Dissociation limit in Kohn-Sham density functional theory

TL;DR

This paper analyzes the dissociation limit of diatomic molecules in Kohn-Sham DFT, proving the energy convergence to a minimum over atomic electron distributions and illustrating the effect of exchange strength with numerical results for H₂.

Contribution

It provides a rigorous proof of the dissociation limit in Kohn-Sham DFT and explores how exchange strength influences the energy splitting, supported by numerical examples.

Findings

Energy converges to a minimum over atomic electron distributions at dissociation

The minimum may differ from symmetric splitting depending on exchange strength

Numerical results for H₂ confirm theoretical predictions with Dirac exchange

Abstract

We consider the dissociation limit for molecules of the type $X_2$ in the Kohn-Sham density functional theory setting, where $X$ can be any element with $N$ electrons. We prove that when the two atoms in the system are torn infinitely far apart, the energy of the system convergences to $\min \limits_{\alpha \in [0,N]} \big( I^{X}_{\alpha} + I^{X}_{2N-\alpha} \big)$, where $I^{X}_{\alpha}$ denotes the energy of the atom with $\alpha$ electrons surrounding it. Depending on the "strength" of the exchange this minimum might not be equal to the symmetric splitting $2I^{X}_{N}$. We show numerically that for the $H_2$-molecule with Dirac exchange this gives the expected result of twice the energy of a H-atom $2 I^{H}_1$.