Towards Linear Scaling Kohn-Sham Theory

6 Towards Linear Scaling Kohn-Sham Theory 

However, unlike point charges, the continuous charge distributions that occur in quantum chemistry have varying extents and the applicability of the multipole approximation is not only limited by the distance but also by the extent or diffuseness of the charge distribution. This additional complexity makes a transfer of the concepts of the fast multipole method to applications in quantum chemistry less straightforward. Therefore it should come as no surprise that several adaptations to extend the applicability of the FMM to the Coulomb problem with continuous charge distributions have been suggested. These lead to 

Efforts to tame the unfavorable scaling of electronic structure methods are not limited to density functional theory. For a general summary of the current state of the art see the review by Goedecker, 1999. 

Rather than splitting the physical space into short- and long-range parts as in the above techniques, an alternative is for the Coulomb operator itself to be reformulated and written as a sum of two contributions representing the short- and long-range regimes, 

S(r) will be treated in real space but needs to be applied only in a small neighborhood around the reference charge distribution. L(r), which represents the bulk of the electrostatic interactions (in terms of their number, not in the contribution to the energy), can be treated using various approximations with favorable scaling properties. For example, in the origi- 

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