Conclusions

The algorithm presented in this chapter shows a way to generate a correction for the neglected part of a CASE calculation in only O(N) work, by representing the long-range energy in terms of electronic multipole moments of the system. The most straightforward derivation generates a representation for the energy via a series that is only asymptotically convergent. To overcome this, the series can be approximated by basis functions using knowledge of the derivatives at the origin. Choosing just what type of basis function, however, is still an unsolved problem. A basis function of Gaussians has been tried here and has shown promising results. Most of the neglected energy can be recovered with only a few of these basis functions (and hence only small order moments of the system). Gaussians have the serious drawback of sometimes producing a nonsensical energy through the need for positive exponents to accurately represent G(k).

The most encouraging point from this chapter, though, is that even if the CASE approximation proves to be unworkable in the future, the code written and methods developed in this thesis will still be useful as they will provide the short-range energy for an O(N) Coulomb algorithm.