We may consider the points as electrons repelling each other by electrostatic forces. Therefore this problem is named "Thompson-Problem", because it modells the atom-modell of the later Lord Kelvin.

There are incredibly many articles about this problem, most of them concentrating on results by computational experiments and written by phyisicist or crystallographs. Mathematical papers most deal with estimates on the functional for large N.

[1] Altschuler, E.L.; Williams T.J.; Ratner E.R.; Tipton R.; Stong R.; Dowla F.; Wooten F.:

Possible Global Minimum Lattice Configurations for Thomson's Problem of Charges on a Sphere
Phys. Rev. Lett. 78, 2681–2685

[2] Edmundson, J.R.

The Distribution of Point Charges on the Surface of a Sphere
Acta Cryst. A48 (1992), 60-69

[3] Erber T.; Hockney G.M.:

Equilibrium congurations of n equal charges on a sphere
J. Phys. A (1991)

[4] Föppl, L.:

Stabile Anordnung von Elektronen im Atom
J. Reine Angew. Math. 141 (1912), 251-302

[5] Hardin, R.H.; Sloane, N.J.A.; Smith, W.D.

www.research.att.com/~njas/electrons/

[6] Kuijlaars, A. B. J.; Saff, E. B. :

Distributing many points on a sphere
Mathematical Intelligencer, v19 n1 (1997), pp. 5-11

[7] Kuijlaars, A. B. J.; Saff, E. B. :

Asymptotics for Minimal Discrete Energy on the Sphere
Trans. Amer. Math. Soc 350, 2 (1998), 523-538

[8] Malloy, S.:

The Electrons Applet

[9] cluque@inf.uc3m.es:

Distributing n Charges on a Sphere