Reviews in Computational Chemistry Volume 18

100 Polarizability in Computer Simulations
The shell model has its origin in the Born theory of lattice dynamics, used
in studies of the phonon dispersion curves in crystals. 70,71 Although the Born
theory includes the effects of polarization at each lattice site, it does not
account for the short-range interactions between sites and, most importantly,
neglects the effects of this interaction potential on the polarization behavior.
The shell model, however, incorporates these short-range interactions. 72,73
The earliest applications of the shell model, as with the Born model, were to
analytical studies of phonon dispersion relations in solids. 74 These early applications
have been well reviewed elsewhere. 71,75–77 In general, lattice dynamics
applications of the shell model do not attempt to account for the dynamics of
the nuclei and typically use analytical techniques to describe the statistical
mechanics of the shells. Although the shell model continues to be used in
this fashion, 78 lattice dynamics applications are beyond the scope of this chapter.
In recent decades, the shell model has come into widespread use as a model
Hamiltonian for use in molecular dynamics simulations; it is these applications
of the shell model that are of interest to us here.
The shell model to be described in detail below is essentially identical to
the Drude oscillator model; 79,80 both treat polarization via a pair of charges
attached by a harmonic spring. The different nomenclature results largely
from the use of these models in recent decades by two different scientific communities.
The term Drude model is used more frequently in simulations of the
liquid state, whereas the term shell model is used more often in simulations of
the solid state. As polarizable models become more common in both fields,
the terms are beginning to be used indistinguishably. In this chapter, we will
use the term shell model exclusively to describe polarizable models in which
the dipoles are treated adiabatically; they are always at or near their minimum-energy
conformation. We reserve the term Drude oscillator specifically
for applications where the dipole oscillates either thermally or with a quantum
mechanical zero-point energy, and this oscillating dipole gives rise to a dispersion
interaction. The literature has not been entirely consistent on this point of
terminology, but it is a useful distinction to make.
The shell model describes each polarizable ion or atom as a pair of point
charges separated by a variable distance, as illustrated in Figure 2. These
charges consist of a positive, ‘‘core’’ charge located at the site of the nucleus,
and a negative, ‘‘shell’’ charge. These charges are connected by a harmonic
spring. To some extent, these charges can be justified physically as an effective
(shielded) nuclear charge and a corresponding effective charge in the valence
shell that is responsible for most of the polarization response of the atom. This
interpretation should not be taken literally, however; the magnitude of the
charges are typically treated as adjustable parameters of the model rather
than true shielded charge values. As such, they should be viewed primarily
as an empirical method for representing the dipolar polarization of the site.
The magnitudes of both the core and shell charges are fixed in this
model. The polarization thus occurs via relative displacement of the core