Relativistic materials from an alternative viewpoint

Electrons in materials containing heavy elements are fundamentally relativistic and should in principle be described using the Dirac equation. However, the current standard for treatment of electrons in such materials involves density functional theory methods originally formulated from the Schrödinger equation. While some extensions of the Schrödinger-based formulation have been explored, such as the scalar relativistic approximation with or without spin-orbit coupling, these solutions do not provide a way to fully account for all relativistic effects of electrons, and the language used to describe such solutions are still based in the language of the Schrödinger equation. In this article, we provide a different method for translating between the Dirac and Schrödinger viewpoints in the context of a Coulomb potential. By retaining the Dirac four-vector notation and terminology in taking the non-relativistic limit, we see a much deeper connection between the Dirac and Schrödinger equation solutions that allow us to more directly compare the effects of relativity in the angular and radial functions. Through this viewpoint, we introduce the concepts of densitals and Dirac spherical harmonics that allow us to translate more easily between the Dirac and Schrödinger solutions. These concepts allow us to establish a useful language for discussing relativistic effects in materials containing elements throughout the full periodic table and thereby enable a more fundamental understanding of the effects of relativity on electronic structure.