Relativistic Exponential- Spinor Orbitals and Their Use in Many- Dirac Equation Solution

Abstract

The Dirac–Coulomb type differential equation and its solution; relativistic exponential–type spinor orbitals are introduced. They provide a revised form for operator invariants, namely Dirac invariants, simplifying the treatment of the angular components in calculation of many–electron systems. The relativistic Coulomb energy is determined by employing a spectral solution to Poisson's equation for the one–electron potential, which is expressed in terms of radial functions involving incomplete gamma functions. The computation of incomplete gamma functions poses challenges due to slow convergence rates associated with their series representation. Such difficulties are eliminated by using the bi–directional method along with hyper–radial functions. A new formulation for relativistic auxiliary functions that improve the efficiency in Coulomb energy calculations is presented. These formulations also give rise to the need for inquiring into orthogonal expansions for solutions to Poisson's equation using complete orthonormal sets of exponential orbitals with non–integer principal quantum numbers. They may provide meaningful alternative series representations. © 2025