A family of attenuated Coulomb operators

We discuss a family of computationally useful approximations to the Coulomb operator. These operators, which we term CAP(m), are systematic improvements to our earlier CASE operator. In particular, we have CAP(0)  CASE and CAP(∞)  1r. Because the CAP(m) approximations are all short-ranged, the computational cost of using one to compute the Coulomb energy of N localized charge distributions scales linearly with N. To investigate their accuracy, we have applied a number of CAP(m) approximations to the computation of the hydrogen atom energy and the NaCl Madelung constant. We find that the higher approximations model the original Coulomb operator quite well and the half-integer approximations, though non-vanishing at infinity, are especially accurate.