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WIREs Comput Mol Sci

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Dielectric continuum methods for quantum chemistry

Advanced Review

John M. Herbert

Published Online: Mar 23 2021

DOI: 10.1002/wcms.1519

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Abstract This review describes the theory and implementation of implicit solvation models based on continuum electrostatics. Within quantum chemistry this formalism is sometimes synonymous with the polarizable continuum model, a particular boundary‐element approach to the problem defined by the Poisson or Poisson–Boltzmann equation, but that moniker belies the diversity of available methods. This work reviews the current state‐of‐the art, with emphasis on theory and methods rather than applications. The basics of continuum electrostatics are described, including the nonequilibrium polarization response upon excitation or ionization of the solute. Nonelectrostatic interactions, which must be included in the model in order to obtain accurate solvation energies, are also described. Numerical techniques for implementing the equations are discussed, including linear‐scaling algorithms that can be used in classical or mixed quantum/classical biomolecular electrostatics calculations. Anisotropic models that can describe interfacial solvation are briefly described. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Molecular and Statistical Mechanics > Free Energy Methods

Timings and parallel scalability data for PCM solvers applied to polyalanine helices, using a classical force field description of the solute. (a) Strong‐scaling data for a CG‐FMM algorithm applied to (Ala)₂₅₀, running on 12 shared‐memory cores per node. (b) Weak‐scaling data for (Ala)_n helices of increasing length, versus the number of Lebedev grid points used to discretize the cavity surface. (c) Comparison of timing data for CG‐FMM versus the ddPCM algorithm, for (Ala)_n. The CG‐FMM data in (c) are the same as those in (b), but all ddPCM calculations were run on a single 12‐core node. Data in (a) and (b) are from Ref. 23 and ddPCM data in (c) are from Ref. 190

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(a) Electrostatic solvation energy along an ab initio molecular dynamics trajectory of glycine (PBE0/6‐31+G* level) in implicit water (SwiG/C‐PCM). The simulation starts at t = 0 from the amino acid tautomer (energy data in orange), which is the most stable form of gas‐phase glycine, but in water this species spontaneously transfers a proton to form the zwitterionic tautomer (energy data in blue). (b) Close‐up view of in the region where the proton transfer occurs. Energy fluctuations are smooth despite the bond‐breaking event. Data are from Ref. 22 and the time step is 0.97 fs

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Demonstration of the SwiG‐PCM discretization approach. (a) Geometry optimization of (adenine)(H₂O)₅₂ in a C‐PCM representation of bulk water, using several different algorithms to discretize the vdW cavity surface. (The surface itself is not shown, but the atomistic region appears in the inset.) Optimizations are performed in Cartesian coordinates so the total number of steps is large. (b) Schematic of the SwiG discretization algorithm, in which the surface charges {q_i} are subject to Gaussian blurring and also to a switching function that attenuates the quadrature weights near the cavity surface. (c) Nonelectrostatic solvation energy () as spheres A and B are pulled part. The value of is related to the solvent‐exposed surface area and thus inherits any discontinuities in the surface area function. Panels (a) and (c) are adapted from Ref. 21; copyright 2010 American Chemical Society

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Examples of surface discretization for continuum solvation: (a) tessellation of the molecular surface using the GePol algorithm, (b) Lebedev discretization of the van der Waals surface for a segment of double‐stranded DNA, and (c) Lebedev discretization of the solvent‐excluded (Connolly) surface for a 384‐atom protein. Panel (a) is reprinted from Ref. 15; copyright 2012 John Wiley & Sons. Panel (c) is reprinted from Ref. 25; copyright 2020 Taylor & Francis

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(a) Comparison of electrostatic solvation energies in aqueous solution, , computed using either an ASC‐PCM or else by numerical solution of Poisson's equation using the apbs software. The data set consists of amino acids described using atomic partial charges from a force field, so that there is no outlying charge. The traditional implementation of IEF‐PCM corresponds to X = DAS (see Table 2) but results are also shown for the transpose X = SAD^†. (b) Convergence of for classical histidine as a function of the solvent's dielectric constant, using the SwiG discretization scheme described in Section 3.2. Adapted from Ref. 24; copyright 2011 Elsevier

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Various constructions of the solute cavity surface, using a set of atomic spheres (in gray) whose envelope defines the van der Waals (vdW) surface, shown in black. The solvent‐accessible surface (SAS, in green) is defined either by augmenting the atomic radii by a probe radius (R_A = R_vdW,_A + R_probe) or equivalently as the center point of the probe sphere as it rolls over the vdW surface. (In the example that is shown, R_probe is smaller than any of the vdW radii.) The solvent‐excluded surface (SES, in magenta) is traced out by arcs of the probe that connect points of contact between the probe and the vdW surface. The SES, which has sometimes been called simply the “molecular surface,” eliminates cusps that appear in the vdW surface along seams of intersection between atom‐centered spheres

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Thermodynamic cycle connecting gas‐ and solution‐phase reaction energies for A + B → C. Changes in shape signify that geometries of A, B, and C may be different in solution than they are in the gas phase, in which case the solvation energies should include a term representing the gas‐phase deformation energy

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(a) Zwitterionic tautomer of glycine (⁻O₂CCH₂NH₃⁺) in a molecular van der Waals (vdW) cavity constructed from atom‐centered spheres. Coloring reflects the sign and magnitude of the molecular electrostatic potential evaluated at the cavity surface, φ^ρ(s) for . (b) Schematic illustration of the same molecular cavity (in green) embedded in a dielectric medium (in blue), illustrating how the continuum polarizes in response to the solute's electrostatic potential. The orange probe sphere illustrates how the atomic radii that define the vdW surface might be augmented to afford a “solvent‐accessible surface” (SAS). The region interior to the solute cavity is designated as , and for a sharp dielectric interface one sets ε(r) ≡ ε_in for . If the solute is described using quantum chemistry then the natural choice is ε_in = 1. Outside of the cavity, the permittivity function ε(r) takes the value ε_out, which is usually the static dielectric constant of the solvent, ε_s. Panel (b) is adapted from Ref. 7; copyright 2008 John Wiley & Sons

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Illustrations of anisotropic permittivity functions ε(r) for use in Poisson's equation. (a) A semicontinuum description of chlorate ion at the air water interface, in which the atomistic solute is ClO₃⁻(H₂O)₃₀. The background color shows the function ε(r), interpolating between ε_out = 1 above the Gibbs dividing surface (GDS) and ε_out = 78 below it, with ε_in = 1 inside of the solute cavity. The horizontal line indicates the position of the dividing surface, Å. (b) Periodic water slab bounded on either side by continuum water (ε = 78, shown in purple), with regions characterized by ε > 15 shown in blue. From left to right, the interpolating function is modified (using a “filling threshold” parameter), in order to exclude pockets of high permittivity that encroach into the interstices between the atomistic water molecules. Panel (b) is adapted from Ref. 269; copyright 2019 American Chemical Society

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Solvatochromic shifts in the lowest ¹ππ* state for derivatives of nitrobenzene (PhNO₂) in different solvents, comparing experimental values to ADC(2)/C‐PCM calculations. Solvent effects are described using perturbation to energy (PTE) and perturbation to density (PTD) variants of the perturbation theory state‐specific (ptSS) approach. Also shown are results for an empirically‐scaled version of the nonequilibrium PTD correction. Adapted from Ref. 586; copyright 2015 American Chemical Society

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Flowcharts representing various state‐specific procedures for combining a polarizable continuum model (PCM) or other self‐consistent reaction‐field (SCRF) procedure with a quantum chemistry method that requires a post‐self‐consistent field (SCF) step. (a) Illustration of the perturbation to energy (PTE) and perturbation to density (PTD) schemes, and two different combinations thereof. Forward‐backward arrows (⇄) indicate where the solute density (ρ) and the polarization charge (σ) are iterated to self‐consistency, whereas downward arrows indicate the points at which various contributions to the energy are extracted. (b) Schematic representation of the PTE and perturbation with self‐consistent energy and density (PTED) procedures for an excited‐state (ES) calculation, along with the PTES procedure designed as a lower‐cost approximation to PTED. Panel (a) is adapted from Ref. 587; copyright 2017 The PCCP Owner Societies. Panel (b) is adapted from Ref. 582; copyright 2019 John Wiley & Sons

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Comparison of experimental hydration energies (Minnesota solvation database^300–303) with values computed using CMIRS: (a) all solvation energies, including both neutral molecules as well as ions (with the number of data points indicated in each case), versus (b) results for charge‐neutral solutes only. Reprinted from Ref. 313; copyright 2016 American Chemical Society

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