Slater-type atomic functions

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

International Tables for Crystallography 1992). The function <]/> for Slater-type radial functions can be expressed in terms of a hypergeometric series (Stewart 1980), or in closed form (Avery and Watson 1977, Su and Coppens 1990). The latter are listed in appendix G. As an example, for a first-row atom quadrupolar function (/ = 2) with n, = 2, the integral over the nonnormalized Slater function is... 

Basis Functions. Functions usually centered on atoms which are linearly combined to make up the set of Molecular Orbitals. Except for Semi-Empirical Models where basis functions are Slater type, basis functions are Gaussian type. 

STO-3G. A Minimal Basis Set. Each atomic orbital is written in terms of a sum of three Gaussian functions taken as best fits to Slater-type (exponential) functions. 

Here P and Plm are monopole and higher multipole populations / , are normalized Slater-type radial functions ylm are real spherical harmonic angular functions k and k" are the valence shell expansion /contraction parameters. Hartree-Fock electron densities are used for the spherically averaged core and valence shells. This atom centered multipole model may also be refined against the observed data using the XD program suite [18], where the additional variables are the population and expansion/contraction parameters. If only the monopole is considered, this reduces to a spherical atom model with charge transfer and expansion/contraction of the valence shell. This is commonly referred to as a kappa refinement [19]. 

The OBS-GMCSC method offers a practical approach to the calculation of multiconfiguration electronic wavefunctions that employ non-orthogonal orbitals. Use of simultaneously-optimized Slater-type basis functions enables high accuracy with limited-size basis sets, and ensures strict compliance with the virial theorem. OBS-GMCSC wavefunctions can yield compact and accurate descriptions of the electronic structures of atoms and molecules, while neatly solving symmetry-breaking problems, as illustrated by a brief review of previous results for the boron anion and the dilithium molecule, and by newly obtained results for BH3. 

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 one-electron Kohn-Sham equations were solved using the Vosko-Wilk-Nusair (VWN) functional [27] to obtain the local potential. Gradient correlations for the exchange (Becke fimctional) [28] and correlation (Perdew functional) [29] energy terms were included self-consistently. ADF represents molecular orbitals as linear combinations of Slater-type atomic orbitals. The double- basis set was employed and all calculations were spin unrestricted. Integration accuracies of 10 -10 and 10 were used during the single-point and vibrational frequency calculations, respectively. The cluster size chosen for Ag or any bimetallic was... 

The term STO-3G denotes a minimal Gaussian basis set in which each Slater-type atomic orbital (s, p or d) is approximated by a fixed block of three Gaussian functions ,... 

Hehre WJ, Stewart RF, Pople JA (1969) Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. J Chem Phys 51 2657-2664 Hohenberg P, Kohn W (1964) Density functional theory of the inhomogeneous electron gas. Phys Rev B 136 864-871... 

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 ... 

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