Reviews in Computational Chemistry Volume 18

112 Polarizability in Computer Simulations
limit, and methods constraining the charge based on the bonding network
are an extension of EE models that appear to be successful at controlling
the coherence lengths. 149 There now exist classical models that can describe
the charge transfer reasonably across the full range of a dissociating bond,
but these are currently less well developed. 157
The polarization energy in the EE models can be compared directly to
that in the polarizable point dipole and shell models. Consider the first term
in Eq. [43],
X
X X
w
i 2 a
0 i qi þ 1
2
i 2 a j 2 a
qi qj JijðrijÞ E gp
a
This term represents the energy required to induce charges qi on the atoms of
molecule a in the electric field of its neighbors, relative to the energy of the
isolated molecule. This quantity is simply the polarization energy of the molecule.
The polarization energy of the full system can thus be written
Upol ¼ X X
w 0 i qi þ 1 X X
qi qj JijðrijÞ E
2
gp
" #
a
a
i 2 a
i 2 a j 2 a
which can be compared to Eqs. [9] and [26]. 10
There exist other models that treat polarizability using variable charges
in a way similar to the fluctuating charge model. 22,53,127,143,158 In the Sprik
and Klein 22 model for water, four charge sites are located near the oxygen
atom in a tetrahedral geometry, in addition to the three atom-centered permanent
charges. The tetrahedron of charges is used to represent an induced dipole
moment on the oxygen center. This approach differs from a polarizable point
dipole model in using a dipole of finite extent. It also differs from a shell model
in that the point charges are fixed in the molecular frame. Consequently, the
Sprik–Klein model should perhaps best be considered an entirely different type
of model. The model of Zhu, Singh, and Robinson 158 is similar to the Sprik–
Klein model, but it has no permanent charges. The four charge sites, two on
hydrogen atoms and two on lone-pair positions 1 A˚ from the oxygen atom, are
all variables coupled to the electric field. For both these models, the coupling is
described by the polarizability, a, just as with other dipole polarizable models.
Wilson and Madden 159 described a model for ions in which charge is transferred
between ends of a rigid, rotating rod. In the model of Perng et al., 143
the charge, qi, on an atom is equal to a permanent value, q0 i , plus an induced
part, dqi. The induced charge is dependent on the electrostatic potential at that
site and all the induced charges are coupled through Coulombic interactions,
similar to the fluctuating charge models. In the polarizable point charge (PPC)
model of Svishchev et al., 53 charges are coupled directly to the electric field at
that site, so this model is slightly different from the fluctuating charge model.
½47Š
½48Š