Basis expansion Slater-type functions

The l/REP(r), U ARKP(r), and terms At/f EP(r) in 11s0 of Eqs. (23), (31), and (34) or Eq. (6), respectively, are derived in the form of numerical functions consistent with the large components of Dirac spinors as calculated using the Dirac-Fock program of Desclaux (27). These operators have been used in their numerical form in applications to diatomic systems where basis sets of Slater-type functions are employed (39,42,43). It is often more convenient to represent the operators as expansions in exponential or Gaussian functions (32). The general form of an expansion involving M terms is... 

Up to now we have assumed in this chapter the use of Slater-type orbitals. Actually, use may be made of any type of functions which form a complete set in Hilbert space. Since for practical reasons the expansion (2,1) must be always truncated, it is preferable to choose functions with a fast convergence. This requirement is probably best satisfied just for Slater-type functions. Nevertheless there is another aspect which must be taken into account. It is the rapidity with which we are able to evaluate the integrals over the basis set functions. This is particularly topical for many-center two-electron integrals. In this respect the use of the STO basis set is rather cumbersome. The only widely used alternative is a set of Gaus-slan-type functions (GTF). The properties of Gaussian-type functions are just the opposite - integrals are computed simply and, in comparison to the STO basis set, rather rapidly, but the convergence is slow. 

The Slater-type functions (STF) with the radial part in the form (8.3) and integer n can be used as the basis functions in Hartree-Fock-Roothaan calculations of atomic waveftmctions. The radial dependence of the atomic orbitals is an expansion in the radial Slater-type basis functions ipimp whose indices are I, running over s,p, symmetries, and p counting serially over basis-set members for a given s3Tnmetry ... 

BSSE exists for basis sets of all types and even for approximate or alternative Hamiltonians such as semiempirical forms or density functional methods. It is also not negligible if Slater-type functions are used but the magnitude is often less in these circumstances because the outer regions of the wavefunction are usually better represented than with Gaussians. Likewise for various approximate Hamiltonians, the errors can be large or small. So, for example, since using exchange correlation functions in density functional calculations does not require an expansion in terms of virtual orbitals to obtain some electron correlation, there is a relatively small BSSE for the correlation effect in those calculations. 

BSE = basis set expansion FDA = finite difference approximation raA = finite element approximation GTF = gauss-ian-type function ODE = ordinary differential equation nD = n-dimensional PDE = partial differential equation PW = partial-wave STF = Slater-type function. 

The HF CO method is especially efficient if the Bloch orbitals are calculated in the form of a linear combination of atomic orbitals (LCAO)1 2 since in this case the large amount of experience collected in the field of molecular quantum mechanics can be used in crystal HF studies. The atomic basis orbitals applied for the above mentioned expansion are usually optimized in atoms and molecules. They can be Slater-type exponential functions if the integrals are evaluated in momentum space3 or Gaussian orbitals if one prefers to work in configuration space. The specific computational problems arising from the infinite periodic crystal potential will be discussed later. 

The basis functions are represented as linear combinations of Slater-type orbitals (STO) or here Gaussian-type orbitals (GTO). Expansion coefficients are found by solving the secular problem ... 

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]. 

The simplest level of the nonempirical (ab initio) and semiempirical all-valence calculations is the use of a minimal basis set of AO s where each AO in the expansion of Eq. (2.3) is represented by one function, for example, by a Slater-type orbital (STO) ... 

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... 

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