Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 485
activity, the reproduction of spectra of various nature, just to quote the most common applications)
have been already successfully performed. 83 Many of these extensions have been
made possible by the formulation and the following implementation of efficient procedures
to get analytical derivatives of the free energy in solution with respect to various parameters.
100 To understand the role of this methodological extension in the actual development of
the solvation models we just recall the importance of an efficient computation of analytical
gradients with respect to nuclear coordinates to get reliable geometry optimizations and dynamics.
101
It is here important to recall that such improvements are not limited to BE solvation
methods; for example, Rivail and the Nancy group 102 have recently extended their
multipole-expansion formalism to permit the analytic computation of first and second derivatives
of the solvation free energy for arbitrary cavity shapes, thereby facilitating the assignment
of stationary points on a solvated potential energy surface. Analytic gradients for
SMx models at ab initio theory have been recently described 103 (even if they have been
available longer at the semiempirical level 104 ), and they have been presented also for finite
difference solutions of the Poisson equation and for finite element solutions.
Many other extensions of different importance have been added to the original BE
models in the last few years. Here it may be worth recalling a specific generalization which
also has made BE models preferable in particular applications until now almost completely
restricted to FE and FD methods.
At the beginning of the present section we indicated the simplified system (4), valid
only in the limit of an homogeneous and isotropic dielectric, the commonly studied problem;
actually, this has been the only affordable system until recently, with some exceptions
represented by systems constituted by two isotropic systems with a definite interface. The
situation changed only a few years ago (1997-1998) when BE-ASC methods were extended
to treat macroscopically anisotropic dielectrics (e.g., liquid crystals), 99 ionic solutions (e.g.,
isotropic solutions with nonzero ionic strength), 99 and supercritical liquids. 105
Another important feature included in BE methods, as well in other continuum methods,
are the dynamical effects.
In the previous section devoted to Physical Models, we have recalled that an important
issue of all the solvation methods is their ability to treat nonequilibrium, or dynamical, aspects.
There, for clarity’s sake, we preferred to skip the analysis of this feature, limiting the
exposition to static, or equilibrium problems; here, on the contrary, some comments will be
added.
In continuum models no explicit reference to the discrete microscopical structure of
the solvent is introduced; on one hand, this largely simplifies the problem, but on the other
hand it represents a limit which can effect the quality of the calculations and their extensibility
to more complex problems such as those involving dynamical phenomena. However,
since the first formulations of continuum models, different attempts to overcome this limit
have been proposed.
One of the theories which has gained more attention identifies the solvent response
function with a polarization vector depending on time, P(t), which varies according to the
variations of the field from which it is originated (i.e., a solute and/or an external field). This
vector accounts for many phenomena related to different physical processes taking place inside
and among the solvent molecules. In practice, strong approximations are necessary in
order to formulate a feasible model to deal with such complex quantity. A very simple but