A.1 Derivation of CRK from CPHF Equation
143
In the above procedure, we note that the parameter of external perturbation
V a is included in Eq. (6.50), which affects on the MD coefficients (6.52), Fock
matrix (6.53) and the total energy E.
A.1.2 Derivative of Wavefunction
We have learned that the MO coefficients determined by Eq. (6.52) are affected by
the external perturbation parameter. Therefore, the derivative of MO coefficients
with respect to V b is obtained by differentiating Eq. (6.52),
∂
∂V b
q
F pq C qi − ε i
q
s pq C qi
= 0
=
q
F pq
∂C qi
∂V b
− e
q
p| ˆ
n b |q
C qi +
q
r,s
occ
j
{2 (pq|rs) − (pr|qs)}
∂C rj
∂V b
C sj + C rj
∂C sj
∂V b
C qi
− ε i
q
s pq
∂C qi
∂V b
.
(6.55)
In the right hand side of Eq.(6.55), the differential of the Fock matrix ∂F pq /∂V b
with Eq. (6.53) is included in the second and third terms.
Then the derivative of the MO coefficients, ∂C/∂V b , in Eq. (6.55) is expressed
with U b in Eq. (6.21). Consequently, the first, third and fourth terms of the right
hand side of Eq. (6.55) become
• (first term) + (fourth term)
q
F pq
∂C qi
∂V b
− ε i
q
s pq
∂C qi
∂V b
=
j
q
F pq C qj − ε i
q
s pq C qj
U
b
ji
=
j
ε j − ε i
q
s pq C qj U
b
ji
• (third term)
q
r,s
occ
j
{2 (pq|rs) − (pr|qs)}
∂C rj
∂V b
C sj + C rj
∂C sj
∂V b
C qi