A.1 Derivation of CRK from CPHF Equation
141
where R q (b) is the q coordinate of the site b. As a consequence, the partial charge
at the site a changes by
Q a =
site
b
K ab V b = −
b
K ab E q R q (b).
The change in partial charges gives rise to the change in the dipole moment along
the direction p,
μ p =
site
a
Q a R p (a) = −
site
a,b
K ab E q R q (b)R p (a)
=
⎛
⎝ −
site
a,b
K ab R p (a)R q (b)
⎞
⎠ E q
Then the polarizability tensor is defined as the derivative of dipole moment with
respect to the electric field in the following form,
α pq = lim
E→0
μ p
E q
= −
site
a,b
K ab R p (a)R q (b).
(6.32)
Appendix
A.1 Derivation of CRK from CPHF Equation
In this section we derive the CPHF equation (6.22) and CRK in Eq. (6.25). The CRK
is given as a second-order derivative of the total energy, K ab = (∂ 2 E/∂V a ∂V b ), and
here it is formulated in the closed-shell Hartree-Fock theory [7]. Other formulations
in the unrestricted Hartree-Fock [7] and Kohn-Sham DFT [3] are found in literature.
A.1.1 Wavefunction Under External Field
We briefly summarize the Hartree-Fock wavefunction under the external potential
below. The total energy E for the wavefunction in Eq. (6.19) is presented by [13]
E =
| ˆ
H |
=
| ˆ
H 0 + ˆ
H
|