OPTIMIZED MULTICENTER EXPANSIONS

A new approach to multicenter spherical harmonic expansions is present ed, which is based on Fourier transform and variational methods. The i ndividual radial functions are optimized simultaneously over all sites at each order of spherical harmonics; and it is conjectured that the resulting expansions, for arbitrary functions in three dimensions, wil l be more rapidly convergent than any other type. Both iterative and c losed-form solutions are developed. The analogous cases for two- and o ne-dimensional functions are also treated and examples for all three c ases are provided. The one-dimensional case is found to be qualitative ly different. In a (perhaps) surprising theorem it is proved that, giv en N greater-than-or-equal-to 2 distinct points of R and N (not necess arily equivalent) choices of parity, an arbitrary one-dimensional func tion may be exactly decomposed as the sum of N functions each having o ne of the chosen parities at one of the points. The decomposition is n ot unique.