An iterative procedure for finding locally and globally optimal arrangements of particles on the unit sphere

Determination of globally optimal arrangements of N pairwise-interacting particles is an important problem that occurs in a variety of biological, physical, and chemical applications. We propose a numerical procedure to compute putative optimal configurations. The procedure is able to handle a wide class of pairwise potentials. Locally and globally minimal arrangements of particles on the unit sphere, interacting via the Coulombic, logarithmic, and inverse square law, are computed as samples. We present new results for the logarithmic potential consisting of 45 new local minima for N <= 65 and two new global minima (N = 19, 46), as well as results for the inverse square law potential which has not previously been studied. We provide comprehensive tables of all minima found, and exclude all saddle points. The algorithm can perform computations exceeding N = 100 with reasonable execution times.