Theor Chem Acc (2015) 134:118
1 3
stationary orbitals (including both occupied and virtual
orbitals), namely
In the TD-GVB equations, the elements of the vector ˜
Y are
linked to perturbations δU pq (t) and the range of indices of
the ˜
Y pq elements is as follows
where M basis stands for a number of basis set functions
employed in calculations. The value of the ˜
Y pq element can
be interpreted as a contribution to a given excitation from
a transition either between two occupied orbitals (p ≤ N)
or between occupied and virtual orbitals (p > N) [ 20 , 21 ].
Notice at this point that the dimensions of the analogous ˜
Y
vectors present in the TD-APSG and TD-HF equations are,
respectively, higher and lower. The indices of the respective vectors change as follows
Thus, in TD-HF there are only occupied–virtual transitions, while in TD-APSG, all combinations are possible
(also transitions between weakly occupied orbitals). The
˜
W component of the eigenvector in Eq. ( 19 ) comprises
only N elements
and is related to responses of the coeffi cients
c p
. In
the TD-APSG method, on the other hand, the ˜
W vector is longer since it is composed of M basis elements (also
responses of the c p coeffi cients pertaining to weakly occupied orbitals are taken into account)
Since all orbitals in the HF approximation are either
fully occupied (n p = 1) or virtual (n p = 0) the fi rst-order
responses of the occupation numbers are zero and eigenvectors in the TD-HF equations comprise only ˜
Y components. It has been discussed in Ref. [ 18 ] that ˜
W in
TD-APSG is nonzero only for excitations to totally symmetric states. Nonzero contributions from ˜
W p elements or
˜
Y pq when both p, q > N/2 (transition between two weakly
occupied orbitals) indicates a double character of a given
excitation [ 18 , 22 , 23 ].
It is worth mentioning that excitation energies can be
also obtained from the GVB-PP ground-state wavefunction
by employing the extended random phase approximation
(20)
∀ p≤N ϕ
(1)
p (r, t) =
q
δU pq (t)ϕ q (r).
(21)
TD-GVB ˜
Y pq : p = q + 1, . . . , M basis and q = 1, . . . , N,
(22)
TD-APSG ˜
Y pq : p = q + 1, . . . , M basis and q = 1, . . . , M basis ,
(23)
TD-HF ˜
Y pq : p = N/2 + 1, . . . , M basis and q = 1, . . . , N/2,
(24)
TD-GVB ˜
W p : p = 1, . . . , N,
(25)
TD-APSG ˜
W p : p = 1, . . . , M basis .
(ERPA) in the framework of the Rowe’s equations of
motion formalism [ 24 ] by [ 17 ]. ERPA equations are of the
form
where the matrices A + , A − are identical to the pertinent
matrices present in the TD-GVB Equations ( 19 ). ERPA
excitations are only different from their TD-GVB counterparts if the excitation in question is totally symmetric. For
other symmetries, ERPA and TD-GVB solutions coincide
by construction.
Finally, we want to point out to the connection of the
TD-GVB equations with the recently formulated TD-PINO
(time-dependent phase including natural orbital) formalism [ 25 – 27 ]. TD-PINO is an extension of the TD-RDMFT
(time-dependent reduced density matrix functional theory)
formalism to functionals depending not only on the natural
occupation numbers and the natural spinorbitals but also on
the phases of the latter. The phases are not generic features
of the natural spinorbitals, i.e., they cannot be recovered
from diagonalization of the one-electron density matrix.
Rather they should be seen as additional variational parameters of the functional. Since it has been shown that the
GVB-PP energy function ( 10 ) is identical to PNOF5—one
of the phase including natural orbital functionals [ 14 ]—
then, from the perspective of the natural orbital functional
theory, the TD-GVB equations ( 19 ) can be seen as TDPINO equations [ 18 , 26 ] applied to the PNOF5 functional.
In other words, the results discussed in the next section can
be viewed as the TD-PINO (in the adiabatic approximation) excitation energies obtained by employing the PNOF5
functional.
3 Results
The results presented in this section have been obtained in
symmetry adapted orbital basis sets. To stabilize the TDAPSG and TD-GVB equations and avoid obtaining occasionally spurious excitation energies (cf. a discussion in
Ref. [ 18 ]), we have assumed the lower and upper cutoffs
for sums of pairs of occupation numbers that are included
in calculations for LiH and H 2 O molecules. In other
words, the accepted elements ˜
Y pq of the eigenvectors are
such that ∀ p>q 1 × 10 −4 < n p + n q < 1.98 .The matrices
A + , A − , D + , D − are truncated accordingly.
We begin with the 1 Σ +
g and 1 Σ +
u excitation energy
curves for the hydrogen molecule presented in Figs. 1 and
2 . They have been obtained by employing the TD-APSG,
TD-GVB, and TD-HF methods in the aug-cc-pVTZ basis
set [ 28 ]. The TD-APSG approach yields exact (in a given
basis set) excitation energies since it is equivalent to the
FCI method [ 22 , 23 ] for two-electron systems. In the
(26)
A
+ A
− ˜
Y = ω
2 ˜
Y,
222
Reprinted from the journal
1 3
stationary orbitals (including both occupied and virtual
orbitals), namely
In the TD-GVB equations, the elements of the vector ˜
Y are
linked to perturbations δU pq (t) and the range of indices of
the ˜
Y pq elements is as follows
where M basis stands for a number of basis set functions
employed in calculations. The value of the ˜
Y pq element can
be interpreted as a contribution to a given excitation from
a transition either between two occupied orbitals (p ≤ N)
or between occupied and virtual orbitals (p > N) [ 20 , 21 ].
Notice at this point that the dimensions of the analogous ˜
Y
vectors present in the TD-APSG and TD-HF equations are,
respectively, higher and lower. The indices of the respective vectors change as follows
Thus, in TD-HF there are only occupied–virtual transitions, while in TD-APSG, all combinations are possible
(also transitions between weakly occupied orbitals). The
˜
W component of the eigenvector in Eq. ( 19 ) comprises
only N elements
and is related to responses of the coeffi cients
c p
. In
the TD-APSG method, on the other hand, the ˜
W vector is longer since it is composed of M basis elements (also
responses of the c p coeffi cients pertaining to weakly occupied orbitals are taken into account)
Since all orbitals in the HF approximation are either
fully occupied (n p = 1) or virtual (n p = 0) the fi rst-order
responses of the occupation numbers are zero and eigenvectors in the TD-HF equations comprise only ˜
Y components. It has been discussed in Ref. [ 18 ] that ˜
W in
TD-APSG is nonzero only for excitations to totally symmetric states. Nonzero contributions from ˜
W p elements or
˜
Y pq when both p, q > N/2 (transition between two weakly
occupied orbitals) indicates a double character of a given
excitation [ 18 , 22 , 23 ].
It is worth mentioning that excitation energies can be
also obtained from the GVB-PP ground-state wavefunction
by employing the extended random phase approximation
(20)
∀ p≤N ϕ
(1)
p (r, t) =
q
δU pq (t)ϕ q (r).
(21)
TD-GVB ˜
Y pq : p = q + 1, . . . , M basis and q = 1, . . . , N,
(22)
TD-APSG ˜
Y pq : p = q + 1, . . . , M basis and q = 1, . . . , M basis ,
(23)
TD-HF ˜
Y pq : p = N/2 + 1, . . . , M basis and q = 1, . . . , N/2,
(24)
TD-GVB ˜
W p : p = 1, . . . , N,
(25)
TD-APSG ˜
W p : p = 1, . . . , M basis .
(ERPA) in the framework of the Rowe’s equations of
motion formalism [ 24 ] by [ 17 ]. ERPA equations are of the
form
where the matrices A + , A − are identical to the pertinent
matrices present in the TD-GVB Equations ( 19 ). ERPA
excitations are only different from their TD-GVB counterparts if the excitation in question is totally symmetric. For
other symmetries, ERPA and TD-GVB solutions coincide
by construction.
Finally, we want to point out to the connection of the
TD-GVB equations with the recently formulated TD-PINO
(time-dependent phase including natural orbital) formalism [ 25 – 27 ]. TD-PINO is an extension of the TD-RDMFT
(time-dependent reduced density matrix functional theory)
formalism to functionals depending not only on the natural
occupation numbers and the natural spinorbitals but also on
the phases of the latter. The phases are not generic features
of the natural spinorbitals, i.e., they cannot be recovered
from diagonalization of the one-electron density matrix.
Rather they should be seen as additional variational parameters of the functional. Since it has been shown that the
GVB-PP energy function ( 10 ) is identical to PNOF5—one
of the phase including natural orbital functionals [ 14 ]—
then, from the perspective of the natural orbital functional
theory, the TD-GVB equations ( 19 ) can be seen as TDPINO equations [ 18 , 26 ] applied to the PNOF5 functional.
In other words, the results discussed in the next section can
be viewed as the TD-PINO (in the adiabatic approximation) excitation energies obtained by employing the PNOF5
functional.
3 Results
The results presented in this section have been obtained in
symmetry adapted orbital basis sets. To stabilize the TDAPSG and TD-GVB equations and avoid obtaining occasionally spurious excitation energies (cf. a discussion in
Ref. [ 18 ]), we have assumed the lower and upper cutoffs
for sums of pairs of occupation numbers that are included
in calculations for LiH and H 2 O molecules. In other
words, the accepted elements ˜
Y pq of the eigenvectors are
such that ∀ p>q 1 × 10 −4 < n p + n q < 1.98 .The matrices
A + , A − , D + , D − are truncated accordingly.
We begin with the 1 Σ +
g and 1 Σ +
u excitation energy
curves for the hydrogen molecule presented in Figs. 1 and
2 . They have been obtained by employing the TD-APSG,
TD-GVB, and TD-HF methods in the aug-cc-pVTZ basis
set [ 28 ]. The TD-APSG approach yields exact (in a given
basis set) excitation energies since it is equivalent to the
FCI method [ 22 , 23 ] for two-electron systems. In the
(26)
A
+ A
− ˜
Y = ω
2 ˜
Y,
222
Reprinted from the journal