Essentials of Computational Chemistry

406 11 IMPLICIT MODELS FOR CONDENSED PHASES
than intuition that one continuum modeling algorithm is more or less accurate than another
in the computation of this quantity (Curutchet et al. 2003a). One may take as a standard
for comparison numerically converged solutions of the Poisson equation, but the Poisson
equation is itself a model, and not necessarily the optimal one. In order to make comparisons
against experiment, it is necessary to supplement the polarization (and distortion) energies
with terms corresponding to cavitation, dispersion, structural rearrangement, etc. Models that
purport to compute the full free energy of solvation may then be compared one to another
using experimental free energies of transfer as a common yardstick.
11.3 Continuum Models for Non-electrostatic Interactions
Just as the electrostatic component of the free energy of solvation cannot be measured, neither
can the non-electrostatic components. That being said, various experimental systems may
be biased so as to make one component or another likely to heavily dominate the solvation
free energy. For example, the solvation free energies of charged species would be expected
to be dominated by the electrostatic component, and solvation free energies for ions can
be helpful in the assignment of parametric Born radii to atoms. To assess the free-energy
changes associated with cavitation, dispersion, and other physical effects, different neutral
model systems have been studied, and we examine some of these next.
11.3.1 Specific Component Models
Noble gas atoms have no permanent electrical moments, and the lighter ones are amongst the
least polarizable of chemical systems. Thus, their transfer into a solvent may be regarded as
a process reasonably cleanly associated with cavitation, i.e., the introduction of the noble gas
atom is like introducing a vacuum of equivalent size into the solvent. By examining solvation
data for the noble gases and certain other systems, Pierotti (1976) developed a formula for
the cavitation free energy, associated with a spherical cavity volume, that depends on the
radius of the sphere to the first, second, and third powers. Simulation data have been used
to supplement noble-gas experimental data and refine constants appearing in the Pierotti
formula (Höfinger and Zerbetto 2003). By viewing a non-spherical solute as a collection of
atomic spheres where overlapping volumes are only accounted for once, Pierotti’s formula
has been generalized to molecular cavities (Claverie 1978; Colominas et al. 1999).
Dispersion is a considerably more difficult modeling task. As first noted in Section 2.2.4,
dispersion is a purely quantum mechanical effect associated with the interactions between
instantaneous local moments favorably arranged owing to correlation in electronic motions.
In order to compute dispersion at the QM level, it is necessary to include electron correlation
between interacting fragments, which immediately sets a potentially rather high price on
direct computation. More difficult still, however, is that the continuum model by construction
does not include the solvent molecules in the first place.
As a result, some approaches to computing dispersion energy have involved using either
experimental or theoretical data for gas-phase clusters to estimate the strength of dispersion
interactions between different possible solute and solvent functional groups. However,