Y
ψ occ
ψ vir
¼ e
U ψ occ
ψ vir
¼
X 1
m¼0
U
m
m!
!
ψ occ
ψ vir
¼
ψ
0
occ
ψ
0
vir
ð41Þ
in which the sets of relaxed occupied ψ i ; i ¼ 1, occ
f
gand virtual ψ a ; a ¼ 1, vir
f
g
ground state (reference) KS-orbitals are converted into the resulting sets
ψ
0
i ; i ¼ 1, occ
È
É
and ψ
0
a ; a ¼ 1, vir
È
É
of relaxed occupied and virtual excited state
orbitals, respectively. It should be noted that the relaxed orbital set is orthonormal
to second order in R.
We now obtain for a spin conserving transition the excited state KS determinant
which can be written as
Ψ M ¼ ψ
0
1 ψ
0
1 . . . ψ
0
i ψ
0
j . . . ψ
0
n
ð42Þ
The corresponding change in density Δρ M expanded in terms of the unrelaxed
ground state orbitals takes the form
Δρ M ¼ Δρ M U
αα
ð
ÞþΔρ
R
M
ð43Þ
where
Δρ M U
αα
ð
Þ ¼
X
vir α
ð Þ
a
X
occ α
ð Þ
i
ΔP ai U
αα
ð
Þ ϕ
α
a 1
0
ð Þϕ
α
i 1
ð Þ þ ϕ
α
a 1
0
ð Þϕ
α
i 1
ð Þ
Â
Ã
þ
X
vir α
ð Þ
ab
ΔP ab U
αα
ð
Þϕ
α
a 1
0
ð Þϕ
α
b
À
1
Á þ
X
occ α
ð Þ
ij
ΔP ij U
αα
ð
Þϕ
α
i 1
0
ð Þϕ
α
j
À
1
Á
ð44Þ
is the change in density caused by U
αα alone and equivalent to (32a) but expressed
in terms of unrelaxed ground state orbitals. On the other hand
Δρ
R
M ¼
X α, β
σ
X
vir σ
ð Þ
a
X
occ σ
ð Þ
i
T
1
ð Þσσ
ai
ϕ
α
a 1
0
ð Þϕ
α
i 1
ð Þ þ ϕ
α
a 1
0
ð Þϕ
α
i 1
ð Þ
Â
Ã
þ
X
vir σ
ð Þ
ab
T
2
ð Þσσ
ab ϕ
σ
a 1
0
ð Þϕ
σ
b
À
1
Á þ
X
occ σ
ð Þ
ij
T
2
ð Þσσ
ij
ϕ
σ
i 1
0
ð Þϕ
σ
j
À
1
Á
ð45Þ
is the change in density caused by the relaxation. Here
T
1
ð Þαα
ai
¼ R
αα
ai þ
X
occ α
ð Þ
j
ΔP
αα
ij R
αα
aj ; T
1
ð Þββ
ai
¼ R
ββ
ai
ð46aÞ
82
T. Ziegler et al.