Consider N point charges on (the surface of) a unit conducting sphere,
interacting only through their mutual Coulomb forces. What is the
configuration of the charges for which the Coulombic energy 
is minimized? This
question was originally asked by Thomson for 
,[4] and has since been investigated by many authors
[2,7,8,9,10].
Somewhat surprisingly, it turns out that the configuration of minimum
energy is not the configuration which places the charges at furthest
distance from each other, or the configuration of greatest symmetry.
For example, for 8 charges, the configuration of minimum energy is not a
cube, but a twisted noncubic rectangular parallelpiped[2].
For Thomson's problem, we take the function g in Eq. (1) equal
to 
. For those charges which change coordinates,
we assign new spherical coordinates as 
(
);
here 
, and 
are random numbers between 0 and 1, 
reduces the maximum angular change in proportion to the cooling
schedule, and the additions are done with the appropriate periodicity.
We use 
with
; the results are not very sensitive to small changes from
this value of 
. At each temperature step we consider 100
configurations (iterations); then we lower the temperature T used in
Eq. (1), the Metropolis algorithm, and 
, by a factor
of 0.9. Finally, we use the final configuration from CGO as input to a
conjugate gradient descent algorithm to reduce the energy as far as
possible, using double--precision arithmetic. We repeated the entire
procedure from five different starting configurations for each number of
charges.
Our results using CGO for 
confirm the minimum energy
values found previously[7]. We find improvements in
previously--found minimum energy values for most values of N between
66 and 100. We list values we believe to be the minimum energies for
in table i.
For some numbers of charges the minimum energy obtained by CGO is lower than that obtained by using SA[8], by Monte Carlo simulation[9], or by finding the configuration of minimum energy with a given symmetry[10]. In particular, these results demonstrate that the symmetry of the configuration of minimum energy is not always obvious a priori. The output from CGO is typically well within one tenth of one percent of the final minimum energy after applying the conjugate gradient, and this performance of CGO relative to final minimum energy value does not deteriorate with increasing N. We think that CGO will be readily applicable to other ionic and molecular structure problems.