128
6 Charge Response Kernel for Electronic Polarization
The least square fitting results in the following definition of ˆ
n a and Q nuc
a in Eq. (6.9),
ˆ
n a =
site
b
(A
−1 ) ab ˆ
B b ,
(6.13)
Q
nuc
a =
site
b
(A
−1 ) ab C b e,
(6.14)
where A, B, C are given by
A ab =
grid
n
1
|R(a) − R G (n)|
·
1
|R(b) − R G (n)|
,
(6.15)
ˆ
B a =
grid
n
1
|r − R G (n)|
·
1
|R(a) − R G (n)|
,
(6.16)
C a =
grid
n
nuc
c
Z c
|R N (c) − R G (n)|
·
1
|R(a) − R G (n)|
.
(6.17)
Equations (6.13) and (6.14) determine the partial charge Q a and its operator ˆ
Q a in
Eq. (6.9).
[Problem 6.1] Derive ˆ
n a and Q nuc
a in Eqs. (6.13) and (6.14) by minimizing L in
Eq. (6.12).
Note 1. If the locations of the site a coincide with those of nuclei, Q nuc
a
in
Eq. (6.9) is identical to the nuclear charge at the site in Eq. (6.14).
Note 2. The above procedure of least square fitting is often accompanied with
some extra constraints on the total charge or the dipole moment of the molecule,
e.g.
site
a
Q a = Q (constant),
site
a
Q a R(a) = μ (constant),
where the values Q and μ in the right hand side are given as the constraint
conditions. Such constraints can be readily implemented in the least square fitting
of Problem 6.1 using the Lagrange multipliers [9].