142
6 Charge Response Kernel for Electronic Polarization
=
N e /2
i=1
2
ψ i | ˆ
h|ψ i
+
N e /2
i=1
N e /2
j =1
2
ψ i ψ i |ψ j ψ j
−
ψ i ψ j |ψ i ψ j
+
site
a
Q
nuc
a V a
=
occ
i
2h ii +
occ
i,j
2J ij − K ij
+
site
a
Q
nuc
a V a .
(6.49)
In Eq. (6.49), the one-electron operator ˆ
h includes the perturbation,
ˆ
h = ˆ
h 0 − e
site
a
ˆ
n a V a
(6.50)
where ˆ
h 0 is the conventional one-electron operator in Eq. (6.4), and the second term
indicates the perturbation by the external electrostatic potential.
The Hartree-Fock theory provides a way to determine the wavefunction and
total energy E through the variational principle. It determines the MO coefficients
C pi of Eq. (6.19) by minimizing the energy E of Eq. (6.49), under the constraint of
the following orthonormal conditions among the MOs,
ψ i |ψ j
=
AO
p,q
C pi C qj s pq = δ ij , where s pq = p|q .
(6.51)
This variational procedure derives the following general eigenvector problem, called
the Hartree-Fock-Roothaan equation, for the MO coefficients,
AO
q
F pq C qi = ε i
AO
q
s pq C qi ,
(6.52)
where ε i corresponds to the eigenvalue, called the i-th canonical orbital energy. F pq
is the Fock matrix given as follows,
F pq =
p| ˆ
h|q
+
1
2
AO
r,s
{2 (pq|rs) − (pr|qs)} D rs .
(6.53)
D rs in Eq. (6.53) is called the density matrix,
D rs = 2
occ
i
C ri C si .
(6.54)