Copula methods for modeling pair densities in density functional theory

Area of Science:

• Quantum chemistry
• Statistical mechanics
• Computational physics

Background:

• Accurately determining pair densities is crucial for understanding many-body systems.
• Existing methods often struggle with complex interactions and dissociation processes.
• Copula theory offers a promising framework for relating marginal and joint probability distributions.

Purpose of the Study:

• To develop a new method for approximating the density-to-pair-density map using copula theory.
• To extend copula theory to multi-dimensional marginals for broader applicability.
• To provide accurate approximations of the copula for dissociating systems.

Main Methods:

• Extension of copula theory to multi-dimensional marginals.
• Derivation of analytical formulas for the exact copula in scaling limits.
• Numerical computation of the copula for 1D systems with 2-4 particles.
• Development of approximations for two-particle systems across equilibrium and dissociation.

Main Results:

• Demonstrated that any pair density can be described by the single-particle density and a copula.
• Derived exact copula formulas applicable in scaling limits.
• Successfully computed copulas for dissociating 1D systems.
• Proposed accurate copula approximations for two-particle systems.

Conclusions:

• The proposed copula-based approach provides a powerful tool for approximating pair densities.
• This method offers a unified framework applicable from equilibrium to dissociation.
• The findings have implications for quantum chemistry and statistical mechanics calculations.