Reviews in Computational Chemistry Volume 18

negligible errors in such properties as the electric field generated outside the
molecule. Unfortunately, there are some physical effects that this idealization
obscures, such as the environment-dependent polarizability.
All polarizable models share the ability to polarize, by varying their
charge distribution in response to their environment. In addition, shell models
and EE models with charge-dependent radii have the ability to modify their
polarizability—the magnitude of this polarization response—in response to
their local environment. Consequently, it is reasonable to expect that shell
models and mechanically coupled EE models may be slightly more transferable
to different environments than more standard PPD and EE models. To date, it
is not clear whether this expectation has been fully achieved. Although some
shell-based models for both ionic and molecular compounds have been
demonstrated to be transferable across several phases and wide ranges of
phase points, 73,96,99,243 it is not clear that the transferability displayed by these
models is better than that demonstrated in PPD- or EE-based models. And
even with an environment-dependent polarizability, it has been demonstrated
that the basic shell model cannot fully capture all of the variations in ionic
polarizabilities in different crystal environments. 85
Computational Efficiency
Comparison of the Polarization Models 129
One significant difference between the different methods of incorporating
polarization is their computational efficiency. For energy evaluations, the
electronegativity equalization-based methods are considerably more efficient
than the dipole or shell models. Dipole-based methods require evaluation of
the relatively CPU-expensive dipole–dipole interactions (Eq. [7]). The
charge–charge interactions used in shell models are much cheaper, by about
a factor of three. But this advantage is eliminated by the need to represent
each polarizable center by two point charges, thus quadrupling the total number
of interactions that need to be computed. Methods based on electronegativity
equalization typically represent each polarizable site by a single charge
(either point or diffuse), and energy evaluations are thus three-to-four times
faster than with the other models, for direct summation. Semiempirical methods
have 4–10 basis functions per atom, and each energy evaluation requires
solving large matrices, thereby decreasing the computational efficiency of these
models. 144,172–176 In the simpler two-state empirical model, the additional
computational requirements are comparable to the EE models. 183
Energy evaluation for any collection of point charges and dipoles can be
accelerated significantly by using fast-multipole 244,245 or particle-mesh 246,247
methods. The computational advantages of these methods are proportionally
much greater for the dipole-based models, because they avoid the direct
evaluation of a more expensive interaction. In large systems, the overhead
associated with using dipoles can be reduced to about a third more than
the cost of using point charges. Algorithms for performing conventional, 66