T
2
ð Þαα
ab
¼
X
occ α
ð Þ
i
R
αα
ai R
αα
bi þ
X
occ α
ð Þ
i
X
occ α
ð Þ
j
ΔP
αα
ij U
αα
ð
ÞR
αα
ai R
αα
bj
T
2
ð Þββ
ab
¼
X
vir β
ð Þ
a
R
ββ
ai R
ββ
bi
ð46bÞ
T
2
ð Þαα
ij
¼
X
vir α
ð Þ
a
R
αα
ai R
αα
aj À
X
occ α
ð Þ
l
X
vir α
ð Þ
a
ΔP
αα
il U
αα
ð
ÞR
αα
al R
αα
aj
T
2
ð Þββ
ij
¼ À
X
vir β
ð Þ
i
R
ββ
ai R
ββ
aj
ð46cÞ
We obtain for the excitation energy
ΔE M ¼ ΔE M U
αα
ð
ÞþΔE
R
M
ð47Þ
where
ΔE M U
ð Þ ¼
X
vir α
ð Þ
a
ε
α
a ΔP
αα
aa U
αα
ð
Þ
2 À
X
occ α
ð Þ
i
À ε
α
i ΔP
αα
ii U
αα
ð
Þ
2
þ
X
vir α
ð Þ
ab
X
occ α
ð Þ
i
ΔP
αα
ai U
αα
ð
ÞΔP
αα
bj U
αα
ð
Þ K a α i α b α j α þ K a α i α b α j α
Â
Ã
þ
1
2
X
occ α
ð Þ
ijkl
ΔP
αα
ij U
αα
ð
ÞΔP
αα
kl U
αα
ð
ÞK i α j α k α l α
þ
1
2
X vir
abcd
ΔP
αα
ab U
αα
ð
ÞΔP
αα
cd U
αα
ð
ÞK a α b α c α d α
þ
X
vir α
ð Þ
ab
X
occ α
ð Þ
ij
ΔP
αα
ab U
αα
ð
ÞΔP
αα
ij U
αα
ð
ÞK a α b α i α j α
þ 2
X
vir α
ð Þ
abc
X
occ α
ð Þ
k
ΔP
αα
ab U
αα
ð
ÞΔP
αα
ck U
αα
ð
ÞK a α b α c α k α
þ 2
X
vir α
ð Þ
ijk
X
occ α
ð Þ
c
ΔP
αα
ij U
αα
ð
ÞΔP
αα
ck U
αα
ð
ÞK i α j α c α k α
ð48Þ
is the excitation caused by U
αα alone and equivalent to ΔE M for the SCF-CV(1)DFT scheme of (28) but expressed in terms of canonical and unrelaxed ground state
orbitals. Further
Constricted Variational Density Functional Theory Approach to the. . .
83