1 3
Theor Chem Acc (2015) 134:118
DOI 10.1007/s00214-015-1718-3
REGULAR ARTICLE
Excitation energies from time-dependent generalized valence
bond method
Koushik Chatterjee
1 · Katarzyna Pernal
2
Received: 1 July 2015 / Accepted: 1 September 2015 / Published online: 18 September 2015
© Springer-Verlag Berlin Heidelberg 2015
describing bond breaking or forming processes since the
valence bond wavefunction has a proper form in the dissociation limit [ 2 ]. Generalized valence bond (GVB) method
is a variational counterpart to VB in which orbitals are
optimized self-consistently [ 3 – 6 ]. In the GVB, wavefunction orbitals comprising a breaking bond form a GVB pair
where the orbitals φ a and φ b are overlapping. The remaining
electrons are treated at the Hartree–Fock (HF) level. A special
case of the GVB wavefunction is obtained if all electron pairs in
a molecule are described by distinct, strongly orthogonal GVB
functions formed from the (orthogonal) natural orbitals [ 6 , 7 ]
A method based on the ansatz for the wavefunction comprised of the antisymmetrized product of the GVB pairs
( 2 ) is known as perfect-pairing GVB (GVB-PP) [ 3 , 6 ].
Unlike in the full GVB wavefunction, which allows all
possible spin couplings of different orbitals, GVB-PP is
more restrictive, allowing only perfect-pairing (singlet)
coupling of the spins. The immediate advantage of this
restriction is a gain in computational effi ciency of the
GVB-PP optimization compared to the full GVB ansatz.
The GVB-PP wavefunction retains a desired property of
the GVB approximation, namely it accounts for a left–
right correlation between electrons in each pair and allows
for bond breaking.
The GVB-PP wavefunction can also be derived from of
the strongly orthogonal geminals theory as a special case of
the APSG (antisymmetrized product of strongly orthogonal
geminal) ansatz [ 7 – 10 ] if each APSG geminal is restricted
to be expanded in a two-dimensional subset of orbitals.
(1)
ψ
GVB
pair = [φ 1 (1)φ 2 (2) + φ 2 (1)φ 1 (2)](αβ − βα),
(2)
ψ
GVB−PP
pair
=
c 1 ϕ 1 (1)ϕ 1 (2) + c 2 ϕ 2 (1)ϕ 2 (2)
(αβ − βα).
Abstract A generalized valence bond perfect-pairing (GVBPP) wavefunction has been extensively used in computational
chemistry methods due to its multiconfi gurational character,
which captures in an inexpensive way static electron correlation. GVB-PP has been mostly applied to ground states and
not much is known about its performance in predicting electronic spectra of molecules. Here, we present the formalism
based on the time-dependent linear response theory which
provides excitation energies from the GVB-PP ground-state
wavefunction. The accuracy of the excitation energy dissociation curves parallels that of the TD-HF method around groundstate equilibrium geometries of the investigated molecules.
For stretched-bond molecules, when TD-HF breaks down, the
proposed TD-GVB method remains reliable. TD-GVB may
therefore serve as a useful and inexpensive tool for exploring
potential energy surfaces of excited states of molecules.
Keywords Generalized valence bond · Time-dependent
linear response · Excitation energy
1 Introduction
Valence bond (VB) theory has been developed since the
early days of quantum chemistry in parallel with molecular orbital theory [ 1 ]. The VB is particularly well suited for
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Katarzyna Pernal
pernalk@gmail.com
1
Faculty of Chemistry , Lodz University of Technology ,
ul. Zeromskiego 116 , 90-924 Lodz , Poland
2
Institute of Physics , Lodz University of Technology ,
ul. Wolczanska 219 , 90-924 Lodz , Poland
219
Reprinted from the journal
Theor Chem Acc (2015) 134:118
DOI 10.1007/s00214-015-1718-3
REGULAR ARTICLE
Excitation energies from time-dependent generalized valence
bond method
Koushik Chatterjee
1 · Katarzyna Pernal
2
Received: 1 July 2015 / Accepted: 1 September 2015 / Published online: 18 September 2015
© Springer-Verlag Berlin Heidelberg 2015
describing bond breaking or forming processes since the
valence bond wavefunction has a proper form in the dissociation limit [ 2 ]. Generalized valence bond (GVB) method
is a variational counterpart to VB in which orbitals are
optimized self-consistently [ 3 – 6 ]. In the GVB, wavefunction orbitals comprising a breaking bond form a GVB pair
where the orbitals φ a and φ b are overlapping. The remaining
electrons are treated at the Hartree–Fock (HF) level. A special
case of the GVB wavefunction is obtained if all electron pairs in
a molecule are described by distinct, strongly orthogonal GVB
functions formed from the (orthogonal) natural orbitals [ 6 , 7 ]
A method based on the ansatz for the wavefunction comprised of the antisymmetrized product of the GVB pairs
( 2 ) is known as perfect-pairing GVB (GVB-PP) [ 3 , 6 ].
Unlike in the full GVB wavefunction, which allows all
possible spin couplings of different orbitals, GVB-PP is
more restrictive, allowing only perfect-pairing (singlet)
coupling of the spins. The immediate advantage of this
restriction is a gain in computational effi ciency of the
GVB-PP optimization compared to the full GVB ansatz.
The GVB-PP wavefunction retains a desired property of
the GVB approximation, namely it accounts for a left–
right correlation between electrons in each pair and allows
for bond breaking.
The GVB-PP wavefunction can also be derived from of
the strongly orthogonal geminals theory as a special case of
the APSG (antisymmetrized product of strongly orthogonal
geminal) ansatz [ 7 – 10 ] if each APSG geminal is restricted
to be expanded in a two-dimensional subset of orbitals.
(1)
ψ
GVB
pair = [φ 1 (1)φ 2 (2) + φ 2 (1)φ 1 (2)](αβ − βα),
(2)
ψ
GVB−PP
pair
=
c 1 ϕ 1 (1)ϕ 1 (2) + c 2 ϕ 2 (1)ϕ 2 (2)
(αβ − βα).
Abstract A generalized valence bond perfect-pairing (GVBPP) wavefunction has been extensively used in computational
chemistry methods due to its multiconfi gurational character,
which captures in an inexpensive way static electron correlation. GVB-PP has been mostly applied to ground states and
not much is known about its performance in predicting electronic spectra of molecules. Here, we present the formalism
based on the time-dependent linear response theory which
provides excitation energies from the GVB-PP ground-state
wavefunction. The accuracy of the excitation energy dissociation curves parallels that of the TD-HF method around groundstate equilibrium geometries of the investigated molecules.
For stretched-bond molecules, when TD-HF breaks down, the
proposed TD-GVB method remains reliable. TD-GVB may
therefore serve as a useful and inexpensive tool for exploring
potential energy surfaces of excited states of molecules.
Keywords Generalized valence bond · Time-dependent
linear response · Excitation energy
1 Introduction
Valence bond (VB) theory has been developed since the
early days of quantum chemistry in parallel with molecular orbital theory [ 1 ]. The VB is particularly well suited for
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Katarzyna Pernal
pernalk@gmail.com
1
Faculty of Chemistry , Lodz University of Technology ,
ul. Zeromskiego 116 , 90-924 Lodz , Poland
2
Institute of Physics , Lodz University of Technology ,
ul. Wolczanska 219 , 90-924 Lodz , Poland
219
Reprinted from the journal