Table Of ContentCONTRIBUTORS
Ernest R. Davidson
Bruce S. Hudson
Larry E. McMurchie
W. A. Wassam, Jr.
Lawrence D. Ziegler
E X C I T ED S T A T ES
V O L U ME 5
Edited by EDWARD C. LIM
Department of Chemistry
Wayne State University
Detroit, Michigan
1982
ACADEMIC PRESS
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Copyright © 1982, by Academic Press, Inc.
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82 83 84 85 9 8 7 6 5 4 3 2 1
Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
Ernest R. Davidson (1), Department of Chemistry, University of Wash
ington, Seattle, Washington 98195
Bruce S. Hudson (41), Department of Chemistry, University of Oregon,
Eugene, Oregon 97403
Larry E. McMurchie (1), Department of Chemistry, University of
Washington, Seattle, Washington 98195
W. A. Wassam, Jr.* (141), Department of Chemistry, Wayne State Uni
versity, Detroit, Michigan 48202
Lawrence D. Ziegler* (41), Department of Chemistry, University of
Oregon, Eugene, Oregon 97403
•Present address: Department of Chemistry, Cornell University, Ithaca, New York
14853.
tPresent address: Laser Physics Branch, Naval Research Laboratory, Washington, D. C.
20375.
vii
Contents of Previous Volumes
Volume 1
Molecular Electronic Radiationless Transitions
G. Wilse Robinson
Double Resonance Techniques and the Relaxation Mechanisms
Involving the Lowest Triplet State of Aromatic Compounds
M. A. El-Sayed
Optical Spectra and Relaxation in Molecular Solids
Robin M. Hochstrasser and Paras N. Prasad
Dipole Moments and Polarizabilities of Molecules in Excited
Electronic States
Wolfgang Liptay
Luminescence Characteristics of Polar Aromatic Molecules
C. 7. Seliskar, O. S. Khalil, and 5. P. McGlynn
Interstate Interaction in Aromatic Aldehydes and Ketones
Anthony J. Duben, Lionel Goodman, and Motohiko Koyanagi
Author Index-Subject Index
Volume 2
Geometries of Molecules in Excited Electronic States
K. Keith Innes
Excitons in Pure and Mixed Molecular Crystals
Raoul Kopelman
Some Comments on the Dynamics of Primary Photochemical Processes
Stuart A. Rice
ix
χ CONTENTS OF PREVIOUS VOLUMES
Electron Donor-Acceptor Complexes in Their Excited States
Saburo Nagakura
Author Index-Subject Index
Volume 3
Two-Photon Molecular Spectroscopy in Liquids and Gases
W. Martin McClain and Robert A. Harris
Time-Evolution of Excited Molecular States
Shaul Mukamel and Joshua Jortner
Product Energy Distributions in the Dissociation of Polyatomic
Molecules
Karl F. Freed and Yehuda B. Band
The Mechanism of Optical Nuclear Polarization in Molecular Crystals
Dietmar Stehlik
Vibronic Interactions and Luminescence in Aromatic Molecules with
Nonbonding Electrons
E. C. Lim
Author Index-Subject Index
Volume 4
Resonance Raman Spectroscopy-A Key to Vibronic Coupling
Willem Siebrand and Marek Z. Zgierski
Magnetic Properties of Triplet States
David W. Pratt
Effect of Magnetic Field on Molecular Luminescence
S. H. Lin and Y. Fujimara
Time-Resolved Studies of Excited Molecules
Andre Tramer and Rene Voltz
Subject Index
Ab Initio Calculations
of Excited-State Potential
Surfaces of Polyatomic
Molecules
ERNEST R. DAVIDSON
and
LARRY E. McMURCHIE
Department of Chemistry
University of Washington
Seattle, Washington
I. Introduction 2
II. Simple Methods 3
A. Single-Excitation Configuration Interaction 3
B. One-Configuration Methods 4
C. Multiconfiguration SCF 5
III. Configuration Interaction 6
A. Zeroth-Order Effects 6
B. First-Order Effects 9
IV. Examples 11
A. BH2 11
B. CH2 12
C. NH2 13
D. Water 13
E. Methane 16
F. Ethane 16
G. HNO 17
H. HCN 18
I. HCO 18
J. N02 19
K. Ozone 20
L. Acetylene 23
M. Ethylene 24
N. N2H2 27
O. Formaldehyde 28
P. Formamide 29
Q. Formic Acid 30
1
Copyright © 1982 by Academic Press, Inc.
EXCITED STATES VOL 5 All rights of reproduction in any form reserved.
ISBN 0-12-227205-6
2 ERNEST R. DAVIDSON AND LARRY E. McMURCHIE
R. Ketene 31
S. Butadiene 31
T. Glyoxal 32
U. Acrolein 33
V. Benzene 34
W. Pyrrole 35
References 35
I. Introduction
The simplest view of electronic excited states of closed-shell molecules is
that they are formed by the promotion of one electron from an occupied to
an empty ("virtual") orbital. Implicit in this language is the assumption that
the ground-state wave function is a Slater determinant made from Hartree-
Fock self-consistent-field (SCF) molecular orbitals. Also implicit is the use
of these same orbitals to describe the excited state.
Within this approximation the excitation energy is given simply by
AE = (sa - Ji)a - St
3
for (i, a) and
AE = (sa - Ji)a - ε, + 2Kia
1
for (i9a) excitations. Here ε, and s a are canonical Hartree-Fock orbital
energies from the ground-state calculation and J i aand Ki aare Coulomb and
exchange integrals. That is (Roothaan, 1951),
N/2
Ρφ« = εαφα, P(x,x')= Σ Φί(χ)Φ*(χ')
i=l
F = h + 2f - Jf, /(r J = Jp(rl9 r2)r^ dx2
2 2 2 1
h = -(h/2m) V - e ΣΑΖΑ^\ ^φ = jρίτ^φ^)^ dz2
Ji a= / ^ ( r J l 2^ ) ! 2^ 1^ ^^
1
Ki a= $ΦΑ*ΐ)Φα(τΐ)*Φα(Τ2)ΦΑ*2)*^2 dxXdx2
According to Koopmans' theorem (Koopmans, 1934), —st is the ionization
energy in this approximation and — sa is the electron affinity of the neutral
ground state.
This approximation is almost always too simplistic to be of more than
qualitative interest. Even when it happens to work well, there generally are
AB INITIO CALCULATIONS OF POTENTIAL SURFACES 3
large sources of error which just happen to cancel. Further, its qualitative
usefulness tends to disappear if more than a minimum basis set of valence
orbitals is employed or if the excitation involves Rydberg states, ionic
states, multiple excitations, or localized excitations. Additional complications
arise in studying excited-state potential surfaces because varying nuclear
coordinates can produce extensive changes in the wave function, which
makes uniform accuracy hard to achieve.
II. Simple Methods
Many ab initio computational schemes have been suggested in an effort
to achieve quantitative accuracy without undue loss of simplicity. Unfor
tunately, none of these schemes actually work with uniform, predictable
reliability.
A. Single-Excitation Configuration Interaction
The simplest scheme is simply a configuration interaction (CI) or per
turbation calculation employing ground-state orbitals within a minimum
basis set. If all configurations formed by single excitations (and possibly
double excitations) are considered, the result corresponds formally to the
usual Pariser, Parr, Pople (Parr, 1963) or CNDO/S (Del Bene and Jaffe,
1967a) calculation. Unfortunately, the error in the excitation energy is usually
2-3 eV, and the wave functions of closely spaced states are often qualitatively
incorrect. For high-spin excited states of small molecules, the major sources
of error in this approach to ΔΕ are the ground-state correlation energy and
the limited basis set. Inclusion of double-zeta-plus-polarization and Rydberg
basis functions (Dunning and Hay, 1977) usually gives good term energies
(i.e., energies relative to ionization) for high-spin couplings of small molecules.
The absolute excitation energies will still be more than 1 eV too low unless
double excitations are included, because the ground-state correlation energy
is larger than that of excited states. For large molecules an additional source
of error may be present if the excitation tends to localize in a way which
cannot be simply described using delocalized ground-state orbitals. Low-
spin excited states are more prone to localize than are high-spin states.
A variant of this CI procedure uses the same list of configurations but
with energies and coefficients determined by the equations-of-motion (EOM)
method (Rose et a/., 1973). This method, in some difficult cases, has given
better results than single-excitation CI calculations. On the whole, however,
it does not produce reliable potential surfaces and has not gained wide
acceptance as a predictive tool in the absence of data.
4 ERNEST R. DAVIDSON AND LARRY E. McMURCHIE
B. One-Configuration Methods
An alternative to simple CI is an improved one-configuration approxi
mation. Canonical Hartree-Fock virtual orbitals are computed in the field
of the neutral molecule and correspond to anion orbitals. These are very
different from the orbitals occupied in a spectroscopic excited state, which
has a "hole" in a ground-state orbital. Consequently, even for qualitative
accuracy, improved virtual orbitals (IVO) must be defined which are appro
priate for excited states (Hunt and Goddard, 1969). The simplest such orbitals
can be found just by recomputing the virtual orbitals using a modified
Fock operator corresponding to a hole in some ground-state orbital. A dis
advantage of this definition, as far as simplicity is concerned, is that the
virtual orbitals are different for each hole.
Because the canonical occupied orbitals correspond to cation states rather
than excited states, the IVO method gives good term energies for Rydberg
states but not for valence states. Since the ground-state SCF wave function
is unchanged by a unitary transformation of the occupied orbitals, it is
possible to define an improved set of occupied orbitals which provide a
somewhat better representation of the "hole." This can even be done in a
self-consistent manner for each excited state (Morokuma and Iwata, 1972).
For an accurate description of the excited state, it is often convenient to
begin with fully relaxed orbitals. That is, the best one-configuration descrip
tion of the excited state is sought without any restrictions on the orbitals
(other than perhaps symmetry and spin restrictions) (Davidson and Sten-
kamp, 1976). For high-spin states of small molecules, this usually leads to an
improved first approximation to the excited-state wave function. Unfor
tunately for these states, this "orbital relaxation energy" is of the same
magnitude and opposite sign as the "differential correlation error." Con
sequently, an improved excited-state wave function leads to a worse estimate
of the excitation energy.
There are some circumstances in which the relaxed orbital configuration
may also be a worse description of the excited state as measured by its overlap
with the exact wave function. For example, the Is hole state of F 2 leads to a
localized Is hole on one fluorine atom (Bagus, 1965; Martin and Davidson,
1977). While this gives a distinctly improved energy, the wave function is
3
qualitatively incorrect. Similarly, for the wr* state of glyoxal (Nitzsche and
Davidson, 1978a) and many other molecules, the SCF description of the
excitation gives localized half-filled orbitals with broken symmetry even for
symmetrical nuclear configurations.
Excited singlet states are even more of a problem since excited singlets
involving valence virtual orbitals have larger correlation errors than those
involving Rydberg orbitals. Consequently, the SCF method may well lead to
