Model systems analytical geometry derivatives

Equations [3] and [4] subject to two appropriate botmdary conditions constitute the Poisson-Boltzmann equation. A number of important, and at one time misimderstood, assumptions have been used in its derivation. These are discussed in the last part of this review. Here we apply the PB equation or its DH approximation to three model systems for which analytical solutions are readily formd. While only a handful of systems possess (exact) analytical solutions, there exists at least one such solution for each of the three common one-dimensional geometries (planar, cylindrical, and spherical) and a wide range of approximate analytical solutions can be formd. 

For molecular systems in the vacuum, exact analytical derivatives of the total energy with respect to the nuclear coordinates are available [22] and lead to very efficient local optimization methods [23], The situation is more involved for solvated systems modelled within the implicit solvent framework. The total energy indeed contains reaction field contributions of the form ER(p,p ), which are not calculated analytically, but are replaced by numerical approximations Efp(p,p ), as described in Section 1.2.5. We assume from now on that both the interface Y and the charge distributions p and p depend on n real parameters (A, , A ). In the geometry optimization problem, the A, are the cartesian coordinates of the nuclei. There are several nonequivalent ways to construct approximations of the derivatives of the reaction field energy with respect to the parameters (A1 , A ) ... 

Despite the large number of analytical solutions available for the diffusion equation, their usefulness is restricted to simple geometries and constant diffusion coefficients. The boundary conditions, which can be analytically handled, are equally simple. However, there are many cases of practical interest where the simplifying assumptions introduced when deriving analytical solutions are unacceptable. For example, the diffusion process in polymer systems is sometimes characterized by markedly concentration-dependent diffusion coefficients, which make any analytical result inapplicable. Moreover, the analytical solutions being generally expressed in the form of infinite series, their numerical evaluation is no trivial task. That is, the simplicity of the adopted models is not necessarily reflected by an equivalent simplicity of evaluation. 

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