6.5 Solutions to Problems
139
since the sum of the site charges is constant,
a
ˆ
Q a = const., due to the charge
conservation. (This condition is often imposed explicitly as a constraint in the least
square fitting for ˆ
Q a .) Therefore, in such case
0 | ˆ
H | m
= 0 holds for any
m( = 0).
The above argument concludes that the CRK K ab has always a zero eigenvalue,
with the corresponding eigenvector V ,
V =
⎛
⎜
⎜
⎜
⎝
1
1
. . .
1
⎞
⎟
⎟
⎟
⎠
.
This fact is readily understandable since uniform change in the site potentials has
no influence on the electron distribution. Accordingly
Q a =
b
K ab
V 0 = 0
is derived, which indicates that the sum of the CRK elements in a row (or a column)
is inevitably zero.
6.5.3 Energy and Force with Polarizable Model
[Problem 6.3] By differentiating Eq. (6.29) with respect to the coordinate R(ai),
derive the formula of the force F (ai) in Eq. (6.30). We assume that Q 0
a and K ab are
invariant under the molecular vibration for simplicity.
(Hint) Make use of the self-consistent relations of Q ai and V ai in Eqs. (6.27)
and (6.28).
The formula of the force F (ai) is derived by differentiating the potential energy
U in Eq. (6.29) by the site coordinate R(ai),
F (ai) = −
∂U
∂R(ai)
= −
∂
∂R(ai)
⎧
⎨
⎩
i
>j
a
b
Q ai Q bj
|R(ai) − R(bj )|
−
1
2
i
a
b
K ab V ai V bi
⎫
⎬
⎭
139
since the sum of the site charges is constant,
a
ˆ
Q a = const., due to the charge
conservation. (This condition is often imposed explicitly as a constraint in the least
square fitting for ˆ
Q a .) Therefore, in such case
0 | ˆ
H | m
= 0 holds for any
m( = 0).
The above argument concludes that the CRK K ab has always a zero eigenvalue,
with the corresponding eigenvector V ,
V =
⎛
⎜
⎜
⎜
⎝
1
1
. . .
1
⎞
⎟
⎟
⎟
⎠
.
This fact is readily understandable since uniform change in the site potentials has
no influence on the electron distribution. Accordingly
Q a =
b
K ab
V 0 = 0
is derived, which indicates that the sum of the CRK elements in a row (or a column)
is inevitably zero.
6.5.3 Energy and Force with Polarizable Model
[Problem 6.3] By differentiating Eq. (6.29) with respect to the coordinate R(ai),
derive the formula of the force F (ai) in Eq. (6.30). We assume that Q 0
a and K ab are
invariant under the molecular vibration for simplicity.
(Hint) Make use of the self-consistent relations of Q ai and V ai in Eqs. (6.27)
and (6.28).
The formula of the force F (ai) is derived by differentiating the potential energy
U in Eq. (6.29) by the site coordinate R(ai),
F (ai) = −
∂U
∂R(ai)
= −
∂
∂R(ai)
⎧
⎨
⎩
i
>j
a
b
Q ai Q bj
|R(ai) − R(bj )|
−
1
2
i
a
b
K ab V ai V bi
⎫
⎬
⎭