Global Solutions of Hartree-Fock Theory and their Consequences for Strongly Correlated Quantum Systems

TL;DR

This paper introduces a semidefinite programming approach to find global solutions of Hartree-Fock theory, providing bounds that improve understanding of strongly correlated quantum systems and challenge standard methods.

Contribution

It develops a novel SDP-based method for globally solving Hartree-Fock equations, revealing significant energy differences in strongly correlated molecules.

Findings

SDP approach yields lower energies than standard methods in dissociation regions

Global solutions are guaranteed when upper and lower bounds coincide

Results impact the interpretation of electron correlation in quantum chemistry

Abstract

We present a density matrix approach for computing global solutions of Hartree-Fock theory, based on semidefinite programming (SDP), that gives upper and lower bounds on the Hartree-Fock energy of quantum systems. Equality of the upper- and lower-bound energies guarantees that the computed solution is the globally optimal solution of Hartree-Fock theory. For strongly correlated systems the SDP approach requires us to reassess the accuracy of the Hartree-Fock energies and densities from standard software packages for electronic structure theory. Calculations of the H$_{4}$ dimer and N$_{2}$ molecule show that the energies from SDP Hartree-Fock are lower than those from standard Hartree-Fock methods by 100-200 kcal/mol in the dissociation region. The present findings have important consequences for the computation and interpretation of electron correlation, which is typically defined relative to the Hartree-Fock energy and density.