# 11.2.2 Polarizable Continuum Models

(June 30, 2021)

The multipolar expansion model is based on exact formulas for the solvation energy of a point multipole in a spherical cavity, which is a crude approximation except (or perhaps even) for small molecules, and the Kirkwood-Onsager model has been largely superseded by the more general class of “apparent surface charge” SCRF solvation models, typically known as PCMs. These models improve upon the multipolar expansion method in two ways. Most importantly, they provide a much more realistic description of molecular shape, typically by constructing the “solute cavity” (i.e., the interface between the atomistic region and the dielectric continuum) from a union of atom-centered spheres, an aspect of the model that is discussed in Section 11.2.2.2. In addition, the exact electron density of the solute (rather than a multipole expansion) is used to polarize the continuum. Electrostatic interactions between the solute and the continuum manifest as an induced charge density on the cavity surface, which is discretized into point charges for practical calculations. The surface charges are determined based upon the solute’s electrostatic potential at the cavity surface, hence the surface charges and the solute wave function must be determined self-consistently.

## 11.2.2.1 Formal Theory and Discussion of Different Models

The PCM literature has a long history and there are several different models in widespread use; connections between these models have not always been appreciated. ,,, Chipman , has shown how various PCMs can be formulated within a common theoretical framework; see Ref. Herbert:2021b for a review. The PCM takes the form of a set of linear equations,

 $\mathbf{Kq}=\mathbf{Rv}\;,$ (11.2)

in which the induced charges $q_{i}$ at the cavity surface discretization points [organized into a vector $\mathbf{q}$ in Eq. (11.2)] are computed from the values $v_{i}$ of the solute’s electrostatic potential at those same discretization points. The form of the matrices $\mathbf{K}$ and $\mathbf{R}$ depends upon the particular PCM in question. These matrices are given in Table 11.3 for the PCMs that are available in Q-Chem.

The oldest PCM is the so-called D-PCM model of Tomasi and coworkers, but unlike the models listed in Table 11.3, D-PCM requires explicit evaluation of the electric field normal to the cavity surface. This is undesirable, as evaluation of the electric field is both more expensive and more prone to numerical problems as compared to evaluation of the electrostatic potential. Moreover, the dependence on the electric field can be formally eliminated at the level of the integral equation whose discretized form is given in Eq. (11.2). As such, D-PCM is essentially obsolete, and the PCMs available in Q-Chem require only the evaluation of the electrostatic potential, not the electric field.

The simplest PCM that continues to enjoy widespread use is the conductor-like model, C-PCM. , Originally derived by Klamt and Schüürmann based on arguments invoking the conductor limit ($\varepsilon\rightarrow\infty$), this model can also be derived as an approximation to more formally correct models. , Over the years, the dielectric-dependent factor

 $f_{\varepsilon}=\frac{\varepsilon-1}{\varepsilon+x}$ (11.3)