It is shown that the multivalley problem for donors in Si and GaP is intrinsically a two-band problem. One-band multivalley equations for Ge and two-band multivalley equations for Si and GaP are derived by evaluating all terms to the same order. Intervalley kinetic-energy matrix elements, for which conflicting accounts appeared recently, are evaluated explicitly and shown to provide an excellent measure of the validity of the theory. The retention of Umklapp terms in potential-energy matrix elements, which has recently been demonstrated to be important for quantitative calculations, gives rise to additional terms, not considered previously, which are of the same order as other terms already included. It also leads to the requirement that the impurity pseudopotential must be internally consistent with the host-crystal pseudopotential. As a consequence, when an empirical-pseudopotential band-structure calculation is used, the screened point-charge potential no longer corresponds specifically to the isocoric impurity. Overall, the construction of appropriate impurity pseudopotentials is a more sensitive task than in earlier theories that neglected Umklapp terms. © 1979.