ΔE
R
M ¼
X α, β
σ
X
vir σ
ð Þ
a
T
2
ð Þσσ
aa ε
σ
a À
X
vir σ
ð Þ
i
T
2
ð Þσσ
ii
ε
σ
i
!
þ
X α, β
σ
X α, β
μ
X
vir σ
ð Þ
a
X
occ σ
ð Þ
i
X
vir μ
ð Þ
c
T
1
ð Þσσ
ai
T
1
ð Þμμ
bj
K a σ i σ b μ j μ þ K a σ b μ i σ j μ
h
i
þ
X α, β
σ
X
occ σ
ð Þ
k
X
vir σ
ð Þ
c
T
1
ð Þσσ
ck
X
vir α
ð Þ
ab
ΔP
αα
ab U
αα
ð
ÞK a α b α c σ k σ þ
X
occ α
ð Þ
ij
ΔP
αα
ij U
αα
ð
ÞK i α j α c σ k σ
"
#
ð49Þ
is the relaxation contribution to the excitation energy. The total energy for Ψ M is
given as E M ¼ E 0 ρ
0
ð Þ þ ΔE M U
ð Þ þ ΔE
R
M where E 0 (ρ
0 ) is the ground state energy
expressed in terms of unrelaxed orbitals. The expression for ΔE
R
M is derived after
orthogonalization of Ψ M to the ground state to second order in R.
We optimize ΔE M of (49) by first performing a Taylor expansion from the
starting point reference (U
0,αα , R
0,αα , R
0,ββ ) to U
αα
; R
αα
; R
ββ
À
Á ¼ U
0, αα
þ ΔU
αα ,
À
R
0, αα
þ ΔR
αα , R
0, ββ
þ ΔR
αα
Þ:
E M U
αα
;R
αα
;R
ββ
À
Á ¼ E M U
0,αα
;R
0,αα
;R
0,ββ
À
Á
þ
X
ai
dE M
dU
αα
ai
0
ΔU
αα
ai þ
X α,β
σ
X
ai
dE M
dR
σσ
ai
0
ΔR
σσ
ai þ
1
2
X
ai
X
bj
d
2 E M
dU
αα
ai dU
αα
bj
!
0
ΔU
αα
ai ΔU
αα
bj
þ
1
2
X
ai
X
bj
X α,β
σ
X α,β
τ
d
2 E M
dR
σσ
ai dR
ττ
bj
!
0
ΔR
σσ
ai ΔR
ττ
bj þ
X
ai
X
bj
X α,β
σ
d
2 E M
dU
αα
ai dR
σσ
bj
!
0
ΔU
αα
ai ΔR
σσ
bj
þO
3
½ Š
ð50Þ
Here the subscript “0” indicates that the derivative is evaluated at the reference
(U
0,αα ,R
0,αα ,R
0,ββ ). We can alternatively write the expansion in terms of energy
gradients and energy Hessians as
E M U
αα
; R
αα
; R
ββ
À
Á ¼ E M U
0,αα
; R
0,αα
; R
0, ββ
À
Á þ ΔU
! αα
ΔR
! αα
ΔR
! ββ
g
! U
αα
g
! R
αα
g
! R
ββ
0
B
B
@
1
C
C
A
þ
1
2 ΔU
! αα
ΔU
! αα
ΔU
! αα
H
U
αα ,U
αα
H
U
αα ,R
αα
H
U
αα ,R
ββ
H
R
αα ,U
αα
H
R
αα ,R
αα
H
R
αα ,R
ββ
H
R
ββ ,U
αα
H
R
ββ ,R
αα
H
R
ββ ,R
ββ
0
@
1
A
H
R
ββ ,R
ββ
ΔR
! αα
ΔR
! ββ
0
B
@
1
C
A þ O
3
½ Š
ð51Þ
where the expressions for the gradients g
! U
αα
, g
! R
αα
, g
! R
ββ
and Hessians H
U
αα ,U
αα ,
H
R
αα ,U
αα , etc. can be obtained by a comparison between (50) and (51). Specific
84
T. Ziegler et al.