ΔE T ¼
X
occ=2
i¼1
sin
2
 η
0
γ
0
i
à ε i
v β À ε i
oα
ð
Þ
þ
1
2
X
occ=2
i¼1
X
occ=2
j¼1
sin
2
η
0
γ
0
i
h i
sin
2
 η
0
γ
0
i
Ã
K i
oα i
oα j
oα j
oα þ K i
v β i
v β j
v β j
v β À 2K i
oα i
oα j
v β j
v β
À
Á
þ
X
occ=2
i¼1
X
occ=2
j¼1
sin η
0
γ
0
i
h i
cos η
0
γ
0
i
h i
sin
Â
η
0
γ
0
j
Ã
cos η
0
γ
0
j
h i
K i
oα i
v β j
oα j
v β
ð29Þ
Here the indices i
v β and j
v β refer to β-spin orbitals with the spatial parts ϕ
v β
i and ϕ
v β
i ,
respectively. More details are given in Sect. 2.4. From (28) and (29) we can readily
express ΔE S using (27).
In perturbative all order constricted variational DFT (P-CV(1)-DFT) [27, 64]
we make use of the U matrix optimized to second order according to (14), which is
also the U obtained by ATDDFT-TD (CV(2)-DFT-TD). With this U we are able to
generate Ψ M of (17) and Ψ T of (23) by means of (2a) and (2b). From that we can
calculate ΔE T and ΔE S by means of (27)–(29).
2.3.1 Application of Perturbative All Order Constricted Variational
Density Functional Theory to π ! π*
We have carried out ATDDFT-TD, (CV(2)-DFT-TD), and P-CV(1)-DFT calculations [27] on π ! π* transitions in the series of polyenes depicted in Fig. 1 using
LDA. In the following we simplify the notation by referring to ATDDFT,
ATDDFT-TD, CV(2)-DFT-TD, and P-CV(1)-DFT-TD as TDDFT, TDDFT-TD,
CV(2)-TD, and CV(1)-TD, respectively, throughout Sect. 2.3.
The results are displayed in Table 1. We have divided the π ! π* transitions into
a group A where each excitation is dominated by a single orbital replacement (γ max
> 1:0 ) and a group B where the excitation is best described by several orbital
replacements (γ max < 1:0). The group B transitions generally consist of two orbital
replacements involving the HOMO ! LUMO + 1 and the HOMO À 1 ! LUMO
transitions. It can be seen that the group B results for P-CV(1)-TD with a root
Ethene
E-Butadiene
E-Hexatriene
E-Octatriene
Cyclopropene
Cyclopentadiene
Norbonadiene
Naphthalene
Fig. 1 Molecules used in the study of π ! π* transitions based on P-CV(1)-DFT
70
T. Ziegler et al.
X
occ=2
i¼1
sin
2
 η
0
γ
0
i
à ε i
v β À ε i
oα
ð
Þ
þ
1
2
X
occ=2
i¼1
X
occ=2
j¼1
sin
2
η
0
γ
0
i
h i
sin
2
 η
0
γ
0
i
Ã
K i
oα i
oα j
oα j
oα þ K i
v β i
v β j
v β j
v β À 2K i
oα i
oα j
v β j
v β
À
Á
þ
X
occ=2
i¼1
X
occ=2
j¼1
sin η
0
γ
0
i
h i
cos η
0
γ
0
i
h i
sin
Â
η
0
γ
0
j
Ã
cos η
0
γ
0
j
h i
K i
oα i
v β j
oα j
v β
ð29Þ
Here the indices i
v β and j
v β refer to β-spin orbitals with the spatial parts ϕ
v β
i and ϕ
v β
i ,
respectively. More details are given in Sect. 2.4. From (28) and (29) we can readily
express ΔE S using (27).
In perturbative all order constricted variational DFT (P-CV(1)-DFT) [27, 64]
we make use of the U matrix optimized to second order according to (14), which is
also the U obtained by ATDDFT-TD (CV(2)-DFT-TD). With this U we are able to
generate Ψ M of (17) and Ψ T of (23) by means of (2a) and (2b). From that we can
calculate ΔE T and ΔE S by means of (27)–(29).
2.3.1 Application of Perturbative All Order Constricted Variational
Density Functional Theory to π ! π*
We have carried out ATDDFT-TD, (CV(2)-DFT-TD), and P-CV(1)-DFT calculations [27] on π ! π* transitions in the series of polyenes depicted in Fig. 1 using
LDA. In the following we simplify the notation by referring to ATDDFT,
ATDDFT-TD, CV(2)-DFT-TD, and P-CV(1)-DFT-TD as TDDFT, TDDFT-TD,
CV(2)-TD, and CV(1)-TD, respectively, throughout Sect. 2.3.
The results are displayed in Table 1. We have divided the π ! π* transitions into
a group A where each excitation is dominated by a single orbital replacement (γ max
> 1:0 ) and a group B where the excitation is best described by several orbital
replacements (γ max < 1:0). The group B transitions generally consist of two orbital
replacements involving the HOMO ! LUMO + 1 and the HOMO À 1 ! LUMO
transitions. It can be seen that the group B results for P-CV(1)-TD with a root
Ethene
E-Butadiene
E-Hexatriene
E-Octatriene
Cyclopropene
Cyclopentadiene
Norbonadiene
Naphthalene
Fig. 1 Molecules used in the study of π ! π* transitions based on P-CV(1)-DFT
70
T. Ziegler et al.