Class-modelling methods potential functions

The most popular classification methods are Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), Regularized Discriminant Analysis (RDA), K th Nearest Neighbours (KNN), classification tree methods (such as CART), Soft-Independent Modeling of Class Analogy (SIMCA), potential function classifiers (PFC), Nearest Mean Classifier (NMC) and Weighted Nearest Mean Classifier (WNMC). Moreover, several classification methods can be found among the artificial neural networks. 

KNN)13 14 and potential function methods (PFMs).15,16 Modeling methods establish volumes in the pattern space with different bounds for each class. The bounds can be based on correlation coefficients, distances (e.g. the Euclidian distance in the Pattern Recognition by Independent Multicategory Analysis methods [PRIMA]17 or the Mahalanobis distance in the Unequal [UNEQ] method18), the residual variance19,20 or supervised artificial neural networks (e.g. in the Multi-layer Perception21). 

Two broad classes of technique are available for modeling matter at the atomic level. The first avoids the explicit solution of the Schrodinger equation by using interatomic potentials (IP), which express the energy of the system as a function of nuclear coordinates. Such methods are fast and effective within their domain of applicability and good interatomic potential functions are available for many materials. They are, however, limited as they cannot describe any properties and processes, which depend explicitly on the electronic structme of the material. In contrast, electronic structure calculations solve the Schrodinger equation at some level of approximation allowing direct simulation of, for example, spectroscopic properties and reaction mechanisms. We now present an introduction to interatomic potential-based methods (often referred to as atomistic simulations). 

The main handicap of MD is the knowledge of the function [/( ). There are some systems where reliable approximations to the true (7( r, ) are available. This is, for example, the case of ionic oxides. (7( rJ) is in such a case made of coulombic (pairwise) interactions and short-range terms. A second example is a closed-shell molecular system. In this case the interaction potentials are separated into intraatomic and interatomic parts. A third type of physical system for which suitable approaches to [/( r, ) exist are the transition metals and their alloys. To this class of models belong the glue model and the embedded atom method. Systems where chemical bonds of molecules are broken or created are much more difficult to describe, since the only way to get a proper description of a reaction all the way between reactant and products would be to solve the quantum-mechanical problem at each step of the reaction. 

In conclusion, the current formalism captures extremely rich dynamics in a vast class of differential equations modeling biochemical systems and relates these dynamics to the underlying structure of the biochemical networks. The methods have the potential of lending transparency to the functioning of what now appear to be arbitrarily complex networks. 

The method used here for considering conformal solution models for fluids with molecular anisotropies is based on the method used by Smith (4) for treating isotropic one-fluid conformal solution methods as a class of perturbation methods. The objective of the method is to closely approximate the properties of a mixture by calculating the properties of a hypothetical pure reference fluid. The characterization parameters (in this case, intermolecular potential parameters) of the reference fluid are chosen to be functions of composition (i.e., mole fractions) and the characterization parameters for the various possible molecular pair interactions (like-like and unlike-unlike). In principle, all molecular anisotropies (dipole-dipole, quadrupole-quadrupole, dipole-quadrupole, and higher multipole interactions, as well as overlap and dispersion interactions ) can be included in the method. Here, the various molecular anisotropies are lumped into a single term, so that the intermolecular potential energy uy(ri2, on, a>2) between Molecules 1 and 2 of Species i and / can be written in the form... 

The Finnis-Sinclair analytic functional form was introduced at about the same time as two other similar forms, the embedded-atom method > and the glue model." ° However, the derivation of the Finnis-Sinclair form from the second-moment approximation is very different from the interpretation of the other empirical forms, which are based on effective medium theory as discussed later. This difference in interpretation is often ignored, and all three methods tend to be put into a single class of potential energy function. In practice, the main difference between the methods lies in the systems to which they have been traditionally applied. In developing the embedded-atom method, for example, Baskes, Daw, and Foiles emphasized close-packed lattices rather than body-centered-cubic lattices. Given that angular interactions are usually ig-... 

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