In this article we describe the implementation of a scheme based upon the McMurchie–Davidson recursion relationships to calculate two-electron integrals over Gaussian-type functions in extended systems. Features illustrating the computational efficiency of the approach are highlighted together with additional aspects of our program to compute the ab initio band structures of polymers. The algorithm is applied in calculation of the band structure and Sum Over States polarizability of polyethylene and polysilane using standard basis sets augmented with polarization functions. The role of polarization functions is seen to be small in polyethylene, but in polysilane the silicon d-orbitals make significant contributions to the bonding along the backbone of the chain. This increase in bonding character results in a reduction in the calculated polarizability in going from the unpolarized to polarized basis sets. © 1993 John Wiley & Sons, Inc.