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Copula methods for modeling pair densities in density functional theory

Geneviève Dusson, Claudia Klüppelberg, Gero Friesecke

TL;DR

This work introduces a generalized copula framework to model the density-to-pair-density map in density functional theory by separating marginals (one-body density) from the dependence structure (pair density) via Brenier maps and optimal transport. It establishes a multidimensional Sklar-type theorem, derives exact copulas for standard DFT models (Hartree, LDA, SCE), and provides dissociation-limit results showing universal or subsystem-based structures. Numerical 1D studies for 2–4 electrons validate the theory, reveal multiscale copula features, and demonstrate that simple, learned (e.g., sigmoid) copula models can accurately reproduce pair densities and Coulomb energies, offering a promising path toward robust DFT descriptions in strongly correlated regimes. The results suggest that copula-based representations can foreground inter-electronic correlations and enable machine-learned, physically constrained copulas to improve beyond conventional functionals in challenging 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.

Copula methods for modeling pair densities in density functional theory

TL;DR

This work introduces a generalized copula framework to model the density-to-pair-density map in density functional theory by separating marginals (one-body density) from the dependence structure (pair density) via Brenier maps and optimal transport. It establishes a multidimensional Sklar-type theorem, derives exact copulas for standard DFT models (Hartree, LDA, SCE), and provides dissociation-limit results showing universal or subsystem-based structures. Numerical 1D studies for 2–4 electrons validate the theory, reveal multiscale copula features, and demonstrate that simple, learned (e.g., sigmoid) copula models can accurately reproduce pair densities and Coulomb energies, offering a promising path toward robust DFT descriptions in strongly correlated regimes. The results suggest that copula-based representations can foreground inter-electronic correlations and enable machine-learned, physically constrained copulas to improve beyond conventional functionals in challenging 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.

Paper Structure

This paper contains 19 sections, 5 theorems, 97 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notation and context
  3. Generalized copula
  4. The copulas for some standard DFT models
  5. Independent electrons a.k.a. Hartree functional
  6. Local Density Approximation
  7. Strictly correlated electrons
  8. Dissociating systems
  9. Wavefunction and pair density
  10. Copula for 1D systems
  11. Generalized copula for a class of 3D systems
  12. Representability constraints on the copula
  13. Numerically computed copulas of quantum-mechanical ground states
  14. Two-particle system.
  15. Three-particle systems.
  16. ...and 4 more sections

Key Result

Theorem 1

For any probability density $f$ on $\mathbb{R}^{dN}=\mathbb{R}^d \times ... \times \mathbb{R}^d$ with marginal densities $f_1,...,f_N$ on $\mathbb{R}^d$, there exists a probability density $c$ on the $N$th power $D^{N}$ of the reference region $D$, which we propose to call generalized copula density where $T_i$ is the Brenier map transporting $f_i$ to $f_0$. The generalized copula density is uniqu

Figures (11)

  • Figure 1: Copula for strictly correlated electrons for two- to four-particle systems
  • Figure 2: Hypothetical behavior of the Brenier map for two symmetric points with respect to the reflection plane.
  • Figure 3: Right column: exact copula of ground state for a two-particle system dissociating into two one-electron densities. The other columns show related quantities. Nuclei positions for top row: $(-1,1)$, second row: $(-2,2)$, bottom row: $(-3,3)$.
  • Figure 4: Right column: exact copula of ground state for a three-particle system dissociating into three one-electron densities. The other columns show related quantities. Nuclei positions for top row: $(-2, 0, 2)$, second row: $(-3.5, 0, 3.5)$, bottom row: $(-7, 0, 7)$.
  • Figure 5: Right column: exact copula of ground state for a three-particle system dissociating into a one-electron density and a two-electron density. The other columns show related quantities. Nuclei positions for top row: $(-3, -1, 2)$, second row: $(-4, -2, 3)$, bottom row: $(-5, -3, 4)$.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Theorem 1: Generalized Sklar's theorem
  • Theorem 2: Pair density of dissociated systems
  • Theorem 3: Copula of dissociated systems
  • Theorem 4: Generalized copula of dissociated systems
  • Theorem 5