6.3 Polarizable Model with CRK
133
Non-empirical Modeling One of the most significant advantages of the CRK
modeling as a polarizable model is that the parameters of polarization are fully
determined by ab initio molecular orbital or DFT calculations, based on the common
scheme to the ESP charge. Once we employ a method of ESP charge calculation, the
extension to the CRK is unique and straightforward with no empirical parameter.
Accuracy of the CRK model is entirely conformed to the electronic structure
calculation, and it is possible to systematically improve the accuracy by improving
the electronic structure calculation.
Fully non-empirical character of the CRK model is quite advantageous to general
applications. For example, unstable species or radicals can be readily treated with
common accuracy to stable molecules [7, 8], though these unstable species are not
suitable to empirical modeling due to scarcity of available experimental properties.
It is also straightforward to obtain the conformational dependence of CRK for a
vibrating molecule, by performing the CRK calculations for a deformed molecule
from its equilibrium conformation. Accurate description of the conformational
dependence is a key requisite for MD calculations of vibrational Raman or SFG
spectra.
Dipole and Polarizability Next we describe the instantaneous dipole and polarizability of a molecule using the CRK model. We treat an arbitrary vibrating and
rotating molecule, and assume that the instantaneous coordinate of its site a is
R(a) in the space-fixed coordinates. Then the permanent dipole vector μ 0 and the
polarizability tensor α are represented in the same space-fixed coordinates by
μ
0
p =
site
a
Q
0
a R p (a),
(6.31)
α pq = −
site
a,b
K ab R p (a)R q (b),
(6.32)
where the suffixes p, q denote x ∼ z in the space-fixed coordinates. In the
expressions of Eqs. (6.31) and (6.32), it is noteworthy that Q 0
a and K ab are invariant
with respect to the molecular rotation and thus considered as scalar properties. These
scalar characters are advantageous to describe the vector and tensor elements of μ
and α in an arbitrary coordinate system. Once can determine the values of Q 0
a and
K ab as a function of internal coordinates, the vector/tensor elements of μ 0
p /α pq are
readily expressed with these values and the coordinates of sites {R p (a)}.
[Problem 6.4] Explain the expression of the polarizability α pq in Eq. (6.32).
Recall that the polarizability is the derivative of dipole moment with respect to
spatially uniform electric field.
In summary, the CRK model offers a general scheme to describe the polarizable
force field as well as instantaneous dipole and polarizability in a unified manner.
Therefore, it is particularly suitable to calculate the vibrational SFG spectroscopy
by MD simulation.
133
Non-empirical Modeling One of the most significant advantages of the CRK
modeling as a polarizable model is that the parameters of polarization are fully
determined by ab initio molecular orbital or DFT calculations, based on the common
scheme to the ESP charge. Once we employ a method of ESP charge calculation, the
extension to the CRK is unique and straightforward with no empirical parameter.
Accuracy of the CRK model is entirely conformed to the electronic structure
calculation, and it is possible to systematically improve the accuracy by improving
the electronic structure calculation.
Fully non-empirical character of the CRK model is quite advantageous to general
applications. For example, unstable species or radicals can be readily treated with
common accuracy to stable molecules [7, 8], though these unstable species are not
suitable to empirical modeling due to scarcity of available experimental properties.
It is also straightforward to obtain the conformational dependence of CRK for a
vibrating molecule, by performing the CRK calculations for a deformed molecule
from its equilibrium conformation. Accurate description of the conformational
dependence is a key requisite for MD calculations of vibrational Raman or SFG
spectra.
Dipole and Polarizability Next we describe the instantaneous dipole and polarizability of a molecule using the CRK model. We treat an arbitrary vibrating and
rotating molecule, and assume that the instantaneous coordinate of its site a is
R(a) in the space-fixed coordinates. Then the permanent dipole vector μ 0 and the
polarizability tensor α are represented in the same space-fixed coordinates by
μ
0
p =
site
a
Q
0
a R p (a),
(6.31)
α pq = −
site
a,b
K ab R p (a)R q (b),
(6.32)
where the suffixes p, q denote x ∼ z in the space-fixed coordinates. In the
expressions of Eqs. (6.31) and (6.32), it is noteworthy that Q 0
a and K ab are invariant
with respect to the molecular rotation and thus considered as scalar properties. These
scalar characters are advantageous to describe the vector and tensor elements of μ
and α in an arbitrary coordinate system. Once can determine the values of Q 0
a and
K ab as a function of internal coordinates, the vector/tensor elements of μ 0
p /α pq are
readily expressed with these values and the coordinates of sites {R p (a)}.
[Problem 6.4] Explain the expression of the polarizability α pq in Eq. (6.32).
Recall that the polarizability is the derivative of dipole moment with respect to
spatially uniform electric field.
In summary, the CRK model offers a general scheme to describe the polarizable
force field as well as instantaneous dipole and polarizability in a unified manner.
Therefore, it is particularly suitable to calculate the vibrational SFG spectroscopy
by MD simulation.