Mueller's Exchange-Correlation Energy in Density-Matrix-Functional Theory

TL;DR

This paper provides a comprehensive mathematical analysis of Mueller's density-matrix-functional theory, highlighting its convexity, unique densities, and existence of minimizers, offering insights into its properties and differences from Hartree-Fock theory.

Contribution

It systematically investigates Mueller's functional, proving convexity, existence of minimizers, and properties of the orbitals, advancing the mathematical understanding of this density-matrix approach.

Findings

Mueller's functional is convex, unlike Hartree-Fock.

Minimizers have unique densities, which is physically desirable.

Existence of minimizers is established for N ≤ Z.

Abstract

The increasing interest in the Mueller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock functional, but with a modified exchange term in which the square of the density matrix \gamma(X, X') is replaced by the square of \gamma^{1/2}(X, X'). After an extensive introductory discussion of density-matrix-functional theory we show, among other things, that this functional is convex (unlike the HF functional) and that energy minimizing \gamma's have unique densities \rho(x), which is a physically desirable property often absent in HF theory. We show that minimizers exist if N \leq Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of \gamma, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.