136
6 Charge Response Kernel for Electronic Polarization
A eff = −
i
site
a
site
b
K abi
⎡
⎣
j
site
c
[G
−1
] ai,c j R(c
j
)
⎤
⎦ ⊗
⎡
⎣
j
site
c
[G
−1
] bi,cj R(cj )
⎤
⎦ .
(6.44)
Equations (6.42) and (6.44) are the CRK expressions of the dipole and effective
polarizability, and are equivalent to Eq. (5.28) in Chap. 5.
6.5 Solutions to Problems
6.5.1 ESP Charge
[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).
The condition of minimizing L is
∂L
∂Q a
= 2
grid
n
φ model (R G (n)) − φ(R G (n))
∂φ model (R G (n))
∂Q a
= 0
= 2
site
b
grid
n
Q b
|R(b) − R G (n)|
·
1
|R(a) − R G (n)|
− 2
grid
n
⎧
⎨
⎩
−e
AO
p,q
D pq
p|
1
|r − R G (n)|
|q
+
nuc
c
Z c e
|R N (c) − R G (n)|
⎫
⎬
⎭
·
1
|R(a) − R G (n)|
.
Therefore, the optimized values of {Q a } satisfy the following equation,
site
b
A ab Q b = −e
AO
p,q
D pq
p| ˆ
B a |q
+ C a e,
(6.45)
where
A ab =
grid
n
1
|R(a) − R G (n)|
·
1
|R(b) − R G (n)|
,
(6.15)
6 Charge Response Kernel for Electronic Polarization
A eff = −
i
site
a
site
b
K abi
⎡
⎣
j
site
c
[G
−1
] ai,c j R(c
j
)
⎤
⎦ ⊗
⎡
⎣
j
site
c
[G
−1
] bi,cj R(cj )
⎤
⎦ .
(6.44)
Equations (6.42) and (6.44) are the CRK expressions of the dipole and effective
polarizability, and are equivalent to Eq. (5.28) in Chap. 5.
6.5 Solutions to Problems
6.5.1 ESP Charge
[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).
The condition of minimizing L is
∂L
∂Q a
= 2
grid
n
φ model (R G (n)) − φ(R G (n))
∂φ model (R G (n))
∂Q a
= 0
= 2
site
b
grid
n
Q b
|R(b) − R G (n)|
·
1
|R(a) − R G (n)|
− 2
grid
n
⎧
⎨
⎩
−e
AO
p,q
D pq
p|
1
|r − R G (n)|
|q
+
nuc
c
Z c e
|R N (c) − R G (n)|
⎫
⎬
⎭
·
1
|R(a) − R G (n)|
.
Therefore, the optimized values of {Q a } satisfy the following equation,
site
b
A ab Q b = −e
AO
p,q
D pq
p| ˆ
B a |q
+ C a e,
(6.45)
where
A ab =
grid
n
1
|R(a) − R G (n)|
·
1
|R(b) − R G (n)|
,
(6.15)