Intrinsically Smooth Discretization

An issue with the PCM formalism introduced in Section 11.2.2.1 is that the electrostatic energy is in general a discontinuous function as the solute atoms are displaced, because the number and size of the surface tesserae may change as a function of solute geometry. A similar problem is suffered by finite-difference Poisson-Boltzmann solvers, and the solution in those cases (in order to achieve stable forces for MD simulations, for example) is tight thresholding and/or some kind of interpolation between grid points [80-83]. 

The situation is simpler in the case of PCMs, where only the cavity surface (and not the whole of three-dimensional space) needs to be discretized. A switching function of the form 

A solution to this problem is to use Gaussian blurring of the surface charges [41, 87], in which each discretization charge q,- is replaced by a Gaussian function 

The width parameters are chosen so as to approximate a uniform surface charge in the case of a single point charge centered in a spherical cavity [87], and are fixed parameters once the number of Lebedev discretization points per sphere is specified. The matrix 

The matrix elements of D require some care. Off-diagonal elements can be computed from Sy according to [42] 

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An issue with all of these discretization schemes—except possibly the genuine isodensity surface that is not considered in this work—is the fact that the solvation energy is a discontinuous function of the atomic coordinates, because discretization points appear and disappear as the overlap between atomic spheres changes. (In principle, the energy also loses rotational invariance upon discretization, but we fund that this problem is not serious [42]). The discontinuity problem, which is shared by finite-difference Poisson-Boltzmann solvers, has recently been resolved in the context of PCMs, with the development of intrinsically smooth discretization algorithms [42, 70, 76, 87]. These are discussed in Section 11.4.1. 

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