On the Geometric Potential and the Relationship between the Exact Electron Factorization and Density Functional Theory

TL;DR

This paper explores the geometric potential in the exact electron factorization (EEF) and its connection to density functional theory (DFT), providing new insights into the physical meaning of one-electron potentials.

Contribution

It establishes a detailed relationship between EEF and DFT, interpreting key potentials like the Pauli and Hartree-exchange-correlation in geometric terms.

Findings

The Pauli potential is linked to the geometric potential $v^{ m G}$, not just fermionic-bosonic differences.

The geometric potential $v^{ m G}$ measures environmental change and has a clear physical interpretation.

Analysis of diatomic molecules illustrates the behavior of $v^{ m G}$ and its significance in electronic structure.

Abstract

There are different ways to obtain an exact one-electron theory for a many-electron system, and the exact electron factorization (EEF) is one of them. In the EEF, the Schr\"odinger equation for one electron in the environment of other electrons is constructed. The environment provides the potentials that appear in this equation: A scalar potential $v^{\rm H}$ representing the energy of the environment and another scalar potential $v^{\rm G}$ as well as a vector potential that have geometric meaning. By replacing the interacting many-electron system with the non-interacting Kohn-Sham (KS) system, we show how the EEF is related to density functional theory (DFT) and we interpret the Hartree-exchange-correlation potential as well as the Pauli potential in terms of the EEF. In particular, we show that from the EEF viewpoint, the Pauli potential does not represent the difference between a fermionic and a bosonic non-interacting system, but that it corresponds to $v^{\rm G}$ and partly to $v^{\rm H}$ for the (fermionic) KS system. We then study the meaning of $v^{\rm G}$ in detail: Its geometric origin as a metric measuring the change of the environment is presented. Additionally, its behavior for a simple model of a homo- and heteronucler diatomic is investigated and interpreted with the help of a two-state model. In this way, we provide a physical interpretation for the one-electron potentials that appear in the EEF and in DFT.