Copula methods for modeling pair densities in density functional theory

TL;DR

This paper introduces a novel copula-based approach to model pair densities in density functional theory, enabling more accurate descriptions of electron interactions by linking single-particle densities with copulas.

Contribution

It extends copula theory to multi-dimensional marginals and provides analytical, numerical, and approximate methods for modeling pair densities in quantum systems.

Findings

Derived analytical formulas for copulas in scaling limits.

Numerically computed copulas for systems with 2-4 particles.

Proposed accurate copula approximations for different physical regimes.

Abstract

We propose a new approach towards approximating the density-to-pair-density map based on copula theory from statistics. We extend the copula theory to multi-dimensional marginals, and deduce that one can describe any (exact or approximate) pair density by the single-particle density and a copula. We present analytical formulas for the exact copula in scaling limits, numerically compute the copula for dissociating systems with two to four particles in one dimension, and propose accurate approximations of the copula between equilibrium and dissociation for two-particle systems.