Convergence of direct space exchange lattice sums in polymer band structure calculations

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Abstract
A study of the convergence properties associated with direct lattice summations of the exchange contributions to the quantities needed in a restricted Hartree–Fock calculation for a polymer, leads to a detailed analysis of the analytic and asymptotic properties of the Fock–Dirac density matrix in the LCAO representation. The results, obtained simply by means of Fourier analysis, provide an important characterization of the restricted Hartree–Fock method as applied to chain-like systems. As a by-product we obtain a general proof in direct space that a partially filled band, which influences drastically the analyticity properties of the density matrix in reciprocal space, leads to a logarithmically diverging derivative of the orbital energy at the Fermi level. This in turn gives a vanishing density of states at that level. This result, well known for the electron gas model, is thus an inherent property of the restricted Hartree–Fock approximation.