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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EE2~~~~E~ ( Hydrogenic Atoms in Molecules, HAM/1, unpublished), in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
