Slater-type functions

The coefficients and the exponents are found by least-squares fitting, in which the overlap between the Slater type function and the Gaussian expansion is maximised. Thus, for the Is Slater type orbital we seek to maximise the following integral ... 

In the Clementi and Roetti tables, the radial wave function of all orbitals in each electron subshell j is described as a sum of Slater-type functions ... 

N2 = 2. For carbon, the subshells are expansions of 6, respectively, 4, Slater-type functions, that is, V = 6, v2 = 4. Because of the spherical averaging of pc and pv, the occupancies of orbitals with the same n and l values are the same, regardless of their m values. In other words, the electrons in a subshell are evenly distributed among the orbitals with different values of the magnetic quantum number m. 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

The calculations are not all at exactly the same bond length R. The basis set is indicated after the slash in the method. R, L, C, and T are basis sets of Slater-type functions. The aug-cc-pVDZ and aug-cc-pVTZ basis sets [360] are composed of Gaussian functions. SCF stands for self-consistent-field MC, for multiconfiguration FO, for first-order Cl, for configuration interaction MR, for multireference MPn, for nth-order Mpller-Plesset perturbation theory and SDQ, for singles, doubles, and quadruples. 

Here cw- is the coefficient with which the /-th basis function Xt(r Q (usually a Slater-type function, or STF for short) enters the jU-th orbital of the a-th configuration, Q is an adjustable parameter (usually the STF s exponential parameter), and TV /the fixed but otherwise arbitrary number of basis 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... 

The entries in the first columns of the two tables are total energies of atoms given by the SCF calculations with the minimal basis set of Slater-type orbitals (STO, see Section 2.B.). The entries in the second columns are energies given by the calculations in which each STO was replaced by two Slater-type functions with the exponents so optimized to give the minimum total energy. From Tables 2.1 and... 

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. 

Goldfield et used a very similar model to study in more detail potential energy surfaces for the reactions of M (Li, Na, K, Rb, Cs) with Brj. The three valence orbitals were represented as ns Slater-type functions and the three VB structures were written ... 

Apart from the expansion into Gauss-type functions the use of Slater-type functions has been discussed (Grant and Quiney 1988), although the analytic evaluation of integrals becomes as hopeless as in the nonrelativistic theory. Therefore, these STFs are only a good choice for atoms, linear molecules, or for four-component density functional calculations, where integrals over the total electron density are evaluated numerically. 

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