Editors Prof. Dr. Gaston Berthier Prof. Dr. Hans H. Jaffe Universite de Paris Department of Chemistry Institut de Biologie University of Cincinnati Physico-Chi m iq ue Cincinnati, Ohio 45221/USA Fondation Edmond de Rothschild 13, rue Pierre et Marie Curie F-75005 Paris Prof. Joshua Jortner Institute of Chemistry Prof. Dr. Michael J. S. Dewar Tel-Aviv University Department of Chemistry 61390 Ramat-Aviv The University of Texas Tel-Aviv/Israel Austin, Texas 78712/USA Prof. Dr. Hanns Fischer Physikalisch-Chemisches Institut Prof. Dr. Werner Kutzelnigg der UniversiUit Zurich Lehrstuhl fUr Theoretische Chemie Ramistr.76 der Universitat Bochum CH-8001 ZUrich Postfach 102148 0-4630 Bochum 1 Prof. Kenichi Fukui Kyoto University Dept. of Hydrocarbon Chemistry Kyoto/Japan Prof. Dr. Klaus Ruedenberg Department of Chemistry Prof. Dr. George G. Hall Iowa State University Department of Mathematics Ames, Iowa 50010/USA The University of Nottingham University Park Nottingham NG7 2RD/Great Britain Prof. Jacopo Tomasi Prof. Dr. Jurgen Hinze Dipartimento di Chimica e Fakultat fUr Chemie Chimica Industriale Universitat Bielefeld Universita di Pisa Postfach 8640 Via Risorgimento, 35 0-4800 Bielefeld I-Pisa Lecture Notes in Chemistry Edited by G.Berthier M.J.S. Dewar H.Fischer K.Fukui G.G.Hall J.Hinze H.H.Jaffe J.Jortner W. Kutzelnigg K. Ruedenberg J. Tomasi 38 E. Lindholm o L. Asbrink Molecular Orbitals and their .E nergies, Studied by the Semiempirical HAM Method Springer-Verlag Berlin Heidelberg New York Tokyo 1985 Authors E. Lindholm L. Asbrink Physics Department, Royal Institute of Technology S-100 44 Stockholm ISBN-13: 978-3-540-15659-8 e-ISBN-13: 978-3-642-45595-7 001: 10.1007/978-3-642-45595-7 Library of Congress Cataloging in Publication Data. Lindholm, E. (Einar), 1913-Molecular orbitals and their energies, studied by the semiempirical HAM method. (Lecture notes in chemistry; 38) 1. Molecular orbitals. I. Asbrink, L., 1944-11. Title. III. Series. QD461.L724 1985541.2'285-14797 ISBN-13: 978-3-540-15659-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Preface This treatment of molecular and atomic physics is primarily meant as a textbook. It is intended for both chemists and physicists. ·It can be read without much knowledge of quantum mechanics or mathematics, since all such details are explained-. It has developed through a series of lectures at the Royal Institute of Technology. The content is to about 50 % theoretical and to 50 % experimental. The reason why the authors, who are experimentalists, went into theory is the following. When we during the beginning of the 1970's measured photo electron spectra of organic molecules, it appeared to be impossible to understand them by use of available theoretical calculations. To handle hydrocarbons we ( together with C. Fridh ) constructed in 1972 a purely empirical procedure, SPINDO [1] which has proved to be useful, but the extension to molecules with hetero atoms appeared to be difficult. One of us ( L.A.) proposed then another purely ~~E!E!~~! EE2~~~~E~ ( Hydrogenic Atoms in Molecules, HAM/1, unpublished), in f5..y which the Fock matrix elements were parametrized using Slater's shielding concept. The self-repulsion was compensated by a term "-1". The §~~2~~_~ff2E~, HAM/2 [2] , started from the total energy E:. of the molecule. The atomic parts of L used the Slater shielding constants, and the bond parts of E. were taken from SPINDO. The Fock matrix elements Fpv were then obtained from E in a conventional way. ['3] , The ~h!E~_~g2E!:' HAM/3 started from i'ltomic spectro scopy instead. Detailed studies by one of us ( L.A.) produced expressions for the atomic shielding efficiencies, which gave high accuracy in atomic calculations. HAM/3 has been used to calculate ionization energies, excitation energies and electron affinities, and many examples are found in the papers listed in Sec.H.6. The HAM/3 computer program has been submitted to Quantum Chemistry Program Exchange ( QCPE ) [~] and all calculations IV described in this treatment can therefore easily be reproduced using the QCPE program. The method obtained a very important support when D.P. Chong in Vancouver, Canada, compared the HAM/3 results with his very [SJ reliable calculations of ionization energies and found good agreement. He has afterwards contributed very much to the develop- ment of the method, and he has recently adapted the QCPE [b]. program to a personal computer In spite of its results the HAM method had been criticized [7.8] . It had never been shown that the exchange integral in the Hartree-Fock method can be replaced by a term "-1". In 1979 we ( together with C. Fridh ) could show that exploitation of the idempotency of density matrices can give the desired proof. [9]. The paper was published together with the critic It remained then only to deduce the HAM method from the Hartree-Fock method. After many discussions this was solved in 1979 ( by L.A.) as described in Sec.D. We found later ( in 1980 that pair-correlation energies can be included in this proof. This deduction has recently (1985) been improved when it appeared to be possible to deduce the treatment of correlation from density functional theory. The !2Y~!h_~!!2~!' HAM/4, could now start. A large number of atoms have been studied as described in Sec.E. and new theoretical methods to handle the multiplet splitting in atomic spectroscopy have been presented. A determination of the molecular parameters would certainly now result in a theoretical procedure, capable of high accuracy in calculation of ionization energies, excitation energies, electron affinities and other properties of small and large molecules. Outside Sweden, the HAM/3 method has been used in several laboratories after it had been submitted to QCPE in 1980. Its use has predominantly been restricted to interpretations of photo electron spectra. We believe, however, that applications to UV spectroscopy and electron affinities will be more important for the future, since these fields are less developed and more important for a further application on chemical reactions. The study of electron affinities is therefore stressed in two chapters. v It is a pleasure to acknowledge our gratitude to our coworker C. Fridh and to G. Ahlgren, G. Bieri, S. de Bruijn, J.-L. Calais, D.P. Chong, O. Edqvist, K.F. Freed, O. Goscinski, I. Lindgren, S. Ljunggren, B.I. Lundqvist, S. Lundqvist, R. Manne, R.G. Parr, P. Sand, L.E. Selin and A. Svensson for cooperation or discussions, and to the Swedish Natural Science Research Council, the Bank of Sweden Tercentenary Foundation and Knut och Alice Wal-lenbergs Stiftelse for econOmic support. Stockholm February 1985 E. Lindholm L. Asbrink References 1 • C. Fridh, L. Asbrink and E. Lindholm, Chem.Phys.Letters 12., 282 (1972). 2. L. Asbrink, C. Fridh and E. Lindholm, in: Chemical Spectro- scopy and Photochemistry in the Vacuum-Ultraviolet, C. Sandorfy, P. Ausloos and M.B. Robin (eds.), Reidel, Dordrecht (1974). 3. L. Asbrink, C. Fridh and E. Lindholm, Chem.Phys.Letters 52, 63, 69, 72 (1977). 4. L. Asbrink, C. Fridh and E. Lindholm, QCPE 1£, 393 (1980) ( Quantum Chemistry Program Exchange, Indiana University, Bloomington, Indiana ). 5. D.P. Chon'l, Theoret.Chim.Acta .21, 55 (.1979). 6. D.P. Chong, QCPE QCMP005 (1985). 7. S. de Bruijn, Chem.Phys.Letters 52,76 (1977). 8. S. de Bruijn, Theoret.Chim.Acta 50, 313 (1979). 9. L. Asbrink, C. Fridh, E. Lindholm and S. de Bruijn, Chem.Phys.Letters 66, 411 (1979). Contents Contents A. The LCAD model : "LCAD" •..•....••.••....•.•••.•...•.......... 1. Molecular orbitals 2. The LCAD formalism 2 3. The normalization and orthogonality of orbitals 3 4. How to interpret the print-out from a calculation 5 5. The charge in fA--.) 7 6. The charges on atoms and in bonds 9 7. The idempotency of density matrices 10 B. Hartree-Fock total energy: "HF" ....••....••.....•..•....•.. 11 1. The Hamilton operator 11 2. The wavefunctions in Hartree-Fock theory 12 3. The total energy in Hartree-Fock theory 12 4. The total energy in LCAD Hartree-Fock theory 14 5. Self-repulsion 14 C. Density functional theory "Density functional theory" ...•. 17 1. Correlation 17 2. Correlation energy 17 3. Exact energy expression 18 4. Exchange-correlation energy 19 5. Density functional theory: Kohn-Sham orbitals 22 6. Introducing Kohn-Sham orbitals 23 7. Introducing LCAD 25 8. Pair-correlation energies 27 9. Semiempirical methods 29 10.Comment·on semiempirical theories 30 1 1.Conventional CI method to handle correlation 31 12.Proof for Gunnarsson-Lundqvist E xc 33 D. Total energy of molecules and atoms : "HAM" •••••.••..•..•..• 36 1. Rearrangement of the total energy expression 37 2. Shielding efficiencies ~~~ in the one-center terms 43 3. The one-center energies in a molecule 44 4. Further study of the shielding efficiencies 44 E . Atoms : " At oms" .•.•••..••.••.•..•.••••..•.•••.••............ 4 7 1. The simple atom 47 2. The energies of the spin-configurations 49 3. Comments on the shielding efficiencies 51 4. Previous work on shielding efficiencies 52 5. Total energies of atoms and atomic ions in HAM/3 53 Contents VII 6. The multiplet split in atomic spectroscopy 62 7. The average state 63 8. Energies of terms ---- energies of average states 63 9. The physical meaning of the parameters 71 10. The semiempirical methods HAM/3 and HAM/4 73 F. Molecules : "Molecules" •••••••••••••••.••••••••••••••••..•.. 76 1. Interpretation of the energy expression for a molecule 76 2. Local dipoles 82 3. The final expression for the total energy 84 4. The par,ametrization of HAM/3 85 G. Solving the Schrodinger equation "SCF" ••••••••••....••..•. 87 1. Variational calculus 87 2. Deduction of Roothaan~s equations 88 3. The Fock matrix elements 91 4. Solving the Roothaan equations 93 5. Some useful relations for the eigenvalue 96 6. Comparison with the Hartree-Fock method 96, 7. The eigenvalue € ~ in Hartree-Fock and HAM 97 8. Molecules with a small HOMO-LUMO gap 100 H. Ionization and photoelectron spectroscopy: "PES" •....•••.• 103 1. Calculation of ionization energy in the HAM model 103 2. Treatment of ionization energies in Hartree-Fock 108 3. Calculation of ionization energies in ab-initio work 109 4. Experimental methods for study of ionization 110 5. Ionization of molecules: some results 114 6. Further studies 133 I. Excitation and UV spectroscopy : "uv" .••••.•.•..•.•......•. 142 1. Calculation of excitation energy in the HAM model 142 2. A primitive CI method to find singlet~triplet splitting 145 3. Calculation of intensitites 150 4. Semiempirical methods to calculate excitation 152 5. Rydberg transitions 157 6. Calculation of excitation energies in ab-initio work 160 7. Experimental methods for study of excitation 163 8. Excitation of molecules: some results 167 9. Degenerate excited configurations will interact: CI 171 10. Excitation of linear molecules 175 VIII Contents J. Negative ions and electron affinities: "EA" ••••••••••••••• 187 1. Calculation of electron affinities in the HAM model 187 2. Experimental methods for determination of EA's 189 3. Electron affinities of molecules: some results 197 4. The relation between the PES, UV and EA results 208 5. Other calculations of electron affinities 210 6. 0"* orbitals 212 K. Studies of 1 s electrons : "ESCA" •••••••••• '. • • • • • • • • • • • • • • •• 219 1. Calculation of 1s ionization energies in the HAM model 219 2. Experimental methods in ESCA 222 3. ESCA energies: some results 223 4. Excitation of 1s electrons, studied in electron impact 225 5. Excitation of 1s electrons, studied spectroscopically 232 L. Shake up in PES and EA : "Shake up" ••••••.••••••••••.•••••• 235 1. Shake up in PES 235 2. Calculation of the. PES shake-up energy 235 3. Shake ups in PES: some results 242 4. Discussion of calculations of shake up in PES 24'8 5. Shake up in EA 256 6. Shake ups in EA in small molecules: some results 261 7. The UV spectrum of the naphthalene anion 263 8. Shake ups in EA in larger molecules 265 M. Total energy: "Total energy" ••••••••••••••••••••••••••••.• 272 1. The total energy of a molecule 272 2. Heat of formation 273 3. Check of ,the transition state method. 275 4. Doubly charged ions 276 N. Dipole moments : "Dipole moment" 277 1. Calculation of dipole moment 277 2. Dipole moment of HCN 278 o. Chemical reactions : "Reactions" 280 1. Can a HAM model be used? 280 2. Dissociation of cyclobutane 281 3. The internal rotation of ethylene 285 Abstract IX The HAM method is obtained in the following way. Density functional theory starts from the total electronic ener gy [ of a molecule, which is the sum of kinetic energy, attraction to nuclei, electron-electron repulsion, exchange energy (Ex) and corre lation energy (Ee ). All these terms are functionals of the total electronic density (!. lY..: ' Kohn and Sham introduced orbitals defined by 'lit ~ = ~ 'Y~ E and determined them from by variational calculus. It appears that 1:: c. can be written (1) c.o~~ where the pair-correlation energy is denoted eAvl~ If this is combined with the Hartree-Fock-LCAO total energy expression, we find E 1>: E. = - i ~v ) 'il ~ ¢v d t: ¢: ~ ~v ) <P~ ~ Za r;l d-r: (4) + ~ L ~y Pi f (p.v/16") - i (f-(jl~v) L , .•: v].,u- (J (b) The solutions of the corresponding Kohn-Sham one-particle self consistent equation are the Kohn-Sham orbitals. These orbitals give E , the correct energy they are orthonormal and their density matrix is idempotent. In eq.(S) the self-repulsion is eliminated by the exchange integral. This is handled in HAM by adding the idempotency relation fPSIP - 2fP =0 after mUltiplication with a factor. This trick does not change the total energy E since the relation is zero, but after some algebra the HAM total energy appears in a new formulation