Distance-dependent function

More recently, the concept of the molecular lipophilicity potential (MLP), initially introduced by Audry and coworkers (2), has attracted increasing attention. The method involves mapping the local lipophilicity at points in 3D space around a chemical compound through the use of a parametrized fragmental system coupled to an empirical distance function. Despite the absence of any physical basis for the distance-dependent functions introduced, and the difficulties inher-... 

This model has been, and still is, widely used especially for some specific applications. An older use is in the description of dielectric saturation effects around ions. The origin is the Debye model, not completely satisfying and thus subjected over the years to many variants. The spherical symmetry of the problem suggests the use of a distance dependent function s(r). The functions belonging to this family are often called sigmoidal functions because their spatial profile starts from a low value and increases monotonically to reach... 

The SAMI parametrization [74] further extends the number of two-electron integrals included in the treatment. They are calculated first for the AOs taken as in the STO-3G Gaussian basis set, but then scaled using the distance dependent functions containing adjustable parameters. The SAMI method has been parametrized for the elements H, Li, C, N, O, F, Si, P, S, Cl, Fe, Cu, Br, and I. Unfortunately, this parametrization was never thoroughly published and studied. The same applies to the PM5 method [75] which is implemented only by a commercial software, without adequate explanation.2 Further refinement of the system of correcting Gaussian contributions to the interatomic interaction functions has been proposed in [71],... 

The energy eigenvalues of a particle in a spherical cell of diameter R can be obtained from Schrodinger s equation. It is convenient to write the wave function as a product of an angle-dependent and a distance-dependent function, the latter of which is given by ... 

This is in contrast to the soft-sphere approximation where some distance-dependent function such as the second term in the Lennard-Jones equation (3.166) is used. 

HINT), is a hydrophobic field calculated by -> Leo-Hansch hydrophobic fragmental constants scaled by surface area and a distance-dependent function [Kellogg et al., 1991 Kellogg and Abraham, 1992a Abraham and Kellogg, 1993]. Hydropathy is a term used in structural molecular biology to represent the hydrophobicity of amino acid side chains. 

The algorithm of an ANN imitates the information processing in the human brain, whose capabilities in image processing are undefeated so far. The self-organizing map consists of artificial neurons that are characterized by weight vectors with the same dimensionality as the descriptor set The artificial neurons are connected by a distance-dependent function. 

Here qt and qj are point atomic charges and/GB is a distance-dependent function. The summation is over all pairs of atoms in the solute. The fGB function is chosen to satisfy certain boundary conditions for a distance ry = 0,/GB = Rh and for a long distance ry = oo, foB = ry- These conditions turn Eq. (31) into the Born formula when ry = 0 and would recover the Coulomb interactions for ry = oo. Indeed, the total electrostatic energy of interaction between charges at large distance is given by... 

The simple distance-dependent dielectric has no physical basis and so it is not generally recommended, except when no alternative is possible. More sophisticated distance-dependent functions can also be employed. Many of these have an approximately sigmoidal shape in which the relative permittivity is low at short distances and then rises towards the bulk value at long distances. One example of such a function is [Smith and Pettit 1994] ... 

Values for the distance dependent functions M(d) and V(d) for flat plates... 

The distance dependent functions for tails and loops attached to spheres... 

Hesselink et al. have also evaluated the distance dependence function V(d) [see equation (11.26)] for the elastic free energy contribution to steric stabilization. The result obtained for tails is the same as that published by Meier (1967)... 

The distance dependent functions for the elastic repulsion of spheres and flat plates... 

Fig. 153. The universal distance dependent function for the free energy of interaction according to the selfconsistent field approach (after Joanny et al., 1979).

Two groups have developed hydrophobic fields for CoMFA 3D-QSAR applications. The HINT (hydrophobic interaction) program calculates the hydrophobic field of a molecule using Hansch-Leo hydrophobic constants scaled by surface area and a distance-dependent function. The MLP program... 

This model is appropriate for a homogeneous system with only one dielectric constant and no ionic strength. By making e a distance-dependent function e(r), the model can be adapted to account for thef variation in dielectric to approximately represent a systerfi with low dielectric solutes immersed in high dielectric solvent. However, to properly account for dielectric discontinuities and ionic strength effects, other models, such as a continuum solvent Poisson-Boltzmann (PB) model must be used to compute electrostatic forces. 

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