Evaluation of atomic integrals for hybrid Gaussian type and plane-wave basis functions via the McMurchie-Davidson recursion formula

A convenient formalism is developed for the evaluation of atomic integrals composed of a hybrid Gaussian type function and plane-wave (GTF-PW) basis set, based upon the recursion scheme proposed by McMurchie and Davidson [L. E. McMurchie and E. R. Davidson, J. Comput. Phys. 26, 218 (1978)] which was originally for Gaussian type basis functions. We show that revisions of recursion relations in the original article are necessary in order to allow systematic production of overlap, kinetic energy, nuclear attraction, and electron repulsion integrals in compact forms. Involving easy calculation of complex incomplete gamma functions, the recursion relations enable the use of hybrid GTF-PW basis functions with arbitrarily large angular momentum. This basis function can be applied to the first-principle calculation for solids involving localized electron orbitals.