Discrete variational methods basis functions

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 pair of relationships which are useful in manipulating the quantities involved in the use of the discrete variational method when used in conjunction with a basis-expansion approach. Any function which is a linear expansion of the basis functions and for which we have a physical space representation evaluated at our choice of coordinates may have its expansion coefficients determined by multiplication by the inverse of... 

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. 

Implementations have been realized using Gaussian functions (GTO s) ([38, 39] and Slater-type orbitals (STO s) [5, 40, 41], and numerical basis sets [42, 43, 44]. The auxiliary basis may be avoided by the use of a purely numerical representation of the potential on a grid (usually called DVM - Discrete Variational Method [45, 5]), by certain approximations for the potential (Multiple Scattering concept within the so-called mufl5n-tin approximation - [46]), the linear combination of muffin-tin orbitals [47, 3], and in connection with the pseudopotential concept the application of plane-wave basis expansions - see, e.g.. Ref. [112]. 

A different approach was developed by Baerends, Ellis, and Ros (1973). In addition to adopting the Slater potential for the exchange, their approach had two distinct features. The first was an efficient numerical integration procedure, the discrete variational method (DVM), which permitted the use of any type of basis function for expansion, not only Slater-type orbitals or Gaussian-type orbitals, but also numerical atomic orbitals. The second feature was an evaluation of the Coulomb potential from... 

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... 

According to the LCAO methods used for theoretically investigating of B32 type Zintl phases, the wave functions are also expanded in spherical harmonics. However, the basis functions are properly chosen atomic orbitals with radial parts that are not exact solutions for the crystal potential used in the one-electron equation. In the first detailed charge analysis of LiAl by Zunger the discrete variational energy-band 1 s, 2 s, 2 p and Alls,2s,2p,3s,3p and 3d orbitals were used for these investigations. 

The discrete variational (DV) method numerically calculates the basis atomic orbitals using the following wave equation for the radial atomic orbital function Rja r) in spherical coordinates... 

Another real-space approach is to use a finite-element (FE) basis. " The equations that result are quite similar to the FD method, but because localized basis functions are used to represent the solution, the method is variational. In addition, the FE method tends to be more easily adaptable than the FD method a great deal of effort has been devoted to FE grid (or mesh) generation techniques in science and engineering applications. Other real-space-related methods include discrete variable representations, Lagrange meshes,and wavelets. ... 

For spectral methods, the solution/(, z) throughout the domain Q is represented via a polynomial trial function expansion. The discretization procedure thus consists in finding the approximate solution in a reduced subspace, i.e./ (, z) e (i2) c X( 2). Up to this point, in the variational analysis, no explicit definition of X i2), the space used for expressing/(, z), have been made. One possible representation of X (i7) can be obtained in terms of the two-dimensional Lagrange basis functions (12.442), i.e. X i2) = span fj (, z),..., z). Thus, in this particular... 

In brief, the basis of the finite element method is the representation of a body or a structure by an assemblage of subdivisions called finite element. Simple fimctions are chosen approximate the distribution or variation of the actual displacements over each finite element. The unknown magnitudes or amplitudes of the displacement fimctions are the displacements at the nodal points. Hence, the final solution will yield the approximate displacements at the discrete locations in the body, the nodal points. A displacement function can be expressed in various forms, such as polynomials and trigonometric fimctions. Since polynomials offer ease in mathematical manipulations, they have been employed in finite element applications. 

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