Kirkwood model

The spherical cavity, dipole only, SCRF model is known as the OnMger model.The Kirkwood model s refers to a general multipole expansion, if the cavity is ellipsoidal the Kirkwood—Westheimer model arise." A fixed dipole moment of yr in the Onsager model gives rise to an energy stabilization. 

The charge of a number of proteins has been measured by titration. The early experimental work focused on the determination of charge as a function of pH later work focused on comparing the experimental and theoretical results the latter obtained from the extensions of the Tanford-Kirkwood models on the electrostatic behavior of proteins. Ed-sall and Wyman [104] discuss the early work on the electrostatics of polar molecules and ions in solution, considering fundamental coulombic interactions and accounting for the dielectric properties of the media. Tanford [383,384], and Tanford and Kirkwood [387] describe the development of the Tanford-Kirkwood theories of protein electrostatics. For more recent work on protein electrostatics see Lenhoff and coworkers [64,146,334]. 

In contrast, the Onsager—Kirkwood model provides a polarizability in polar liquids larger than that in vapors.21 This is a result of the increase of the dipole moment by the strong electric field, which is generated when a molecule with a permanent dipole moment is introduced in a polarizable medium (Onsager), and the correlation between the orientations of neighboring molecules (Kirkwood).21... 

Simple Liquids. The correlation parameters (151) and (152) are accessible to numerical calculation for simple models of radial interaction between the molecules. In a number of cases, evaluations for simple liquids can be usefully made using the Kirkwood model of hard spheres of diameter d and volume v = Trd lS, when... 

Kinetic balance, of basis sets, 214 Kirkwood model, solvation, 395 Kirkwood-Westheimer model, solvation, 395... 

Fig. 4.3 Plot of the molar polarization according to the Kirkwood model against lOOO/r, where T is the absolute temperature for water ( ) and acetonitrile ( ) in the temperature range 0-50°C.

The above examples illustrate that continuum models such as the Kirkwood model are reasonably successful in describing the static permittivity, provided one has an independent means of estimating the correlation parameter Unfortunately, these estimates are available for only a few polar solvents, so that gK must be considered an independent parameter. The version of Kirkwood s theory presented here only considers orientational polarization. When distortional polarization, that is, the effect of molecular polarizability, is included, interpretation of experimental results is less clear. Since the approach taken here involves continuum concepts, it is necessarily limited. In the following section, a simple model based on a molecular description of a polar liquid is presented. 

The Kirkwood model refers to a general multipole expansion in a spherical cavity, while the Kirkwood-Westheimer model arises for an ellipsoidal cavity. ... 

The charge distribution of the molecule can be represented either as atom-centred partial charges or as a multipole expansion. For a neutral molecule, the lowest order approximation considers only the dipole moment. This may be a quite poor approximation, and fails completely for symmetric molecules that do not have a dipole moment. For obtaining converged results, it is often necessarily to extend the expansion up to order six or more, i.e. including dipole, quadrupole, octupole, etc., moments. Furthermore, only for small and symmetric molecules can the approximation of a spherical or ellipsoidal cavity be considered realistic. The use of the Bom/ Onsager/Kirkwood models should therefore only be considered as a rough estimate of the solvent effects, and quantitative results can rarely be obtained. 

The cavity size in the Bom/Onsager/Kirkwood models strongly influences the calculated stabilization. Unfortunately, there is no consensus on how to choose the cavity radius. In some cases, the molecular volume is calculated from the experimental density of the solvent and the cavity radius is defined by equating the cavity volume to the molecular volume. Alternatively, the cavity size may be derived from the (experimental) dielectric constant and the calculated dipole moment and polarizability. In any case, the underlying assumption of these models is that the molecule is roughly spherical or ellipsoidal, which is only generally true for small compact molecules. 

If, according to the original work, C S,A) is taken from the Onsager-Kirkwood model, then ... 

At a higher computational level, the Kirkwood model gave [M]d+ 11.8° for the enantiomer. At a still higher level, Random Phase Approximation calculations were used to obtain an extensive catalog of transitions for (R,R)-( —)-31, which reproduced the... 

There were several attempts to generalize the Debye function like the Cole/Cole formula (Cole and Cole 1941) (symmetric broadened relaxation function), the Cole/Davidson equation (Davidson and Cole 1950, 1951), or the Fuoss/Kirkwood model (asymmetric broadened relaxation function) (Fuoss and Kirkwood 1941). The most general formula is the model function of Havriliak and Negami (HN function) (Havriliak and Negami 1966,1967 Havriliak 1997) which reads... 

The complete continuum approach was employed in the Kirkwood model on an ab initio level with the basis set of the floating Gauss functions in 1976 [17]. Around that time, a similar formalism for taking the solvent into account was included in the CNDO/2 method [18]. However, such calculational schemes did not gain wide acceptance by reason of excessive expenditure of computer time, difficulties in evaluating some integrals and overall drawbacks inherent in the macroscopic approximation. Eventually some simplified techniques were developed, each of which takes usually one of the components in Eq. (3.8) into account. Next the simplest of these will be considered. 

The factors fU i" (J, 1), called the reaction-field factors, appear to be dependent on the dielectric constant of the solvent, and on the shape of the cavity only. They can be determined fully analytically in the cases of a single center expansion and of a cavity with a regular shape, such as a sphere, a spheroid or an ellipsoid. The case of the sphere is particularly simple and corresponds to the well-known Kirkwood model of solvation the reaction-field factors are scalar quantities which depend on I and on the radius a of the sphere ... 

When the two solutes to be considered are immersed in an ionic solution, the use of FE (or FD) methods based on a straightforward application of the linearized Poisson-Boltzmann equation is not advisable. It is much better to resort to the modified formulation of the Kirkwood model outlined in Section 6.2 in which the single solute is surrounded by a second sphere containing the pure solvent, and then by the bulk of the solvent containing the ionic solution. In this case the use of a BEM/FEM combined procedure is preferable. 

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