Reviews in Computational Chemistry Volume 18

94 Polarizability in Computer Simulations
(in internal coordinates). These effects are often incorporated into the fixed
charge distribution and are not explicitly included in the static field, which
is calculated using only charges from different molecules. 15,19–37
Some water models use a shielding function, SðrÞ, that changes the
contribution to Ei from the charge at j, 20,26,30,31,37
Ei ¼ X
j 6¼ i
SðrijÞqjrij
jrijj 3
The shielding function differs from 1 only at small distances and accounts
for the fact that at small separations the electric field will be modified by
the spatial extent of the electron cloud. For larger molecules, the interactions
from atoms that are directly bonded to atom i and are separated by two bonds
or less (termed 1–2 and1–3 bonded interactions) do not typically contribute
to Ei. 32,38
In the most general case, all the dipoles will interact through the dipole
field tensor. The method of Applequist et al. 39,40 for calculating molecular
polarizabilities uses this approach. One problem with coupling all the dipoles
with the interaction given by Eq. [4] is the ‘‘polarization catastrophe’’. As
pointed out by Applequist, Carl, and Fung 39 and Thole, 41 the molecular polarization,
and therefore the induced dipole moment, may become infinite at
small distances. The mathematical origins of such singularities are made
more evident by considering a simple system consisting of two atoms (A and
B) with isotropic polarizabilities, aA and aB. The molecular polarizability,
which relates the molecular dipole moment (l ¼ l A þ l B) to the electric field,
has two components, one parallel and one perpendicular to the bond axis
between A and B,
½14Š
a jj ¼½aA þ aB þð4aAaB=r 3 ÞŠ=½1 ð4aAaB=r 6 ÞŠ ½15Š
a? ¼½aA þ aB ð2aAaB=r 3 ÞŠ=½1 ðaAaB=r 6 ÞŠ ½16Š
The parallel component, a jj, becomes infinite as the distance between the two
atoms approaches ð4aAaBÞ 1=6 . The singularities can be avoided by making the
polarizabilities sufficiently small so that at the typical distances between the
atoms (>1A ˚ ) the factor ð4aAaBÞ=r 6 is always less than one. The Applequist
polarizabilities are in fact small compared to ab initio values. 41,42 Applequist’s
atomic polarizabilities were selected to optimize the molecular polarizabilities
for a set of 41 molecules (see Table 1). Note that careful choice of polarizabilities
can move the singularities in Eqs. [15] and [16] to small distances, but not
eliminate them completely, thus causing problems for simulation techniques
such as Monte Carlo (MC), which tend to sample these nonphysical regions
of configuration space.