Reviews in Computational Chemistry Volume 18

minimization can be replaced by more sophisticated minimization techniques,
102 such as conjugate gradients. 103 Depending on the convergence criterion
used, these iterative methods typically require between 3 and 10
iterations. 77,99,101,103
The dynamic approach to solving for the shell displacements was first
proposed by Mitchell and Fincham. 90 In this method, the mass of each atom
or ion is partitioned between the core and the shell. The mass of the shell
charge is typically taken to be less than 10% of the total particle mass, and
often as light as 0.2 amu. 82,90,97,104 No physical significance is attributed to
the charge mass, as it is not meant to represent the mass of the electronic
degrees of freedom whose polarization the shell charge represents. Rather, it
is a parameter chosen solely for the numerical efficiency of the integration
algorithm. Choosing a very light shell mass allows the shells (i.e., the dipole
moments) to adjust very quickly in response to the electric field generated
by the core (nuclear) degrees of freedom. In the limit of an infinitely light shell
mass, the adiabatic limit would be recovered. The choice of shell mass also has
a direct effect on the characteristic frequency of the oscillating shell model. For
a particle with total mass M, a fraction f of which is attributed to the shell and
1 f to the core, the reduced mass will be m ¼ f ð1 f ÞM, resulting in an
oscillation frequency of
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o ¼
f ð1
k
f ÞM
Shell Models 105
An overly small shell mass would thus result in high oscillation frequencies,
requiring the use of an exceedingly small time step for integration of the
dynamics—an undesirable situation for lengthy simulations. In practice, the
shell mass is chosen to be (1) light enough to ensure adequate response times,
(2) heavy enough that reasonable time steps may be used, and (3) away from
resonance with any other oscillations in the system.
The dynamic treatment of the charges is quite similar to the extended
Lagrangian approach for predicting the values of the polarizable point dipoles,
as discussed in the previous section. One noteworthy difference between these
approaches, however, is that the positions of the shell charges are ordinary
physical degrees of freedom. Thus the Lagrangian does not have to be
‘‘extended’’ with fictitious masses and kinetic energies to encompass their
dynamics.
With an appropriate partitioning of the particle masses between core and
shell, this dynamic method for integrating the dynamics of the shell model can
become more efficient than iterative methods. The lighter masses in the system
require time steps 2–5 times smaller than those required in an iterative shell
model simulation (or a nonpolarizable simulation). 82,90,97,99,104 But because
the iterative methods require 3–10 force evaluations per time step to achieve
½31Š