Hartree-Fock formalism for solids. I. Reciprocal-lattice expansion for the Hartree-Fock exchange term

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Abstract
Using the linear-combination-of-atomic-orbitals formalism, the Hartree-Fock (HF) exchange operator has been expressed as a double Fourier integral. In the case of Gaussian basis functions, the Fourier coefficients of the exchange have been calculated analytically in terms of error functions of complex argument. When this Fourier expansion is used to compute matrix elements of the exchange between two Bloch functions, all of the direct lattice sums and the Fourier integral disappear, leaving only a double reciprocal-lattice sum in the expression for the exchange-matrix element. The convergence of this double Fourier series is comparable to the convergence of the Fourier series for the Coulomb term, which has been previously investigated by the authors. In the double Fourier series for the exchange term, as in the case of the Coulomb term, only the first few Fourier coefficients change significantly from the overlapping atomic potential (OAP) to the self-consistent result, and therefore the number of integrals that need to be stored for the self-consistent iterations is greatly reduced. Thus the double reciprocal-lattice expansion for the exchange term enables one to perform a self-consistent HF calculation for crystals with large atoms if the first iteration can be obtained with the OAP. In the case of the OAP, the number of integrals to be computed in direct space is greatly reduced over the number required in the self-consistent case, and therefore the exchange term can be calculated in direct space for crystals with heavy atoms.