Reformulation in mathematical programming: An application to quantum chemistry

This paper concerns the application of reformulation techniques in mathematical programming to a specific problem arising in quantum chemistry, namely the solution of Hartree-Fock systems of equations, which describe atomic and molecular electronic wave functions based on the minimization of a functional of the energy. Their traditional solution method does not provide a guarantee of global optimality and its output depends on a provided initial starting point. We formulate this problem as a multi-extremal nonconvex polynomial programming problem, and solve it with a spatial Branch-and-Bound algorithm for global optimization. The lower bounds at each node are provided by reformulating the problem in such a way that its convex relaxation is tight. The validity of the proposed approach was established by successfully computing the ground-state of the helium and beryllium atoms.