ϕ
o α
i ¼
X
occ=2
j
À
W
βα
Á
ji ϕ j
ð25Þ
and
ϕ
v β
i ¼
X
a¼vir
a¼vir=2þ1
À
V
βα
Á
ai ϕ a
ð26Þ
With the excitation energy of Ψ T given by ΔE T and that of Ψ M by ΔE M , we can
write the singlet transition energy as [40]
ΔE S ¼ 2ΔE M À ΔE T
ð27Þ
provided that U
αα
¼ U
βα . This implies that V
αα
¼ V
βα , W
αα
¼ W
βα , and γ ¼ γ
0 . As
a result, ϕ
o α
i of (19) and (25) become identical as do the spatial parts of ϕ
v α
i in (20)
and ϕ
v β
i in (26). Straightforward manipulations [27, 28] allow us finally to write
down the mixed state transition energy to all orders in U in a compact and closed
form as
ΔE M ¼
X
occ=2
i¼1
sin
2
 ηγ i
à ε i
vα À ε i
oα
ð
Þ
þ
1
2
X
occ=2
i¼1
X
occ=2
j¼1
sin
2
ηγ i
½ Šsin
2
 ηγ j
Ã
K i
oα i
oα j
oα j
oα þ K i
vα i
vα j
vα j
vα À 2K i
oα i
oα j
vα j
vα
À
Á
þ
X
occ=2
i¼1
X
occ=2
j¼1
sin ηγ i
½ Šcos ηγ i
½ Šsin
Â
ηγ j
Ã
cos ηγ j
 Ã
K i
oα i
vα j
oα j
vα þ K i
oα i
vα j
vα j
oα
À
Á
þ 2
X
occ=2
i¼1
X
occ=2
j¼1
sin ηγ i
½ Šsin ηγ i
½ Šsin
 ηγ j
Ã
cos ηγ j
 Ã
K i
vα i
vα j
oα j
vα
À 2
X
occ=2
i¼1
X
occ=2
j¼1
sin ηγ i
½ Šsin ηγ i
½ Šsin
 ηγ j
Ã
cos ηγ j
 Ã
K i
oα i
oα j
oα j
vα
ð28Þ
for spin-conserving transition from a close shell ground state. More details are
given in Sect. 2.4. Here the indices i
o α , j
o α , i
v α , and j
v α refer to α-spin orbitals with the
spatial parts ϕ
o α
i , ϕ
o α
j , ϕ
v α
i , and ϕ
v α
j , respectively. The expression for the
corresponding triplet transition energy reads
Constricted Variational Density Functional Theory Approach to the. . .
69