^
a
{ ^ i
Â
à 2
ð Þ ¼
X
b
1
4
X
kld
kd
lb
À
Á
dk
al
À
Á
ε kl, bd ε kl, da
þ
X
k
M kb M ka
ε k, b ε k, a
!
^
b
{ ^ i
þ
X
j
1
4
X
mcd
md
jc
À
Á
ci
dm
À
Á
ε m j, cd ε im, cd
þ
X
d
M jd M di
ε j, d ε i, d
!
^
a
{ ^ j :
ð164Þ
(It should be noted that we have used the linked-cluster theorem to eliminate
contributions from disconnected diagrams. For a proof for the EOM of the one- and
two-particle the Green’s function, see [55].)
We may now proceed to calculate
ÀΠ
2
ð Þ
sr, q p ω
ð Þ ¼ ^
p
{ ^
q
T
{
1
1
ð Þ
P
1
ð Þ, À1
ω
ð Þ T
{
1
^
r
{ ^
s
0
ð Þ
þ ^
p
{ ^
q
T
{
1
0
ð Þ
P
1
ð Þ, À1
ω
ð Þ T
{
1
^
r
{ ^
s
1
ð Þ
þ ^
p
{ ^
q
T
{
1
1
ð Þ
P
0
ð Þ, À1
ω
ð Þ T
{
1
^
r
{ ^
s
1
ð Þ
þ ^
p
{ ^
q
T
{
1
0
ð Þ
P
2
ð Þ, À1
ω
ð Þ T
{
1
^
r
{ ^
s
0
ð Þ :
ð165Þ
The only new contributions which arise at this level are from the block P
(2) ,
which is given by
P
2
ð Þ
¼ Γ
2
ð Þ
1, 1 À Γ
1
ð Þ
1, 2 Γ
0
ð Þ, À1
2, 2
ω
ð ÞΓ
1
ð Þ
2, 1 :
ð166Þ
(We are anticipating the ω-dependence of the various Γ-blocks which are
derived below.) Because the block Γ
ð2Þ
1;1 is affected by the orthonormalization
procedure, it may be useful to provide a few more details. Expanding order-byorder,
52
M.E. Casida and M. Huix-Rotllant