Π
À1
s
ω
ð Þ À Π
0þ1þ2
ð
Þ , À1
ω
ð Þ ¼
P
1þ2
ð
Þ
ω
ð Þ
ÀΓ
1þ2
ð
Þ
1, 1
ÀΓ
1þ2
ð
Þ
1, 1
P
1þ2
ð
Þ
ω
ð Þ
0
B
@
1
C
A:
ð175Þ
Note that the off-diagonal (ph,hp)- and (hp,ph)-blocks are frequencyindependent and that the diagonal blocks are given by (166). Ignoring localization
for the moment, we may now cast the present Kohn–Sham based second-order
polarization propagator approximation (SOPPA/KS) into the familiar form of
(27) with
A ia, jb ω
ð Þ ¼ δ i, j δ a, b ε a, i þ P
1þ2
ð
Þ
ia, jb ω
ð Þ
B ia, b j ω
ð Þ ¼ À Γ
1þ2
ð
Þ
1, 1
ia, b j
:
ð176Þ
Localization – see (142) –complicates these formulae by mixing the P
1þ2
ð
Þ
ω
ð Þ
and Γ
1þ2
ð
Þ
1, 1
terms,
A ia, jb ω
ð Þ ¼ δ i, j δ a, b ε a À ε i
ð
Þ
þ Λ s
ð Þ hp, hp ω
ð ÞP
1þ2
ð
Þ
ω
ð Þ Λ
{
s
À Á
hp, hp
ω
ð Þ
h
i
ia, jb
þ Λ s
ð Þ hp, ph ω
ð ÞP
1þ2
ð
Þ
ω
ð Þ Λ
{
s
À Á
ph, hp
ω
ð Þ
h
i
ia, jb
À Λ s
ð Þ hp, ph ω
ð ÞΓ
1þ2
ð
Þ
Λ
{
s
À Á
hp, hp
ω
ð Þ
h
i
ia, jb
À Λ s
ð Þ hp, hp ω
ð ÞΓ
1þ2
ð
Þ
Λ
{
s
À Á
ph, hp
ω
ð Þ
h
i
ia, jb
B ia, b j ω
ð Þ ¼ Λ s
ð Þ hp, hp P
1þ2
ð
Þ
ω
ð Þ Λ
{
s
À Á
hp, ph
h
i
ia, b j
þ Λ s
ð Þ hp, ph P
1þ2
ð
Þ
ω
ð Þ Λ
{
s
À Á
ph, ph
h
i
ia, b j
À Λ s
ð Þ hp, ph ω
ð ÞΓ
1þ2
ð
Þ
Λ
{
s
À Á
hp, ph
ω
ð Þ
h
i
ia, b j
À Λ s
ð Þ hp, hp ω
ð ÞΓ
1þ2
ð
Þ
Λ
{
s
À Á
ph, ph
ω
ð Þ
h
i
ia, b j
:
ð177Þ
Of course, this extra complication is unnecessary if all we want to do is to
calculate improved excitation energies and transition amplitudes by means of
DFT-based many-body perturbation theory. It is only needed when our goal is to
study the effect of localization on purely TDDFT quantities such as the xc-kernel
and the TDDFT vectors X and Y.
MBPT Insights About and Corrections to TD-DFT
55