G spinors

Ishikawa and coworkers [15,24] have shown that G-spinors, with orbitals spanned in Gaussian-type functions (GIF) chosen according to (14), satisfy kinetic balance for finite c values if the nucleus is modeled as a uniformly-charged sphere. 

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]. 

Relativistic generalization for G-spinor basis functions of the well-known McMurchie-Davidson algorithm [3] for direct evaluation of interaction integrals. 

Other G-spinor parameters depend on the component index /3. Thus - l = j + f3r]i,l2. 

The G-spinor representation (12), when substituted into (5), results in a time-dependent charge-current density... 

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 matrices J, K and B are direct, exchange and Breit interaction matrices, of which only the first is block diagonal. Their matrix elements are linear combinations of interaction integrals over G-spinors. 

Goulomb interaction integrals over molecular orbitals can be written as a sum of similar interaction integrals with G-spinor overlap densities and... 

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. 

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. 

In molecular calculations it is convenient to express G-spinors in terms of Spherical Gaussian-type Functions (SGTF) [107]... 

The spinor structure has been completely absorbed into the Eq coefficients to give a formula which is no more complicated in structure than its nonrelativistic equivalent, although a relativistic calculation inevitably involves more terms. However, the Eq coefficients preserve G-spinor symmetry, and this can be exploited to eliminate redundant calculations. The range of summation over the Cartesian indices s t,u in (210) is finite, and is restricted so that = j + t + u< Amaxf where Amax is given in Table 2. 

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