# Four-component relativistic range-separated density-functional theory: Short-range exchange local-density approximation

## TL;DR

This paper extends range-separated density-functional theory to a four-component relativistic framework, combining wave-function and density functional methods for accurate short-range exchange calculations in high-density relativistic systems.

## Contribution

It introduces a relativistic short-range exchange local-density approximation based on a relativistic homogeneous electron gas, incorporating relativistic effects into short-range exchange energies.

## Key findings

- Relativistic effects significantly impact short-range exchange energies at high densities.
- The developed approximation is systematically improvable using Padé approximants.
- The formalism effectively combines wave-function and density functional approaches for relativistic systems.

## Abstract

We lay out the extension of range-separated density-functional theory to a four-component relativistic frame-work using a Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. This formalism combines a wave-function method for the long-range part of the electron-electron interaction with a density(-current) functionalfor the short-range part of the interaction. We construct for this formalism a short-range exchange local-densityapproximation based on calculations on a relativistic homogeneous electron gas with a modified Coulomb-Breitelectron-electron interaction. More specifically, we provide the relativistic short-range Coulomb and Breit ex-change energies per particle of the relativistic homogeneous electron gas in the form of Pad e approximantswhich are systematically improvable to arbitrary accuracy. These quantities, as well as the associated effectiveCoulomb-Breit exchange hole, show the important impact of relativity on short-range exchange effects for highdensities.