  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.
