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

The problem of determination of globally optimal arrangements of N pairwise-interacting particles arises in a variety of biological, physical, and chemical applications. At the same time, the important related question of finding all, or many, local minima of the corresponding energy functions, and the study of structure of these minima, has received relatively little attention.

A computational procedure is proposed to compute locally optimal and putative globally optimal arrangements of N particles constrained to a sphere. The procedure is able to handle a wide class of pairwise potentials, and can be generalized to other kinds of surfaces and interactions.

As computational examples, locally and globally energy-minimizing arrangements of particles on the unit sphere, interacting via the Coulombic, logarithmic, and inverse square law potentials, are computed. 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 saddle points. The algorithm can perform computations exceeding N = 100 with reasonable execution times.