Theor Chem Acc (2015) 134:118
1 3
breaking so the ground state of the water molecule with
the elongated bond is qualitatively correctly described
unlike in the HF approximation. Thus, in the case of
the first two 1 A
and the first 1 A
excitation energy the
TD-HF curves break down when R increases, whereas
the TD-GVB and TD-APSG curves stay together and
they parallel the CCSD reference. The second 1 A
excitation, on the other hand, does not include contributions from the orbitals involved in bond breaking but
from orbitals localized on the OH bond of a constant
length. The TD-GVB method does not outperform the
TD-HF approach for this excitation.
4 Conclusions
The purpose of this paper was to fi ll a gap and formulate
time-dependent linear response equations for the GVBPP wavefunction and to show performance of the resulting TD-GVB method in predicting excitation energies of
molecules with stretched bonds. Although we have only
investigated a few very small systems, the obtained results
allow one to formulate general conclusions about quality
of the TD-GVB excitations. Each geminal in the GVBPP ansatz includes only two orbitals in its expansion as
opposed to APSG geminals which comprise an unrestricted
number of orbitals. Thus, the GVB-PP wavefunction (and
the corresponding reduced density matrices) misses more
dynamic correlation than its APSG counterpart. However,
this seems to have little effect on the quality of excitation energies obtained from the linear response methods.
Each excitation involves transitions from strongly occupied orbitals (bonding, antibonding, lone pairs) to weakly
occupied orbitals (including antibonding orbitals). By confronting the structure of the eigenvectors in the TD-GVB
equations [Eq. ( 21 )] and in the TD-APSG [Eq. ( 22 ], it is
clear that in both methods such transitions are accounted
for. The main difference between the two approaches is
that orbitals which are purely virtual in the GVB method
(their occupation is exactly 0) are nearly virtual in APSG,
i.e., the corresponding natural occupation numbers are
greater than 0 (typically their values are of the order of
10 −3 or less). Apparently, whether the orbital is exactly or
nearly virtual affects the excitation energies to only a small
extent and the accuracy of the TD-GVB, and TD-APSG
excitations is observed to be comparable. Computationally, TD-GVB is more advantageous then the TD-APSG
method due to the smaller computational effort in obtaining
the GVB-PP ground state than that of the APSG and lower
dimensionality of the TD-GVB eigenproblem comparing
to TD-APSG. Unfortunately, when applied to molecules in
ground-state equilibrium geometries, the TD-GVB does not
lead to obtaining excitations of substantially better accuracy than the uncorrelated TD-HF method. When bonds
are stretched, however, TD-GVB benefi ts from the correct
description of a ground state and the method is much more
reliable than TD-HF.
Taking into account that (1) solving the TD-GVB equations can be used to correct ground-state GVB-PP energies [ 31 – 33 ], (2) the TD-GVB equations lead to obtaining
often reasonably accurate excitation energies, and fi nally
that (3) the calculations can be done at modest computational cost, the TD-GVB approximation may be a useful
tool in exploring potential energy surfaces of excited states
of molecules.
Finally, we want to stress out that since the GVB-PP
approximation is equivalent to PNOF5 [ 13 , 14 ]—one of
the natural orbital functionals—the results presented in
this paper can be seen as examples of performance of the
PNOF5 functional in the framework of the recently proposed TD-PINO theory [ 26 ].
Acknowledgments This work has been supported by the National
Science Centre of Poland under grant DEC-2012/07/E/ST4/03023.
K.C. acknowledges fi nancial support under Grant DEC-2014/13/N/
ST4/03990.
Appendix
The matrices A + , A − , D + , D − and E + , E − appearing in the
TD-GVB Eq. ( 19 ) are determined by the optimal natural
orbitals
ϕ p (r)
and the expansion coeffi cients {c p } resulting from the optimization of the GVB-PP energy expression, Eq. ( 10 ). Their elements read
(27)
∀ p>q
r>s
A +
pq,rs = (c p + c q ) −1 (A pq,rs + B pq,rs )(c r + c s ) −1 ,
(28)
∀ p>q
r>s
A −
rs,pq = (c p − c q ) −1 (A pq,rs − B pq,rs )(c r − c s ) −1 ,
(29)
∀ p>q
r
D
+
pq,r =
B pq,rr
2(c p + c q )c r
,
(30)
∀ pq E
+
pq =
B pp,qq
4c p c q
,
225
Reprinted from the journal
1 3
breaking so the ground state of the water molecule with
the elongated bond is qualitatively correctly described
unlike in the HF approximation. Thus, in the case of
the first two 1 A
and the first 1 A
excitation energy the
TD-HF curves break down when R increases, whereas
the TD-GVB and TD-APSG curves stay together and
they parallel the CCSD reference. The second 1 A
excitation, on the other hand, does not include contributions from the orbitals involved in bond breaking but
from orbitals localized on the OH bond of a constant
length. The TD-GVB method does not outperform the
TD-HF approach for this excitation.
4 Conclusions
The purpose of this paper was to fi ll a gap and formulate
time-dependent linear response equations for the GVBPP wavefunction and to show performance of the resulting TD-GVB method in predicting excitation energies of
molecules with stretched bonds. Although we have only
investigated a few very small systems, the obtained results
allow one to formulate general conclusions about quality
of the TD-GVB excitations. Each geminal in the GVBPP ansatz includes only two orbitals in its expansion as
opposed to APSG geminals which comprise an unrestricted
number of orbitals. Thus, the GVB-PP wavefunction (and
the corresponding reduced density matrices) misses more
dynamic correlation than its APSG counterpart. However,
this seems to have little effect on the quality of excitation energies obtained from the linear response methods.
Each excitation involves transitions from strongly occupied orbitals (bonding, antibonding, lone pairs) to weakly
occupied orbitals (including antibonding orbitals). By confronting the structure of the eigenvectors in the TD-GVB
equations [Eq. ( 21 )] and in the TD-APSG [Eq. ( 22 ], it is
clear that in both methods such transitions are accounted
for. The main difference between the two approaches is
that orbitals which are purely virtual in the GVB method
(their occupation is exactly 0) are nearly virtual in APSG,
i.e., the corresponding natural occupation numbers are
greater than 0 (typically their values are of the order of
10 −3 or less). Apparently, whether the orbital is exactly or
nearly virtual affects the excitation energies to only a small
extent and the accuracy of the TD-GVB, and TD-APSG
excitations is observed to be comparable. Computationally, TD-GVB is more advantageous then the TD-APSG
method due to the smaller computational effort in obtaining
the GVB-PP ground state than that of the APSG and lower
dimensionality of the TD-GVB eigenproblem comparing
to TD-APSG. Unfortunately, when applied to molecules in
ground-state equilibrium geometries, the TD-GVB does not
lead to obtaining excitations of substantially better accuracy than the uncorrelated TD-HF method. When bonds
are stretched, however, TD-GVB benefi ts from the correct
description of a ground state and the method is much more
reliable than TD-HF.
Taking into account that (1) solving the TD-GVB equations can be used to correct ground-state GVB-PP energies [ 31 – 33 ], (2) the TD-GVB equations lead to obtaining
often reasonably accurate excitation energies, and fi nally
that (3) the calculations can be done at modest computational cost, the TD-GVB approximation may be a useful
tool in exploring potential energy surfaces of excited states
of molecules.
Finally, we want to stress out that since the GVB-PP
approximation is equivalent to PNOF5 [ 13 , 14 ]—one of
the natural orbital functionals—the results presented in
this paper can be seen as examples of performance of the
PNOF5 functional in the framework of the recently proposed TD-PINO theory [ 26 ].
Acknowledgments This work has been supported by the National
Science Centre of Poland under grant DEC-2012/07/E/ST4/03023.
K.C. acknowledges fi nancial support under Grant DEC-2014/13/N/
ST4/03990.
Appendix
The matrices A + , A − , D + , D − and E + , E − appearing in the
TD-GVB Eq. ( 19 ) are determined by the optimal natural
orbitals
ϕ p (r)
and the expansion coeffi cients {c p } resulting from the optimization of the GVB-PP energy expression, Eq. ( 10 ). Their elements read
(27)
∀ p>q
r>s
A +
pq,rs = (c p + c q ) −1 (A pq,rs + B pq,rs )(c r + c s ) −1 ,
(28)
∀ p>q
r>s
A −
rs,pq = (c p − c q ) −1 (A pq,rs − B pq,rs )(c r − c s ) −1 ,
(29)
∀ p>q
r
D
+
pq,r =
B pq,rr
2(c p + c q )c r
,
(30)
∀ pq E
+
pq =
B pp,qq
4c p c q
,
225
Reprinted from the journal