Reviews in Computational Chemistry Volume 18

116 Polarizability in Computer Simulations
methods. 10,168 Finally, the parameters themselves can be directly calculated
using density functional theory (DFT) methods. 169,170
As presented, the EE approach given by Eq. [43] is a simple mathematical
model resulting from a Taylor series; it can be given a more rigorous foundation
using electronic density functional theory. 169 Using DFT, and making
simplifying approximations for the exchange and kinetic energy functionals,
expressions analogous to Eq. [43] can be derived. 130,131,171 This approach
has been termed chemical potential equalization (CPE). 130 Efforts like CPE
or even parameterizations of fluctuating charge models using electronic structure
calculations represent a step away from empirical potential models toward
ab initio simulation methods. However, even with a sophisticated treatment of
the charges, empirical terms in the potential such as the Lennard–Jones interaction
still remain. A standard method is to set the Lennard–Jones parameters
so that the energies and geometries of important dimer conformations (e.g.,
hydrogen-bonded dimers) are close to ab initio values. 10,144,145,166 In some
cases, the remaining potential parameters have been taken from existing force
fields. 146,150 One interesting extension of the fluctuating charge model has
been developed by Siepmann and co-workers. 147 In their model, the
Lennard–Jones size parameter becomes a variable that is coupled to the charge
on a given atom. The size of the atom increases as the atom becomes
more negatively charged and obtains greater electronic density. This increase
in size is thus consistent with physical intuition. Other models in which some
of the remaining potential parameters are treated as variables are described in
the next section.
SEMIEMPIRICAL MODELS
A number of quantum polarizable models have been developed. 144,172–177
These treatments of polarizability represent a step toward full ab initio methods.
The models can be characterized by a small number of electronic states or
potential energy surfaces, which are coupled to each other. For the purposes of
this tutorial, our description is of the method of Gao. 173,174 In his method,
molecular orbitals, f A, for each molecule are defined as a linear combination
of N b atomic orbitals, w m,
f A ¼ XN b
m ¼ 1
cmAw m
As is standard in semiempirical methods, 178 the molecular orbitals are orthonormal,
so the overlap matrix, SAB, is assumed to be diagonal. The molecular
wave function, a, is a Hartree 144 or Hartree–Fock 173 product of the molecular
orbitals. For a (closed-shell) molecule with 2M electrons, there will be M
doubly occupied molecular orbitals. The wave function of a system comprised
½54Š