S-spinors

Table 8 Second-order many-body perturbation theory corrections to beryllium-like ions using non-relativistic (E ), Dirac-Coulomb (E ) and Dirac-Coulomb-Breit (E ) hamiltonians, obtained using the atomic precursor to BERTHA, known as SWIRLES. Basis sets are even-tempered S-spinors of dimension N= 17, with exponent sets, Xi generated by Xi = abi-i, with a = 0.413, and p = 1.376. Angular momenta in the range 0 < / < 6 have been included in the partial wave expansion of each second-order energy, and the total relativistic correction toE has been collected as Ef. All energies in hartree.

S-spinors. It is unfortunate that L-spinors are virtually useless for treating... 

G-spinors are appropriate for distributed charge nuclear models, and are much the most convenient for relativistic molecular calculations. Whereas neither L-spinors nor S-spinors satisfy the matching criterion (140) for finite c (although they do in the nonrelativistic limit), G-spinors are matched according to (140) for all values of c. The radial functions can be written... 

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]. 

We now have a collection of integrals over basis functions which must be evaluated in order to construct the Fock matrix. For S-spinors, these can be deduced from formulae given by Kim [95,37]. The main difference is that Kim was not aware of the importance of kinetic matching adaptation of his formulae is routine. The integrals involved are all related to the gamma function or the error function [83, Chapters 6 and 7]. We therefore concentrate here on G-spinors, which can be applied both to atomic and to molecular calculations. 

For the relativistic case there are three analogous choices of expansion functions to those discussed above. The hydrogenic functions have their analogue in the L-spinors obtained from the solution of the Dirac-Coulomb equation [4]. Again their use is mainly restricted to analytic work in atomic calculations, due to the difficulties in evaluating the integrals [5]. The analogue of the STO is the S-spinor which may be written in the form... 

For the main group elements, the lowest spinor is usually one of the valence spinors, which means that at least the valence pseudospinors for the atom will not be approximated, and these are the most important spinors in a molecular calculation. However, for the transition metals, it is often important to have both the ns and the (n -I- l)s spinors in the valence space. The pseudopotential must then be chosen for the ns pseudospinor, and this choice will affect the (n -I-1 )s pseudospinor. 

---