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

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.