Lattice sum calculations for 1/r(p) interactions via multipole expansions and Euler summation

A method is developed here for doing multiple calculations of lattice sums when the lattice structure is kept fixed, while the molecular orientations or the molecules within the unit cells are altered. The approach involves a two-step process. In the first step, a multipole expansion is factored in such a way as to separate the geometry from the multipole moments. This fac torization produces a formula for generating geometry constants that unique ly define the lattice structure. A direct calculation of these geometry con stants, for all but the very smallest of crystals, is computationally impra ctical. In the second step, an Euler summation method is introduced that al lo rvs for efficient calculation of the geometry constants. This method has a worst case computational complexity of O((log N)(2) /N), where N is the number of unit cells. If the lattice sum is rapidly converging, then the co mputational complexity can be significantly less than N. Once the geometry constants have been calculated, calculating a lattice sum for a given molec ule becomes computationally very fast. Millions of different molecular orie ntations or molecules can quickly be evaluated for the given lattice struct ure. (C) 2000 John Wiley & Sons, Inc.