Reformulation in mathematical programming: An application to quantum chemistry

  • Authors:

  • Leo Liberti;Carlile Lavor;Nelson Maculan;Marco Antonio Chaer Nascimento

  • Affiliations:

  • LIX, ícole Polytechnique, F-91128 Palaiseau, France;Department of Applied Mathematics (IMECC-UNICAMP), State University of Campinas, CP 6065, 13081-970 Campinas-SP, Brazil;COPPE, Universidade Federal do Rio de Janeiro, UFRJ, CP 68511, Rio de Janeiro, RJ 21945-970, Brazil;Departamento de Físico-Química, Instituto de Química, Universidade Federal do Rio de Janeiro, UFRJ, Rio de Janeiro, RJ 21949-970, Brazil

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

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.