Reviews in Computational Chemistry Volume 18

is treated using variable charges. The total energy is then
E ¼ Epol þ 1 X
2
N XN i ¼ 1 j 6¼ i
and the forces on the nuclear coordinates are provided by the derivative of E
with respect to the positions.
The charges can be found through Mulliken population analysis, 180
which, because the overlap matrix is diagonal, is
qa ¼ K Za 2 XM
a ¼ 1
Semiempirical Models 119
where the sum over m is over atomic orbitals centered on atom a, and K is an
empirical scaling parameter correcting for errors in the Mulliken charges (K is
about 2). The Lennard–Jones parameters are assumed to be independent of the
electronic states and all applications to date have been for rigid molecular geometries,
so the models do not need to include nonbonded interactions.
The electronic structure of molecules can be described at the semiempirical
level using, for example, the Austin model (AM1) 181 or at the ab initio
level with a Gaussian basis set. 182 Other quantum theoretical methods can be
used, however, as illustrated the method of Kim and co-workers 175,176 who
use a ‘‘truncated adiabatic basis’’ consisting of the ground and first few excited
states of the isolated molecule. For water, these methods introduce about 7–10
basis functions per molecule. 144,176 The wave function coefficients in these
models are found using an iterative method. 144,172–176 An interesting variant
of the empirical valence bond (EVB) approach has recently been introduced by
Lefohn, Ovchinnikov and Voth. 177 In this approach, as applied to water, there
are only three EVB states per molecule, and all potential parameters, rather
than being derived from ab initio or semiempirical methods, are parameterized
against experimental data.
Another method for treating polarizability is to have more than one
potential surface with different electronic properties coupled together. This
method is applicable to systems that can be represented by a few electronic
states, like those with resonance. Each of these states can have its own potential
energy parameters. One such model was developed for the peptide
bond. 183 The peptide bond can be described as consisting of the resonance
structures of two states, one with a N C single bond and no formal charges
and the other with a N C double bond and formal charges on the nitrogen
and oxygen. Each of these states is coupled to the environment, which in
turn can shift the energies of the states. The potential parameters for these
states can be different, but in the peptide bond model only the charges of
the peptide group atoms and the dihedral force constant for rotations about
X
m
Eij
c 2 ma
!
½67Š
½68Š