We use moment methods to construct a converging hierarchy of optimization
problems to lower bound the ground state energy of interacting particle
systems. We approximate the infinite dimensional optimization problems in this
hierarchy by block diagonal semidefinite programs. For this we develop the
necessary harmonic analysis for spaces consisting of subsets of another space,
and we develop symmetric sum-of-squares techniques. We compute the second step
of our hierarchy for Riesz s-energy problems with five particles on the
2-dimensional unit sphere, where the s=1 case known as the Thomson problem.
This yields new sharp bounds (up to high precision) and suggests the second
step of our hierarchy may be sharp throughout a phase transition and may be
universally sharp for 5-particles on S2. This is the first time a
4-point bound has been computed for a continuous problem
