A general procedure is described for computing matrix elements of operators that occur in a realistic nuclear Schrödinger Hamiltonian, in a basis of orbitals with radial factors of the form N aγ2na+la exp ( -γaγ2), where na is an arbitrary nonnegative integer and γa is an arbitrary positive number. The method is suitable for efficient large-scale computation of these matrix elements, needed when orbitals of physical interest (such as Hartree-Fock orbitals) are expressed as linear combinations of basis functions of the kind indicated. The analysis of matrix elements of a tensor operator provides a new method of reduction to linearly independent reduced matrix elements, the number of which is smaller than it is in the usual analysis. Thus, the number of independent parameters in the corresponding empirical theory of complex nuclear spectra is reduced. The specific operator forms considered here are those present in the asymptotic one-pion exchange potential, except that the functional forms of the potentials V(γ) are not prescribed beyond the requirement that they should be integrable.