Standard Poisson-Boltzmann equation

A solvated MD simulation is performed to determine an ensemble of conformations for the molecule of interest. This ensemble is then used to calculate the terms in this equation. Vm is the standard molecular mechanics energy for each member of the ensemble (calculated after removing the solvent water). G PB is the solvation free energy calculated by numerical integration of the Poisson-Boltzmann equation plus a simple surface energy term to estimate the nonpolar free energy contribution. T is the absolute temperature. S mm is the entropy, which is estimated using... 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

Solvation energies were computed at the double-c level using a self-consistent reaction field approach based on numerical solutions of the Poisson-Boltzmann equation 58-60). These were computed at the optimized gas-phase geometry utilizing an appropriate dielectric constant for comparison to the experimental conditions (e = 37.5 for acetonitrile e = 20.7 for acetone). The standard set of optimized radii in Jaguar were employed Mo (1.526 A), W (1.534 A), H (1.150 A), C (1.900 A), O (1.600 A). Vibrational analyses using analytical frequencies were computed at the double-q level, ensuring all stationary points to be minima. 

The need for computationally facile models for dynamical applications requires further trade-offs between accuracy and speed. Descending from the PB model down the approximations tree. Figure 7.1, one arrives at the generalized Born (GB) model that has been developed as a computationally efficient approximation to numerical solutions of the PB equation. The analytical GB method is an approximate, relative to the PB model, way to calculate the electrostatic part of the solvation free energy, AGei, see [18] for a review. The methodology has become particularly popular in MD applications [10,19-23], due to its relative simplicity and computational efficiency, compared to the more standard numerical solution of the Poisson-Boltzmann equation. 

The sign of u is determined by that of the zeta potential, which in turn depends on that of the surface charge density at the plane of shear. Equation 3.37 depends in an essential way on the stipulation that x) (and 0 are solutions of the Poisson equation (Eq. 3.26). It does not require that x) be a solution of the Poisson-Boltzmann equation. If that condition is also imposed, then standard DDL theory may be applied to relate the surface charge density at the plane of shear to... 

This chapter explores the number of mean field constructions for ions whose structure goes beyond the point charge description, the representation used in the standard Poisson-Boltzmann (PB) equation. The structural details omitted within a point charge picture can be related to electrostatic structure of an ion,... 

In this section we present in more detail the key assumptions behind the use of the Poisson-Boltzmann equation and discuss some of the ways in which these assumptions have been relaxed. We also give brief introductions to several of the popular alternative approaches to standard PB theory. 

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