Since you are asking about an algorithmic perspective, I wanted to point out that a closely related variation (with logarithmic potential instead of hard core potential) is the subject of one of Smale's "[Mathematical Problems for the
Next Century][1]." In Problem 7, Smale asks if there is a polynomial time algorithm ([in a particular model of computation over real numbers][2]) to place $N$ points on the sphere so that the total energy is $O(\log N)$ above the minimum energy.

Now, for specific finite values of $N$, Neil Sloane has a table of best known spherical codes with references: http://neilsloane.com/packings/

  [1]: https://dx.doi.org/10.1007%2Fbf03025291
  [2]: https://mathoverflow.net/questions/186226/the-link-and-equivalence-between-variant-definition-of-computation-model-and-com