Kinetic Formulation of the Kohn-Sham Equations for ab initio Electronic Structure Calculations

TL;DR

This paper presents a novel kinetic theory approach to density functional theory, deriving Kohn-Sham equations from a Boltzmann kinetic equation and validating it through numerical calculations of atomic and molecular systems.

Contribution

It introduces a kinetic formulation of the Kohn-Sham equations, connecting quantum electronic structure calculations with kinetic theory and providing a new computational framework.

Findings

Excellent agreement with literature values for exchange and correlation energies.

Successful computation of molecular geometry of methane.

Validation of the approach through numerical simulations.

Abstract

We introduce a new approach to density functional theory based on kinetic theory, showing that the Kohn-Sham equations can be derived as a macroscopic limit of a suitable Boltzmann kinetic equation in the limit of small mean free path versus the typical scale of density gradients (Chapman-Enskog expansion). To derive the approach, we first write the Schr\"odinger equation as a special case of a Boltzmann equation for a gas of quasi-particles, with the potential playing the role of an external source that generates and destroys particles, so as to drive the system towards the ground state. The ions are treated as classical particles, using the Born-Oppenheimer dynamics, or by imposing concurrent evolution with the electronic orbitals. In order to provide quantitative support to our approach, we implement a discrete (lattice) model and compute, the exchange and correlation energies of simple atoms, and the geometrical configuration of the methane molecule. Excellent agreement with values in the literature is found.