There is considerable literature on this question, and closely related variations. See:
- The Thompson problem: Which configurations of $n$ electrons on a sphere minimize the electrostatic potential energy?
- The Tammes problem: Which configurations of $n$ points on a sphere maximizes the smallest distance between any two points? Sometimes phrased as packing $n$ congruent circles on a sphere:
![DiskPacking][1]
(Image from [Paul Sutcliffe](http://www.maths.dur.ac.uk/~dma0pms/my-projects/proj4_1213.html).)
According to
Musin, Oleg R., and Alexey S. Tarasov. "The Tammes problem for $N=14$." arXiv:1410.2536 Abstract (2014).
the Tammes problem is solved exactly for
- For $n=3,4,6,12$ by L. Fejes Toth (1943).
- For $n=5,7,8,9$ by Schütte and van der Waerden (1951).
- For $n=10,11$ by Danzer (1963).
- For $n=24$ by Robinson (1961).
- For $n=13, 14$ by Musin and Tarsov (2014).
[![N=14.][2]][2]
Fig.1 from Musin and Tarsov: $n=14$.