Basis sets Slater-type functions

Plane Waves and Atomic-like Basis Sets. Slater-type Functions... 

Basis set Slater type and Gaussian type basis functions. 

If one includes functions with n - / even in (1.1) (i.e. one uses set b) the basis is formally overcomplete. However the error decreases exponentially with the size of the basis [2,16]. Unfortunately for this type ofbasis the evaluation of the integrals is practically as difficult as for Slater type basis functions, such that basis sets of type (b) have not been used in practice. 

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. 

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... 

Fig. 10 Computed and experimental ROA for methyloxirane. Top Two spectra computed with Slater-type basis sets and nonhybrid functionals, from [84]. Bottom Computation with a Gaussian-type basis set and the B3LYP hybrid functional, and experimental spectrum. Data to prepare the plots were taken from [83] and [84]...

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. 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

Fig. 19.8. Differences between the isotropic Compton profile computed from theory and various experiments. The theoretical data is from a multi-reference configuration interaction (MRCI) calculation in a (5s5p4d3f) basis set of Slater-type functions [104]. Experimental data (+), 25 keV electron impact at 12° [109] ( ), average of Ag Ka and Mo Ka X-ray scattering [108,105] ( ), 160 keV y-ray scattering [105] (O), 160 keV y ray scattering reanalyzed [105,106] (A), 60keV y-ray scattering [107]. The dotted lines enclose the band of uncertainty in the experimental data.

Ref. [104]. r is a (5s5p4d3f) basis set of Slater-type functions. SCF is self-consistent field, MRCI is multi-reference configuration interaction. 

Some other versions of the DFT method like the Beijing Density Functional method (BDF) (see the chapter of C. van Wuellen in this issue) were also used for small compounds of the heaviest elements like 111 and 114 [115-117]. There, four-component numerical atomic spinors obtained by finite-difference atomic calculations are used for cores, while basis sets for valence spinors are a combination of numerical atomic spinors and kinetically balanced Slater-type functions. The non-relativistic GGA for F is used there. 

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