Spinor basis sets

We have now at our disposal a range of criteria for designing spinor basis sets. The functions are all of the form (96) with basis functions 

L-spinors, [87], are related to Dirac hydrogenic functions in much the same way as Sturmian functions [93,94] are related to Schrodinger hydrogenic functions. The radial parts are given by [87, III] 

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 

G-spinors satisfy the analytic boundary conditions (137) for jc 0 and (138) for tc 0. A G-spinor basis set consists of functions of the form of (147-149) with suitably chosen exponents Xm, m = 1,2. d - The choice of sequences Xfn which ensure linear independence of the G-spinors and a form of completeness is discussed in [86]. It is often sufficient to use the GTO exponents from nonrelativistic calculations, of which there are many compilations in the literature perhaps augmented with one or two functions with a larger value of A to improve the fit around the nucleus. 

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Abstract. BERTHA is a 4-component relativistic molecular structure program based on relativistic Gaussian (G-spinor) basis sets which is intended to make affordable studies of atomic and molecular electronic structure, particularly of systems containing high-Z elements. This paper reviews some of the novel technical features embodied in the code, and assesses its current status, its potential and its prospects. 

Relativistic charge-current densities expressed in terms of G-spinor basis sets for stable and economical numerical calculations [2]. 

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

The behaviour of (103) in the nonrelativistic limit, c —provides the final ingredient needed for selection of spinor basis sets. We shift the origin of energy by writing e = E — c, so that (103) becomes... 

It is easy to show that (141) preserves charge conjugation symmetry in the free particle problem. But perhaps what is just as important is that spinor basis sets satisfying (141) listed below generate no spurious solutions for JC > 0 of the sort reported by [39,41,42,91,92]. 

Unfortunately contraction of basis sets creates further problems with overcompleteness, now also involving large component > 0 functions. The contractions lead to a duplication of the space e.g. a 2pi/2 and a 2pz/2 contraction gives twelve spin-orbitals from which only six 2-spinors are needed, and the rest should be removed. This doubling further exacerbates the problems described above for the small component. Again, the problem is averted if one works in a 2-spinor basis set. 

The main problem to be addressed in the choice between scalar and 2-spinor basis sets is that of linear dependence, which can affect the convergence of the SCF procedure. Several issues arise from the basis set choice that have consequences for linear dependence. The first of these is the choice of RKB or UKB in the representation of the small component of the j = I - 1/2 spinors. UKB has more severe linear dependence problems than RKB, due to the overrepresentation of the small component. However, RKB requires some manipulation of the integrals, because the nl and (n + 2)1 integrals are not generated as part of the same angular momentum shell. This issue affects both scalar and 2-spinor basis sets. 

One consequence of the choice between 2-spinor and scalar basis sets is that the results of DHF calculations with the two are not equivalent. The extra functions in the small-component space affect the spinor eigenvalues and hence the total energy and the ionization potential. These differences are likely to be minor for most of chemistry. If contracted basis sets are used, however, the duplication of large-component functions for the spin-orbit components in the scalar basis set provides extra flexibility that is not present for a 2-spinor basis set, and the valence properties could be signiflcantly different. 

The large-component basis must be slightly larger than for the nonrelativistic calculation of the same quality. (This applies also to 2-spinor basis sets.)... 

There is a 1 1 ratio of large- and small-component functions, that is, the restricted kinetic balance relation applies between the primitive large- and small-component 2-spinor basis sets. 

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