144
6 Charge Response Kernel for Electronic Polarization
=
q,r,s
occ
j
{4 (pq|rs) − (pr|qs) − (ps|qr)}
∂C rj
∂V b
C sj C qi
=
q,r,s
occ
j
k
{4 (pq|rs) − (pr|qs) − (ps|qr)} C rk U
b
kj C sj C qi
where (pq|rs) = (χ p χ q |χ r χ s ). We have employed Eq. (6.52) to derive the
first+fourth terms, and the symmetry relation of the two-electron integral, (pq|rs) =
(pq|sr), to derive the third term. By summarizing the above new expressions,
Eq. (6.55) becomes
j
ε j − ε i
q
s pq C qj U
b
ji − e
q
p| ˆ
n b |q
C qi
+
q,r,s
occ
j
k
{4 (pq|rs) − (pr|qs) − (ps|qr)} C rk U
b
kj C sj C qi = 0.
We further convert this equation into the MO representation by operating
p C pl
on both sides of the equation,
j
ε j − ε i
p,q
C pl s pq C qj U
b
ji − e
p,q
p| ˆ
n b |q
C pl C qi
+
p,q,r,s
occ
j
k
{4 (pq|rs) − (pr|qs) − (ps|qr)} C pl C rk U
b
kj C sj C qi
=
j
ε j − ε i
δ lj U
b
ji − e
ψ l | ˆ
n b |ψ i
+
occ
j
k
H likj U
b
kj = 0.
(6.56)
H likj has been defined in Eq. (6.23),
H likj = 4
ψ l ψ i |ψ k ψ j
−
ψ l ψ k |ψ i ψ j
−
ψ l ψ j |ψ k ψ i
,
(6.23)
ψ i ψ j |ψ k ψ l
=
ψ i (r)ψ j (r)
e 2
|r − r |
ψ k (r
)ψ l (r
)drdr
=
AO
p,q,r,s
C pi C qj C rk C sl
χ p χ q |χ r χ s
,
and
ψ l | ˆ
n b |ψ i
=
AO
p,q
C pl C qi χ p | ˆ
n b |χ q .
(6.24)