A new formulation of the Hartree–Fock–Roothaan method for electronic structure calculations on crystals

Abstract

We present a general formulation of the Hartree–Fock–Roothaan method of electronic structure calculations for systems subject to periodic boundary conditions and apply this method to crystals. The derivation of the method does not involve any divergent or conditionally convergent infinite series. The final result for the Hartree–Fock energy per unit cell consists of only absolutely convergent series and can be written in a form whose structure is almost identical to that for the nonperiodic Hartree–Fock energy. A Fock matrix that consists of only absolutely convergent series is also defined. An important feature of the method is that the Ewald potential, which has been used in the past to eliminate divergences in series involving the expectation value of the Coulomb interaction, is introduced in a physically reasonable way at an early stage of the formulation of the quantum mechanical problem. In the final result, the Ewald potential is used not only to express the Coulomb energy, but also to express the exchange energy as an absolutely convergent series, thereby eliminating the problem of slow convergence, or lack of convergence, of the series for the exchange energy. The numerical implementation of this method, which is not discussed in this paper, requires calculation of standard one- and two-electron matrix elements of the electronic kinetic energy and the Coulomb interaction, as well as certain easily calculated moments of basis function overlap charge densities. No integrals involving matrix elements of the Ewald potential between basis functions are required for evaluation of either the energy or the Fock matrix. Instead, Ewald interactions must be evaluated only for point multipoles. The methods used here to formulate the Hartree–Fock problem can be extended to formulate Mo/ller–Plesset perturbation theory and coupled cluster theory for crystals.