The on-the-surface instance of your problem is sometimes known as the *Thomson problem*, after J.J. who posed the problem over a century ago. There is [a detailed Wikipedia page](http://en.wikipedia.org/wiki/Thomson_problem) on the topic. The $n=5$ case was only settled in 2010—"a rigorous computer-assisted proof" established that the *triangular bipyramid* is the unique minimizer of the Coulomb potential. > Richard Evan Schwartz. "The 5 Electron Case of Thomson's Problem." 2010. [arXiv link](http://arxiv.org/abs/1001.3702) > "In the [triangular bipyramid], two points are antipodal points on [the sphere] and the remaining 3 points form an equilateral triangle on the equator midway between the two antipodal points": <br /> <img src="https://i.sstatic.net/Koi2S.png" width="300" /> <br /> <sup>(Image from brakke @[this link](http://www.shapeways.com/model/428975/triangular-dipyramid-70mm.html))</sup> <br /> Many other candidates for minimal energy configurations have been identified but not rigorously proven. For example, for $n=14$, it seems that the *gyroelongated hexagonal dipyramid* is the minimal energy configuration (combinatorially).