Self-consistent electric field

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

It was observed in the previous section that a certain limit case of non-reactive binary ion-exchange is described by the porous medium equation with m = 2 in other words, a weak shock is to be expected at the boundary of the support. Recall that this shock results from a specific interplay of ion migration in a self-consistent electric field with diffusion. Another source of shocks (weak or even strong in the sense to be elaborated upon below) may be fast reactions of ion binding by the ion-exchanger. 

This book treats a selection of topics in electro-diffusion—a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context [1]—[3]. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors [4]. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to ion separation and semiconductor device technology. Some aspects of electrodiffusion are relevant for electrophysiology. 

Optimization of Electrostatics (Self-Consistent Electric Field)... 

This model, as previously discussed, allowed large charge to be built up at various atomic positions in the molecule. The reason for this is that the atomic potentials do not act to counter this charge depletion or growth as would be expected in a self-consistent (electric) field method. Although this degraded the accuracy of EHT theory when applied to ionic situations, it simply was not possible to calculate the structure of transition metal complexes with this scheme. 

The atomic orbitals that are used constitute was is known as the basis set and a minimum basis set for compounds of second-row elements is made up of the 2s, 2p, 2py, and 2p orbitals of each atom, along with the 1 orbitals of the hydrogen atoms. In MO calculations, an initial molecular structure and a set of approximate MOs are chosen and the molecular energy is calculated. Iterative cycles of calculation of a self-consistent electrical field (SCF) and geometry optimization are then repeated until a... 

In addition to the basic structural aspects captured by force fields for water, it may be necessary to model the polarizable nature of water (the tendency for molecules to line up, essentially instantaneously, in the direction of a prescribed or self-consistent electric field). 

The Self-Consistent Reaction Field (SCRF) model considers the solvent as a uniform polarizable medium with a dielectric constant of s, with the solute M placed in a suitable shaped hole in the medium. Creation of a cavity in the medium costs energy, i.e. this is a destabilization, while dispersion interactions between the solvent and solute add a stabilization (this is roughly the van der Waals energy between solvent and solute). The electric charge distribution of M will furthermore polarize the medium (induce charge moments), which in turn acts back on the molecule, thereby producing an electrostatic stabilization. The solvation (free) energy may thus be written as... 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

A classical description of the molecule M in Figure 14.9 can be a force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or more sophisticated electronic structure methods, i.e. FIF, DFT, MCSCF, MP2, CCSD, etc. When a quantum description of M is employed, the calculated electric moments induce charges in the dielectric medium, which in turn acts back on the molecule, causing the wave function to respond and thereby changing the electric moments, etc. The interaction with the solvent model must thus be calculated by an iterative procedure, leading to various Self-Consistent Reaction Field (SCRF) models. 

ABSTRACT. The effect of a macroscopic phase (solvent or adsorbent) interacting with a molecule can be taken into account in quantum chemical computations hy means of a self-consistent reaction field approach. The macroscopic phase is represented by a continuum with macroscopic dielectric properties. The molecule undergoes an electric potential arising from the polarization by the molecular charge distribution of the macroscopic pliase. 

The extension of continuum solvation modes to evaluate vibrational frequencies of molecular systems in solution was pioneered by RivaU and co-workers in the 1980s [150] by exploiting a semiempirical QM molecular model coupled with a continuum description of the medium. Further extension to ab initio QM methods, including the treatment of electron correlation effects and electrical and mechanical anharmonicities, was then proposed [151-153] in the framework of the polarizable continuum model (PCM). Wang et al. [154] used an ab initio self consistent reaction field (SCRF) Onsager model to compute vibrational frequencies at different levels of... 

An alternative to using explicit water molecules in modeling solvation phenomena is the use of solvation models. In these models, each solute molecule is assumed to exist in a solvent. The solvent is almost structureless it surrounds the solute, approaching everywhere to within a finite distance, known as the solvent accessible surface. The electric field of the solute induces an electric response in the solvent, on the solvent accessible surface. This respon.se, in turn, modifies the electronic structure of the solute. Because of this, these methods are known as Self-consistent Reaction Field models. 

The work done increases the energy of the total system and one must now decide how to divide this energy between the field and the specimen. This separation is not measurably significant, so the division can be made arbitrarily several self-consistent systems exist. The first temi on the right-hand side of equation (A2.1.6) is obviously the work of creating the electric field, e.g. charging the plates of a condenser in tlie absence of the specimen, so it appears logical to consider the second temi as the work done on the specimen. 

Even when we discuss the electric field of light without reference to any particular charge, we must be aware of these differences. When that field interacts with a charge, as in light scattering, we will be in trouble unless a self-consistent set of units has been employed. 

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