Computational convenience

The venerable bireciprocal potential consists of a repulsive tenn A/t and an attractive temi -B/r with n > m. This potential fiinction was introduced by Mie [118] but is usually named after Leimard-Jones who used it extensively. Almost invariably, / = 6 is chosen so that the attractive tenn represents the leading dispersion tenn. Many different choices of n have been used, but the most connnon is n = 12 because of its computational convenience. The Leimard-Jones (12,6) potential can be written in tenns of the well depth (s) and either the minimum position or the zero potential location (a) as... 

Many factors affect dry deposition, but for computational convenience air quaUty models resort to using a single quantity called the deposition velocity, designated or to prescribe the deposition rate. The deposition velocity is defined such that the flux T of species i to the ground is... 

Usually called simply a i< -value, it adds nothing to thermodynamic knowledge of X T.E. However, its use may make for computational convenience, allowing formal ehmination of one set of mole fractions [yi] or (x,) in favor of the other. Moreover, it characterizes lightness of a constituent species. For a light species, tending to concentrate in the vapor phase, K> 1 for a heavy species, tending to concentrate in the liquid phase, K<1. 

This matrix method is both flexible and computationally convenient. It can handle complemented events. such as required for event trees as well as distributions by simply... 

A time-dependent Schrodinger equation with H replaced by/(H) can be used to infer dynamics information about H, with f H) being chosen for computational convenience. 

The spin functions have the important property that they are orthonormal, i. e., = = 1 and = = 0. For computational convenience, the spin orbitals themselves are usually chosen to be orthonormal also ... 

Although in principle one could choose a set of arbitrary values for the solvent coordinates sm, solve the eigenvalue equation (2.23), and compute the free energy (2.12), in practice a preliminary aquaintance with the equilibrium solvation picture for the target reaction system serves as a computationally convenient doorway for the calculations in the nonequilibrium solvation regime. We show this below in the section dedicated to an illustration of the method for a two state case reported in BH-II. 

The molecule of the difference scheme over the time steps is given in Figure 3.19. The half-step may not have in general the significance of an intermediate time step and is more a matter of computational convenience. 

Another important issue associated with tribological simulations involves the definition of the system to be studied. For example, a simple tribological system consists of two atomically flat, defect-free surfaces sliding past one another. Because of computational convenience, it is common... 

As a final point, we note that typical surfaces are usually not crystalline but instead are covered by amorphous layers. These layers are much rougher at the atomic scale than the model crystalline surfaces that one would typically use for computational convenience or for fundamental research. The additional roughness at the microscopic level from disorder increases the friction between surfaces considerably, even when they are separated by a boundary lubricant.15 Flowever, no systematic studies have been performed to explore the effect of roughness on boundary-lubricated systems, and only a few attempts have been made to investigate dissipation mechanisms in the amorphous layers under sliding conditions from an atomistic point of view. 

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. 

This is a quantity which can be easily constructed given a set of molecular orbitals (the coefficients C ) and a precalculated set of atomic orbital integrals. At this point, the Hartree-Fock equations have been reduced to a matrix eigenvector problem, FC = SCe, but not in a computationally convenient form. Following the analysis leading to equation 84, we first define the transformed Fock matrix as... 

A (truncated) multipole expansion is a computationally convenient single-center formalism that allows one to quantitatively compute die degree to which a positive or negative test charge is attracted to or repelled by die molecule that is being represented by the multipole expansion. This quantity, die molecular electrostatic potential (MEP), can be computed exactly for any position r as... 

In the case of a-K2Cr04, the estimation was carried out on a crude assumption that the interatomic charge could be assigned equally to the Cr and the four O atoms. Staudenmann et al. (69) have proposed a different method to define the effective region of an atom, where an atomic region is represented by an assembly of many small parallel piped volume elements for computational convenience. This method may enable us to define the effective atomic region more precisely for covalent molecules. From a practical point of view, however, it seems to be difficult to define an exact atomic region in this way. 

For computational convenience, the potential < Vs (r p) > is discretized and represented by a set of point charges qt that simulate the electrostatic potential generated by the continuous solvent distribution... 

The second simplification, which is introduced for computational convenience, is the strong orthogonality (SO) constraint, by which all the orbitals in... 

The pseudopotential method has various advantages. Eliminating the core electrons from the problem reduces the number of particles that must be considered in the Kohn-Sham (KS) equations for the effective one-electron potential. For example, a pseudopotential calculation for bulk silicon (with 10 core and 4 valence electrons) requires the calculation of 4 occupied bands at each k-point, while an all-electron approach would require the calculation of 14 occupied bands. More importantly, the smooth spatial variation of the pseudopotential and pseudowavefunction allows the use of computationally convenient and unbiased basis, such as plane wave basis sets or grids in space. 

Erlang- and phase-type distributions provide a versatile class of distributions, and are shown to fit naturally into a Markovian compartmental system, where particles move between a series of compartments, so that phase-type compartmental retention-time distributions can be incorporated simply by increasing the size of the system. This class of distributions is sufficiently rich to allow for a wide range of behaviors, and at the same time offers computational convenience for data analysis. Such distributions have been used extensively in theoretical studies (e.g., [366]), because of their range of behavior, as well as in experimental work (e.g., [367]). Especially for compartmental models, the phase-type distributions were used by Faddy [364] and Matis [301,306] as examples of long-tailed distributions with high coefficients of variation. 

Another limitation of most work to date is that relatively few cases of theoretically determined MWD s have been checked against experimentally determined values. In many instances, experimentally determined kinetic parameters from the literature have not even been used parameters were chosen instead for computational convenience and gave misleading results. The works of Hamielec and co-workers represent notable exceptions to the above. Using carefully controlled batch experiments, MWD determination by gel permeation chromatography (GPC), Hamielec et al. (25) obtained good agreement with theoretical calculations based on parameters at low viscosities. Duerksen et al. (17) did the same for CSTR s. 

Different solvation methods can be obtained depending on the way the (Vs(r p)) xj tern1 is calculated. So, for instance, in dielectric continuum models ( Vs(r p)) x is a function of the solvent dielectric constant and of the geometric parameters that define the molecular cavity where the solute molecule is placed. In ASEP/MD, the information necessary to calculate Vs(r, p))[Xj is obtained from molecular dynamics calculations. In this way (Vs(r p))[Xj incorporates information about the microscopic structure of the solvent around the solute, furthermore, specific solute-solvent interactions can be properly accounted for. For computational convenience, the potential Vs(r p)) X is discretized and represented by a set of point charges, that simulate the electrostatic potential generated by the solvent distribution. The set of charges, is obtained in three steps [26] ... 

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