Classical Continuum Electrostatics

Continuum electrostatics approximations in which the solvent is represented as a featureless dielectric medium are an increasingly popular approach for the electrostatic 

The main effect of the permanent charges of the solute is to polarize the solvent. The induced charge density pind in the solvent is related to the solvent polarization density P(r) [83, 84] 

At any point r, the polarization P(r) and the total electrostatic field Etot(r) are assumed to be linearly related, 

In practical cases, it is the solute charges that are modeled explicitly, and treated as permanent source charges. In contrast, the whole solvent medium is usually treated as a continuum, without any explicit, permanent, source charges. (This is reasonable for a solvent made of small, neutral molecules ionic liquids would obviously need a different treatment.) Since there are no permanent charges in the solvent, 

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In this chapter we provide an introductory overview of the imphcit solvent models commonly used in biomolecular simulations. A number of questions concerning the formulation and development of imphcit solvent models are addressed. In Section II, we begin by providing a rigorous fonmilation of imphcit solvent from statistical mechanics. In addition, the fundamental concept of the potential of mean force (PMF) is introduced. In Section III, a decomposition of the PMF in terms of nonpolar and electrostatic contributions is elaborated. Owing to its importance in biophysics. Section IV is devoted entirely to classical continuum electrostatics. For the sake of completeness, other computational... 

IV. CLASSICAL CONTINUUM ELECTROSTATICS A. The Poisson Equation for Macroscopic Media... 

So-called solvation/structural forces, or (in water) hydration forces, arise in the gap between a pair of particles or surfaces when solvent (water) molecules become ordered by the proximity of the surfaces. When such ordering occurs, there is a breakdown in the classical continuum theories of the van der Waals and electrostatic double-layer forces, with the consequence that the monotonic forces they conventionally predict are replaced (or accompanied) by exponentially decaying oscillatory forces with a periodicity roughly equal to the size of the confined species (Min et al, 2008). In practice, these confined species may be of widely variable structural and chemical types — ranging in size from small solvent molecules (like water) up to macromolecules and nanoparticles. 

Finally, it is worth emphasizing that for classical solutes s 0 and Eq. (11.17) represents an exact solution to the continuum electrostatics problem. To emphasize this point, we have performed numerical comparisons of MM/PCM calculations versus results obtained from the "adaptive Poisson-Boltzmann solver" (APBS) [5], which represents a recent implementation of the three-dimensional finite-difference approach. (The solvent s ionic strength was set to zero in the APBS calculations.) Results for amino acids, plotted in Fig. 11.2, show sub-kcal/mol differences in most cases, and differences of < 0.1 kcal/mol for the "X = DAS" version of SS(V)PE... 

The discussion of continuum electrostatics in Section 11.2.1 was limited to solution of Poisson s equation, which can be achieved exactly (for classical solutes) or to a good approximation (for QM solutes) using PCMs. In biomolecular applications, however, the objective is usually solution of the Poisson-Boltzmann equation [4, 33]. For low concentrations of dissolved ions, the latter is often replaced by the linearized Poisson-Boltzmann equation (LPBE),... 

An alternative description of the metal that can be profitably coupled to the continuum electrostatic approach is given by the jellium model. In this model, the valence electrons of the conductor are treated as an interacting electron gas in the neutralizing background of the averaged distribution of the positive cores (some discreteness in this core distribution may be introduced to describe atomic motions). The jellium is described at the quantum level by the Density Functional Theory (DFT) and it represents a solid bridge between classical and full QM descriptions of such interfaces further improvements of the jellium which introduce a discrete nature of the lattice are now often used. ... 

These methods treat the solute at the classical discrete level, and the solvent is represented as a classical continuum dielectric. The free energy of solvation is computed as the addition of steric and electrostatic components (equation 2). The steric term can be partitioned into cavitation and van der Waals contributions, while some authors determine it as a whole from empirical relationships with the molecular surface (see refs. 20,21 and 23 for a more detailed explanation). The electrostatic contribution is computed from the classical theory of polarizable fluids [19-23,26], which assumes that the solvent is a dielectric continuum which reacts against the solute charge distribution. 

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

Continuum models are rooted in classical electrostatics, and its applications in the analysis of the dielectric constants of polar liquids. A key relationship is Poisson s equation,... 

Compared to all other intermolecular interactions, the Coulomb interaction is described by a simple law, i.e.. Equation 15.2. A theory for Coulombic interaction, therefore, uses the concepts and laws that have been developed in classical electrostatics. However, it is worth pointing out that the dielectric constant is a macroscopic property and it is therefore, in principle, not correct to describe the solvent as a dielectric continuum on the molecular level. Nevertheless, experience has shown that it is in fact a useful approximation. 

Continuum models are the most efficient way to include condensed-phase effects into quantum-mechanical calculations, and this is typically accomplished by using the self-consistent reaction field (SCRF) approach for the electrostatic component. Therefore it is very common to replace the quantal problem by a classical one in which the electronic energy plus the coulombic interactions of the nuclei, taken together, are modeled by a classical force field—this approach usually called molecular mechanics (MM) (Cramer and Truhlar, 1996). 

At the heart of all continuum solvent models is a reliance on the Poisson equation of classical electrostatics to express the electrostatic potential as a function of the charge density and the dielectric constant. The Poisson equation, valid for situations where a surrounding dielectric medium responds in a linear fashion to the embedding of charge, is written... 

As for the QM/MM description also for PCM, non-electrostatic (or van der Walls) terms can be added to the Vent operator in this case, besides the dispersion and repulsion terms, a new term has to be considered, namely the energy required to build a cavity of the proper shape and dimension in the continuum dielectric. This further continuum-specific term is generally indicated as cavitation. Generally all the non-electrostatic terms are expressed using empirical expressions and thus their effect is only on the energy and not on the solute wave function. As a matter of fact, dispersion and repulsion effects can be (and have been) described at a PCM-QM level and included in the solute-effective Hamiltonian 7/eff as two new operators modifying the SCRF scheme. Their definition can be found in Ref. [17] while a recent systematic comparison of these contributions determined either using the QM or the classical methods is reported in Ref. [18]... 

Solvent continuum models are now routinely used in quantum mechanical (QM) studies to calculate solvation effects on molecular properties and reactivity. In these models, the solvent is represented by a dielectric continuum that in the presence of electronic and nuclear charges of the solute polarizes, creating an electrostatic potential, the so-called reaction field . The concept goes back to classical electrostatic schemes by Martin [1], Bell [2] and Onsager [3] who made fundamental contributions to the theory of solutions. Scholte [4] and Kirkwood [5] introduced the use of multipole moment distributions. The first implementation in QM calculations was reported in a pioneer work by Rivail and Rinaldi [6,7], Other fundamental investigations were carried out by Tapia and Goscinski [8], Hilton-McCreery et al. [9] and Miertus et al. [10], Many improvements have been made since then (for a review,... 

The atomic radii may be further refined to improve the agreement between experimental and theoretical solvation free energies. Work on this direction has been done by Luque and Orozco (see [66] and references cited therein) while Barone et al. [67] defined a set of rules to estimate atomic radii. Further discussion on this point can be found in the review by Tomasi and co-workers [15], It must be noted that the parameterization of atomic radii on the basis of a good experiment-theory agreement of solvation energies is problematic because of the difficulty to separate electrostatic and non-electrostatic terms. The comparison of continuum calculations with statistical simulations provides another way to check the validity of cavity definition. A comparison between continuum and classical Monte Carlo simulations was reported by Costa-Cabral et al. [68] in the early 1980s and more recently, molecular dynamics simulations using combined quantum mechanics and molecular mechanics (QM/MM) force-fields have been carried out to analyze the case of water molecule in liquid water [69],... 

It is also possible to combine the supermolecule and continuum approaches by using specific solvent molecules to capture the short-range effects (i.e., those involving specific noncovalent interactions between solute and solvent) and a reaction field to treat longer range effects.33-35 Alternatively, structures along the gas phase reaction coordinate can be immersed in a box of hundreds (or more) of explicit solvent molecules that are treated using force field approaches.36,37 Each type of method - the SCRF, solvent box, and supermolecule approaches - tests the importance of particular features of the solvent on the reactivity of the solute dielectric constant, multiple specific classical electrostatic interactions, and specific local directional noncovalent interactions, respectively. 

We focus first on the outer-shell contribution of Eq. (7.8), p. 145. That contribution is the hydration free energy in liquid water for a distinguished water molecule under the constraint that no inner-shell neighbors are permitted. We will adopt a van der Waals model for that quantity, as in Section 4.1. Thus, we treat first the packing issue implied by the constraint Oy [1 i>a (7)] of Eq. (7.8) then we append a contribution due to dispersion interactions, Eq. (4.6), p. 62. Einally, we include a contribution due to classic electrostatic interactions on the basis of a dielectric continuum model. Section 4.2, p. 67. 

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