Flowchart of the algorithm (upper panel). Block A represents the self-consistent iterative method for ɛ and ϕ described in Sec. III (lower panel). Loops 1 and 2 are proposed (but not analyzed herein) to allow solute-conformation/liquid-density relaxations, and charge relaxation, respectively.
Left panel: Quantities defined in Eq. (6) to solve the Poisson equation in a two-dimensional non-uniform grid. Right panel: Representation of the three classes of sub-cells obtained upon cell refinement according to Eq. (7): interior (light grey), side (dark grey), and vertex (white) sub-cells require different treatment in the context of non-uniform grids. Extension to three-dimensions is straightforward.
Grey scale representation of (a) molecular charge density (white: negative charge, black; positive charge), and (b) liquid density (white: larger density, black: ρ = 0; heights and locations of the water density peaks were calculated in Ref. 18). (c) Non-uniform grid with local resolution determined by the liquid density; the highest resolution is at the solute/liquid interface. The cells have sides of length h = 1 Å, and using ζ ∼ 0.1 yields n = 2–13 [cf. Eq. (7)]. Coordinates x and y are in Å. For visualization purpose all quantities defined on the non-uniform grid are interpolated onto a high-resolution uniform grid using random Renka–Cline interpolation with the OriginLab software.
Grey-scale representation of (a) dielectric ɛ(r) (white: larger values, peaks of up to ɛ ∼ 150 are observed; black: ɛ = 1), and (b) potential ϕ(r) (white: negative field; black: positive) obtained self-consistently as illustrated in Fig. 1 (cf. Sec. III). Coordinates x and y are in Å. (c) Convergence of the calculation (upper panel) for two relaxation methods used to solve Eq. (6): weighted Jacobi with ω = 1 (strict Jacobi, solid circles) and ω = 2/3 (open circles), and Gauss-Seidel (squares; red-black GS yields identical results). Convergence of the calculation for the Na+–Cl− ion pair at a fixed separation of 2.6 Å is also shown for comparison (lower panel).
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