132
6 Charge Response Kernel for Electronic Polarization
where Q 0
a is the partial charge at the site a in the isolated condition free from the
external potential. Equation (6.28) shows that the site partial charge Q ai varies in
response to the external potential up to the first order. 2 Equations (6.27) and (6.28)
provide a self-consistent scheme to determine V bi and Q ai simultaneously. For
an actual condensed system, the site coordinates R(aj ) changes with molecular
motions, and accordingly Eqs. (6.27) and (6.28) are solved to obtain the instantaneous values of V bi and Q ai at each time t during the MD simulation. The effect of
electronic polarization is thereby implemented through fluctuation of partial charges
during the time evolution.
In the CRK model, the whole electrostatic potential energy of the condensed
system is given by
U =
i
>j
a
b
Q ai Q bj
|R(ai) − R(bj )|
−
1
2
i
a
b
K ab V ai V bi .
(6.29)
The first term of the right hand side stands for the intermolecular site-site Coulomb
interactions, and the second term stands for the reorganization energy, as detailed
in Appendix A.2. The second term accounts for destabilization energy due to the
deformation of electronic polarization. Note that this term is always positive, since
the CRK K ab is a non-positive definite matrix (see Problem 6.2). The force acting
on the site a of molecule i, F (ai), is derived from Eq. (6.29) by
F (ai) = −
∂U
∂R(ai)
=
j ( =i)
b
Q ai Q bj (R(ai) − R(bj ))
|R(ai) − R(bj )|
3
.
(6.30)
We notice that Eq. (6.30) does not include the derivatives ∂Q/∂R or ∂V /∂R. The
expression of F (ai) in Eq. (6.30) apparently coincide with the expression obtained
by differentiating the first term of U in Eq. (6.29) simply by R(ai), as if the site
charges Q ai and Q bj were fixed. This feature greatly simplifies the calculation
of Eq. (6.30), by virtue of the self-consistent conditions of Eqs. (6.27) and (6.28).
Appendix A.2 further discuss this feature.
[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).
2 Equation (6.28) treats a single species for simplicity. Extension to multi-species systems is
straightforward, and shown in Eq. (6.34) in Sect. 6.4.
6 Charge Response Kernel for Electronic Polarization
where Q 0
a is the partial charge at the site a in the isolated condition free from the
external potential. Equation (6.28) shows that the site partial charge Q ai varies in
response to the external potential up to the first order. 2 Equations (6.27) and (6.28)
provide a self-consistent scheme to determine V bi and Q ai simultaneously. For
an actual condensed system, the site coordinates R(aj ) changes with molecular
motions, and accordingly Eqs. (6.27) and (6.28) are solved to obtain the instantaneous values of V bi and Q ai at each time t during the MD simulation. The effect of
electronic polarization is thereby implemented through fluctuation of partial charges
during the time evolution.
In the CRK model, the whole electrostatic potential energy of the condensed
system is given by
U =
i
>j
a
b
Q ai Q bj
|R(ai) − R(bj )|
−
1
2
i
a
b
K ab V ai V bi .
(6.29)
The first term of the right hand side stands for the intermolecular site-site Coulomb
interactions, and the second term stands for the reorganization energy, as detailed
in Appendix A.2. The second term accounts for destabilization energy due to the
deformation of electronic polarization. Note that this term is always positive, since
the CRK K ab is a non-positive definite matrix (see Problem 6.2). The force acting
on the site a of molecule i, F (ai), is derived from Eq. (6.29) by
F (ai) = −
∂U
∂R(ai)
=
j ( =i)
b
Q ai Q bj (R(ai) − R(bj ))
|R(ai) − R(bj )|
3
.
(6.30)
We notice that Eq. (6.30) does not include the derivatives ∂Q/∂R or ∂V /∂R. The
expression of F (ai) in Eq. (6.30) apparently coincide with the expression obtained
by differentiating the first term of U in Eq. (6.29) simply by R(ai), as if the site
charges Q ai and Q bj were fixed. This feature greatly simplifies the calculation
of Eq. (6.30), by virtue of the self-consistent conditions of Eqs. (6.27) and (6.28).
Appendix A.2 further discuss this feature.
[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).
2 Equation (6.28) treats a single species for simplicity. Extension to multi-species systems is
straightforward, and shown in Eq. (6.34) in Sect. 6.4.