Basis Set Expansion of Molecular Spinors

We should mention that relativistic electronic structure calculations on solid-state systems have been carried out as well. However, our focus here will not be on the special case of translational symmetry as we rather concentrate on molecules and molecular aggregates. The general principles may, of course, be transferred to the special case of crystals, and we may refer the reader for further details to Refs. [390,391,537-544], where four-component and also approximate relativistic Hamiltonians are considered. 

In order to describe the one-electron spinors of molecules that enter the Slater determinants to approximate the total electronic wave function it is natural to be inspired by the fact that molecules are composed of atoms. It should be noted that this at first sight obvious fact is not obvious at all as we describe only the ingredients, i.e., the elementary particles (electrons and atomic nuclei) and not individual atoms of a molecule. To define what an atom in a molecule shall be is not a trivial issue in view of the continuous total electron distribution p r) of a molecule. Nonetheless, we may utilize what we have 

For the relativistic description of molecules this means that each molecular spinor ipi r) entering a Slater determinant is to be expanded in a set of four-component atomic spinors (r). 

Each atomic spinor tp r) = tp r,RA) has its center at the position of the nucleus Ra of some atom A. In a first step, we include only those atomic spinors (r) which would be considered in an atomic Dirac-Hartree-Fock calculation on every atom of the molecule. Of course, if a given atom occurs more than once in the molecule, a set of atomic spinors of this atom is to be placed at every position where a nucleus of this t) e of atom occurs in the molecule. The number of basis spinors m is then smallest for such a minimal basis set. In this case, it can be calculated as the number of shells s per atom times the degeneracy d of these shells times the number of atoms M in the molecule, m = s A) x d s) x M. 

In practice, the number of basis spinors m increases even further for a minimal basis set because we have no analytic expression for the atomic spinors of a many-electron atom available. Hence, the atomic spinors themselves need to be expanded in terms of known basis functions (p r,RA)- 

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