Electrostatic equations

Since electrostatic effects dominate the thermodynamic cycle as shown in Figure 10-2, major development efforts have focused on the calculation of electrostatic energy for transferring the neutral and charged forms of the ionizable group from water with dielectric constant of about 80 to the protein with a low dielectric constant (see later discussions). This led to the development of continuum based models, where water and protein are described as uniform dielectric media, and enter into the linearized Poisson-Boltzmann (PB) electrostatic equation,... 

Combination of the electrostatic equations with the chemical equations then completely defines the interface. 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The main, currently used, surface complexation models (SCMs) are the constant capacitance, the diffuse double layer (DDL) or two layer, the triple layer, the four layer and the CD-MUSIC models. These models differ mainly in their descriptions of the electrical double layer at the oxide/solution interface and, in particular, in the locations of the various adsorbing species. As a result, the electrostatic equations which are used to relate surface potential to surface charge, i. e. the way the free energy of adsorption is divided into its chemical and electrostatic components, are different for each model. A further difference is the method by which the weakly bound (non specifically adsorbing see below) ions are treated. The CD-MUSIC model differs from all the others in that it attempts to take into account the nature and arrangement of the surface functional groups of the adsorbent. These models, which are fully described in a number of reviews (Westall and Hohl, 1980 Westall, 1986, 1987 James and Parks, 1982 Sparks, 1986 Schindler and Stumm, 1987 Davis and Kent, 1990 Hiemstra and Van Riemsdijk, 1996 Venema et al., 1996) are summarised here. 

The range of application of the integral equation method is not limited to the standard dielectric model. It encompasses the cases of anisotropic dielectrics [8] (liquid crystals), weak ionic solutions [8], metallic surfaces (see ref. [28] and references cited therein),. .. However, it is required that the electrostatic equation outside the cavity is linear, with constant coefficients. For instance, liquid crystals and weak ionic solutions can be modelled by the electrostatic equations... 

One can associate with any linear electrostatic equation with constant coefficient, formally denoted by Le V = 4ttp (Le is a differential operator with constant coefficients), a function Ge(r) called the Green kernel of the operator LJ4tt and defined by... 

At the end of the optimization, when both the derivatives of the functional with respect to the nuclear coordinates and the PCM charges are nil, the electrostatic equations for the dielectric are satisfied and the equilibrium DPCM charges are obtained. 

In the solvation theory a reformulation of electrostatic Equations (1.119) is expedient. The solute charge density p(r) serves as an input variable, i.e. the driving force. The target of a computation is the scalar solvent response potential [Pg.97]

A completely different approach has also been proposed to compute dQJdx [14,15] instead of finding the derivatives of Equation (3.7), one can differentiate the basic PCM electrostatic equations and then find the solutions to the new equations. By the repeated application of the divergence theorem, this procedure leads to the following expression for the free energy gradients ... 

The electrostatic equations for a molecule in a cavity surrounded by a dielectric continuum are solved by considering the conditions the potential must fulfill [59], The electrostatic potential must obey Poisson s equation inside the cavity ... 

In the water trimer induction nonadditivity provides a dominant contribution, which effectively overshadows all the other terms. Its mechanism is simple. For instance, in a cyclic water trimer the multipoles of A inductively alter the multipoles at B, which, in turn, inductively alter the multipoles at C, which then alter those on A, and so on, until the self-consistency is reached. Various formulations of this simple model were implemented in the simulations since the 1970s [84-87,63,64,50]. To include the many-body induction effects of point charges interacting with a set of polarizable atomic centers the following classical electrostatics equation is solved iteratively... 

When dissolved in nonpolar solvents such as benzene or diethyl ether, the colourless (2a) forms an equally colourless solution. However, in more polar solvents [e.g. acetone, acetonitrile), the deep-red colour of the resonance-stabilized carbanion of (3a) appears (1 = 475... 490 nm), and its intensity increases with increasing solvent polarity. The carbon-carbon bond in (2a) can be broken merely by changing from a less polar to a more polar solvent. Cation and anion solvation provides the driving force for this heterolysis reaction, whereas solvent displacement is required for the reverse coordination reaction. The Gibbs energy for the heterolysis of (2a) correlates well with the reciprocal solvent relative permittivity in accordance with the Born electrostatic equation [285], except for EPD solvents such as dimethyl sulfoxide, which give larger values than would be expected for a purely electrostatic solvation [284]. 

We may therefore make use of well-known solutions of electrostatic equations in order to relate resistance measured by a point probe to volume resistivities. 

M. Neves-Petersen, S. Petersen, Protein electrostatics a review of the equations and methods used to model electrostatic equations in biomolecules - applications in biotechnology, Biotechnol. 

II) Transition Stern-diffuse layer. All electrostatic equations (sec. 3.6c, flg. 3.20) remain unaltered after changing all charges and potentials Into their respective stationary values. The current passing from the Stem to the diffuse layer is determined by the diffusion coefficient for normal transport, x As argued In sec. 2.2c, Is probably lower than D(bulk) but of the same order. However, special cases are possible, say systems with a very high Stem layer... 

Since one goal of classical electrostatic applications to protein reactivity is to incorporate available detailed structural information, the system is usually quite complicated and analytical solutions to the various electrostatic equations are rarely available. Numerical methods for solving these equations rapidly and accurately are therefore a non-trivial requirement. Various methods of calculation are briefly discussed. A key component of such calculations is to have reliable input parameters and data. These typically include the position, size, and charge distribution of all the atoms or groups being explicitly treated, and parameters describing the electrostatic... 

A common source of non-linearity is the Boltzmann term in the PB equation (15). In this case Eqn. (18) is invalid, and the integration in Eqn. (17) become tedious since the electrostatic equations have to be re-solved many times. Variational expressions for the total electrostatic energy for the non-linear PB model are available (Reiner and Radke 1990 Sharp and Honig 1990a Zhou 1994)... 

More recently, an alternative way to proceed has been proposed such procedure avoids computing any geometrical derivatives of the surface charges induced matrices. The new approach consists in differentiating first the electrostatic equation ... 

The effect of an applied alternating E-field may be analyzed as an ordinary capacitor coupled system (Chapter 3). The tissue of interest may be modeled as a part of the dielectric, perhaps with air and other conductors or insulators. The analysis of simple geometries can be done according to analytical solutions of ordinary electrostatic equations as given in Chapter 6. Real systems are often so complicated that analysis preferably is done with the Finite Element Method (FEM) (Chapter 9). 

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