Efficient evaluation of Coulomb integrals in a mixed Gaussian and plane‐wave basis

An efficient way of calculation is presented for matrix elements between two plane waves interacting with a molecular Coulombic field. In concurrence with the absolute value of the momentum transfer vector, K = k1 − k2, the most effective method of calculation is selected. The case of K = 0 requires special treatment. For 0 < |K| ≤ 0.3, it is profitable to evaluate the integrals by means of the multipole expansion, and for |K| > 0.3 the density fitting can be applied. For the large |K| the electronic part of the integral is much smaller than the nuclear part and the integral may be approximated by the nuclear contribution only. Some examples for testing the accuracy and time saved are presented. The primary purpose of this paper is to accelerate electron scattering calculations, but it also may be profitable for the electronic structure theory in attempts to use mixed Gaussian and plane‐wave basis sets.