I was planning to get to this later but I am not sure I'll have time today. To force it to work I can do
pts = Apply[{2 \[Pi] #1, ArcCos[2 #2 - 1]} &, RandomReal[1, {10, 2}],
1];
energy[p_] :=
Module[{cart},
cart = Apply[{Sin[#1]*Cos[#2], Sin[#1]*Sin[#2], Cos[#1]} &, p, 1];
Total[Outer[Exp[-Norm[#1 - #2]] &, cart, cart, 1], 2]]
fvars = Table[{x[i], y[i]}, {i, 1, Length@pts}];
NMinimize[energy[fvars], Flatten@fvars]
(*{32.2548, {x[1] \[Rule] -2.0039, y[1] \[Rule] -0.778507,
x[2] \[Rule] -1.13768, y[2] \[Rule] -1.5639, x[3] \[Rule] 1.13769,
y[3] \[Rule] 0.00689281, x[4] \[Rule] 2.00391,
y[4] \[Rule] -0.778497, x[5] \[Rule] -2.00391,
y[5] \[Rule] 0.792293, x[6] \[Rule] -9.32847\[Times]10^-6,
y[6] \[Rule] 1.20086, x[7] \[Rule] 1.1377, y[7] \[Rule] -1.5639,
x[8] \[Rule] 3.14158, y[8] \[Rule] 2.10272, x[9] \[Rule] 2.0039,
y[9] \[Rule] 0.792295, x[10] \[Rule] -1.13769,
y[10] \[Rule] 0.00689875}}
*)
*)
But I wouldn't do it this way.
