Discrete variational methods computations

Within the density functional theory (DFT), several schemes for generation of pseudopotentials were developed. Some of them construct pseudopotentials for pseudoorbitals derived from atomic calculations [29] - [31], while the others make use [32] - [36] of parameterized analytical pseudopotentials. In a specific implementation of the numerical integration for solving the DFT one-electron equations, named Discrete-Variational Method (DVM) [37]- [41], one does not need to fit pseudoorbitals or pseudopotentials by any analytical functions, because the matrix elements of an effective Hamiltonian can be computed directly with either analytical or numerical basis set (or a mixed one). 

A suitable computational approach for the investigation of electronic and geometric structures of transactinide compounds is the fully relativistic Dirac-Slater discrete-variational method (DS-DVM), in a modem version called the density functional theory (DFT) method, which was originally developed in the 1970s (Rosdn and Ellis 1975). It offers a good compromise between accuracy and computational effort. A detailed description can be found in Chapter 4 of this book. 

This means, of course, that we can use basis functions chosen on scientific grounds rather than have to compromise and use functions of convenience. In a word the Discrete Variational Method (DVM) (as this technique has come to be known) enables the use of Slater functions in molecular calculations. In fact, the method is essentially independent of the form of the basis functions since the evaluation of any likely function is computationally trivial. 

In order to make a correct analysis of such an experimental spectrum, an appropriate theoretical calculation is indispensable. For this purpose, some of calculational methods based on the molecular orbital theory and band structure theory have been applied. Usually, the calculation is performed for the ground electronic state. However, such calculation sometimes leads to an incorrect result, because the spectrum corresponds to a transition process among the electronic states, and inevitably involves the effects due to the electronic excitation and creation of electronic hole at the core or/and valence levels. Discrete variational(DV) Xa molecular orbital (MO) method which utilizes flexible numerical atomic orbitals for the basis functions has several advantages to simulate the electronic transition processes. In the present paper, some details of the computational procedure of the self-consistent-field (SCF) DV-Xa method is firstly described. Applications of the DV-Xa method to the theoretical analysises of XPS, XES, XANES and ELNES spectra are... 

The well-known inaccuracy of numerical differentiation precludes the direct calculation of pressure by the insertion of the computed velocity field into Equation (3.6). This problem is, however, very effectively resolved using the following variational recovery method Consider the discretized form of Equation (3.6) given as... 

Continuous penalty method - to discretize the continuity and (r, z) components of the equation of motion, Equations (5.22) and (5.24), for the calculation of r,. and v. Pressure is computed via the variational recovery procedure (Chapter 3, Section 4). 

Later, Kuppermann and Belford (1962a, b) initiated computer-based numerical solution of (7.1), giving the space-time variation of the species concentrations from these, the survival probability at a given time may be obtained by numerical integration over space. Since then, this method has been vigorously followed by others. John (1952) has discussed the convergence requirement for the discretized form of (7.1), which must be used in computers this turns out to be AT/(Ap)2normalized forms of r and t. Often, Ar/(Ap)2 = 1/6 is used to ensure better convergence. Of course, any procedure requires a reaction scheme, values of diffusion and rate coefficients, and a statement about initial number of species and their distribution in space (vide infra). 

The extension of Gillespie s algorithm to spatially distributed systems is straightforward. A lattice is used to represent binding sites of adsorbates, which correspond to local minima of the potential energy surface. The discrete nature of KMC coupled with possible separation of time scales of various processes could render KMC inefficient. The work of Bortz et al. on the n-fold or continuous time MC CTMC) method can lead to computational speedup of the KMC method, which, however, has been underutilized most probably because of its difficult implementation. This method classifies all atoms in a finite number of classes according to their transition probability. Probabilities are computed a priori and each event is successful, in contrast to the Metropolis method (and other null event algorithms) whose fraction of unsuccessful (null) events increases drastically at low temperatures and for stiff problems. In conjunction with efficient search within a class and dynamic variation of atom coordi-nates, " the CPU time can be practically independent of lattice size. After each event, the time is incremented by a continuous amount. 

Most of the studies cited are based on the linearity or the availability of analytical solutions for the systems considered. Many engineering problems, mostly in the chemical engineering field, are non-linear and results must be obtained through computations. Several numerical studies have been reported in the literature on distributed systems. One of the early computational results on distributed optimization was given by Denn ei al. (1966). The solution of the linearized variational equations by Green s function that leads to both the necessary conditions and computational scheme based on the method of steepest ascent, was obtained. The computational difficulties due to the discretization scheme of the catalyst decay problem was overcome by Jackson (1967). Computational results based on steepest ascent for the optimal control for a non-linear distributed systems were also reported by Denn (1966). 

The mathematical modelling of the T-H-M phenomena uses initial - boundary value problems for differential or variational equations involving the physical principles. We shall assume that these problems are discretized by the finite element or similar methods. What we want to point out is that the numerical solution can be computationally very expensive due to... 

These methods can be applied for problems in which differential equations describe any continua. The methods have different approaches for performing calculations. FEM assumes a variational formulation, BEM a formulation by means of integrals, and FDM uses differential equations. Due to the mathematical formulation, it is irrelevant whether the computational problem comes from mechanics, acoustics, or fluid mechanics. In aU three methods, the discretization of the structure is common and is required in order to derive the mathematical formulations for the desired tasks. 

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