G-spinors basis sets

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. 

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. 

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. 

Since this chapter is meant to focus on 4-component type methods I will give some attention to the recent developments that reduce the time spent in evaluation of matrix elements over g. It not trivial to reduce the amount of work associated with the separate upper and lower component basis sets since the norm of the small component wave function may be rather large for heavy atoms. One possibility is to use the locality of the small components of the wave function to replace long-range interactions by a classical interaction [27]. If we distinguish between the upper (large) two and lower (small) two components of the basis 4-spinors... 

The choice and generation of basis sets has been addressed by many authors [190,192,528,554-563]. While we consider here only the basic principles of basis-set construction, we should note that this is a delicate issue as it determines the accuracy of a calculation. Therefore, we refer the reader to the references just given and to the review in Ref. [564]. In Ref. [559] it is stressed that the selection of the number of basis functions used for the representation of a shell riiKi should not be made on the grounds of the nonrelativistic shell classification nj/j but on the natural basis of j quantum numbers resulting in basis sets of similar size for, e.g., Si/2 and pi/2 shells, while the p /2 basis may be chosen to be smaller. As a consequence, if, for instance, pi/2 and p /2 shells are treated on the tijli footing, the number of contracted basis functions may be doubled (at least in principle). The ansatz which has been used most frequently for the representation of molecular one-electron spinors is a basis expansion into Gauss-type spinors. 

It should be remembered, of course, that Lie groups have an independent existence apart from their role in the theory of f electrons in the lanthanides. Some of the bizarre properties that turn up in the f shell might well derive from isoscalar factors that receive a ready explanation in another context. An example of this is provided by the vanishing of the spin-orbit interaction when it is set between F and G states belonging to the irreducible representation (21) of G2. In the f shell, (21) is merely a 64-dimensional representation of no special interest. However, for mixed configurations of p and h electrons, it fits exactly into the spinor representations (iiiHi 2) of SO(14) with dimensions 2 (Judd 1970). These are the analogs of the spinor representations of eq. (130), and (21) describes the quasiparticle basis of the configurations (p-t-h)". The spinor representations of SO(3) and (Hiii) of SO(ll) provide the quasi-particle bases for the p and h shells respectively and their SO(3) structures, namely S 1,2 and 512+ 912+ is/2> when coupled, must yield the L structure of (21) of G2, namely D-l-F-l-G-l-H-t-K-t-L. In this context, the F and G terms of (21) are associated with the different irreducible representations Sj,2 and S9/2. It is this property,... 

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