  We present a density matrix approach for computing global solutions of
restricted open-shell Hartree-Fock theory, based on semidefinite programming
(SDP), that gives upper and lower bounds on the Hartree-Fock energy of quantum
systems. While wave function approaches to Hartree-Fock theory yield an upper
bound to the Hartree-Fock energy, we derive a semidefinite relaxation of
Hartree-Fock theory that yields a rigorous lower bound on the Hartree-Fock
energy. We also develop an upper-bound algorithm in which Hartree-Fock theory
is cast as a SDP with a nonconvex constraint on the rank of the matrix
variable. Equality of the upper- and lower-bound energies guarantees that the
computed solution is the globally optimal solution of Hartree-Fock theory. The
work extends a previously presented method for closed-shell systems [S.
Veeraraghavan and D. A. Mazziotti, Phys. Rev. A89, 010502R (2014)]. For
strongly correlated systems the SDP approach provides an alternative to the
locally optimized Hartree-Fock energies and densities with a certificate of
global optimality. Applications are made to the potential energy curves of C2,
CN, Cr2, and NO2.
