148
6 Charge Response Kernel for Electronic Polarization
where the partial charge Q ai is fixed during this process. Therefore, the reorganization energy U reorg is represented with the sum of the above two contributions,
U
reorg
= U
(i)
+ U
(ii)
=
i
a
Q
0
a V ai +
1
2
i
a,b
K ab V ai V bi −
i
a
Q ai V ai
(6.58)
= −
1
2
i
a,b
K ab V ai V bi .
(6.57)
where the last expression is derived using Eq. (6.28).
A.2.2 Variational Principle of Polarization
The above derivation of the reorganization energy in Eq. (6.58) allows us to define
the whole electrostatic potential energy of the condensed system to be
U (Q, V ) =
1
2
i
j ( =i)
a,b
Q ai Q bj
|R(ai) − R(bj )|
+
i
a
Q
0
ai V ai +
1
2
i
a,b
K ab V ai V bi −
i
a
Q ai V ai .
(6.59)
We confirmed that U in Eq. (6.59) coincides with U in Eq. (6.29) with the help of
Eq. (6.28).
Let us suppose that U in Eq. (6.59) is a function of {Q ai } and {V ai }. Then the
conditions to minimize U (Q, V ) are given as follows.
∂ U (Q, V )
∂Q ai
=
j ( =i)
b
Q bj
|R(ai) − R(bj )|
− V ai = 0,
(6.60)
∂ U (Q, V )
∂V ai
= Q
0
ai +
b
K ab V bi − Q ai = 0.
(6.61)
Interestingly, these Eqs. (6.60) and (6.61) are identical to the self-consistent conditions of Eqs. (6.27) and (6.28), respectively. This argument indicates that the
self-consistent conditions of Q ai and V ai are regarded as variational principle
to minimize the whole energy U (Q, V ) in Eq. (6.59). The minimum energy of
U (Q, V ) coincides with the actual energy U in Eq. (6.29).