Use of graphics processing units for efficient evaluation of derivatives of exchange integrals by means of Fourier transform of the 1/r operator and its numerical quadrature

Abstract

In this paper, we propose an efficient way for evaluation of derivatives of exchange integrals. We propose an approach in which we factorize the non-local exchange kernel into a sum of separable terms. We exploit a discretized Fourier transform for the 1/r operator, and we devise a method that allows us to employ a manageable number of plane-wave functions in the Fourier expansion while still keeping necessary accuracy. Resulting formulas are amenable for efficient evaluation on graphics processing units (GPU). We discuss the GPU implementation for derivatives of two-electron repulsion integrals of the (gk|gk) type in the hybrid Gaussian and plane-wave basis. Derivatives of such integrals are needed for computation of cross sections in vibrationally inelastic electron scattering by polyatomic molecules. Speedup and accuracy achieved are demonstrated for cross sections of selected vibrational modes of cyclopropane, benzene and adamantane. The proposed factorization method is general and may be applied to any type of exchange integrals. We note briefly on its possible application to exchange integrals and their derivatives in quantum chemical computational methods.