6.3 Polarizable Model with CRK
131
(Hint) Suppose that the zero-th order wavefunction and the total energy for the
ground state are 0 and E 0 , respectively, with no external perturbation V = 0.
Then, by adding the perturbation V , the total energy E is expanded in a series of
perturbation in the following manner,
E = E 0 + E
(1)
+ E
(2)
+ · · · = E 0 +
a
Q a V a +
1
2
a,b
K ab V a V b + · · · ,
(6.26)
where
Q a =
∂E
∂V a
V =0
and K ab =
∂ 2 E
∂V a ∂V b
V =0
.
Recall that the second-order perturbation energy E (2) for the ground state is always
negative.
6.3 Polarizable Model with CRK
The CRK represents the electronic polarization using the ESP site charges, and can
be readily utilized as a polarizable model. Here we summarize the merits of CRK
modeling and its application to the SFG calculations.
Force Fields Most of current molecular simulations for polyatomic molecules
adopt force fields based on the interaction site model of molecules, where intermolecular interactions are constructed with site-site interactions. The CRK modeling is also based on the interaction site model, and readily incorporated in the MD
simulation. It allows for varying partial charges at the interaction sites, and thereby
representing the electronic polarization in MD simulation [8, 11].
Let us suppose a molecule in condensed phase, such as solution or interface.
The electrostatic potential V bi at the site b of molecule i is presented by the
intermolecular Coulombic interactions from surrounding molecules j ( = i) in the
following form,
V bi =
molecule
j ( =i)
site
a
Q aj
|R(bi) − R(aj )|
,
(6.27)
where Q aj and R(aj ) are the partial charge and coordinates of the site a of molecule
j . The partial charge Q ai of molecule i under the electrostatic potential V bi is
represented with the CRK by
Q ai = Q
0
a +
site
b
K ab V bi ,
(6.28)
131
(Hint) Suppose that the zero-th order wavefunction and the total energy for the
ground state are 0 and E 0 , respectively, with no external perturbation V = 0.
Then, by adding the perturbation V , the total energy E is expanded in a series of
perturbation in the following manner,
E = E 0 + E
(1)
+ E
(2)
+ · · · = E 0 +
a
Q a V a +
1
2
a,b
K ab V a V b + · · · ,
(6.26)
where
Q a =
∂E
∂V a
V =0
and K ab =
∂ 2 E
∂V a ∂V b
V =0
.
Recall that the second-order perturbation energy E (2) for the ground state is always
negative.
6.3 Polarizable Model with CRK
The CRK represents the electronic polarization using the ESP site charges, and can
be readily utilized as a polarizable model. Here we summarize the merits of CRK
modeling and its application to the SFG calculations.
Force Fields Most of current molecular simulations for polyatomic molecules
adopt force fields based on the interaction site model of molecules, where intermolecular interactions are constructed with site-site interactions. The CRK modeling is also based on the interaction site model, and readily incorporated in the MD
simulation. It allows for varying partial charges at the interaction sites, and thereby
representing the electronic polarization in MD simulation [8, 11].
Let us suppose a molecule in condensed phase, such as solution or interface.
The electrostatic potential V bi at the site b of molecule i is presented by the
intermolecular Coulombic interactions from surrounding molecules j ( = i) in the
following form,
V bi =
molecule
j ( =i)
site
a
Q aj
|R(bi) − R(aj )|
,
(6.27)
where Q aj and R(aj ) are the partial charge and coordinates of the site a of molecule
j . The partial charge Q ai of molecule i under the electrostatic potential V bi is
represented with the CRK by
Q ai = Q
0
a +
site
b
K ab V bi ,
(6.28)