Essentials of Computational Chemistry

404 11 IMPLICIT MODELS FOR CONDENSED PHASES
It may seem that using the Poisson equation to determine the effective Born radii, as
described above, defeats the purpose of developing GB as an alternative to PB. Actually,
the solution of the Poisson equation for a single charge is vastly simpler than for a
complete charge distribution, but the procedure is still computationally intensive, and subject
to possible numerical noise. An analytical approximation to this procedure, known as pairwise
descreening (PD), has been described by Hawkins, Cramer, and Truhlar (1995), and has
been shown to increase computational efficiency with very little cost in accuracy. Gallicchio
and Levy (2004) have described a modified PD algorithm that has improved sensitivity to
conformational changes in biological macromolecules and emphasized its potential utility in
docking calculations. A procedure that is similar in spirit to the PD approach but also takes
advantage of the molecular connectivity that must be defined in a force-field calculation (and
is thus limited to such applications) has also been described (Qiu et al. 1997).
Note that the GB approach describes the charge distribution of the solute using atomcentered
atomic partial charges. In that sense it may be called a distributed monopole
representation. A key issue, obviously, is how those partial charges are computed. In forcefield
GB implementations, all models to date simply use the partial charges already defined
for the atom types for use in solving the charge–charge interaction term in the molecular
mechanics energy, and parameters in the GB model, like the Coulomb radius, are optimized
with respect to this choice (see, for example, Cheng et al. 2000, Onufriev, Case, and Bashford
2002, and Zhang et al. 2003). For quantum mechanical calculations, the charges may
in principle be determined from any one of the many methods described in Section 9.3.
However, it must be kept in mind that at the QM level, the calculation is of the SCRF
variety. That is, the atomic partial charges will be free to change as the wave function
polarizes in response to the surrounding dielectric medium.
In order to implement the reaction field conveniently into the SCF equations, it is helpful
if the partial charges have a relatively simply dependence on elements of the density matrix.
Thus, for instance, early versions of the QM SCRF GB solvation models of Cramer and
Truhlar (so-called SMx models, where x is essentially a version number) used Mulliken
charges. As noted in Section 9.3, however, Mulliken charges provide a rather poor approximation
of the molecular charge distribution. Later generations of these models, to include
the most modern versions SM5.42 and SM5.43, use the CM2 and CM3 Class IV charge
models, respectively, to assign the atomic partial charges (hence the ‘.42’ and ‘.43’ suffixes
in the model names). As the charge models are designed to predict ‘good’ partial atomic
charges irrespective of the underlying wave function, there is a leveling of the electrostatics
across methods and SMx models for different levels of theory tend to have very similar
parameters. The parameters themselves are primarily the Coulomb radii ρk, asdefinedin
Fig 11.7. A GB SCRF implementation has also been reported for an SCC-DFTB model (Xie
and Liu 2002).
11.2.3 Conductor-like Screening Model
When the Poisson equation is solved using a boundary element approach, the charges on the
tesselated molecular surface are determined so that they provide an equivalent representation