Premium
Lattice sum calculations for 1/ r p interactions via multipole expansions and Euler summation
Author(s) -
Steiger Don,
Glaser Rainer
Publication year - 2000
Publication title -
journal of computational chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.907
H-Index - 188
eISSN - 1096-987X
pISSN - 0192-8651
DOI - 10.1002/1096-987x(20010130)22:2<208::aid-jcc8>3.0.co;2-v
Subject(s) - multipole expansion , euler's formula , lattice (music) , mathematics , euler summation , physics , computational chemistry , mathematical physics , mathematical analysis , quantum mechanics , backward euler method , chemistry , euler equations , semi implicit euler method , acoustics
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 factorization produces a formula for generating geometry constants that uniquely define the lattice structure. A direct calculation of these geometry constants, for all but the very smallest of crystals, is computationally impractical. In the second step, an Euler summation method is introduced that allows 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 computational complexity can be significantly less than N . Once the geometry constants have been calculated, calculating a lattice sum for a given molecule becomes computationally very fast. Millions of different molecular orientations or molecules can quickly be evaluated for the given lattice structure. © 2000 John Wiley & Sons, Inc. J Comput Chem 22: 208–215, 2001