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 maximize 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). Added (8Sep15): Exact radius for $n=10$ by Sugimoto & Tanemura.
- For $n=24$ by Robinson (1961).
- For $n=13, 14$ by Musin and Tarasov (2014).
[![N=14.][2]][2]
Fig.1 from Musin & Tarasov: $n=14$.
**Added** (*8Sep15*): The exact radius for $n=10$ was just found:
Teruhisa Sugimoto, Masaharu Tanemura. "Exact value of Tammes problem for N=10." Sep 2015. arXiv 1509.01768 Abstract.
[![STFig1b][3]][3]
Fig.1b from Sugimoto & Tanemura.
Added (31Dec2017) in response to a question by @R_Berger: For $n=20$, the best arrangement for the Tammes problem is not the dodecahedron's vertices. The optimal is unknown, but this beats the dodecahedron:
[![DodecaTammes][4]][4]
Coordinates from [Neil Sloane link](http://neilsloane.com/packings/dim3/), due to R.H. Hardin, N.J.A. Sloane & W.D. Smith (1994).