Testing a Heterogeneous Polarizable Continuum Model against Exact Poisson Boundary Conditions

The polarizable continuum model (PCM) is a computationally efficient way to incorporate dielectric boundary conditions into electronic structure calculations, via a boundary-element reformulation of Poisson's equation. This transformation is only rigorously valid for an isotropic dielectric medium. To simulate anisotropic solvation, as encountered at an interface or when parts of a system are solvent-exposed while other parts are in a nonpolar environment, ad hoc modifications to the PCM formalism have been suggested, in which a dielectric constant is assigned separately to each atomic sphere that contributes to the solute cavity. The accuracy of this "heterogeneous" PCM (HetPCM) method is tested here for the first time, by comparison to results from a generalized Poisson equation solver. The latter is a more expensive and cumbersome approach to incorporate arbitrary dielectric boundary conditions, but one that corresponds to a well-defined scalar permittivity function, ε(r). We examine simple model systems for which a function ε(r) can be constructed in a manner that maps reasonably well onto a dielectric constant for each atomic sphere, using a solvent-exposed dielectric constant ε_solv = 78 and a range of smaller values to represent hydrophobic environments. For nonpolar dielectric constants ε_nonp ≤ 2, differences between the HetPCM and Poisson solvation energies are large compared to the effect of anisotropy on the solvation energy. For ε_nonp = 4 and ε_nonp = 10, however, HetPCM and anisotropic Poisson solvation energies agree to within 2 kcal/mol in most cases. As a realistic use case, we apply the HetPCM method to predict solvation energies and pK_a values for blue copper proteins. The HetPCM method affords pK_a values that are more in line with experimental results as compared to either gas-phase calculations or homogeneous (isotropic) PCM results.