For the CV(1)-scheme we note that the singlet excitation energy for a transition
involving a single promotion i ! a such as
1 A 1g 1g !
1 B 2u 2u according to (27)–
(29) has the simple form [27]:
ΔE
CV 1
ð Þ
S
i ! a
ð
Þ¼ε a À ε i þ
1
2
K aaaa þ
1
2
K iiii þ 2K aaii À K iiaa
ð30Þ
Here i is the HOMO π and a the LUMO π* in the current study. Further, a bar “
– ”
indicates an orbital of β-spin. For CV(2)-TD we obtain for the same transition
according to (14) and (27)
ΔE
CV 2
ð Þ
S
i ! a
ð
Þ¼ε a À ε i þ 2K aiai À K aiai
ð31Þ
For HF these two expressions are identical [27] because K aaaa ¼ K iiii ¼ 0 and
K aiai ¼ ÀK aaii . However, for any of the popular functionals this is not the case.
Thus, the two expressions give rise to different excitation energies for the same
functional. In the study at hand [27] on the
1 A 1g 1g !
1 B 2u 2u transition the sum of
the K-integrals in (30) is larger than the sum of the K-integrals in (31) by 0.5 eV,
giving rise to the better performance of CV(1)-TD compared to CV(2)-TD for the
1 A 1g 1g !
1 B 2u 2u transition. We note that acenes have also been well described by
the variational DFT-based spin-restricted ensemble referenced Kohn–Sham
(REKS) method [57].
1
2.4 Self-Consistent All Order Constricted Variational
Density Functional Theory
Using the U matrix from TDDFT-TD or CV(2)-TD to calculate ΔE M of (28) or ΔE T
of (29), as is done in P-CV(1)-TD, might be a good approximation. However,
ultimately, one would want to use a U matrix which actually minimizes ΔE M of (28)
or ΔE T of (29). Such a procedure leads us to self-consistent CV(1)-DFT (SCF-CV
(1)-DFT) [27] which we discuss next. We note that the U matrix can also be
organized as a vector U
!
with pairs “ai” of occupied and virtual orbitals as running
numbers. The two formulations are used interchangeably in the following.
From the occupied excited state orbitals of (17) and (23) we can express the
electron density and spin matrices.
2,3 Starting with a spin-conserving transition
from a close shell ground state, we can, without loss of generality, assume that it
1 See the chapter “Ensemble DFT approach to excited states of strongly correlated molecular
systems” by M. Filatov.
2 See Sect. 3.1 from part S1 of supporting information in Ziegler et al. [27].
3 See Sect. 3.3 from part S1 of supporting information in Ziegler et al. [27].
Constricted Variational Density Functional Theory Approach to the. . .
75
involving a single promotion i ! a such as
1 A 1g 1g !
1 B 2u 2u according to (27)–
(29) has the simple form [27]:
ΔE
CV 1
ð Þ
S
i ! a
ð
Þ¼ε a À ε i þ
1
2
K aaaa þ
1
2
K iiii þ 2K aaii À K iiaa
ð30Þ
Here i is the HOMO π and a the LUMO π* in the current study. Further, a bar “
– ”
indicates an orbital of β-spin. For CV(2)-TD we obtain for the same transition
according to (14) and (27)
ΔE
CV 2
ð Þ
S
i ! a
ð
Þ¼ε a À ε i þ 2K aiai À K aiai
ð31Þ
For HF these two expressions are identical [27] because K aaaa ¼ K iiii ¼ 0 and
K aiai ¼ ÀK aaii . However, for any of the popular functionals this is not the case.
Thus, the two expressions give rise to different excitation energies for the same
functional. In the study at hand [27] on the
1 A 1g 1g !
1 B 2u 2u transition the sum of
the K-integrals in (30) is larger than the sum of the K-integrals in (31) by 0.5 eV,
giving rise to the better performance of CV(1)-TD compared to CV(2)-TD for the
1 A 1g 1g !
1 B 2u 2u transition. We note that acenes have also been well described by
the variational DFT-based spin-restricted ensemble referenced Kohn–Sham
(REKS) method [57].
1
2.4 Self-Consistent All Order Constricted Variational
Density Functional Theory
Using the U matrix from TDDFT-TD or CV(2)-TD to calculate ΔE M of (28) or ΔE T
of (29), as is done in P-CV(1)-TD, might be a good approximation. However,
ultimately, one would want to use a U matrix which actually minimizes ΔE M of (28)
or ΔE T of (29). Such a procedure leads us to self-consistent CV(1)-DFT (SCF-CV
(1)-DFT) [27] which we discuss next. We note that the U matrix can also be
organized as a vector U
!
with pairs “ai” of occupied and virtual orbitals as running
numbers. The two formulations are used interchangeably in the following.
From the occupied excited state orbitals of (17) and (23) we can express the
electron density and spin matrices.
2,3 Starting with a spin-conserving transition
from a close shell ground state, we can, without loss of generality, assume that it
1 See the chapter “Ensemble DFT approach to excited states of strongly correlated molecular
systems” by M. Filatov.
2 See Sect. 3.1 from part S1 of supporting information in Ziegler et al. [27].
3 See Sect. 3.3 from part S1 of supporting information in Ziegler et al. [27].
Constricted Variational Density Functional Theory Approach to the. . .
75