Advantages of Slater-type spinor orbitals in the Dirac–Hartree–Fock method. Results for hydrogen-like atoms with super-critical nuclear charge

Abstract

This work presents the formalism for evaluating molecular SCF equations, adapted to four-component Dirac spinors, which in turn reduce to Slater-type orbitals with non-integer principal quantum numbers in the non-relativistic limit. If Slater-type spinor orbitals are used in the algebraic approximation to solve the Dirac equation, the “catastrophe” previously noted for atomic numbers $$Z>137$$, in Dirac equation resolution with a potential corresponding to a point-charge no longer applies. It is observed that, ground-state energy of hydrogen-like atoms reaches the negative-energy continuum $$\left( -mc^2 \right)$$ with super-critical nuclear charge $$Z_{c}$$, about $$Z_{c}=160$$. The difficulty associated with finding relations for molecular integrals over Slater-type spinors which are non-analytic in the sense of complex analysis at $$r = 0$$, is eliminated. Unique numerical accuracy is provided by solving the molecular integrals through Laplace expansion of the Coulomb interaction and prolate spheroidal coordinates. New convergent series representation formulae are derived. The technique draws on previous work by the author. The general formalism is presented in this paper.