. The MCSCF method treats a linear combination of Slater determinants !Pi, ymZCSCF

=

atY/i a

(1 7)

where the !Pt are formed from ordinary LCAO MOs, but optimizes both the linear expansion coefficients at and the LCAO coefficients (except that it may happen that the ai are fixed by some symmetry or spin requirement). Various prescriptions exist for solving the complex variational problem: for example, a method has been recently proposed for treating a combination of all double excitations plus the ground but in general such calculations are prohibitive for molecules of interest to chemists. Because the variation principle is involved, certain matrix elements disappear as in ordinary SCF theory, and such relations have recently been referred to as generalized A major review of the MCSCF method has been given by Brillouin Wah1;39the practical limit on the number of configurations seems to be around 50 at present, but energy results compare extremely favourably with those of ‘traditional’ CI calculations invoking many more configurations, and other calculated properties are encouraging. 36 37 38 39

A. D. McLean and M. Yoshimine, personal communication to S. Green, reported in ref. 6. D. B. Cook, Mol. Phys., 1975, 30, 733. B. Levy and G. Berthier, Internat. J. Quantum Chem., 1968, 2, 307. A. C . Wahl and G . Das, Adv. Quantum Chem., 1970, 5 , 261.

84

meoretical Chemistry

UHFMethods.A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this Ieads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,4Owhere two sets of singly occupied orbitals are constructed instead of the doubly occupied set, The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem: one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minirnizati~n,~’ arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. A major difficulty is that there is no guarantee that the projected wavefunction will give better expectation values than the UHF wavefunction. Ideally the energy should be minimized after spin projection, which is an MCSCF-type of problem. Also, the amount of correlation introduced is unknown. In spite of these difficulties, Smeyers and Delgado-Barri~~~ have found that both the total energy and dipole moment variation with bond length in LiH can approach the best CI results using a very simple UHF function with spin projection. Many-body Perturbation Theory. A more radical way to recover the correlation energy is by the use of many body perturbation theory. Here again the wavefunction is expressed in terms of a basis set expansion, but the expansion for the energy and other expectation values are dealt with directly. Normally these expressions are complicated sets of integrals, but here the problem can be simplified by the use of diagrams. This enables the dominant terms to be identified and cancelling terms to be eliminated. Sometimessets of diagrams can be combined and their total contribution found. It is also easier to see where approximations can be made. These techniques were originally applied to nuclear ‘matter’ and the electron gas; both are infinite systems where plane waves form a suitable basis. In finite problems such as atomic or molecular calculations, the occupied and unoccupied Hartree-Fock orbitals are normally chosen. Although a lot more effort is required to get the starting functions, the problem is then somewhat easier because the troublesome divergences inherent in infinite systems do not occur. It is desirable to choose a basis set where most of the excited states form a continuum; then the summations over excited states may be replaced by integrals, which are easier to deal with. Most of the calculations to date have been done on atoms, when the angular co-ordinatescan be dealt with simply and sums over states become integrals over radial quantum numbers. For molecules the lack of spherical symmetry causes computational difficulties,although themulticentre problem has been surmounted in more orthodox HF calculations and there seems to be no additional difficulty in the many-body methods. Current calculationshave been 40 41

42

J. A. Pople and R. K. Nesbet, J . Chem. Phys., 1954, 22, 571. A. T. Amos and G. G. Hall, Proc. Roy. SOC.,1961, A 2 6 3 , 4 8 3 ; A. T. Amos and L. C. Snyder, J . Cltem. Phys., 1964, 41, 1773. There is a trivial error in some of the formulae in the latter reference, a serious error in some of the formulae in the former! Y . G . Smeyers and G. Delgado-Barrio, Internat. J . Qiintitrtni Client., 1974, 8, 733.

The Calculation of Elecfricand Magnetic Properties

85

in a one-centre, or united-atom basis, for example Kelly’s calculation on H,43 and that of Das et aL4*on HF. In the latter calculation, 92% of the correlation energy was obtained and the field at the fluorine nucleus was satisfactory. Even better results would be expected for molecules where the departures from spherical symmetry are less. In contrast, one-centre Hartree-Fock calculations usually give rather poor results compared with multicentre ones. First-order properties have not been greatly exploited by this method, which has proved fruitful for second-order properties (notably polarizability). There is no reason why it should not be so applied. Constrained Variations.The frequently observed fact that a wavefunction gives a good energy but poor expectation values for other operators suggests introducing an additional constraint into the variational procedure: that the expectation value of some other operator is equal to some predetermined value. The constraints may be either theoretical, e.g. that the net forces on the nuclei are zero at the equilibrium geometry, or empirical if the constraint is an experimental value. The hope is that by improving one property a significant improvement will also be obtained for other operators, without changing the energy very much from the unconstrained value. Mukherji and K a r p l ~ swho , ~ ~ were probably the first to use empirical constraints, showed how such constraints can be incorporated using Lagrangian multipliers. They used a limited basis set LCAO-MO-SCF function due to R a n d and varied the wavefunction so that the values of the dipole moment and the deuteron quadrupole coupling constant in DF were equal to the experimental ones, within a small degree of error. The energy was raised by only 0.004 hartree while significant improvements in other properties were noted - for example, the diamagnetic susceptibility and the paramagnetic susceptibility -and the calculated force on the proton decreased by ca. 50%. Rasiel and Whitman4s started from a three-configuration wavefunction for LiH due to Robinson,47which gave the lowest energy then available but a dipole moment in error by 12%. Constraining the dipole moment to equal the experimental value raised the energy by a mere 0.14eV, whilst the forces on the nuclei, the proton shielding constant, diamagnetic susceptibility, and gauge invariance of the wavefunction all improved. The disadvantage of this sort of approach is the need to know some experimental property in advance. Apart from any ‘unaesthetic’ aspect, the method cannot be used for anything but the experimental geometry of a wellcharacterized molecule. A second method of constrained variation involves the hypervirial theorems. The virial theorem is really a special case of a more general set of hypervirial theorems characterized by the equation where tt’ is a function of the co-ordinate and momenta operators. These hold for exact wavefunctions but not for approximate ones and can therefore form the basis of

43

H. P. Kelly, Phys. Rev. Letters, 1969, 23, 455.

44

T. Lee, N. C. Dutta, and T. P. Das, Phys. Rev. Letters, 1970, 25, 204. A. Mukherji and M. Karplus, J. Chem. Phys., 1963, 38,44.

*5 48

4‘

Y.Rasiel and D. R. Whitman, J. Chem. Phys., 1965, 42, 2124. J. M. Robinson, Ph.D. thesis, University of Texas, 1957.

86

Theoretical Chemistry

a constraint method. Epstein and Hirschfelder4*have shown that an approximate wavefunction can always be made to obey any hypervirial theorem by a variational method where there is a certain mode of variation depending on @',and is miniN

mized. If

1.v=$C ('&*it--k*bi) the ordinary virial theorem is obtained and the i= 1

variational mode is scaling of the electron co-ordinates. For the Hellman-Feynman N

theorem, @'= C

and the mode is uniform translation of the electron co-ordinates

i= 1

in any direction.49 Byers Brown has further developed the constrained variational method of Rasiel and Whitman and shown the relation to the hypervirial theorem constraints. He developed a perturbation theory for the constrained variational method and discussed the use of first- and second-order properties for c o n ~ t r a i n t sHe . ~ ~also produced formulae for the effect of a constraint on the value of the energy and other expectation values. The method was applied to LiH using Robinson's wavefunction and gave excellent agreement with Rasiel and Whitman's result. In a second calculation the total forces at the nuclei are constrained to equal zero. This costs only 0.09 eV and improves the accuracy of the dipole by a factor of two. The method was also applied50to the mixed basis set calculation of Matsen and Browne.sl The perturbation series converges very rapidly, but in this case there is a negligible change in other calculated properties, presumably because the starting wavefunction was of good quality. The method is easy to use because the only requirements are the MOs, the orbital energies, and matrix elements over the given constraint operator and over certain constraint formulae given in their first paper.4g Chong and Rasie152later compared the method of Byers Brown with that of Rasiel and Whitman, and found that the Byers Brown formulation was the best for practical purposes of computation. Further work has been done by C h ~ n and g ~ by ~ Chen,64who introduces weighting operators which are related to the hypervirial theorems but form stronger constraints. A summary of the results of Byerx Browns0and of Chong and Rasie152for LiH is shown in Table 6. The properties calculated by Browne and Matsen s1 are also given for comparison. Column 2 contains the unconstrained results of Chong and RasielS2 using Robinson's47 wavefunction. The Chong and Rasiel results where the dipole moment is constrained to equal the experimental value are shown in column 3, whilst column 4 shows the results when the total force is constrained to be zero. The symbols for molecular properties have their usual meaning. Local Energy Methods. The effort needed to evaluate the integrals required in the variation and perturbation methods is large, and increases rapidly with increasing moIecuIar complexity: since the early days of quantum chemistry there has been J. 0. Hirschfelder, J . Chern. Phys., 1960, 33, 1462; S. T. Epstein, Phys. Rev., 1961, 123, 1495; C. A. Coulson, J. Chern. Phys., 1962,36, 941. 49 W. Byers Brown, J . Chem. Phys., 1966,44, 567. 50 D. P. Chong and W. Byers Brown, J. Chem. Phys., 1966, 45, 392. 51 J. C. Browne and F. A. Matsen, Phys. Rev., 1964, A135, 1227. s2 D. P. Chong and Y. Rasiel, J . Chem. Phys., 1966, 44, 1819. 53 D. P. Chong, J. Chem. Phys., 1967,47,4907; M. L. Benston and D. P. Chong, MoZ. Phys., 1967, 48

12, 487. 54

J. C . Y . Chen, Phys. Letters, 1965, 16, 269; J . Chem. Phys., 1965, 43, 3673.

The Calculationof Electric and Magnetic Properties

87

Table 6 Summary of constrained variational calculations on LiH, bond length = 3.046a0 Property PIDebYe

@>/am. a Q/buckinghamb

106 ~(Li)/cm3mol-l lo6 ~(H)/cm3mol-l a(Li)/p.p.m. dH)/p.p.m. a

Unconstrained accurate wavefn.50 5.889 0 . 01425 -4.1488 -21 .1731 - 24.6905 107.751 39.220

F is the total force operator.

Constraint on Simple Dipole Total wavefn.52 moment force Experiment 4.162 5.882 5.396 5.882 0.0367 0.9687 0 0 - 5.5486 - 7.0842 - 7.2497 -26.956 - 27.281 -25.785 -32.956 -30.195 -31.925 105.62 105.55 105.47 33,251 33.909 33.618 -

Quadrupole moments calculated at the centre of mass.

interest in developing methods which avoid the integral problem. For an exact eigenfunction, fiY = EY

(19)

1 pH!P = E

(20)

which may be trivially rewritten

the eigenvalue Eis of course a constant, independent of co-ordinates. For an approximate eigenfunction it turns out that the local energy E(r), where

is far from constant, even when Y gives quite good expectation values.55Conversely, however, the constancy of the local energy is usually taken as a measure of the goodness of an approximate wavefunction. This is usually accomplished by minimiz57 where ing the mean-square deviation W of the local energy from its mean (22)

and

The gi are weighting factors. The number of points used in the sums is ideally small, and the relationship of the above expressions to those encountered in numerical integration should be obvious. The weighting factors g f and the points themselves ri are in principle arbitrary, but it appears in practice that the results are unreliable unless they are carefully chosen. A review of some local energy calculations for small systems has been given by Frost et aZ.,56but the method in all its different variants has so far enjoyed little application to molecular properties5’ 55 56

57

J. H. Bartlett, Phys. Rev., 1955, 98, 1067. A. A. Frost, R. C. Kellogg, and E. C. Curtiss, Rev. Mod. Phys., 1960, 32,313. J. Goodisman, ‘Diatomic Interaction Potential Theory’, Academic Press, New York, 1973, and references therein.

4

88

‘Ilteoretical Chemistry

Bond Properties.-An interesting application of the calculation of molecular properties is the calculation of bond properties; chemistry is after all concerned with the bond concept, and most chemists are interested in the bond contributions to molecular properties, even though these bond properties themselves cannot generally be directly found from experiment. Although standard SCF MOs are of a delocalized nature, several methods are available for the transformation to an equivalent set of Zocalizedorbitals which correspond closely to the chemist’s ideas of inner shells, lone pairs, covalent bonds, and the like.58Such localization procedures are valuable for a variety of reasons, not the least being that the resulting localized MOs can be used directly to calculate bond properties. Recently for example, dipole and quadrupole moments of C-C and C-H bonds have been reported, calculated from a variety of wavefunctions ranging from semi-empirical through the Frost floating spherical gaussian model 6 o to quite accurate SCF-HF.61y 62 For a doubly occupied localized MO w ( r )we define the associated charge density as jg

Pt(r) =

5 Z,rS(R, -r) -2wi*(r)wi(r)

(24)

where Zai is the nuclear charge contribution to the ith bond orbital from the ath nucleus. England and and Pritchard and Kern61 associate two units of positive charge with each localized orbital, the charge being divided equally amongst the nuclei associated with the ith bond. The bond dipole and quadrupole moments can be easily calculated from pi = JPi(r)rdY;

6i = +JPi(r)(3rr-r21)dV

(25)

the latter definition depending on the choice of origin unless the bond dipole is zero. It is conventional to choose the bond midpoint, and Table 7 summarizes some typical results. In particular, the C-H bond dipole moments are calculated to be much larger than earlier calculations, for example those of C o ~ l s o nwould , ~ ~ have suggested. It has been shown, however, that the early calculations were in error because of the essentially crude nature of the wavefunctions employed,81 but that still leaves the dilemma that, although the ‘experimental’ C-H bond dipole is unknown, it is generally accepted that a value of ca. 0.5 D is appropriate. This dilemma has not yet been resolved but, as Amos remarks,6oone will simply have to accept that the calculated value is the size it is, and learn to use it accordingly. As one might expect, the Frost model gives rather poor bond quadrupole moments owing to the diffuse nature of spherical gaussian charge distributions.Amos et aZ. have additionally extended the calculations to second-order bond ~ r o p e r t i e s . ~ ~ 4 Second-order Properties Whereas first-order properties can be obtained directly from the ground-state wave53

j ! ’ (i(’

61 6z

63 R4

C. Edmiston and K. Ruedenberg, Reo. Mod. Phvs., 1963, 35, 457; H. Weinstein, R. Pauncz, and M. Cohen, Adu. Atomic Mol. Phys., 1971, 7, 97; K. Ruedenberg, in ‘Modern Quantum Chemistry’, (Istanbul Lectures), Part 1, Academic Press, New York, 1965, etc. W. England and M. S . Gordon, J. Amer. Chem. Soc., 1971, 93, 4649; 1972, 94, 5168. A. T. Amos, R. J. Crispin, and R. A. Smith, Theor. Chim. A d a , 1975, 39, 7. R. H. Pritchard and C. W. Kern, J. Arner. Chem. SOC.,1969,91, 1631. S. Rothenberg, J . Chem. Phys., 1969, 57, 3389; J. Amer. Chem. SOC.,1971, 93, 68. C. A. Coulson, Trans. Faraday SOC.,1942, 38, 433. A. T. Amos and R . J. Crispin, Chem. Phys. Letters, 1973, 22, 580; MoI. Phys., 1976, 31, 147.

The Calculationof Electric and Magnetic Properties

89

Tam 7 Typical calculated bond dipoles and quadrupoles. Dipole moments are given in Debyes, quadrupole moments in Buckinghams. The positive direction for dipole is defined to be from the heavy atom outward$,andquadrupole moments are calculated with respect to the mid-point of the bond Quadrupole

CH4 CH

Dipole

II

L

Comment

Re$

-2.08 -2.02 -1.86 -2.03

2.19 1.32 -

-1.09 -0.66 -

Frost model Accurate SCF HF Accurate SCF HF Semi-empirical

60

-2.17 -1.97 -1.95 0 0 0

2.21 1.33

-1.11 -0.66

Frost model Accurate SCF HF Semi-empirical Frost model Accurate SCF HF Accurate SCF HF

60 61 59 60 61

-

65a

62 59

C2H6

CH

cc

-

5.28 4.09 4.45

-2.64 -2.04 -2.23

6%

function, higher-order electromagnetic properties depend on the influence of external perturbations. In general these properties are defined by expanding the total energy as a Taylor series in the perturbation A,;

and we are chiefly interested in the third term. In the electrical case where A,= Fathe second-order energy becomes =3FaFpag,where a is the polarizability tensor. In the magnetic case Aa =Ha, H2) =+HaH~xaBHF and xHF is the high-frequency part of the diamagnetic susceptibility. A further possibility is that the nuclei may have a non-zero magnetic moment ~ C NThe . corresponding is + ~ N , H ~ Uwhere ~ B ~ uHF * is the high-frequency part of the diamagnetic shielding tensor. Polarizability will be dealt with first because it is the easiest of the three properties to calculate and has certainly received the most attention. Many of the conclusions also apply to x and u, which are dealt with in much less detail. In each section we have tried to pick out the most important methods and consider them in detail at the expense of the less useful methods. Thus, for example, although the variational technique of Karplus and Kolker is simpler than the other uncoupled Hartree-Fock perturbation methods, it is not a very useful technique for calculating polarizabilities. It is very useful for calculations of magnetic susceptibility, however, where many other techniques are inappropriate. Polarizability.-The hamiltonian for a molecule in a uniform electricfield is given by H = @ +@), where H(l)= - &F, F being the electric field vector. Developing a normal Rayleigh-Schrodinger perturbation scheme, the second-order contribution to the energy is

65

(a) R. M.Pitzer, J. Chem. Phys., 1967,46,4871; (b) W. E.Palke and W. N.Lipscomb, J. Arner. Chern. SOC.,1966,88,2384.

4*

90

Theoretical Chemistry

From equation (8), the second-order energy is +xF2 resorting to scalar notation. Cancelling F 2we find

Normally the electronic state of interest is the ground state, when

Here a,,is the static polarizability where the perturbation is time-independent. For time-dependent fields (e.g. optical fields) the perturbing hamiltonian is H ( ’ )= -,ua(F,t)Fa(f), and the problem is treated in a similar way using time-dependent perturbation theory. Direct calculation from equation (29) requires a knowledge of all excited-state wavefunctions. If these are known, equation (29) becomes a sum over all discrete states and an integration over continuum states. For systems of more than two electrons, excited-state wavefunctions are difficult to come by! Even less is known about the continuum states, but for the diamagnetic susceptibility their contribution is thought to be of the same order of magnitude as that of the discrete states. For other systems the direct use of equation (29) is clearly a non-starter. The infinitesum may be avoided by means of the closure method ;C 1 Y,,>
L*

A. Dalgarno, Adu. Phys., 1962, 11, 281. P. W. Langhoff, M. Karplus, and R. P. Hurst, Phys. Rec., 1965, A139, 1415. M. Karplus and H. J. Kolker, J. Chetn. Phys., 1963, 38, 1263; 39, 2011,

The Calculation of Electric and Magnetic Properties

91

method thus shobs poorly on all counts. Uncoupled calculations by method (d) have been performed by Yoshimine and Hurst for a large number of atoms and small molecules and by Langhoff and Hurst 7 0 for hyperpolarizabilities. Uncoupled methods also have the advantage that they involve only the ground-state wavefunction. More recently, Caves and Karplus 71 have used diagrammatic techniques to investigate Hartree-Fock perturbation theory. They developed a double perturbation expansion in the perturbing field and the difference between the true electron repulsion potential and the Hartree-Fock potential, V. This is compared with a solution of the coupled Hartree-Fock equations. In their interesting analysis they show that the CPHF equations include all terms first order in Vand some types of terms up to infinite order. They propose an alternative iteration procedure which sums an additional set of diagrams and thus should give results more accurate than the CPHF scheme. Calculations on H, and Be confirmed these conclusions. Subsequently Chang, Pu, and Das 72 carried out a many-body calculation of d~ for the ground state of lithium and compared the results with Hartree-Fock theory. Their Bruckner-Goldstone method is illustrative of many other calculations. The n-particle hamiltonian n

n

i= 1

is altered by replacing the two-body potentials v i j with single-particle ones, &, and Bruckner Goldstone perturbation theory develops using the difference, i.e. fi(0)

=

i=

f i + ;I,= 1

n

1

n

c fiji=lj>l

and

jji

= 2;

i=l

ci

(31)

The other principal feature is the use of second-quantization formalism: H(0)

ml)

=

x

=

zn

(32)

Enqn+l]n

W q 1 VInzHhp+q*+q?nqn-

~,~,m,n

c


PI 4

on which the diagrammatic methods depend. Hartree-Fock orbitals are used as the complete sets of states. Their results are excellent, and they emphasize that all the techniques discussed by Langhoff, Karplus, and Hurst neglect correlation effects, whereas these present no additional problem to the BG method. An alternative expression for the second-order energy is =

(YV(O)l~(l)l

‘PCU)

(33)

from which equation (27) can be obtained by expanding YJfll) in all the excited states of Y ( O ) . Clearly Y(l) depends on the size of the perturbation parameter A. A second general approach to the calculation of E(,) and hence a is to calculate P(l)directly for particular values of 3,. This involves solving the SCF equations in the presence of the perturbation, and the method is usually called the finite field technique (FFT). 69

70 ‘I1

72

M.Yoshimine and R. P. Hurst, Phys. Rev., 1964, A135, 612. P. W. Langhoff and R. P. Hurst, Phvs. Rev., 1965, A139, 1415. T. C. Caves and M. Karplus, J. Chem. Phys., 1969,50, 3649. E. S. Chang, R. T. Pu, and T. P. Das, Phys. Reo., 1968, A174, 16.

92

Theoretical Chemistry

This method was first proposed by Cohen and Roothaan 73Q for the calculation of atomic polarizabilities, and used soon afterwards by Schweig74within a semiempirical framework for large conjugated dye molecules. The majority of polarizability calculations use the FFT, perhaps primarily because it is easy to incorporate into standard SCF computer programs; in the presence of a perturbation A which is a sum of one-electron operators, the Hartree-Fock SCF hamiltonian ~ z F = ~ + G ( Rbecomes )~ h + d + G ( R ) , and the SCF equations are solved by any standard technique. Thus, all that is involved is to add an extra array into the Hartree-Fock hamiltonian matrix hF every iteration. The method can be extended to higher polarizabilities, and a review by Pople ef ~ 1 1 . ’gives ~ ~ a good introduction to the method, including a discussion of the computational errors likely to be involved. McWeeny, however, has developed a self-consistent perturbation theory which applies directly to the density matrix R without the intermediate calculation of the perturbed orbitals inherent in the coupled Hartree-Fock technique. His theory applies to a general oneelectron perturbation and yields directly, for a one-deteminant wavefunction of the closed-shell 75= or ‘restricted HF-open-shell’75b types, the expectation values of all observables of the perturbed system to any order of perturbation theory, whilst allowing h* to change so as to maintain self-consistency to the same order. The cost of such calculations is of the same order of magnitude as for FFT calculations, however, since McWeeny’s method involves the repeated construction of a matrix G(d’),where A’ is related to A above, and thus requires one pass over the twoelectron integral file per iteration, as does the FFT. This is the timeconsuming step for real The advantage of the self-consistent perturbation method is that the orders of perturbation theory are rigorously separated, so the calculation of (e.g.) higher polarizabilities is straightforward. With the relatively recent development of experimental techniques capable of exploring non-linear optical effects,these quantities are becoming more important. Ditchfield ef ~ 1 have . compared ~ ~ the finite field technique (FFT) with the conventional sum over states (SOS) formula for H2, which is possible since good excited states for this molecule are available. Real spin-independent perturbations, such as pa(F)Fa can mix the closed-shell ground state only with excited singlets. However, in FFT some double excitations are also introduced. Whether the results of the two methods are significantly different depends on the importance of these particular doubly excited states. Thus where CI is especially important in the wavefunction, the FFT will give a different result to the SOS. For H2, however, the FFT a is the closest to the accepted value. Spin-dependent perturbations act differently on electrons with different spins. In this case the singly excited states are triplets. Similar arguments apply concerning the importance of CI, and this time where the nuclear spin-spin coupling constant is calculated, the FFT results are very poor, while SOS is reasonable though only after CI has improved the wavefunction. The calculated value 73 74

75 76

( a ) H. D. Cohen and C. C. J. Roothaan, J . Chern. Phys., 1965,43, 534; (b) J. A. Pople, J. W. McIver, jun., and N. S. Ostlund, ibid., 1968, 49, 2961. A. Schweig, Chem. Phys. Letters, 1967, 1, 163. ( a ) R. McWeeny and G. Diercksen, J. Chem. Phys., 1966,44, 3544; (6) R. McWeeny and G. Diercksen, ibkf., 1969, 49* 4852. R. Ditchfield, N. S. Ostlund, J. N. Murrell, and M.A. Turpin, Mol. Phys., 1970, 18,433.

93

The Calculationof Electric and Magnetic Properties

using only the virtual states in the SOS formula underestimates J(H,H) as seriously as F R overestimates it. Calculations of Higher Polarizabilitiesand Semiempirical Calculations.4'Hare and Hunt 7 7 have calculated the first hyjxrpolarizabilitiesof some first-row diatomics by the uncoupled method. They find that a,and to a greater extent p, are very sensitive to the basis set. Table 8 shows the effect of basis set on a for LiCI. Gupta et 01.'~ have Tab& 8 The eflect of basis set on the polarizability of LiCI. All calculatwns are for internuclear separation of 3.825 a.u. (a.u. of polarizability= 1.4818 x cm3). The basis sets used are Dunning's basis sets Basis set details Li (s), CI (8, P)

1a.u.

al/a.u.

12.48

8.47

-467.013821

19.81

11.09

-467.034665

23.78

23.09

-467.042588

OL II

53 primitives, 30 basis functions

Li (s, P), c1 (s, P) 59 primitives, 36 basis functions Li (s, PI, C' (s, P, 4

71 primitives, 48 basis functions

Elhartree

reported a coupled Hartree-Fock calculation of the quadrupole polarizability of various open-shell ions and atoms; the paper is significant because it is one of the very few in the literature dealing with such topics. McLean and Yoshimine79have employed a particularly useful method for the a, p, y, A, B, and C tensors of equation (10). They solved the SCF equations in the presence of an axially symmetrical non-uniform field produced by point charges placed on the axis. They calculated the all-z components for several diatomics. The point charges are placed at a distance such that Y is negligible at the charges and the field they produce can then be regarded as external to the molecule. They chose values so that the expansions for pz and Tzzcould be truncated down to terms involving only these tensors. The dipole and quadrupole moments wefe then computed for each charge value and the set of simultaneous equations solved. Y~~~~and Bzz:zzare even more sensitive to the basis set than the other quantities, and the value of computing these elements seems doubtful. McLean and Yoshimine80have also computed these tensor elements for a number of linear molecules in their usual extensive STO basis sets. The method is also explained in detail elsewhere.81 Schweig has extended his highly empirical calculations to all components of p and y for many large organic molecules.82He has introduced the perturbation term 83 and also calculates the non-zero elements of the A, B, and C tensors for some aromatic rings and the amine-substituted compounds. Schweig has also related a,p, and J. M. O'Hare and R. P. Hunt, J. Chem. Phys., 1967,46,2356. A. Gupta, H. P. Roy, and P. K. Mukhcjae, Intermt, J , Quantum Chem., 1975, 9, 1. 79 A. D. McLean and M. Yoshimine, J. Chem. Phys., 1 9 6 7 , 4 , 3682. 80 Technical Note 438, Amer. Nat. Bur. Standards, 1967. *I A. D. McLean and M. Yoshimine, I.B.M. J . Res. Deaelop., 1968, 12. 206. 0% A. Schweig, Chem. Phys. Letters, 1967, 1, 195. 89 A. Schweig, Mol. Phys., 1968, 14, 533. 77 '8

94

Theoretical Chemistry

y to solvent effects on electronic spectra;84 p and y are found to have a negligible effect. In a more conventional CND0/2 calculation, Hush and Williams * jcalculated the parallel and perpendicular components of cz for a series of first-row diatomics, and also calculated !z for all first-row atoms using a similar formula. In view of the errors in their formulae as published, it is not clear whether their results are correct or not, although our own experience is that CND0/2 gives in general very low values for a. Hush and Williams8G have also extended their CND0/2 calculations to hyperpolarizabilities ( p-elements only) for several linear molecules together with HzO, NH3, and CH,. The results are unimpressive where comparison with experiment is possible. Meyer and Schweig8 7 have published an extensive comparison of MINDO/ 1, MIND0/2, and CND0/2 values of a for a large number of molecules and they also report a comparison with a6 irritio results. They find that MINDO/l is the method to be preferred of the semi-empirical methods. Perhaps not surprisingly, it is found that adding polarization functions ( p on H ) within the CINDO scheme gives better agreement with experiment,88although the cynic could claim that adding more parameters is bound to give better answers! Perhaps more significantly, it is interesting to find that, if the free-atom values are subtracted from CNDO-calculated a values, then this quantity compares well with the corresponding experimental value; it appears that the 'interaction' term is fairly well calculated but the atomic contributions to a are poorly calculated by CNDO. A few seconds thought will convince the reader that this conclusion is r e a s ~ n a b l e . ~ ~ Various semi-empirical methods hake been compared for all properties in an important review by Klopman and o'Lear~,~O whilst Adams et 0 1 . ~have ~ compared the finite field, the variation, and the second-order infinite sum methods for the calculation of a in DNA bases. They find that the variation-perturbation method gives the most reliable results, but as their calculations were at the iterative extended Hiickel level there is no guarantee as to the generality of their conclusions. Magnetic Susceptibility.-In a magnetic field H a magnetic material acquires magnetization M given by M = X-H (35)

because the electrons move in a way that can be considered as an intramolecular current. The magnetic moment of a single molecule, equivalent to these currents, is written PM = X * H (36) where x is the magnetic susceptibility of the molecule, the quantity of interest to us. In general there are three contributions to the magnetic susceptibility tensor:

x 8'

=

xp

+

x=

+

XHP

(37)

A. Schweig, Mol. Phys., 1968, 15, 1.

N . S. Hush and M. L. Williams, Chem. Phys. Letters, 1967, 5, 507. N. S. Hush and M. L. Williams, Theor. Chirn. Actu, 1972, 25, 346. 87 H. Meyer and A. Schweig, Theor. Chinr. Acta, 1973, 29, 375. a* See, for example, J. J. C. Teixeira-Dias and P. J. Saare, J.C.S. Faraday 11, 1975, 71, 906. 80 N. S. Hush and M. L. Williams, Chern. Phys. Letters, 1970, 6, 163. 90 G. Klopman and B. O'Leary, Fortschr. Chem. Forsch., 1970, 15, Heft 4. 91 S. Adams, S. Nir, and R. Rein, Internat. J . Quantum Chern., 1975, 9, 701.

85 *6

95

The Calcirlation of Electric and A4agiietic Properties

The first component x p is the paramagnetic susceptibility, which is due to the coupling of unpaired electrons with the magnetic field and is therefore absent for closed-shell molecules. When present it is much larger than the other two terms and is thus the predominant effect. xp is dependent on temperature and is particularly important in transition-metal chemistry. It will not be discussed further here. The second and third terms are always present. They are sometimes called the Langevin and High Frequency parts of the magnetic susceptibility, respectively, sometimes the diamagnetic and temperature-independent paramagnetic susceptibilities. The second term can also be called the Van Vleck paramagnetism. We will use the Langevin/Nigh Frequency notation throughout. Thus for closed-shell molecules

x

= XL

+

XHF

(38)

and it is this susceptibility that occurs in the second-order energy equation E ( 2 ) = $H. x * H. The x tensor is symmetrical in all indices and for a linear molecule there are only two unique components, x 11 and XI. A full quantum-mechanical treatment introduces the magnetic field H into the hamiltonian via the vector potential A. Work usually proceeds in the Coulomb gauge where V .A =O. Expanding the total energy and the magnetic dipole in powers of H and comparing terms leads to expressions for x in atomic units

Since /lm is related to the orbital angular momentum vector L the second term is often written in terms of this operator. The first term is the Langevin term; it is the expectation value of a one-electron operator C ( i t 2 + $ i z ) , and can therefore be obtained directly from the ground-state wavefunction. It has formed the basis for a quite successful additivity scheme, Pascal’s rules, which work well except for the conjugated hydrocarbons where non-additivity is ascribed to ring currents. Nevertheless, as it depends on the square of the electron co-ordinates, xL will be sensitive to the basis set used in variational calculations. The second high-frequency term involves a sum over all discrete states and an integration over the continuum states; the difficulties involved have been outlined before. Little is known about the continuum states, but what few calculations there are for simple systems 92 suggest that they may be at least as important as the discrete states. For this reason early calculations were done in the closure approximation, notably by Van Vleck in the 1930’s. The difficulties of calculating xHF have been reviewed by W e l t n e ~ .Experimentally ~~ xHF may be obtained from rotational magnetic moments. For linear molecules these can be obtained from molecular-beam experiments, which also measure the anisotropy 11- xI directly. The anisotropies may also be derived from crystal data, the Cotton-Mouton effect and, recently, Zeeman microwave studies principally by Flygare et al.94 One important simplification occurs for atoms: xHFis zero if the nucleus is taken 92

93 94

L. C. Snyder and R. G . Pam, J . Chern. Phys., 1961, 31, 837; H. F. Hameka, ibid., 1964, 40, 3 127. W. Weltner, J. Clietn. Phys., 1958, .28, 477. See, for example, W. H. Flygare and R. C. Benson, Mul. Phys., 1971, 20, 225.

96

Theoretical Chemistry

as the origin of the vector potential, beccuse of !he spherical symmetry. In this case the eigenstates in the summation C < O ~ & ~ k ) ( k ~are ~ z orthogonal ~O> and so the only k#O

term which contributes is when k=O, but this is excluded from the sum. In this case x is a simple one-electron property. There is one further interesting point before turning to specific methods for x . xL is we would negative and usually larger than the positive xHF. However, if have a paramagnetic closed shell. This was first predicted for BH by Stevens and L i p s c ~ r n b but , ~ ~has yet to be checked experimentally. Gauge Inuariance. An additional complication arises in the case of magnetic properties not present in the electrical case. Because V . B = O , B can be written B= V x A. However, there are an infinite number of ways of defining A since A' = A

+

rd,

(40) where 6 is any twice-differentiable scalar function, is also a solution. A choice of A is called a 'gauge', and equation (40)is a 'gauge transformation'. It is clear that a change of origin R +R' is in general a gauge transformation because O$(R)# V$(R'). Thus A is origin-dependent. Physical observables should not depend on the choice of origin; they should be gauge-itzoariant. For a calculation using a complete basis set calculated magnetic constants would not depend on the choice or origin of the vector potential A, but use of any finite basis set gives calculated properties that are not truly gauge-invariant and the extent to which they are so is an indication of the quality of such calculations. In practice, calculations of zHFare based on the uncoupled Hartree-Fock, the finite field, and the self-consistent perturbation methods. Some workers use gaugeinvariant atomic orbitals (GIAOs). A full review of the gauge invariance of SCF wavefunctions has been given by E p ~ t e i n . ~ ~ Karplus and Kolker's 97 simple uncoupled method uses a variational technique which minimizes the second-order energy in the field. An effective hamiltonian is introduced of which the Hartree-Fock wavefunction from an ordinary SCF calculation is assumed to be an eigenfunction. The functional then involves matrix elements over the ground state only. There is some self-consistency error because the groundstate wavefunction is not an eigenfunction in the presence of the perturbing field, but it was estimated that this will lead to only a small error in the calculations on diatomics with which they are concerned. The authors also calculated x for LiH at several separations and estimated vibrational corrections to to be small. Their method gives reasonable agreement for the average susceptibilities but rather poor anisotropies. They noted that xHFis very sensitive to the choice of basis set in the ground-state function. This method has recently been applied at CNDOJ2level by SadIej.s8 Stevens and Lipscomb's method, for diatomics, avoids the self-consistency error by writing the first-order perturbed wavefunction in terms of the n-orbitals, since the ground-state wavefunction involves only +orbitals. Thus for linear molecules the 95 s6

D7

R. M. Stevens and W. N. Lipscomb. J. Chem. Phys., 1965, 42, 3666. S. T.Epstein, J . Chem. Phys., 1965, 42, 2897. M. Karplus and H.J. Kolker. J- Chem. Phys.. 1962, 36. 2275. A. Sadlej, Mof. Phys., 1971, 20, 593.

The Calcultion of Electric and Magnetic Properties

97

first-order perturbed wavefunction, which determines x, is written in terms of 2p, and 3pn orbitals which are then highly optimized. Their calculated polarizabilities are in good agreement with experiment. Results for x are also generally good except for HF, which tbey ascribe to the fact that the experimental value for the molecule was determined in the liquid state. It is worth noting though that both Karplus and Kolker, and Hameka, obtained a reasonable value for HF. Moccia et al.@*-105 have calculated magnetic susceptibilities for some small polyatomic molecules using STO basis sets that are large and contain polarization functions. It seems that a very good basis set is needed for zHF,and the decrease in accuracy and gauge invariance as molecular complexity increases bears witness to this. Ditchfield lo6,lo' has reported calculations at the ST0/4-31 level, but concluded that the use of SCF perturbation theory with such basis sets is inadequate for calculating absolute and relative magnetic susceptibilities. The main source of error is in the calculation of xHF, but there is a marked improvement on adding polarization functions. Table 9 shows representative results at the two levels of sophistication. Table 9 Magnetic susceptibility results calculated using SCF pertubation theory. The units are loA6J mol-l throughout. The origin of the vector potential was taken either as the centre of mass or at the starred atom Calculated

Experimental

Molecuk Large STO spd basis set *LiH LiH* *BH BH* H2O NH3 CH3F

x=

XHF

XHF

-20.47 -23.81 -18.02 -37.78 -15.59 -20.47 -60.02

12.83 15.85 36.77 56.30 1.21 3.83 32.64

12.71 16.05 1.504 3.89 38.05

ST0/4-3 1G basis set H2O NH3 CH 3F

-15.0 -20.7 -60.30

0.9 1.8 16.6

1.504 3.89 38.05

Both of the previous methods calculate xL and zHPseparately. These are of opposite sign and numerically of the same order of magnitude, and so errors in both could make the total error in the sum large. It is therefore desirable to calculate the G. P. Arrighini, M. Maestro, and R. Moccia, J . Chem. Phys., 1968, 49, 882. G. P. Arrighini, C. Guidotti, C. Maestro, R. Moccia. and 0. Salvetti, J. Chem. Phys., 1968,49, 2224. l0L G. P. Arrighini, C. Guidotti, M. Maestro, R. Moccia, and 0. Salvetti, J . Chern. Phys., 1969.51, 480. lo2 G. P. Arrighini, P. Giovanni, C. Guidotti, and 0. Salvetti, J. Chem. Phys., 1970, 52, 1037. G. P. Arrighini, M. Maestro, and R. Moccia, J . Chem. Phys., 1970, 52, 6411. lo4 G. P. Arrighini, M. Maestro, and R. Moccia, Chem. Phys. Letters, 1970, 7 , 351. 105 G. P. Arrighini, J. Tomasi, and C. Petrongolo, Theor. Chim. Acra, 1970, 18, 341. 106 R. Ditchfield, D. P. Miller, and J. A. Pople, Chem. Phys. Letrers, 1970, 6, 573. 107 R. Ditchfield, in 'Molecular Structure and Properties', ed. G. Allen, MTP International Review of Science, Physical Chemistry Series 1, Volume 2, Buttenvorths, London, and University Park Press, Baltimore, 1972. g9

Theoretical Chcrnistry

98

total magnetic susceptibility, x, directly. The use of GIAOs should achieve this as well as giving gauge invariance even in a limited basis set. Basically the AOs are multiplied by a complex gauge-dependent factor, but as always there is a serious drawback to an attractive scheme, this time in the difficulty of the integral evaluation. Most calculations so far have been for the magnetic shielding constant where there are fewer terms. One exception lo* starts with the field-free wavefunction, the AOs being multiplied by a complex factor and a perturbation expression being used for the susceptibility. The final wavefunction is not self consistent and the results are poor when compared with those of Karplus and Kolker, and of Stevens and Lipscomb. A number of semi-empirical calculations using GIAOs have also appeared in the literature, mainly from Pople's g r 0 ~ p . l ~ ~ Finally the interested reader is referred to the comprehensive article by Ditchfield lo7 for further details regarding the measurement and the calculation, at all levels of sophistication, of magnetic susceptibilities. Magnetic Shielding.-A magnetic field H induces intramolecular currents which depend on the susceptibility. These induced currents produce an induced field H i n d which modifies the external field. The effective local field at a nucleus A is therefore H e f f A= H iHindA

(411

The induced field is related to the external field by the equation HindA

=

-QA*H

(42)

where the magnetic shielding tensor elements depend on the electronic environment of the nucleus A. In high-resolution n.1n.r. experiments the molecules of the sample are rotating rapidly; the components of the shielding tensor are molecule-fixed so that one measures an effective shielding parameter OA given by C A = $ ( V A , ~ ~ + C T A , ~ OA+) ~ + and so we recover the familiar equation Herrh = H ( 1 - O A )

(43)

As in the m e of susceptibilities, there can be in general three contributions to the

magnetic shielding a: the paramagnetic term again dominates when present, but for a closed-shell singlet-state molecule there are just two contributions a = aL

+

OHF

(44)

where, in atomic units,

is the Lamb termlog and aH* the high-frequency part. For atoms aHFis zero and aL=5(azzL+guyL+ozzL)is just the average value of the operatorx l / r t , so calculations are particularly simple, All the remarks made previously about one-electron properties apply to the calculation of crL. Calculations of have been reported by Karplus and Kolkerllo and by Stevens and Lipscomb,'ll the latter being in better agreement with experiment, although a marked gauge dependence is still evident H. F. Hamcka, Physica, 1970, 53, 1265, and references therein. W. E. Lamb, Phys. Rev., 1941, 60,817. no M. Karplus and H.J. Kolker, J . Chem. Phys., 1964, 41, 1259. R. M. Stevens and W . N. Lipscomb, J . Chern. Phys., 1964, 40,2238, lo*

lo9

99

The Calculation of Electric and Magnetic Properties

even when using such high-quality basis sets. Calculated values of the spin-rotation constants, however, are in good agreement with experiment for LiH and for HF. Ditchfield et aZ.loS~ 112 have reported values of CT calculated at the ST0/4-31G level using SCF perturbation theory: the results are in moderate agreement with experiment but some important trends across series of molecules are correctly reproduced. They conclude that their method will satisfactorily calculate the chemical shift of a particular nucleus in a group of molecules of comparable size provided that the position of that nucleus relative to the origin of the vector potential does not markedly change.lo7GIAOs have also been used successfully to calculate chemical shifts for both carbon and hydrogen, Representative calculations are shown in Table 10, and again the interested reader is referred to Ditchfield's review.lo7 Table 10 Representatiue calculations of 13C magnetic shielding constants relative to 13Cin CH, calculated using SCFperturbation theory and CIAO A alp.p.m .

Molecule CHJF C2Hz CaH4 CH2O

112

Calculated -74.5 -90.5 -152.8 -250.9

R. Ditchfield, D. P. Miller, and J. A. Pople, J . Chem. Phys.,

Obseroed -77.5

-76.0 - 126.0

- 197.0

1971;54, 4861.

4 The Use of Pseudopotentials in Molecular Calculations BY R. N. OIXON AND 1. L. ROBERTSON

1 Introduction During the past few years there has been a rapid increase in the range of molecules for which the methods of quantum chemistry have proved useful. Improvements in computer power and developments in theory have both contributed to these advances, so that ab initiu methods are now routine for molecules which could only have been the subjects of semi-empirical calculations a few years ago. One area of this extension has been to molecules containing atoms of high atomic number, and it is this area which is the subject of this Report. Empirically the electrons in such molecules behave as if they belonged to two distinct groups, the ‘core’ electrons which remain closely associated with the various nuclei and the ‘valence’ electrons which are responsible for the intrinsically molecular properties. These intuitively appealing concepts are difficult to define rigorously since electrons cannot be precisely localized. However, the orbital model allows us to distinguish between those atomic orbitals which are almost invariant to molecular formation, i.e. the ‘core’ orbitals, and those that undergo larger spatial and energetic reorganization, the ‘valence’ orbitals. The theoretical difficulty of making this separation derives from the indistinguishability of electrons and the requirement that the total wavefunction be antisymmetric with respect to permutations of the electronic co-ordinates. One approach has been to abandon a full quantum mechanical description in favour of a simplified model hamiltonian which can be conveniently parameterized in terms of experimental quantities. This is the rationale behind Hiickel theory, CNDO, and other more sophisticated methods such as MINDO. These techniques have been well documented and reviewed elsewhere (Dewar,l Pariser, Parr, and Pople,2 Murrell and Hargett,:’ etc.) and will not be pursued further here. From the computational point of view any treatment which reduces the number of orbitals which are explicitly taken into account is very attractive. In the normal LCAO-MO method the number of integrals to be calculated, stored, and read for each SCF cycle is roughly proportional to the fourth power of the number of basis functions, and for a CI calculation the integral transformation process depends on the number of basis functions to the fifth power. The basis set required for a good

M. J. S. Dewar, D. H. Lo, D. B. Patterson, N. Trinajstic, and G. E. Peterson,

C h m . Comm.,

1971. 238.

J. A. Pople, Trans. Faraduy Soc., 1953.49. 1375. J. N. Murrell and A. J. Hargett, ‘Semi-empirical SCF-MO Theory of Molecules,’ Wiley, London and New York, 1972.

100

The Use of Pseudopotentials in Molecular Calculations

101

representation of the valence orbitals can usually be much smaller than that for an all-electron calculation and the time required is often 10-100 times shorter. In addition, since only the contribution to the energy from the valence orbitals need be calculated, there is a significant increase in numerical precision. (For instance, the total energy of the Fe atom is 1262.40 a.u. if all the electrons are considered, whereas the ‘valence only’ calculation gives an energy of 21.1881 a . ~ . ~ ) The scheme for achieving the core-valence separation which we wish to discuss in this Report is embodied in the idea of a ‘pseudopotential’. The hamiltonian for the electronic part of the wavefunction can be symbolically expressed within the Born-Oppenheimer (‘clamped nucleus’) approximation as H e = Te

+

Pn.e

+ Ve.e

(1)

where e and n refer to electrons and nuclei respectively. However, if we wish to consider only the valence electrons an extra term must be introduced to account for the influence of the core orbitals on the valence space. Symbolically we write A:ff(vaIence only) =

Te + Pn.e +

Pxseudo

+

Pe. e

(2)

The first explanation and use of such a pseudopotential is due to Hellman5 (1935) who used it in atomic calculations.6 More recently the pseudopotential concept was reformulated by Phillips and Weinman’ who were interested in its application to the solid state.&lO Research in both solid- and liquid-state physics with pseudopotentials was reviewed by Ziman,ll and work in the fields of atomic spectroscopy and scattering has been discussed by Bard~1ey.l~ For an earlier review on applications to the molecular environment the reader is referred to Weeks et all3 In this article we shall concentrate on molecular calculations, specifically those of an ab initio nature. Our objective in Section 2 has been to outline the theoretical origins of the pseudopotential approximation, and in Section 3 we have described some of the techniques which have been used in actual calculations. Section 4 attempts to present results from a representative sample of pseudopotential calculations, and our emphasis has been to concentrate on particular molecules which have been the subjects of investigation by the various approaches, rather than to catalogue every available calculation. Finally, in Section 5, we have drawn some conclusions on the relative merits of the different methods and implementations of pseudopotentials. Some of the possible future developments are outlined in the context of the likely progress in quantum chemistry. 2 Core-Valence Separability and the Formal Derivation of

Pseudohamiltonians Evidence from both experimental and theoretical studies on numerous molecular I. L. Robertson, unpublished, Ph.D, thesis, University of Bristol, 1978. H. Hellman, Actu Physicochem. U.R.S.S.,1935, 1, 913; J. Chem. Phys., 1935, 3, 61. 6 H. Hellman, J. Chem. Phys., 1936,4,234; H. Hellman and W. Kassatotschin, Actu Physicochcm. U.R.S.S., 1936,5,23. J. C . Phillips and L. Kleinman, Phys. Rev., 1959,114,287. J. C . Phillips and L. Kleinman, Phys. Rev., 1959, 116, 880. J. C. Phillips and L. Kleinman, Phys. Reu., 1960,117, 460. J. C. Phillips and L. Kleinman, Phys. Rev., 1960,118, 1153. 11 J. M. Ziman, Solid State Phys,, 1971, 24, 1. l2 J. N. Bardsley, Case Studies, Atom. Phys., 1974, 4, 299. l a J. D. Weeks, A. Hazi, and S. A. Rice, Adu. Chem. Phys., 1969,16283. 4

5

Theoretical Chemistry

102

systems strongly supports the 'core-valence' concept; it is the electrons in valence orbitals which are responsible for chemical bonding, whereas the core orbitals are essentially invariant to their electronic environment. However, the indistinguishability of the electrons complicates their formal separation into core and valence sets. Fock, Vesselov, and PetrashenI4 were the first to formulate the approach for treating a subset of the electrons. They showed that if the total wavefunction were expressed as (3)

= AI(@coreQjvslrnre)

where !DC and QV are many-electron functions and 2 is the antisymmetrizing operator, then it is possible for the total energy to be written in terms of the energy for the core and valence sets: Etotai

= Gore

+

(4)

fivI~v>t

However, this is only possible if the core and valence functions are strongly orthogonal, i.e.

\ %(l, 2 . . .

p

. . ~ t ~ ) @ ~ (2',l ' ., . . . rr\')dr/, jc

= 0

for all

14

..

~ ( 1 nr) ( 5 )

The resulting valence hamiltonian fiv includes the influence of the core on the valence electrons. In the simple case where the core function GCconsists of a single (6)

operators :

where Ppv is the permutation (exchange) operator. For a single atom Hvis then :

Unfortunately the necessity of satisfying equation ( 5 ) at all stages in the solution of the valence-electron Schrodinger equation has proved a major obstacle to the implementation of this approach. Much has been written about the necessary and sufficient conditions for separating the energy.l5-lSHowever, for computational convenience one requires a method which eliminates the need to compute the complete set of two-electron integrals over the total (core plus valence) basis set. Any method that requires the explicit orthogonality of the core and valence spaces still requires these integrals. (For an approximate method that uses partial explicit orthogonalization but does not compute ail the integrals, see Horn and M~rre1l.l~) l4

l5 l6 1'

18 lQ

V. Fock, M. Vesselov, and M. Petrashen, Zhur. eksp. tmr. Fiz., 1940, 10, 723. P. 0. Lowdin, Phjbs. Rev., 1969, A357, 139. P. G. Lykos and R. G. Parr, J. Cftem.Phys., 1956, 24, 1166. Y. O h r n and R. McWeeny, Arkic. Fvs., 1966, 31, 461. S. Huzinaga and A. A. Cantu, J. Chem. Ph33s., 1971, 55, 5543. M. Horn and J. N. Murrell, J.C.S. Forodav I , 1974, 70,769.

The Use of Pseudopotentiuls in Moleculur Culculutions

103

Explicit orthogonalityconstraints can be removed by transformingthe hamiltonian so that it only acts on a specific subspace (e.g. the valence space) of the all-electron space. This can be done formally by the use of the projection operator method (see Huzinaga and CantulBor Kahn, Baybutt, and Truhlar20). If the core function is written as in equation (6) a projection operator may be defined for each valence electron p : core

(10)

lMP))<4&)l

=

and the transformed valence hamiltonian is

#Xpv

=

[l-Pc(l)l[~-Pc(2)1

+

Ev[

.. . [i -Pc(n,)lfiv[l -PC(1)] . . . [I -Pc(nv)l nv

? Pc(/c) - X

p=l

111

v

P c ( P ) ~ c ( v+) +

I

(1 1)

where the second term arises because of the renormalization of the energy expression. Clearly this equation is of little practical use, since the hamiltonian now no longer ever reduces to a sum of just one- and two-electron terms but containscomplex many-electron operators. (For further developments of this approach see Freed or Westhaus, Bradford, and Hall. 21) As an approximation to equation (11) it is useful to consider a valence hamiltonian for which only the one-electron operators are transformed and the two-electron interactions remain unchanged. The hamiltonian now simplifies to :

which as Kahn et ~

1point . out ~ ~can be expressed in the general form

(This has enabled these authors to derive a computational means for obtaining the effective core-valence interaction ( D o r e ) as a local one-electron operator.) Now consider the case where both the core and valence spaces can be expressed as single determinants with each orbital doubly occupied :

i.e. the Hartree-Fock case of two closed-shell singlet systems. Following the formu-

lation due to Huzinaga22we shall show that expressions may be derived which are similar to those of Phillips and Kleinman.’ The strong orthogonality correlation ( 5 ) can now be written

<+fil+v) 20 21 2)

=

0

foralli = 1 . . . +nv, k = 1. . . + n c

(15)

L. R. Kahn, P. Baybutt, and D. Trular, J . Chem. Phvs., 1976, 65, 3826. K. F. Freed, Chem. Phys. Letters, 1974, 29, 143; P. Westhaus, E. G. Bradford, and D. Hall, J. Chein. Phys., 1975, 62, 1607; ibid., 1975, 63, 5416; ibid., 1976, 64, 4276. S. Huzinaga, J. Chem. Phys., 1969, 51, 3971.

104

Theoretical Chemistry

and the core and valence orbitals themselves can be chosen to form orthonormal sets:

(+;I+:> (+TI+;)

k, I = 1 . . . 3tIc i , j = I . . +nv

= 6kr = Sij

.

The expression for the energy of the valence electrons is then

where *I2

q0t.I.

(2JL-

=

KkF)

(18)

and J f j ,J k i , Ktj, and K M are the usual matrix elements of the coulomb and exchange operators defined as in equations (7) and (8). The eigenvalue problem can be written as

where the last term arises to allow for the core-valence orthogonality constraint (15) via Lagrange multipliers AM. The eigenvalue matrix ~ j can i be diagonalized by a unitary transformation amongst the set {41} and AM be expressed in terms of projection operators as orbital 'coupling operators' 2 2 to give

With the definition of equation (10) this becomes

This further simplifies to give an effective Fock operator,

on condition that

[E, Pel

= 0

The condition for (23)to be true is that the core orbitals 4; are eigenfunctions of the valence Fock operator, i.e.

(Note that this is just the same condition under which the generalized pseudopotential of Weeks, Hazi, and Rice13 reduced to the Phillips-Kleinman form.') In this case the operator Pis equivalent to the Hartree-Fock operator for the total wavefunction a,+, @c+v

= 2(@c@v)

(25)

and the effective, valenceonly, operator perf, equation (22), can be used without any supplementary core-valence orthogonality constraints. (Naturally the valence orbitals are still required to be orthogonalized to one another.)

The Use of Pseudopotentials in Molecular Calculations

105

It is common for valence-only calculations to use a form of effective hamiltonian which is based on the eigenfunctions for atoms or ions with only one vaZence elect r ~ nThis . ~ is ~ equivalent to choosing a set of core orbitals 4’: which satisfy

[rz + rather than equation (24) which ensures that equation (23) holds. Although this choice of orbitals q4’; possibly simplifies the parameterization of the core-valence interaction (Section 3) it has been found greatly to decrease the accuracy of calculations for systems with several valence electrons, (e.g. Fe based on Fe7+).24 Molecules with Several Atomic Cores.-From the above discussion it is seen that, in principle, the effective hamiltonian for atomic valence electrons is dependent on the valence state of the atom, this dependence arising from the valence contribution to the all-electron Fock operator I? In practice this dependence is very weak unless the atom is multiply ionized, and can usually be safely neglected, so that a single effective hamiltonian can suffice for many valence states. However, for a molecular system in which there is more than one core region additional approximations must be introduced to maintain a simple form of the effective hamiltonian. For two atomic cores defined in terms of orbital sets {$;} and {$:I and a valence set (4’) the equation equivalent to (21) is

Now the only choice for and {#B} which will exactly reduce this equation to the form of equation (22) in deriving peffare those orbitals which would be found by an all-electron calculation on the molecule, i.e.

Clearly any attempt to base peffon such molecularly defined cores defeats the aims of pseudopotential theory. However, the approximate invariance of atomic cores to molecule formation implies that, of the total of ~ N electrons A which could be associated with the centre A in an atomic calculation, are ‘core’ electrons and $zi will contribute to the molecular valence set. Thus we can define a one-centred Fock operator:

E&)

-fnA

= hA(d

+ow+ c

[2jA’(IU)-I?*’(p)I

(29)

2

and equivalently for centre B. PAdiffers from P in that the interactions between the electron p and both the core and valence shells of atom B have been dropped.18 However, the total potential of atom B in the vicinity of the core of atom A will be almost zero if B is a neutral atom, and essentially constant even if B is charged. Thus reasonable choices for the core sets { 4f) and { @} may be derived by using the 23

L. Szasz and G. McGinn, J. Chem. Phys., 1965,42, 2363; ibid., 1966,45,2898; ibid., 1967,47, 3495; ibid., 1968, 48, 2997.

24

C. F. Melius, B. D. Olafsen, and W. A. Goddard, Chem. Phys. Letters, 1974, 28,457.

Theoretical Chemistry

106

separate 'atomic' equations: PA(#;

= &$;

(30)

$4; The Fock operator for the valence orbitals may therefore be approximated by a simple generalization of equation (22), =

PB&

where #

= ji

+

&ore

+ h2t v (2.Q-IZ~)

nv = nk

i

+

n;3

(32)

and &re

=

6:;-1 Gg

=

%LA

cE

(ut-gij +

fnB

cI

(2jf-Q

(33)

This Fock operator has been derived starting from the assumption of a HartreeFock valence function GV[equation (14)]. However, it can be seen that the coupling of the valence electrons has little influence on the core electrons, so that the manyelectron valence hamiltonian may be similarly approximated as

The first and last terms in equation (34) consist of the valence-electron components of the all-electron hamiltonian of equation (l), and the remaining terms constitute the pseudopotential represented symbolically in equation (2). Further, if we assume that the interaction of the two cores A and B can be approximated by a point charge potential (see Kahn et al.24for errors in this assumption),

then the total energy of the system is given by E =

(@v]Aeff]@v> + .?'R! . ~ B+

EA

+

€B

(36)

Before the effective hamiltonian can be used in actual calculations some means must be found for expressing the terms &ore [equation (33)] and the projection operator terms in equations (31) or (34) in a form which is convenient for computing matrix elements; this is the subject of parameterization, which is dealt with in Section 3. Two other formal problems remain at this level. Firstly there is the need to modify equation (29) and, as a result, equations (31) and (34) if the atomic calculations on the separate atoms are of the 'open shell' kind as is usually the case. In order not to bias the later molecular calculation the core operators and projection terms can be derived for some average of all the possible open-she11 configurations,26 although care should be exercised in the choice of the hamiltonian for which the 25

R. N. Dixon, G. G. Balint-Kurti, and P. W. Taker, Mof. Phys., 1976,32, 1651,

The Use of Pseudopotentials in Molecular Calculations

107

orbitals are canonicalized since there is no unique definition (see below). The problem of uniqueness also arises when solving for the eigenfunctions of the effective valence operator. The eigenvalues of the all-electron valence orbitals can be shown to be lower bounds to the eigenvalues of a valence-only calculation.26However, the valence eigenfunctions are not ~ n i q u e Any . ~ ~valence ~ ~ ~ orbital found from a valence-only calculation (in future we refer to these as pseudo-orbitals) can be expressed as the corresponding all-electron orbital 4: plus an arbitrary linear combination of core orbitals 4 ,

The action of the projection operator ( - & ~ + A ) l $ ~ > < $is~to I raise the eigenvalue of the core orbital 4; to the value A. A new lower bound for the eigenvalue for the pseudo-orbital xl can be shown to be the lower of A and er. In practice the core eigenvalues are usually shifted so as to be degenerate with the lowest valence eigenvalues of the same symmetry. The coefficients at in equation (37) can now assume values which allow the pseudo-orbital xp to be nodeless and thus capable of representation by a smaller basis set expansion. Non-uniqueness of the Pseudopotential.-The Phillips-Kleinman pseudopotential contains operators C(sv- &h)l4k)<4kl which shifts the eigenvalues of the core funck

tions { 4;) so that they are degenerate with the lowest valence eigenfunction of the same symmetry. However, if we consider two functions x and x’ which are linear combinations of the same core and valence eigenfunctions,

x

= +v

+ Cak4k k

and

X’ = b v -k C b k $ k k

(38)

then, since

[l-pc]x = 4v and [l-pc]x’ = $v (39) both x and x’ are equally legitimate solutions of the valence electron problem: This arbitrariness was noticed by Cohen and Heine and by Schlosser 28 who suggested that one way of resolving the difficulty would be to place extra conditions on the pseudopotential operator. In particular they proposed that the eigenfunctions Xn2 should be required to have the minimum possible radial kinetic energy:

soa[i(?)I2

dr = minimum

This is intuitively appealing since the resulting eigenfunctions would also tend to be ‘smoother’ since the kinetic energy is proportional to the curvature of the wavefunction. The non-uniqueness problem is more apparent in the case where the pseudopotential is to be defined by localizing the non-local H a r t r s F o c k (HF) potential due to 26

27 2*

P. Coffey, C. S. Ewig, and J. R. Van Wazer, J. Amer. Chem. SOC.,1975,97, 1656; C. S. Ewig and J. R. Van Wazer, J. Chem. Phys., 1975, 63, 4035; C. S. Ewig, P. Coffey, and J. R. Van Wazer, Znorg. Chem., 1975, 14, 1848. V. Heine, Solid State Phys., 1970,24, 1 ; V. Heine and M. H. Cohen, ibid., p. 37. M. H. Cohen and V. Heine, Phys. Reo., 1961,122, 1821; H. Schlosser, J. Chem. Phys., 1970, 53, 4035.

5

Theoretical Chemistry

108

the core electrons; for instance the equation for a single valence electron in an orbital q5i can be written [ - f ~ 2

-

+

local] di =

Eidi

(42)

or equivalently:

where :

If the functions #i have nodes at r o the resulting potential has poles of order l / ( r- ro). One way of avoiding this 29 is to use the nodeless orbitals arising from the G1 method (Goddard 30) to define UlOc81. Alternatively smooth valence orbitals can be constructed by adding linear combinations of core functions to the valence orbitals. The resulting ‘coreless HF orbitals’, +fHF, are then used to construct the potential. However, there is in general no unique way of generating and this pseudopotential is also not unique:

#FHF

In order to ensure that the orbitals are ‘smooth’ and that there is a prescription for choosing them Kahn and Goddard20 have minimized, with respect to the coefficients bnz, the functional

subject to

thereby minimizing a weighted mean of the projection on to the core functions and of the kinetic energy. In the Reporters’ pseudopotential calculations31 a different approach is taken by replacing the Phillips-Kleinman term with an operator, -

c &k I d k ) < + k I - 4

I+V><+V

I - I x><x I 1

(48)

where x is chosen to be a good representation of #v in the valence space but is almost zero in the core region. The effective hamiltonian satisfies the requirement that its lowest eigenvalue should be cV but also ensures that the total energy is an absolute minimum rather than a stationary value 25 and is unique for a particular choice of x. Further comments on the uniqueness problem have been made by Szasz32(SCF with pseudowavefunctions) who replaced the pseudopotential operator by an 29

31 S2

L. R . Kahn and W. A. Goddard, J. Chem. Phys., 1972,56,2685. W. A. Goddard, Phys. Rev., 1967, 157, 81. R. N. Dixon and I. L. Robertson, to be published. L. Szasz, J. Chem. Phys., 1976, 64, 492; L. Szasz and L. Brown, ibid., 1976, 65, 1393.

The Use of Pseudopotentials in Molecular Calculations

109

arbitrary non-local potential :

where Fc is an undetermined function. A unique potential can be generated for particular choices of Fc:e.g. Szasz used Fc =

(50)

-2&+k

However, the resulting effective hamiltonian is non-hermitian and this could cause further problems (see Huzinaga and Cantu18). The derivation of the pseudopotentials discussed above has been developed from equation (12), in which the effect of core projection operators on the valencevalence electron repulsion has been neglected. The error introduced by this approximation can be largely removed by adding to the core repulsion operators the repulsion from the difference in valence electron density in the reference atoms between the all-electron and valence-electron calculations:33 oCOre

=

u(&+ Rv

- Rp)

-

~ ( R fc Rv

- Rp) -

core

f

Ekl$h><+kl

- EVC I +v><$v I- I XV)<X" I I

(51)

where the R are density operators 3 4 for the core, valence, and pseudovalence spaces. [Thus f c =f(Rc).] Open-shell Pseudohami1tonians.-The majority of atoms do not have valence structures which can be represented by the fully closed-shell wavefunction of equation (14), and consequently ab initio pseudopotentials cannot be derived directly from the theory outlined above. Acceptable wavefunctions for such atoms require either more than one determinant or the use of the symmetry-equivalenced or generalized Hartree-Fock method, and usually include partially filled shells. The total all-electron wavefunction may be symbolically expressed in terms of four subspaces, ~

A = E

W&>, {W, {$v2>, {Ad)

(52)

where each { 4) represents a sub-space as follows: { $c) core orbitals (doubly occupied); { &J closed-shell valence orbitals (doubly occupied); { &,I open-shell valence orbitals (partially occupied); { &} virtual orbitals (unoccupied). Then, following Saunders, 36 we can define a general Fock operator, for the case of a single open shell: =

I;:

+

2J(Rc

+

Rv1)

- E(Rc + RvJ + caJ(RvJ

- fd,R(Rvt) (53)

where c, and d, are coefficients determined by the fractional occupation and degeneracy of the open shell. (c, and d, are related to the a, b, andfof R o ~ t h a a n . ~ ~ ) In most cases, the pseudopotential will not be required for a single specific spin coupling and so it is reasonable to average over all possible open-shell configurations 33 34

35 36

R. N. Dixon and J. M. V. Hugo, Mol. Phys., 1975, 29, 953. R. McWeeny and B. T. Sutcliffe, 'Methods o f Molecular Quantum Mechanics', Academic Press, New York, 1969. V. R. Saunders and M. F. Guest, ATMOL 3 User Note No. 9, Atlas Computing Division, Rutherford Laboratory. C. C. J. Roothaan, Reu. Mud. Phys., 1960, 32, 179.

Theoretical Chemistry

110

and use average values C, and a, (where

&=aa)and write (54)

= EaG(Rv2)

FJ(Rv2)-+daE(RvJ

The general matrix representation for the hamiltonian in the A 0 basis is thens6

C

v1

P =

(55)

V2

U

where p3 and p2 are specific forms of the operator pa (i.e. particular choices of the parameters c, and d,) for the closed-shell-open-shell,and open shell-virtual orbitals respectively. If the valence-electron wavefunction is also symbolically expressed, @VE

=

hz),

@({XPi),

(56)

{xu))

where { xp,} signifies closed-shell valence orbitals (doubly occupied), { xp2} openshell valence orbitals (partially occupied), and { xu) virtual orbitals (unoccupied), the matrix representation of the effective hamiltonian is P1

P2

u (57)

We require that the general functional form for the operator is

Pff= R

+

Vtseudo

+

+

G(Rpl)

&(Rpz)

Then if we assume that the following approximations are valid: pv&(RvJ

Pp&(RvZ)

pv&(RvJ

pp&(Rv&

and VI

wherep", = C[&)
ppz,the pseudopotential can

V

be written = G(Rc

+

Rv1

- RpJ

+

CaG(Rv,

-

Rpa)

-

Ekl+k)<+~I

(60)

However, it is known that the open-shell hamiltonian itself is n ~ n - u n i q u eIn . ~par~ ticular this means that the eigenvalues are dependent on the particular form of the hamiltonian for which the orbitals are canonical. Any attempt to use the extended form of the Phillips-Klehman projection operator, which for open-shell cases can be written

The Use of Pseudopotentials in Molecular Calculations

111

must ensure that the eigenvalues are such that eVl = eP1 for ail VI, p1 E ~ ,= for all VZ, pa by using the same definition for the open-shell hamiltonian in the all-electron and valence-only calculations. We can see that the non-uniqueness of the pseudopotential and of the open-shell hamiltonian have similar origins. Following Roothaan 36 the total open-shell hamiltonian pz may be written in terms of the basic operator pa by using projection operators to define the particular form of the operator for each sub-space: P,T = R G(Rc Rv1 Z,RV2) [ A PvrlG[(c3-ca)RvzlPva + Pv&[(c3- c,)Rv,l[Pc + All + PuG[(c2- c,lRv,l~v,

+

+

+ Pv,G[(c2-

+

+

+

(63)

C,)RVZlPU

For a closed-shell system the effective valence hamiltonian (we follow the work of

Weeks, Hazi, and Rice13)can be written in the general form

where again projection operators are used to modify the operator for each sub-space (the core and valence spaces). 3 The Parameterization of Pseudopotentials Having reviewed the theoretical background to the core-valence separation, we now turn to the practical implementation of the theory. Starting from equations (31j (34) we note that the valence pseudo-orbitals are eigenfunctionsof an equation which can be written as [I;

+

@ore

Following Weeks et aZ.13 we write

@Ore

+ evlx, =

EtXZ

(65)

as

where PG*K is the generalized Phillips-Kleinman potential. There are now basically two choices for the parameterization of @ore: (i) to recognize that @ore is primarily a repulsive potential which excludes the valence orbitals from the core space, and hence to attempt to construct an equivalent local potential (that is, a potential which is simply a function of the radial co-ordinates from the nuclei), or (ii) to consider PGPK defined in terms of the set of core projection operators I&>
A

e-ZkT/y

(67)

Theoretical Chemistry

112

which is a simple local potential with two adjustable parameters A and k. This is an example of the first class, although because of its crude functional form we will refer to it and similar functions as ‘model potentials’ Similar to the Hellman approach are the potentials listed below:

due to Szasz and McGinn; 2 3 ocore =

c c -A1 exp( - y vr2 ) 1 m=-1

h>
(Schwartz and Switalski 37);

(Simons 38);

(Barthelat and Durand 39 - PSIBMOL). In the last three cases above the authors have made the radial form of the potential dependent on the angular part of the wavefunction on which it operates, recognizing that the potential experienced by an electron in, for example, the 3p orbital of chlorine is different from that in the 3s. Such potentials are termed semi-local. This dependence is particularly important when there are valence orbitals in an atom which have angular momenta which are not present in the core, e.g. the 3d orbital of the first row of transition metals. The adjustable parameters in these potentials can be chosen by constraining the eigenvalues and eigenvectors of the effective hamiltonian of equation (65) to agree with those from an all-electron calculation, or by requiring that the calculations reproduce some appropriate experimental quantities such as ionization or excitation energies. In the former case the potential has an ab initio origin; in the latter it is said to be semi-empirical. The potential due to Fues40 [equation (70)] is especially useful in this respect since its eigenvalues and eigenfunctions are analytic functions of its parameters, and this facilitates the fitting procedure. The complexity of the functional form for @ore increases from equation (67) to equation (7 1). To remove the necessity of accepting any particular predetermined functional form Kahn and Goddard29evaluated a semi-local potential by making use of

and solved for olOre(r) as a tabulated numerical function. This method generates Vl(r) directly from equation (65) and in addition can take into account that the 3’ 38 39 4O

M. E. Schwartz and J. D. Switalski, J . Chem. Phys., 1972, 57, 4125. G . Simons, J . Chern. Phys., 1971, 55, 756. J. C. Barthelat, Ph. Durand, and A. Serafini, hlol. P ~ J * J1977, . , 33, 159. E. Fues, Arm. Plrvs., 1926, 80, 367.

The Use of Pseudopotentials in Molecular Calculations

113

e(

valence-valence electron interactions represented by xv) are different from that found in an all-electron calculation, G(&) [see equation (51)]. This method assumes that the xnl are approximately known before the calculation is started and that they are nodeless functions, the so-called 'coreless HF functions'. Having found U;ore(r) numerically, it is fitted by a linear combination of gaussian functions: Ufore(r) -

T]

=

-8

E k

dkzrnkl

e x p ( - ?kzr2)

(73)

for computational convenience in evaluating matrix elements. Moskowitz et aL41 have now used this form of the pseudopotential in a large number of calculations and the parameters and related basis functions have been tabulated for all the atoms from Li to Zn and the halogens. For calculations of atomic ionization potentials and the separations between various spectroscopic states these results are very good; often the valence-electron calculations reproduce the all-electron calculations with a.u. More will be said about this approach later. errors of N Returning to equation (65) it can be seen that the non-local terms in 8cO.e are & and PGPK, whereas?., is a local function representing the screening of the valence electrons from the nuclear charge. Assuming that PGPK can be reduced to the ppK form, a form for Ocore is

where r , is the greater of rl and r2, the fist term is responsible for approximating the nuclear charge density [i.e..f(R,)J,and the second is a non-local pseudopotential represented by projection operators. Dixon and Hugo 33 performed calculations on Na,, HCl, and C1, using this operator in a form where the $&) were approximated by single STO's; the exponents and the coefficientsak were chosen with reference to all-electron ab initio calculations Similarly, Kleiner and McWeeny42used a potential

where no attempt was made to parameterize the j coperator, but the ak and #k were chosen so as to reproduce the experimental eigenvalues. Thus theirs was a semiempirical potential which implicitly included the atomic correlation energy into the calculations. A potential of the form

is in widespread use at present 44 in conjunction with gaussian basis sets and in one case together with other approximations [Kaufman: V N D D 0 4 5 ] .In an 28p

41

42

43 44 45

439

J. W. Moskowitz, S. Topiol, C. F. Melius, M. D. Newton, and J. Jofri, E.R.D.A. Research and Development Report, New York University, Courant Institute of Mathematical Sciences, Department of Chemistry, January, 1976. M. Kleiner and R. McWeeny, Chem. Phys. Letters, 1973, 19, 476. R. Osman, C . S. Ewig, and J. R. Van Wazer, Chem. Phys. Letters, 1976, 39, 27. V. Bonifacic and S. Huzinaga, J. Chem. Phys., 1974, 60, 2779. J. Kaufman and H. Popkie, J. Chem. Phys., 1977,66,4827

114

Theoretical Chemistry

attempt to reduce the interdependence of potential and the valence-electron basis set, Dixon and Robertson 31 expanded the projection operator part of the pseudopotential in a basis of three or four gaussian functions: 48

and this form was successfully used in calculations on Clz, Cu,, and CuCl, and on F2-,Clz-, Br2-, Iz-, F3-, Cis-, BrS-, Is-, and 47 The method of parameterization for the above potential is comparatively complex 46 and involves numerically fitting the sum of the kernels of the 2,and fPK operators to the function in equation (77). For the case of Cu this required the non-linear optimization of twelve exponents (say four for each of l = O , 1, 2) and the linear solution for the 60 coefficients ( A j ) . To avoid this fitting procedure, which inevitably introduces some degree of error and unintentional flexibility into the potential, a new approach has been used in the Reporters’ more recent work for calculations in which iron and its simple compounds have served as test cases. From an all-electron calculation we have available the matrix representations of the operators fC and Rc in the all-electron basis. Keeping the same basis we have sought coefficients Arj for each I value, such that 469

2 c

+ V p=

y. A j . l Igr,r)
(78)

3

where gi,z are the primitive gaussian functions and (79)

(gt.1 Igr,r) = St/,r

This is automatically satisfied if the coefficients are given by &,r

c s;-,fz(4.w +

=

V P d

P,9

s;,l,

(80)

where Kpq.1

=
and VP*J = (gPJ I @IcIg,,r)

The Phillips-Kleinman term is

and the orbitals are expressed in a gaussian basis set:

Ih , l )

=

c

Ct,l

a

(83)

Igv>

Hence the equations (81) can be simplified to: A i j , ~=

Z SF2,lKpp,ZSG!l

P,4

+

C Ci,lCj,t(ev,t-

E ~ J )

k

In this form both kcand PPK can be directly parameterized in terms of the all-electrom basis of the reference ab initio calculation such that their matrix elements merely 4* 47

R. N. Dixon, P. W.Tasker, and G . G.Balint-Kurti, Mol. Phys., 1977, 34, 1455. P. W. Tasker, Mol. Phys., 1977, 33, 51 1.

The Use of Pseudopotentials in Molecular Calculations

115

involve overlap integrals. This work has retained the representation of fcin terms of the repulsion from a set of Is gaussians. This parameterization still requires some exponent optimization, but less than in the method of reference 46. In practice the gaussians have been determined by using a least-squares method to fit the integral of the core charge density, that is by minimizing the functional

for an appropriate range of mesh points r ~The . charge density corrections of equation (51) can be easily incorporated into this parameterization. Although this method of calculating the matrix elements of the core coulomb potential does explicitly involve core-valence two-electron integrals, these are not too numerous and are of the simplest type, only involving 1s gaussians on each centre. Consequently the calculations are much faster than using the complete orbital description of the core charge. It has also proved valuable to include a level shift parameter A in order to project the core eigenvalues to a positive value (see Section 2) rather than to be degenerate with the atomic valence orbitals. Thus the full functional form of the pseudopotential is : ticore = 2 m c + c(Rv-Rp)l-RERc + d(Rv-RR,)I + ?(-a + A)l+k)(4kl

+

r, (V

&v

+

A)[l+v><+vl-

IXv>(Xvll

(86)

The coefficients c and d arise in the case of an open valence shell reference atom. The method of choosing the xv, to ensure that @ore is uniquely defined, is as follows. In all cases there is no difficulty in partitioning the all-electronbasis set, both on the grounds of orbital exponents and orbital coefficients, into two sets. One of these sets, the ‘valence set’, makes the major contributions to the valence-shell orbitals, whereas the core orbitals are almost entirely made up of the ‘core set’. The coefficients of the ‘core set’ in the valence orbitals are present merely to produce corevalence orthogonality. Consequently it is found that by adding a linear combination of core orbitals to each valence orbital a new orbital is generated which has negligible coefficients for the ‘core set’. In this way, after dropping the small coefficients and renormalization, we obtain the desired xv, which functions are smooth within the core space. Table 1 and Figure 1 illustrate this process for the 4s orbital of bromine. It is worth noting that the coefficient of $38 in x~~ is substantial. This magnitude for the coefficient of the outermost core orbital is typical for heavy atoms with many valence electrons and explains the need to modify the core charge density as in equation (51) in order to obtain the correct total coulomb and exchange potentials. The Method of Explicit Corevalence Orthogonality.-Horn and Murrell l9 have proposed an approximate method for introducing core-valence orthogonality in performing valence-electron calculations, and this has been implemented in a number of paper^.^*-^^ The outline of their scheme is as follows: 48

49 60 51

J. N. Murrell and I. G. Vincent, J.C.S. Faraday ZZ, 1975, 71, 890. J. N. Murrell and C. F. Scollery, J.C.S. Dalton, 1976, 9, 81 8. V. R. Saunders, M. F. Guest, and I. G. Vincent, ATMOL User Note No. 13, Atlas Computing Division, Rutherford Laboratory. R. G. Hyde and B. J. Peel,J.C.S. Faraday ZZ, 1973,69, 1593; ibid., 1976,72, 571.

116

Theoretical Chemistry

Table 1 The generation of a 4s pseudo-orbital of the bromine atom by adding a linear combination of core orbitals to the 4s SCF orbital 1 s gaussian exponent 574 300 89 070 20 210 5 736 1899 698.7 277.8 115.2 35.97 15.50 4.771 2,077 0.4211 0.1610

418

428

0.00022 0.00168 0.00884 0.03549 0.11269 0.27182 0.41352 0.27902 0.03138 -0.00751 0.00281 - 0.00139 0.00039 -0.00016

x48

448

$38

-0.oooO7 -0.00053 -0.00281 - 0.0113 1 - 0.03848 -0.10131 -0.21286 -0.18186 0.46510 0.64153 0.08233 - 0.01512 0.00358 -0.00151

O.ooOo3 0.00021 0.00112 0.00453 0.01535 0.04159 0.08898 0.08551 - 0-29547 -0.56433 0.51389 0.74457 0.04180 - 0.01051

-0.m1 -0.oooO6 -0.00034 -0.00139 -0.00471 - 0.01279 - 0.02746 - 0.02685 0.09713 0.19692 -0.22842 -0.52291 0.68770 0.53618

0.00000 0.00000

0.00000 -0.oooO1 -0.oooO3 -0.oooO3 0.m1 0.00027 - 0.OO187 0.00000

- 0.02667 -0.22163 0.65683 0.49654

x d S = -0.00030~1,+0.02873h8+0.35828.33a+0.93317~48. This has been chosen to minimize the first ten coefficients. Only the final four coefficients are retained in valence electron calculations.

0.4

0.2

$ (r) 0 (a.u.)

-0.2

-0.4

0

1

2

3

4

5

r (a.u.)

Figure 1 A comparison between the radial dependence of (a) the near-Hartree-Fock 4s orbital of bromine and (b) a pseudo-orbital obtained as a linear combination of the 4s orbital with the core s-orbitals (see Table 1)

117

The Use of Pseudopotentials in Molecular Calculations

Hartree-Fock all-electron calculations are carried out using a limited basis set for each atomic species required. Two molecular valence-electron basis sets are defined; one is Schmidtorthogonalized to the atomic all-electron core orbitals, and the other is a truncated valence-only basis. All molecular one-electron integrals are evaluated exactly in the orthogonal basis, but the nuclear attraction integrals are scaled by using effective nuclear charges Zeffwhich have been chosen so as to reproduce atomic eigenvalues. All many-centre integrals are evaluated in the non-orthogonal basis. In addition, Ruedenburg’s a p p r o ~ i m a t i o nis~used ~ for some of the three- and fourcentre integrals. The valence-electron molecular orbitals are obtained from the standard SCF equations. Some of the results of this method are quoted in Section 4. However, it has also been used on a number of bigger molecules, for example CBr,, TeH,F, A1&16, and the Group V tri- and penta-halides X Y , and XY,, where X = N , Sb, or As and Y =F, C1, or Br.48-5f This method is probably as accurate as some other simple pseudopotential approaches. However, there appear to be some difficulties in improving it to the standard of some of the recent pseudopotential calculations. Attempts to use larger than minimal basis sets required the inclusion of a Phillips-Kleinman term in addition to the orthogonality procedure in order to prevent collapse of the valence orbitals into the core space. Thus in calculations on A12CI, Vincent53 had to include not only the A1 3s and C13s and 3p shells but also the A1 2p and C12s and 2p shells explicitly in the valenceelectron basis in order to obtain good results. Consequently this calculation was not substantially less expensive in computing time than an equivalent all-electron calculation. Table 2 summarizes the various pseudopotentials which have been proposed. Table 2 Functional forms for the one-electron operators in valence-electron pseudohamiltonians Operator One valence electron pseudohamiltonian fic t; = - p a - nv z (Ev - Q c ) l 4 L ) < 4 L I r

+

k

Applications

Alkali-metal atoms, many others

ReJ 7-10,13

Semi-empirical efective hamiltonians Many Na, K, Rb, Cs, Mg+, 23,38,54 Ca+, Sr+, Ba+ 62 53

54

K. Ruedenburg, C. C. J. Roothaan, and J. Jaunzemis, J. Chem. Phys., 1956,24,201. I. G. Vincent, Ph.D. thesis, University of Sussex, 1976. G. McGinn, J. Chem. Phys., 1969, 50, 1404; ibid., 1969, 51, 5090; ibid., 1970, 52, 3358; ibid., 1970,53, 3635; ibid., 1971, 54, 1671; ibid., 1973, 58, 772.

Diatomic alkali metals 55 Na2, Kz,CsRb, etc. Li, Na, K, Rb, Cs, 56 Be+, Mg+,Ca+, Sr+, Znf, Cd+, Hg+ Alkali-metal trimers 57 CH4 58

h t$

= -+VZ = -+VZ

a)

+

CI CB~(Ini)
-

?;r>Rc

h = -+v -

radius)

l m

7+

l

m

A1 e-Ylr' IZm)(lm r

I

Alkali metals; widely 59 used in the solid state Li, Na, Lit, Na:, Li2, 37,6044 LiH, LiH;, NaH;Z, N2, H20

N2,H20, BeO, LiH, 6 6 6 LiF, Be112, LiH+, Li;, HeH Ab initio effective hamiltonians (i) Semi-local forms

where the last two terms are obtained numerically and then fitted to a linear combination of STOs as Liz, Na2, K2, LiH, 23, 54 NaH, KH G. Simons and P. Mazziotti, J. Chem. Phys., 1970, 52, 2449. G. Simons, Chern. Phys. Lefrers, 1971, 12, 404;ibid., 1973, 18, 315. 57 G. A. Hart and P. L. Goodfriend, Mol. Phys., 1975,29, 1109. 513 J. C. Barthelat and Ph. Durand, Chem. Phys. Leiters, 1975, 27, 191. 59 I. Abarenkov and V. Heine, Phil. Mag., 1964, 9, 451; ibid., 1965, 12, 529 60 M. E. Schwartz and J. D. Switalski, J. Chem. Phys., 1972, 57, 4132. 61 M. E. Schwartz, Chem. Phys. Letters, 1973, 21, 314. 62 J. D. Switalski, J.-T. J. Huang, and M. E. Schwartz, J . Chem. Phys., 1974, 60,2252. J. D. Switalski and M. E. Schwartz, J. Chem. Phys., 1975,62, 1521. 64 N. K. Ray and J. D. Switalski, Theor. Chim. Acta, 1976,41, 329. 65 T. C. Chang, P. Habitz, B. Pittel, and W. H. E. Schwarz, Theor. Chim. Acta, 1974, 34, 263. 66 T. C. Chang, P. Habitz and W. H. E. Schwarz, Theor. Chim. Acro, 1977, 44, 61. 55

56

The Use of Pseudopotentials in Molecular Ccrlculations Operator

119 Applicatioiis

Ref.

where the xCEF are 'coreless Hartree-Fock orbitals' and the last term is fitted to CJeft

= Uzmax(r)

with U(r) =

+

Emax- 1

Z Z udr)llm><1ml 1

m

E A p i e-ajr'

Potential tabulated for 20, 24, Li-Zn; many ato29,41, mic and ionic states; 67 CHsCH3, SiH3CH3, SiH3SiH3

a'

73

39,74

(PSIBMOL) (ii) Non-local projection operator forms nv 2e-ara f; = - tv2- 7 - 7 (E2s - El51 I 415)(+18

+

I

LiH, Li2H+, LiHi

68

C12, Br2, 12, FCI, 26,43 FBr, etc.; Ni(C0)4, Pd(C014, Pt(co)4

F2,

(NOCOR) 44,69,70 45 S. Topiol, M. A. Ratner, and J. W. Moskowitz, Chem. Phys. Letters, 1977, 20, 1. S. Topiol, A. A. Frost, J. W. Moskowitz, and M. A. Ratner, J. Chem. Phys., 1977, 66, 5130. 60 V. Bonifacic and S. Huzinaga, J. Chem. Phys., 1975, 62, 1507; ibid., p. 1509; Chem. Phys. Letters, 1975, 36, 573; J. Chem. Phys., 1976, 65, 2322. 70 D. McWilliams and S. Huzinaga, J. Chem. Phys., 1975,63,4678. 7 1 A. H. Harker, Mol. Phys., 1976,32, 583. 72 H. E. Popkie and J. J. Kaufman, Internat. J. Quantum Chem., 1976,10,47; Chem. PhJw.Letters, 87

68

1977, 47, 55. 73

74

J. C. Barthelat, Ph. Durand, and G. Nicolas, Chem. Phys. Letters, 1974, 27, 191;Theor. Chim. Acta, 1975,38, 283. Ch. Teichteil, J. P. Malrieu, and J. C. Barthelat, Mol. Phys., 1977, 33, 181.

Theoretical Chemistry

120 Operator As above with other NDO approximations

Applications Ref. HF, HCI, F2, Cl2, pyr- 71,72

role, pyridine, nitrobenzene (MODPOT)

where Rc was fitted to 4

Rc z

qi

e-aira

&

and the last term was fitted to a single STO for each I and m: H2S, Na2, NaCl, Cl2 Z a1 I r n v - 2 e-6zr ~ l ~ Ylmrfiv-2 > < e-61.l

x

33

z m

As above but with least-squares fitting of the term g(Rc) (Ev- Ek) I +k><+k I

+

k

-2

Z

l

R

=

3 3

x C. 2 i

j

Aij, t

Ir 1e-air' Yrm)(Yinirle-QP I

Cl, Clz, CU,CUZ,CUCI, 4, 25, 46, F;, Cl;, Br;, I;, Fj, 47 Cl,, Br;, I,, SO

z + f(Rc) + R(Rc + Rv - R p ) - x &k I+k)<+k 1 - E v [ I &>(xv 11 k -+V2-,

10

where Rc z

e-"tr' and the last three terms are Fe, Fe+, Fez+, Feat-, 31 FeH+, KBrz represented as above but in a large gaussian basis (see text) qi

1.

4 Discussion and Comparison of Particular Cases Having described the variety of ways in which the core-valence interaction may be parameterized it is clear that we should examine how they perform in actual calculations. Generally the advances in the complexity of the parameterization have produced commensurate improvements in the accuracy of the results. However, by introducing a large number of parameters the simplicity of the 'core-valence' concept is lost and, in practice, the fitting of the parameters themselves can be expensive in terms of computer time, although they only need to be obtained once for each atom. From the tables of results which follow, two sorts of comparison can be drawn; firstly the pseudopotential calculation involving the valence electrons (VE) can be compared with the all-electron (AE) calculation in the same basis set, and secondly both the AE and the VE calculations can be compared either with experimental quantities or, what is possibly better here, with an AE calculation in a very large ('Hartree-Fock limit') basis so that the eigenvalues may be compared. The former comparison reveals how well the pseudopotential allows the VE calculation to reproduce the AE one, given that they both have essentially the same flexibility in the valence space, and the latter shows how the small basis set calculations may differ

The Use of Pseudopotentials in Molecular Calculations

121

from the large one. The systematic errors introduced by a small basis set in AE calculations have well researched in a number of cases, but experience is still required where a pseudopotential has been introduced. Also of particular interest are those calculations where more than one size of basis set has been used with the same pseudopotential. Ideally a pseudopotential should be parameterized so that it is independent of the valence basis with which it is later used. If this has been shown not to be the case, the parameters must be reoptimized for each basis set. The calculations described below have been selected from the extensive recent literature on pseudopotential calculations as examples which allow these comparisons. The method of Kahn, Goddard, et al. is probably the most exact representation of the ‘local’ class of pseudopotentials, and it is thus interesting to compare their results with those calculated using the Reporters’ own non-local representation, which is one of the more exact of this type. Values of the ionization energies and eigenvahes of the iron atom and its positive ions are presented in Table 3 and for (FeH)+ in Table 4.s4ss1 Attempts were made to make the calculations as comparable Table 3 A comparison of semi-local and non-local pseudopotential calculations of ionization energies of Fe (a.u.) Method All-electron SCFb Semi-local VESCFb Error (VE-AE) Allelectron SCFC Non-local VE SCFC Error (VE-AE)

Fe+49365 0.3132 0.3145 0.0013 0.31085 0.31076 -0.oooO8

Electronic configurations Fez+4s03d6 0.7801 0.7888 0.0087 0.7806 0.7856

Fe3+ 4s03d5 1.8229 1.8540 0.0311 1.8209 1 .8125 -0.0084

0.0050 Each calculation is for a valence state, not a spectroscopic state. The energies are relative to Fe 4s23d6, and are the differences between two separate SCF calculations. Ref. 24. This reference includes comparison of many other excitation energies using AE and semi-local VE calculations. C Refs. 4, 31.

Table 4 A comparison of the accuracies of semi-local and non-local pseudopotential calculations on FeH+ ( R = 3 a.u., energies in a.u.) Method All-electrona large basis Semi-local VEa largebasis Error WE-AE) Semi-local VEQ’ small basis Error(VE-AE) All-electronb Non-local VEb Error WE-AE)

Dt!

Ed8

%?r

EdU

%a

Ecrb

0.0567

- 1.0248

-0.9934

-0.8422

-1.0619

-0.6729

0.0617 0.0050

-1.0247 O.OOO1

-1.0009 -0.0065

-0.8501 -0.0079

-1.0852 -0.0233

-0.6785 -0.0056

0.061 1 0.0044

- 1 .0222 -0.9978 -0.8474 - 1 .0843 - 0.6774

0.1314 0.1343 0.0030

-0.8618 -0.8562 0.0056

0.0026

-0.0044

-0.0052

-0.8498 -0.8403 0.0087

-

-0.0224

-0.0045

-0.6417 -0.6370 -0.0047 d g d g i ; 5C+.* Refs. 4, 31. This is

-

a Ref. 24. This is a GVB calculation for the configuration a ? & an SCF calculation for the singlet valence state did$& Comparisons between (a) and (b) are therefore not meaningful, but within each set the wavefunctions are equivalent.

122

Theoretical Chemistry

as possible by using the same basis sets etc. However, we refer readers to the original references for exact details. In particular, it is noticeable that the total energy for the all-electron calculation on Fe of Kahn et is more than 1 a.u. higher than that of W a c h t e r ~or~ ~of the Reporters; this is presumably due to their use of a different orbital optimization procedure. The semi-local pseudopotential 2 4 used for the calculations of Table 3 was based on a parameterization for the neutral atom. Melius, Olafson, and Goddard also included in their paper some calculations based on the single valence electron ion Fe7+.As expected, this parameterization leads to far worse results, which differ on average from the all-electron results by 0.05 a.u., thus emphasizing the importance of the contribution of valence-valence interaction to the effective potential [equation (51)l. Both methods enable VE calculations to be carried out with almost AE accuracy. In this particular example the Reporters’ calculations appear to give slightly closer agreement between AE and VE values. The good result for the first ionization energy of the Fe atom is, however, not too surprising; we deliberately chose to parameterize the pseudopotential in terms of an AE calculation on the hypothetical, ‘half-ionized’ case with configuration 4s2 3dssSwhich is an average of Fe and Fef following the ‘transition-state’ concept of Slater.75aIn addition the value of A [see equation (86)] was also chosen (A=0.7 a.u.) so as to make the first I.P. correct, although calculations with other values of A (0.2 < A < 2.0) show that the results are not a strong function of A, as expected. Thus it is seen that if the pseudopotential is chosen to be a reasonably exact representation of the ab initio @ore it is possible to come very close to the results of a comparable all-electron calculation, whether the model be a local one or a non-local one. The rest of this section is devoted to the more approximate methods, some of which are quite successful despite their simple functional form. LiH.-We begin by looking at pseudopotential calculations on LiH, since after the single valence electron atoms and molecules this is probably the simplest case on which to perform calculations. The doubly occupied 1s orbital on Li, whose influence on the valence electrons is to be parameterized, is well localized spatially, spherically symmetric (hence only the I = 0 terms are important), and well separated energetically from the valence orbitals. In this particularly favourable case all the results are quite accurate except for the value of Re calculated by Szasz and McGinn 23 which is much too large (Table 5). The calculations of both Chang et aLgSand Schwartzg2 are based on a semiempirical pseudopotential which implicitly takes into account some of the effects of electronic correlation. However, both of these results are much closer to the HF limit calculation of Cade and H U O ’rather ~ than to experiment: e.g. compare the values of De. The calculation of S z a ~ employed z~~ an additional function (ppol)to model the effect of core polarization. This was based on the calculations of atomic polarizability by C a l l a ~ a y The . ~ ~ use of this function does not appear necessary, judging by the other calculations, although very successful 75 75a

l7

A. 5. H. Wachters, J . Chem. Phys., 1970, 52, 1033. J. C. Slater, ‘Quantum Theory of Molecules and Solids IV, The Self-consistent Field for Molecules and Solids’, McGraw-Hill, New York, 1974, p. 51. P. Cade and W. Huo, J. Chem. Phys., 1966,45, 1063. J. Callaway, Phys. Rev., 1958, 112, 322.

123

The Use of Pseudopotentials in Molecular Calculations

Table 5 A comparison of all-electron (AE)and valence-electron( VE)SCF calculations for LiH (all data in a.u.) Basis set and method

Near HF Li4s,5p H 5s,5p]

AE VE

AE VE

VE withoyt pp0l VE with Vpol HF limit AE Experimental -

Re a 3.025 2.97

De 0.0544 0.0570

P

2.360 2.365

(3.015) (3.015)

-

2.359 2.980

3.66 3.60 3.015

0.05280 0.05279 0.056

2.364

3.015

0.0925

2.295

Authors Ref. Chang, Habitz, Pittel, 65 Schwan - 0.3017 Huang, - 0 . 3 0 0 ) Schwartz 62 Switalski, szasz, 23 McGinn -0.30172 Cade, Huo 76 E2 (I

-0.302 -0.303)

1

}

( ) calculation at the experimental bond length.

calculations involving both dipole and quadrupole terms in the polarization have been carried out by Dalgarno and by Child.78 (These are rather outside our definition of pseudopotentials and will not be considered here.) N,.-The exceptionally short bond length of N2(Re 2.0 a.u.) should provide a severe test of the pseudopotential technique for three reasons. Firstly, although the non-orthogonality of the nitrogen dz8orbital to the core 41bon the same centre can be accurately parameterized, the 4 2 8 will also overlap the ($18 orbital on the other centre and may cause ‘collapse’. Secondly, the problem of core polarization may be more acute if the bond length is short and, lastly, it may 110 longer be adequate to treat the ‘core-core’ interaction by the ‘reduced nuclear point charge’ approximation [ZAZBIRAB;equation (35)]. Five sets of data are presented in Table 6 for Nz calculations. The first two sets of results are produced by semi-empirical pseudopotentials. However, whereas the VE results 0fChangeta1,~~agreeverywell with their own AE calculation, thoseof Switalski and Schwartz63 are probably much less accurate. In particular their predicted bond length is too long which implies that their potential is too repulsive. As with LiH, it is noticeable that the semi-empirical pseudohamiltonian value for De is closer to the AE result (either with the same basis or the H F limit basis) than to the experimental value, which weakens the argument in favour of semi-empirical parameterization. Comparing the AE and VE eigenvalue spectra the pseudopotential of Huzinaga’O appears to give the best reproduction, and the ‘explicit orthogonality’ method of Murrellla the worst. Other calculations by Murrell (e.g.the work on AICls by Murrell and Vincents3) suggest that this method is further limited to single-zeta size basis sets (as used on N2)unless Phillips-Kleinman projection operators are used in addition to the orthogonalization technique. F,.-The F, molecule is known to be difficult to treat by the conventional AE single-determinant HF method; it is therefore interesting to look at the results of VE calculations. Table 7 shows six different calculations. Of these the pseudoN

78

A. Dalgarno, C. Bottcher, and G. A. Victor, Chem. Phys. Letters, 1970, 7 , 265; A. C. Roach and M. S. Child, Mol. Phys., 1968, 14, 1.

Table 6 A comparison of all-electron (AE) and valence-electron (VE) SCF calculationsfor N2(a21 data in a.u.) Basis set and method All-electron AE Semi-empirical VE Semi-empirical

VE

Re 2.005 2.02

De 0.1846 0.1846

-1.520 -1.540

-0.771 -0.780

-0.6301 -0.6340

- 0.6325 -0.6342

2.44

-

-1.409

-0.822

-0.5965

-0.5965

-

- 1.5243' - 1.5173a

- 0.7698" -0.7671a -0.721 -0.676 -0.778

- 0.6310' -0.6271a -0.504 -0.545 -0.635

- 0.6246' -0. 6166a -0.592 -0.576 -0.616

EZV*

4-31G AE 2.070 2.059 4-31 G VE 1 C all-electron~ (2.068) Orthogonalized 1 C VE" (2.068) 2.005 Near-HF allelectron

-

- 1.485

0.1964

- 1.475

Experiment

0.364

a 79

2.068

-1.442

Quu

EQvg

-

Calculation at experimental bond length. P. E. Cade, K. D. Sales, and A. C. Wahl, J . Chem. Phys., 1966,44,1973.

** V. H. Dibeler, J. A. Walker, and K. E. McCulloch, J. Chem. Phys., 1969,50,4230; ibid., 1970,53,4414.

ElXU

Authors

} ESchwarz %,: Switalski, Schwartz McWilliams, Huzinaga

} }E&ll

Cade, Sales, Wahl

Ref. 66

63 70 19 79

80

3

k

G

Table 7 A comparison of all-electron (AE) and valence-electron (VE) SCF calculationsfor F2(all data in a.u.) Basis set and method 2C(ST0/4G) AE VE 4-lllG AE MODPOT VE 2C AE PSIBMOL VE 25 ‘d’ AE PSIBMOL VE lt NOCOR VE

Re a (2.68) (2.68) (2.50) (2.50) 2.52 2.51 2.67 2.68 2.495

(1Os6p 1d)/[5s4p1d] AE (4s4p1d) VEC

2.86 2.80

0.0231 0.0364

Experiment

2.679

0.0617

+

8

De -0.0744 -0.0719

-

( ) calculation at a fixed internuclear distance.

E2DB

-92tTU

-1.7580 -1.7351 -1.825 - 1.830 - 1.62590 - 1.6112b - 1.759ob - 1.7628b

-1.4854 -1.4805 - 1.470 - 1.462 - 1. 3613b - 1.34250 - 1.4864b - 1.4886’

- 1.7099

- 1.7148 -

81nu

-0.8201 -0.8156 -0.833 -0.831 -0.607gb -0.58690 -0.8193t’ -0.80960

- 1.5065 - 1.5190

Calculated at R=2.679.

-0.7863 -0.7844

E3ug

-0.7444 -0.7415 -0.767 -0.764 -0.5461

2

Authors

E l ng

-0.6798 -0.6745 -0.651 -0.650

-0.53Wb

-0.4755 -0.8300 -0.8208

3 72

-0.6677 -0.6653

Two-configuration MCSCF (ODC/GVB).

}

Ewig, Van Wazer Kahn, Baybutt, Trular

8 2

2 74

-0.7419b -0.732P

C

Harker

} } &:an?;

Ref. 3 2 71 i;-

26

Q * $.

g. a

(3

20 80

126

Theoretical Chemistry

potentials used by Harker'l and that of K a ~ f m a nare ~~ both based on parameters proposed by H u ~ i n a g aand , ~ ~ thus we can compare the behaviour of the pseudopotential with two slightly different valence basis sets. Similarly, Teichteil et u Z . ~ ~ have carried out two pairs of calculationswith the same pseudopotential but different basis sets. All the pseudopotentials are of the non-local projection operator type except for that of Kahn et aL20 who use a localized form combined with their ODC/GVB method. Harker'l has also performed calculations on Ar, using the same pseudopotential and concluded that although his results for F2 were good, those for Ar, were very basis-set dependent. Comparing the results of Harker71 with those of Kaufman" indicates that even on F2 the eigenvalue spectrum is basis-set dependent. Thus the standard deviation between their two sets of AE eigenvalues is 0.035 a.u., and that between their VE eigenvalues is 0.046 a.u., whereas the deviation between the appropriate AE and VE eigenvalues is 0.011 am. for Harker 71 and 0.004a.u. for Kaufman.72It follows that in this case the basis-set dependence is not inherent in the pseudopotential. A similar analysis of the two pairs of calculations by Teichtei174 reaches the same conclusion. The results of the latter authors also emphasize the importance of 'd' functions in the basis set if the correct bond length is to be achieved. The calculation of Kahn et a1.20on F, is noticeably wrong in this respect: despite the use of a 'd' basis function (with an optimized exponent) and the use of the ODC method, both their AE and VE calculations overestimate the value of Re by about 0.2 a.u. Furthermore the ordering of their InUand 3a, eigenvalues disagrees with all the other calculations. a,.-The diatomic molecule CI,has been treated by many authors and we quote results of nine calculations (Table 8), representing seven different pseudopotential methods. For six calculations we can make more precise comparison than usual by considering the standard deviations of the set of valence eigenvalues of VE calculation from the AE one (both VE and AE use the same valence basis set). The calculation of Popkie and K a ~ f m a nhas ~ ~the smallest standard deviation and the eigenvalue spectrum is very well reproduced. However, their result for the equilibrium bond length is 0.2 a.u. too short compared with their own AE calculation. Indeed all the calculations which use a non-local projection operator parameterization of the pseudopotential, except for the work of Tei~hteil,'~ [i.e. Popkie-Kaufman '* (Huzinaga's method), Van Wazer,2s Dixon and Hugo,33Dixon and Robertson "'3 find the Re value for the VE calculation to be shorter (by -0.1 a.u.) than the corresponding AE work. This suggests that the errors in the above may arise from inaccurate parameterization of the very large coulomb term, since this does not appear explicitly in the pseudopotential of Teichteil. The core-valence coulomb interaction is represented by a function which models the screening of the nuclear charge. Its long-range behaviour will not be apparent from atomic calculations but will show up in multicentre calculations. The value quoted for De represents the difference in energy of the molecule, at its equilibrium separation, from the sum of the atomic energies. As expected, in the HF single-determinant approximation De is negative, implying that the molecule is unbound. Teichteil et ~ 1 1 use . ~ ~configuration interaction to improve the binding energy and although for CI, only two configurations were used the results appear

-

Table 8 A comparison of all-electron (AE) and valence-electron (YE) SCF calculationsfor Clz (all data in a.u.) Basis set and method 4-1 11G MODPOT 25

PSIBMOL 25+'d' PSIBMOL 1C(ST0/3G) Orthogonalized 1C(ST0/3G)

NOCOR 12s9p 3s3pld 25(ST0/3G)

Experiment a

Re 4.20 4.00 4.11 4.15 3.81 3.84 @

AE VE AE VE AE VE

VE (3.96) AE 4.15 VE 3.96 AEb (3.756) VEb 3.81 AE VE VE

De -0.OOO4 O.oo00

-

0.0705

4.13 4.05 4.00

0.0095 -0.0162 -0.0139

3.98

0.0911

( ) calculation at a fixed internuclear distance.

e4ug

840%

82nu

-1.035 -1.038 -1.0122 - 1.0237 -1.OOO4 -1.0149

-0.552 -0.553 -0.5896 -0.5761 -0.5774 -0.5669

-0.549 -0.536 -0.5919 -0.5716 -0.6051 -0.5920

-1.141

-0.999

-0.565

-0.515

-

-

-

-1.2266 - 1.2008

-1.0222 - 1.0290

- 1.2030

- 1.0338

-0.5750 -0.5682

-1.1984 -1.1829

-1.0533 -1.0427

-0.5717 -0.5706 -0.5639

-

-

-

* Two-configuration MCSCF(ODC/GVB).

EarO

e5ug

-1.183 -1.176 -1.2406 - 1.%29 -1.2128 -1.2212

-0.6215 -0.6271 -0.5699 -0.5651 -0.5551 c

-0.465 -0.473 0.4694 -0.4553 -0.4513 -0.4429

-

-0.397 -0.440 -0.430 -0.4539 -0.4522

-0.4755

-0.4831 -0.4770

Authors

Teichteil, Malrieu, Barthelat Hyde, Peel

} %?hazer } E&,

51 26

Dixon, Robertson

n

3 s E

20

5

Trular

}: :;:

$

33 4, 31

I28

Theoretical Chemistry

successful. Large-scale CI in the valence space has also been used in calculations on Br2 (see below). The calculation of Kahn et aL20 also produced a positive value for De by employing their ‘optimized double configuration’ method (ODCIGVB) which is essentially a twoconfiguration MCSCF calculation. In general we conclude that pseudopotentials are not restricted to single-determinant calculations; indeed the Reporters’ own work includes successful calculations on Cu, (CI 1216determinants) and multistructure valence-bond calculations on the di- and tri-halogen negative ions.2s*47 Br, and HBr.-We have already observed the phenomenon of ‘basis-set dependence’ in pseudopotential calculations: i.e. since some pseudopotentials are parameterized with respect to a particular VE basis unpredictable behaviour results when they are used in conjunction with other (perhaps more flexible) basis sets for which they are not optimal. Now we give details of calculations which illustrate the associated problem of ‘pseudopotential parameterization dependence’, i.e. the effect of changing the number of terms used to fit the desired functional form for the pseudopotential. Amongst calculations on all the halogen halides and dihalogens, Ewig, Coffey, and Van Wazer28 performed four calculations on Br, in which both the valence basis sets and the number of terms parameterizing the core-valence coulombic interaction were varied. For the latter, two forms were tried, as follows:

Their results are reproduced in Table 9.

Table 9 Basis-set and pseudopotential dependence of calculated properties of Br, 26 Pseucibpotentialb

Basis seta 1C ST0/3G AE VE AE 25 VE VE Experimentd -

.

Re/A we/Cm-1 I.P./a.u 2.343 398.0 0.327 2.352 355.5 0.359 Vl 2.292 358.4 0.398 2.400 300.1 0.414 Vl 2.265 331.8 v2 0.474 2.283 323.2 0.393 a AE = all-electron; VE = valence electrons only. See text, equation (87). Using Koopmans’s

theorem, I.P. = - E.

d

-

Ref. 81.

In general, we see that the results with the double-zeta basis and the potential V, agree better with the AE calculation in the same basis than does the calculation with V1;to quote Van Wazer, ‘the flexibility of the basis set should be balanced by the flexibility of the functional form of the pseudopotential’. In contrast, calculations on HBr and Br, by Kahn et aL20used a pseudopotential which included the function

81

P. A. Straub and A. D. Mclean, Theor. Chini. Acra, 1974, 32, 227.

The Use of Pseudopotentials in Molecular Calculations

1 29

for the total valenaxore interaction. (See above for more details of this parameterization.) Calculations of the potential curves of HBr and Bra were carried out with five, seven, and nine terms in the expansion and no significant differences were observed; for example, for HBr the difference in the dissociation energy De was 0.0022 a.u. and in the equilibrium bond length 0.10 a.u. for calculations with the five- and nine-term fits, respectively. The pseudopotential used by Van Wazer26 requires the parameters to be fitted with direct reference to an all-electron calculation, whereas the pseudopotential used by Kahn20 is first derived in a numerical form from AE calculations using very large basis sets. The subsequent fitting which is carried out to facilitate the evaluation of matrix elements, and therefore the VE calculations, is not strongly dependent on the pseudopotential parameterization so long as an adequate number of functions is used. Metal Carbonyls: Ni(CO),, Pd(CO),, Pt(CO),.-One of the main reasons for the development of effectivepotentials was the desire to carry out calculations on systems where the number of electrons would prohibit an AE approach. In a pioneering study, Osman et al.43report calculations on the three transition-metal complexes Ni(C0)4, Pd(C0)4, and Pt(C0)4, using the 'NOCOR' pseudopotential method. For Ni(C0)4 these results are in good agreement with an AE calculation (Hillera2). However, a recent AE calculation on Pd(CO)4 (Demuynck83) has shown that they both incorrectly predict the ordering of the orbital energies. The pseudopotential calculation gives the highest occupied orbital as 3e and places the 11 t2second; in the AE SCF work these are reversed, as are some of the deeper lying orbitals (Table 10). Since the ordering of the orbitals is independent of the basis set in the AE cal-

Table 10 A comparison of all-electron and valence-electron SCF calculations of the orbital energies of Pd(CO), (a.u.) Orbital llt2 3e It1

10ta

2e gal 9tz a

Small basisa ( ~ 1 1 5 ) AE -0.396 -0.527 -0.667 -0.673 -0.690 -0.682 -0.703

Large basisa (=2C) AE -0.402 -0.530 -0.651 -0.660 -0.680 -0.697 -0.716

Small basisb NOCOR VE -0.467 -0.300 -0.668 -0.683 -0.684 -0.745 -0.718

Orbital character d,P d 3z

3z

x U U

Ref. 83. b Ref. 43.

culation Demuynck concluded that the error in the VE calculation must be due to the pseudopotential itself. The feature of the NOCOR parameterization most likely to produce this form of error is the function responsible for modelling the corevalence coulomb and exchange interaction (see above). Although the coulombic interaction is just due to the screening of the nuclear charge by the core electrons, and is therefore a spherically symmetric local operator, the exchange term cannot be modelled in this way. The latter is a non-local operator and depends on both the radial and angular parts of the function on which it operates. In some forms of 88

83

I. H. Hillier and V. R. Saunders, Mol. Phys., 1971, 22, 1025. J. Demuynck, Chem. Phys. Lerters. 1977, 45, 74.

130

Theoretical Chemistry

pseudopotential (e.g. Huzinaga,28Dixon 31s 4*) the exchange term is incorporated into the Phillips-Kleinman projection operator; however, the NOCOR procedure (Van Wazer 43) can only hope to parameterizethe I = 0 part of the operator correctly: This leads to errors in the valence eigenvalue spectrum which are symmetry dependent. HgH.-Das and WahlE4have carried out a calculation on the HgH molecule which has many points of interest for the practical implementation of pseudopotentials on heavy-atomic molecular systems. As the nuclear charge increases so does the importance of the relativistic terms in the hamiltonian, and their influence is not only confined to the core orbitals (e.g. the H g 1s) where the kinetic energy of the electron is comparable with its rest mass,but even affects the valence (Hg 6s)orbitals (Grant 8s) and the binding energy of Hg, (Grant and PyperB6). For the HgH system numerical wavefunctions were obtained for Hg using both relativistic (Desclaux’ programme 87 was used) and non-relativistic hamiltonians. The orbitals were separated into three groups: an inner core (1s up to 3 4 , an outer core (4s-4f), and the valence orbitals (5s--6s, 6p). The latter two sets were then fitted by Slater-type basis functions. This definition of two core regions enabled them to hold the inner set constant (‘frozen core’) whilst making corrections to the outer set, at the end of the calculation, to allow some degree of core polarizability. The correction to the outer core was done approximately oia first-order perturbation theory, and the authors concluded that in this case core distortion effects were negligible. Matrix elements for the valence functions were taken with the effective core potential; the coulomb and exchange terms were handled exactly, numerically, without any parameterization and a Phillips-Kleinman projection operator term was also used. Spin-orbit coupling effects amongst the valence orbitals were treated semiempirically using the operator

where Irk is the angular momentum of the ith electron relative to the kth nucleus and si is the spin of the ith electron. A seven-term MCSCF calculation was carried out in the space of the valence orbitals. The potential energy of HgH was calculated with both the relativistic and the non-relativistic core potentials. The results in Figure 2 clearly show that the importance of relativistic effects in weakly bound systems like HgH can be larger than those due to electronic correlation. Relativistic atomic calculationsare now widely available (Desclaux,87Liberman,88 Carlson,8vand Grant*5)and it seems likely that more attempts will be made to combine relativistic core functions with non-relativistic valence functions. 84 86

86 87 88

G. Das and A. C. Wahl, J. Chem. Phys., 1976,64,4672. I. P. Grant, Computer Phys. Comm., 1973,5,263; D. F. Meyers, 1. P. Grant, and N. C. Pyper, J. Phys. ( B ) , 1976,9, 2777. I. P. Grant, N. C. Pyper. and R. B. Gerber, Chem. Ph-vs.Letters, 1977, 49, 479. J. P. Desclaux, Computer Phys. Comm., 1975, 9, 31. D. A. Liberrnan, D. T. Cromer, and J. T. Waber, Computer Phys. Comm., 1971,2, 107. T.C. Tucker, L. D. Roberts, C. W. Nestor, T. A. Carlson, and F. B. Malik, Phys. Rev., 1968, 174, 118.

The Use of Pseudopotentials in Molecular Calculations

131

0.06

0.04

Potentiai energy 0.02

(a.u.)

0

-0.02 2.5

3.0

3.5

4.0

RHqH(a.u.)

Figure 2 A comparison of valence electron calculationsof the potential energy curve of HgH :ad (a) non-relativistic core potential, SCF valence calculation: (b) relativistic core potential, SCF valence calculation; (c) relativistic core poiential, MCSCF valence calculation. In each case the zero of energy is the sum of the appropriate atomic energies calculated in the same manner H,O.-Finally we compare the available pseudopotential calculations on the triatomic system H,O (Table 11). This is interesting partly as a model system on which calculations can be tested before progressing to full potential-energy surface calculations for heavier atoms. Of the semi-empirical pseudopotential calculations Switalski and Schwartz's values3for the HOH angle is better than that of Schwarz.66The best ab initio calculation is due to Barthelat and Durand,68and although the Huzinaga method70appears to give good agreement for the eigenvaluts, the HOH angle is overestimated by as much as Murrell'sl@calculation underestimates it (- 6"). 5 Conclusions

From the examples in the previous section and from the numerous recent applica-

L

w

N

Table 11 A comparison of all-electron and valence-electron SCF calculations on H 2 0 (all data in a.u. except the valence angle 0) Basis set and method 2C(ST0/3G) AE PSIBMOL VE

Re 1.89 1.89

0,

B.E.b

E2al

Elb2

E3al

elbi

102.5" 102.0"

-

-1.2774 - 1.2730

-0.6185 - 0.6167

-0.4577 -0.4532

-0.3922 - 0.3907

105.2"

-

105'

-

-1.301 -1.3384

-0,721 -0.7137

-0.600 -0.5750

-0.532 -0.4984

0.2445 0.2439

- 1 .3554 - 1.3694

- 0.7200

-0.5693

-0.5028

-0.7152

-0.5733

-0.5050

VE VE

I .829 (1 .78)

AE

VE

1.779 1.780

109" 107"

1.805 1.807 1.81

104.5" 110.6' 98"

-

- 1 ,3466 -1.3456 - 1.293

- 0.7058 -0.7013 -0.626

-0.5601 -0.5556 -0.496

- 0.5028

Orthogonalized l C

AE VE VE

Near-HF

AE

1.779

106.6"

0.2551

-1.3513"

-0.7166c

-0.5836C

-0.5077C

1.809

104.5'

0.3496

-

-

-

-

Semi-empirical Semiempirical

+

2c pol 4-3 1G

Experiment

N

( ) caIculation at a fixed internuclear distance. *O

Dl

* Total binding energy of 2H + 0.

C

-0.4992 -0.433

}

Author

Ref.

Serafini Simons Switalski, Schwartz

56 63

66 Schwarz McWilliams, 70 Huzinaga Horn, I9 Murrell Dunning, 90 Pitzer, Aung 91

Calculated at the experimental geometry.

E. Clementi and H.E. Popkie. J. Chem. Phys., 1972, 57, 1077; T. H. Dunning, R. M. Pitzer, and S. Aung, ibid., p. 5044. W. S. Benedict, N. Gailar, and E. K. Plyler, J . Chem. Phys., 1965,24, 1139; JANAF Thermochemical Tables, 2nd Edn., National Bureau of Standards, Washington, 1971.

g

The Use of Pseudopotentials in Molecular Calculations

133

tions of the pseudopotential technique it is clear that successful calculations can be performed. The simple Hellman-type potential, particularly in the semi-local angular momentum dependent form, is capable of good results for the excited states of For molecular calculations the pseudopotential must give a better description of the radial variations of the core functions, and those potentials with two or three terms in the radial part give a satisfactory representation of the core of the alkali-metal or halogen atoms. For the more difficult systems, e.g. the transition metals, where the core is not well separated from the valence shell, the functional form chosen for the pseudopotential must be more flexible. Quantitatively, pseudopotential calculations are capable of reproducing all-electron calculation to an accuracy of about 0.01 a.u. in the eigenvalue spectrum, about 0.05 a.u. in the ionization potentials, and probably to within a few per cent of the dissociationenergy.Naturally, the closer the form for the potential function used in the actual calculations can approximate the theoretical form for the appropriate pseudohamiltonian, the better the numerical results mirror the reference all-electron calculation. However, the older model potentials are physically appealing and are more susceptible to semiempirical techniques. Earlier we noted that a pseudopotential could be expressed either as local or a nonlocal function of the spatial co-ordinates (with semi-local forms as an intermediate class) and some of the most accurate calculations have been performed with potentials which characterize these extremes. In the work on FeH+ the non-local representation4 reproduces the all-electron single-determinant SCF results slightly better than does the CHFEP method.24Moreover, the latter appears to be more difficult to parameterize since it requires the fitting of a numerical potential to an analytic form. Also the matrix elements of the CHFEP are probably more difficult to calculate than those of the projection operators of the Reporters’ non-local potential which are all expressed in terms of overlap integrals. Theoretically the non-local representation of the pseudopotential should be advantageous in molecular systems which are far from spherical symmetry or where there is a large reorganization of charge on molecule formation. Further work is needed before a clear conclusion can be reached concerning the relative utilities of the semi-local and non-local techniques. The accurate parameterization of the effective core potential has shown that the reduction of the pseudopotential to the form of a one-partide operator is adequate. The scaling of the two-body potentials by the use of an operator 65

PI14 =

A/na

(90)

where I is a scaling factor, can be avoided and the use of a series of canonical transformationsz1may be unnecessary in most cases. Earlier work often included terms in the potential to allow for core polarization. However, this is now generally found to be negligible (Das and Wahl 84), although some very accurate semi-empirical calculations have been done using the dipole and quadrupole p~larizabilities.~~ More important in molecular calculations is the interaction between the cores. Work has been done using the change-density functionalsYz0 and the Reporters’ work4 on Cu, using a method due to Gerrattg2estimates that at equilibrium the core-core 439

sa

J. Gerratt, Proc. Roy. SOC.,1976, A350, 363.

Theoretical Chemistry

134

interaction energy is 0.001 a.u. different from the classical coulombic interaction of the reduced nuclear charges (the dissociation energy of Cu, is 0.06 a.u. in the SCF approximation). In comparison with the energy, the evaluation of other properties has received less attention. In principle expectation values should be evaluated using a valence wavefunction which has to be projected against the core eigenfunctions and is thus orthogonal to them, thereby allowing for the antisymmetry effects of equation (3). Alternatively, new valence-only operators could be defined. In practice some operators such as the dipole moment yield acceptable results with the valence pse~do-orbitals.~~ In general the quality of the results is predictable and the use of pseudopotentials does not introduce any new problems, except that operators which sample the wavefunction in the region near the nucleus (in particular the Fermi contact operators and operators of the form l / P ) will clearly give spurious results. The valence pseudo-wavefunction can be improved by the use of standard techniques to correct for electron correlation, and the methods of CIY49l4 MCSCFYa1 ODC/GVB,20and multistructure VB 26* 47 have all been tried with the expected monotonic improvement in the energy (but there have been no references to suggest that the rate of convergence is altered from that of allelectron calculations as more configurations are included). The explicit use of orthogonalization partially to transform the one- and twoelectron integrals’ s* ‘s has generally been confined to near minimal basis set expansions and consequently inaccurate results. However, the combination of orthogonality constraints and a Phillips-meinman-type pseudo-potential has allowed some interesting work on HgH which combines a relativistic core with a non-relativistic valence wa~efunction,~~ and although recent works3sheds some doubt on the validity of this particular technique it is clearly an area in which future research will develop. Another area which is of current interest is the use of ESCA and photoelectron spectra to look at deep-lying energy levels in molecules, and in these cases it is suggested that a pseudopotential be used to eliminate the valence space and allow attention to be focused on the core. Indeed, use of the pseudopotential concept may considerably widen the scope of quantum chemistry and we may certainly say that the ‘nightmare of the inner sheWs4has ended. 469

**#

93 94

I. P. Grant. N. Pyper, and S. Rose, to be published. J. H. Van Vleck and S. Sherman, Reu. Mod. Pii.vs., 1935, 7, 167.

Author Index

Alagana; G., 33 Albrand, J. P., 29 Allen, L. C., 13, 23, 32, 37 Allison, G. B., 6 Almlof, J., 22 Alston, P. V., 65, 66 Altmann, J. A., 27, 28 Altona, C., 4 Amos, A. T., 84, 88 Andrews, G. D., 55 Andrist, A. H., 53 Archibald, R. M., 9 Archie, W. C., 56 Armstrong, D. R., 9, 29 Arrighini, G. P., 97 Atkins, P. G., 70 Aung, S., 132 Aiman, A., 23

Binkley, J. S., 7, 31 Binsch, G., 69 Bird, R. B., 72 Birnstock, F., 5 Bischof, P. K., 55 Bnsell, R., 13 Blint, R. J., 23 Blem, C. E., 4 Bock, H., 23 Bohme, D. K., 32 Bond, A., 24 Bonifacio, V., 113, 119 Botschwina, P., 15 Bottcher, C. J. F., 75, 123 Bounds. D. G.. 72 Bowman, J. D.’, 68 Boys, S. F., 17 Bradford, E. G., 103 Brauman, J. I., 56 Brown, A., 55 Brown, J., 2 Brown, L., 108 Brown, R. D., 59 Browne, J. C., 86 Brundle, C. R., 9, 24 Buckingham, A. D., 74, 76,

Bader, R. F. W., 12, 68,

Buenker, R. J., 25, 26, 27,

Bagus, P. S., 4 Baird, N. C., 27 Bak, B., 17, 33 Baldwin, J. E., 53, 55, 68 Balint-Kurti, R. G. G., 106,

Burgi, H. B., 18 Burden, F. R.,13 Burnelle, L. A., 15 Byers-Brown, W., 86

Aarons, L. J., 10 Abarenkov, I., 118 Abrahamson, E. W., 40 Adams, D. B., 9, 36 Adams; S., 94- . Ahlenius, T., 5 Ahlrichs, R., 3, 6, 8, 20, 27, 30. 32. 37

72

114

Barber, M., 12 Barbier, C., 29 Bardo, R. D., 37 Bardsley, J. N., 101 Barthelat, J. C., 112, 118, 119

Bartlett. J. H.. 87 Basch, H., 9,<11, 12, 16, 19, 24, 25, 26, 27

Basilevsky, M.V., 69 Bauschlicher, C. W., jun., 7, 57

Baybutt, P., 31, 103 Bendazzoli, G. L., 13, 17 Bender, C. F., 7, 22, 25, 35, 57, 78

Benedict, W. S., 132 Benson, R. C., 95 Benston, M. L., 86 Berenfeld, M.M., 69 Bernardi, F., 4, 17, 18, 53 Bernstein, R. B., 39 Berry, R. S., 32 Berson, J. A., 54 Berthier, G., 19, 29, 83 Betowski, L. D., 32 Bingham, R. C., 53, 55

78 53

Cade, P. E., 78, 122, 124 Cadioli, B., 36 Callaway, J., 122 Cantu, A. A., 102 Caramella, P., 67 Carlberg, D., 53 Carlson, T. A., 130 Caves, T. C., 91 Chalvet, O., 60 Chang, E. S., 91 Chang, T. C., 118 Chen, J. C. Y.,86 Cherry, W., 17 Cheung, L. H., 36 Child, M. S., 123 Chipman, D. M., 68 Chong, D. P., 8, 86 Christensen, D. H., 33 Christoffersen, R. E., 22 Cizek, J., 60 Clark, D. T., 9 Clark. S. C.. 14 Claxton T. -A., 5, 7 Clernenii, E., 9, 23, 38, 132 Cohen, H. D., 92 Cohen. M. H.. 88. 107 Coffey; P., 107 Collins, F. S., 47

135

Collins, J. B., 7, 31 Connor, J. A., 12 Conroy, H., 56 Cook, D. B., 83 Cook, R. L., 77 Coulson, C. A., 86, 88 Cowley, A. H., 29 Cremaschi, P., 7 Criegee, R., 20 Crispin, R. J., 88 Cromer, D. T., 130 Csizmadia, I. G., 2, 7, 8, 13, 18, 21, 22, 27, 28, 29, 35, 36

Curtiss, C. F., 72 Curtiss, E. C., 87 Curtiss, L. A., 2, 4 Dacre, P. D., 12 Dalgarno, A., 90, 123 Das, G., 30, 83, 130 Das, T. P., 85, 91 Dauben, W.G., 53 Daudel, R., 2, 60 David, D. J., 36 Davidson, E. R., 78 Davies, D. W., 70 Day, A. C.,47 Deakyne, C. A., 37 Dedieu, A., 31 del Bene, J. E., 15, 19 de Leeuw, F. H., 71 Delgado-Barrib, G., 84 Demuynck, J., 11, 129 Denes, A. S., 22 Desdaux, J. P., 130 Deutsch, P. W., 4 Devaquet, A., 53, 54, 62, 69 Dewar, M.J. S., 47, 50, 51,

52, 53, 55, 56, 58, 59, 100

Dibeler, V. H., 124 Diercksen, G., 92 Diercksen, G. H. F., 22, 38 Ditchfield, R., 5, 15, 55, 92, 97 99

Dixoh, D. A., 56 Dixon, M., 5, 7 Dixon, R. N., 106, 108, 109, 114

Dobson, J. C., 72 Dougherty, R. C., 58 Driessler, F., 3, 6, 21 Duke, B. J., 1, 29 Duke. R. E.. 67 Dunning, T.-H.,jun., 6, 25, 80, 132

Durand, Ph, 112, 118, 119 Durmaz, S., 7 Dutta. N. C.. 85 Dyczmons, V., 31 Dykstra, C. E., 19, 37 Dymanus, A., 71

136 Ehker, C. W.,6 Edmiston, C.,28,48,88 Ehrenson, S., 23 Elder, M.,12 Ellinger, Y.,19 England, W.,88 Epiotis, N.D.,17,53, 54,66 Epstein, J. R., 5 Epstein, S. T., 86,96 Ewig, C. S., 107, 113 Fameth, W. A., 53 Farnell, L.,70 Farrell, L., 2 Faucher, H., 29 Fearey, A. J., 9 Feuer, J., 64 Fink, W.H.,13, 19 Fischbach, U.,27 Fitzpatrick, M.J., 12 Fleming, I., 45 Flygare, W.H.,95 Fock, V., 102 Freed, K. F., 26, 103 Frost, A. A.,46, 87, 119 Fues, E.,112 Fujimoto, H.,32, 37, 47,56,

60,61,62,64,66,68 Fukui, K.,32, 37,47,56,59, 60,61,62,64,66,68 Gailar, N., 132 Gagnaire, D.,29 Gammie, L., 28 Garner, C. D.,1 1 George, J. K., 67 George, T. F.,50, 51 Gerber, R. B., 130 Gerratt, J., 133 Gilcbrist, T. L., 40 Gill, G. B., 40 Giovanni, P., 97 Goddard W. A., 6, 16, 20

22,24,'25, 33, 50, 105, lOd

Gold, L. P.,78 Golden, D.M.,56 Goodfriend, P. L.,118 Goodisman, J., 87 Gordon, M. S., 88 Gordy, W.,77 Gosavi, R. K., 22 Gouyet, J. F., 69 Grant, I. P., 130, 134 Green, S., 71. 82 Griffiths, R. L., 30 Gropen, O., 21, 29 Guest, M.F., 4, 10, 30, 37,

109, 115 Guidotti, C.,97 Gunthard, Hs.H.,20, 35 Gupta, A., 93 Ha, T.-K., 17,20,35, 36 Habitz, P., 118 Haines, W.J., 8, 36 Halevi, E. A.,50 Halgren, T. A., 7, 31 Hall, D.J., 103 Hall, G. G., 84 Hall, J. H.,7 Hall, L. H., 64 Hall, M. B., 10 Hameka, H. F., 95,98

Author Index Harding, L. B., 33 Hargett, A. J., 100 Hariharson, P. C., 2, 20, 31,

38

H.arker, A. H., 119

H'arrison, J. F., 16 H'arrison, S. W.,3 H[art, B. T.,13, 14, 16 H;art, G. A., 1 1 8 HIaugen, J. A.,3 Hlay. P. J., 6

Hlazi, A., 101 Hiehre, W.J., 2, 20, 36, 53, 54, 55 H[eidelburger,C.,14 H[eilbronner, E.,46 H[eine,V.,107, 118 Hlellmann, H.,101 H[erndon, W.C.,63,64 H[erring, F. G., 8 H[erschbach, D. R.,56 Hlillier. I. H..5, 10, 11, 12, 13, 23, 30, 129 H[inchliffe.A., 72 H[inkley, R.,1, 70, 72 H[irano, T.,64 Hhchfelder, J. O., 68, 72, 86 H[offman, R., 40, 43, 60, 64 Hloheisel, C., 31 H[ollister. C.,11, 25 H[opkinson, A.C.,7, 13, 22 H[opkinson, M.J., 15 H[om, M.,102 H[ouk. K. N..63,64, 65, 67 H[owell, J. M..13, 15, 30,33 H[SU,H.-L., 12, 25 H[su, K.,53 Hluang, J.-T. J., 3, 118 HIudson, R. F., 61 H[uestis, D. L., 24 H[ugo, J. M.V., 109 H[unt, W.J., 6,25 H[uo, W.,78, 122 H[urst, R. P.,82, 90,91,93 H[ush, N. S.,94 H[uzinaga, S., 102, 103, 113, 119 Hyde, R. G., 115 Imamura, A., 60,64 Inagaki, S.,56, 64,66 Inukai, T.,65 Iwata, S., 26 Jansen, P., 17 Jarvie, J. O.,28, 29 Jaunzernis, J., 117 Jensen, H. H., 22, 29, 33, 34 Jesson, J. P.,17 Jofri, J., 1 1 3 Johannsen, R.,21, 34 Johansen, H.,12 Johnson, K. H., 8 Jortner, J., 25 Jug, K., 57 Juras, G. E.,8 Kahn, L. R., 103,108 Kaiser, S., 23 Kammcr, W.E.,25 Kaplan, B. E.,25

Karplus, M., 36, 51, 77, 85, 90, 91,96,98 Kassatotschin, W.,101 Kato. S.. 32. 37. 68 Katriel, J., 50 ' Kaufman, J., 113, 119 Keaveny, I., 72 Keil, F., 8, 30, 32 Kellogg, R. C.,87 Kelly, H.P.,85 Kendrick, J., 5, 11, 23 Kern, C.W.,36, 77, 88 Kim, H., 22 Kirschner, S., 51, 52, 55, 56 Kistenmacher, H., 23 Kleiner, M., 1 1 3 Kleinman, L., 101 Klemperer, W., 78 Klopman, G.,61,94 Kochanski, E.,69 Kojima, T.,65 Kolker, H.J., 90,96, 98 Kollman, P. A.,15 Kollmar, H.W.,51, 52, 55 Komornicki, A., 52, 55 Koopmans, T.,4 Kortzeborn, R. N.,33 Kontecky, J., 60 Kowalewski, J., 5 Kraemer, W. P., 22, 38 Krauss, M.,4 Kroto, H.W.,15 Krusic, P. J., 17 Kucsman. A.,21 Kuebler, N.A.,25 Kuhne. H., 20 Kutzelnigg, W.,3, 6, 8, 20, 21, 27, 31, 32 Ladner, R. C., 6 Laidler, K. J., 57 Lamb, W. E.,98 Langhoff, P.W.,90,91 Langlet, J., 47 Lathan, W.A., 2, 20,31, 36, 55. Lavilla, P. E., 4 Led, J. J., 33 Lee, T., 85 Lehn, J. M.,18 Lehr, R. E.,40 Leroy, G., 16 Levin, G., 24 Levine, R. D.,39 Levy, B., 6,26,83 Liberman, D,A., 130 Lide, D. R., 'un., 77 Lievin, J., Linder, P., 5 Lipscomb, W. N., 5, 7, 11, 31, 72, 89,96,98 Lishka, H., 3, 6, 8, 21,27 Liskow, D.H.,35 Lisle. J. B.. 2 Lister, D. G., 13 Lloyd, D.R.,13 Lo, D.H.,55, 100 Lowdin. P. 0..102 Longuet-Higghs, H. C., 40,

12

43

Luskus, L. J., 64 Lykos, P. G., 102 Mabbs, F. E.,11

Author Index

137

Macauley, R., 15 McClean, A. D., 83 McClellan, A. L., 77 McCulloch. K. E.. 124 McGinn. 6.J.. 105. 117 52, 55, R., 30 93, 128 ,92,102, ~

~

Olafsen, C. F., 105 O’Leary, B., 94 Olofson, R. A., 60 Oosterhoff, L. J., 47 Orr, B. J., 76 Ortenburger, I. B., 4 Osman, R., 113 Ostlund, N. S., 92 Ottenbrite, R. M., 66 Ottersen, T., 33, 34 Overill, R. E., 5

~~~

McWilliams, D., 8, 119 Maestro, M.,97 Maier, W. F., 21 Malik, F. B., 130 Malli, G., 56 Malrieu, J.-P., 47, 119 Mangini, A., 18 Marchand, A. P., 40 Marsmann, H., 29 Marynick, D. S., 7 Maryott, A. A., 77 Massa, L. J., 3 Matsen, F. A., 86 Matsuoka, O., 38 Mazziotti, P., 118 Meakin, P., 17 Meinwald, J., 25 Melius, C. F., 105, 113 Merer, A. J., 25 Metiu, H., 51, 55 Meyer, H., 94 Meyer, R., 35 Meyer, W., 3, 4, 6, 15, 25 Meyers, D. F., 130 Meza, S., 24 Mezey, P. G., 21, 28 Miller, D. P., 5, 97, 99 Millie, P., 18 Minato, T., 56 Miron, E., 25 Mislow, K., 28, 32 Moccia, R., 97 Mortola, A. P., 12 Morton, J. R., 10 Morukama. K., 13,26 Moskowitz, J. W., 11,12,25, 76, 113, 119 Mukherjee, P. K., 93 Mukherji, A., 85 Mulder, J. J. C., 47 Mulliken, R. S., 25, 80 Munchaussen, L. L., 67 Munsch, B., 18 Murrell, J. N., 7, 37, 57, 68, 92, 100, 102, 115 Musher, J. I., 9 Musulin, B., 46

Palke, W. F., 5, 8, 89 Palmieri, P., 13, 17 Pan, D. C., 13 Parr, C. A., 6 Parr, R. G., 95, 102 Patterson, D. B., 100 Pauncz, R., 88 Payzant, J. D., 32 Pearson, R. G., 42, 61, 68 Pederson, L., 13 Pedley, J. B., 7 Pedulli, G. F., 17 Peel, B. J., 115 Pendergast, P., 19 Pepperberg, I. M., 7, 31 Perkins, P. G., 9, 29 Petersen C., 12 Petersen: G. E., 100 Petrashen, M., 102 Petrongolo, C., 97 Peyerimhoff, S. D., 25, 26, 27, 53 Pfeiffer, G. V., 3 PhilliDs. J. C.. 101 Phillips; L. F.’, 30 Pincelli, U., 36 Pjnschmidt, R. K., 53 Pmsky, M., 24 Pittel. B.. 118 Pitze;, R: M., 12, 36, 89, 132 PlesniEar, B., 23 Plyler, E. K., 132 Popkie, H. E., 23, 113, 119, 132 Pople, J. A., 2, 5, 7, 9, 15, 20, 31, 36, 38, 55, 80, 84, 92, 97, 99, 100 Poppinger, D., 55, 57 Poshusta, R. D., 3 Preston, K. F., 10 Pritchard. R. H.. 88 Provan, D., 14 . Pu, R. T., 91 Pulay, P., 4, 7, 15, 25 Pullman. A.. 33 Pullman; B.,’ 2 Pyper, N. C., 130, 134

Nagata, C., 59, 60 Nelson R. D., 77 Nesbet: R. K., 80, 84 Nestor, C. W., 130 Neumann. D.. 76 Newton, M. D., 23 55, 113 Nicolas, G., 119 Nieuwpoort, W. C., 11 Nilssen, E. W., 22 Nir, S., 94 Nixon, J. F., 15

Raab, R. E., 75 Radom, L., 2, 7, 11, 20, 31, 36. 38. 80 Raffenetti, R. C., 37 Raftery, J., 72 Ramsden, C. A., 55 Ramunni, G., 53 Randic, M., 68 Rasiel, Y., 85, 86 Ratner M. A., 119 Rauk, 28,29,32, 36 Ray, N. K., 118 Raz, B., 25 Reetz, M. T., 21

bhrn, Y.,102 O’Hare, J. M., 93

k,

Rein, R., 94 Rice, S. A., 101 Richards, W. G., 1,2, 70, 72 Ridard, J., 26 Roach, A. C., 123 Robb, M. A., 8, 36 Robert, J.-B., 29 Roberts, L. D., 130 Roberts, P. J., 13 Robertson, I. L., 101, 108 Robin, M. B., 9, 24,25 Robinson, J. M., 85 ROOS,B., 5, 22, 31, 38 Roothaan, C. C. J., 92, 109, 117 Ros, P., 18 Rose, S., 134 Rosmos, P., 23 Ross, J., 50, 51 Rossi, A. R., 13, 33 Rothenberg, S.,8, 15, 16, 18, RR

R&th, A., 9 Roy, H. P., 93 Ruedenberg, K., 37, 48, 88, 117 Runtz, G. R., 12,72 Ryan, J. A., 26 Saare, P. J., 94 Sabin, J. R., 22 Sadlej, A., 96 Saebo, S., 21 Salem, L., 53, 54, 61, 62, 68 Sales, K. D., 124 Salmon, W. I., 3 Salvetti, O., 97 Sana, M.,16 Sandorfy, C., 15 Sato, A., 65 Sauer, J., 64 Saunders, V. R., 2,4, 11, 12, 109. 115. 129 Schaefer, H. F., 1, 2, 7, 8, 9, 10, 11, 16, 18, 19, 25, 34, 35, - - 37, 57 llinn. R..~-54, Schil____ ~,Schlegel, H. B., 4, 17, 18, 28 Schleyer, P. von. R., 7, 20, 31. 38 Schlos&r, H., 107 Schmidt, W., 54 Schubert, R., 67 Schulman, J. M., 25 Schwartz, M. E., 3, 112, 118 Schwarz, W. H. E., 118 Schwieg, A., 92, 93, 94 Schwenzer, G. M., 9 Scollery, C. F., 115 Scott, P. R., 2, 70 Scrocco, E., 12, 19, 33 Segal, G., 53 Seip, H. M., 22, 29 Serafini, A., 112 Shalhoub, G., 16 Sherman, S., 134 Shillady, D. D., 13, 65, 66 Shingu, H., 59 Sjegbahn, P., 5 Silla, E., 12 Silver, D. M., 51, 56 Simmons, N. P. C., 15 Simonetta, M., 7 Simons, G., 9, 112, 118 ~

Author Index

138 Sims, J., 64,67 Sinanoglu, O., 47 Sink M.L., 8 slate;, J. c.,122 Slingexland, P. J., 4 Smeyers, Y.G., 84 Smith, J. A. S., 7 Smith, R. A., 88 Snyder, J. P., 54 Snyder, L. C., 16, 84, 95 Solomon,P., 3 Sovers, O., 36 Spialter, L.,32 Staemmler, V., 3, 6, 8,21,28 Stafast. H.. 17, 23 Stamper, J: G.;68 Stevens, R. M., 56,72,96,98 Stewart, R. F., 55 Stmgard, A., 22 Stone. A. J.. 43 Storr,' R. C.;40 Straub, P. A., 128 Strausz, 0. P., 22, 28 Streitweiser, A,, jun., 21 Strich, A., 22, 33 Strozier, R. W., 65, 67 Student, P. J., 55 Subra, R., 19 Sullivan, J. H., 42 Sundberg, K. R., 36 Sung, S. S.,60 Sustmann, R., 64,67, 69 Sutcliffe. B. T., 48, 70, 109 Swenson, J. R., 27 Switalski, J. D., 112, 118 Szasz, L., 105, 108 Talaty, E. R., 9 Tamaka, K., 32 Tanner, A. C., 5 Tasker, P. W., 106, 114 Taylor, M.W.,29 Teichteil, Ch., 119 Teixeira-Dias, J. J. C., 94 Tel, L. M.,18, 35, 36 Theodorakopoulos, G., 21, 28

Thomson, C.,1, 14, 15,72 Thuraisingham, R. A., 68 Tiecco, M.,18 Tomasi, J., 12, 19, 97 Topiol, S., 113, 119 Tossel, J. A., 11 Townshend, R. E., 53 Trager, W. F., 15 Trinajstic, N., 100 Trindle, C., 13, 47 Trular, D. J., 103 Tucker, T. C., 130 Turpin, M. A., 92 Turro, N. J., 53 Ungemach, R. M., 9 Ungemach, S. R., 10 Vaccani, S., 20 Van der Hart, W.J., 47 van der Lugt, W. Th. A. M., 47 Van Vleck, J. H., 134 van Wachen, R., 71 Van Wazer, J. R., 29, 30, 33, 107, 113 Veillard, A., 11, 3 1, 33 Verhaegen, G., 16 Vesselov, M.,102 Vestin. R.. 5 Victor; G.>A.,123 Vidal, B., 37 Vincent, I. G., 115, 117 Waber, J. T., 130 Wachters, A. J., 11, 122 Wade, L. E., 52 Wadt, W. R.,20, 24 Wagner, E. L., 28, 29 Wahl, A. C., 30,83, 124, 130 Wahlgren, U., 1 I , 24 Walch, S. P. 16 Walker, J. A., 124 Walker, T. E. H., I , 2, 70 Walker, W., 32 Wang, P. S. C., 49 Wang, S. J., 33

Watts, C. R., 64 Weber, J., 21 Weeks, J. D., 101 Weirnann, L. I., 22 Weiner, P., 55 Weinstein, H., 88 Weltner, W., 95 Wendoloski, J. J., 6 Werner, H.-J., 6 Westhaus, P., 103 Whangbo, M.-H., 17, 18 Wharton, L., 78 Whitman, D. R., 85 Whitten, J. L., 26 Wiberg, K. R., 6 Wilcott, M. R., 54 Wild, U. P., 36 Wilhite, D. L., 32 Williams, D. R., 68 Williams, J. E., jun., 15, 21 Williams, M. L.,94 Willis, M.R., 40 Wilson, E. B., 49 Wipff, G., 18 Winter, N. W., 22 Wislsff-Nilssen, 34 Wolfe, S., 4, 17, 18, 28, 29, 35, 36 Wood, M. H., 12, 27 Woodward, R. B., 40,43,60 Wright, J. S.,68 Yamabe, S., 37, 56 Yarkony, D. R., 16, 18, 34 Yates, K., 27, 28 Yates, R. L.,53 Yanezawa, T., 59, 60 Yoshimine, 21, 38, 82, 83, 91, 93

Young, R. H., 8

Zahradnik, R., 60 Zeiss, G. D., 60 Zetik, D. F., 3 Ziman, 3. M., 101 Zimmermann, H. E., 46, 50 Zurawski, B., 20, 27