  Determination of \emph{optimal} arrangements of $N$ particles on a sphere is
a well-known problem in physics. A famous example of such is the Thomson
problem of finding equilibrium configurations of electrical charges on a
sphere. More recently however, similar problems involving other potentials and
non-spherical domains have arisen in biophysical systems. Many optimal
configurations have previously been computed, especially for the Thomson
problem, however few results exist for potentials that correspond to more
applied problems. Here we numerically compute optimal configurations
corresponding to the \emph{narrow escape} and \emph{narrow capture} problems in
biophysics. We provide comprehensive tables of global energy minima for
$N\leq120$ and local energy minima for $N\leq65$, and we exclude all saddle
points. Local minima up to $N=120$ are available online.
