Combining Variables in Optimization

In Newton's method one needs to add the coefficients of an equivalence class to compute the potential energy. So one can do the following.

The multiples in the potential function also affect the results for the optimization parameters in Newton's method in nD. In the case of a triangular configuration the parameters can be varied independently by the formulas are the same. This is why all the lengths are equal to one up to a configuration on 4 "atoms" or particles. The extra link for 5 particles results in a more compact cluster.

For the problem with 7 atoms the upper and middle sets of 3 atoms are not in the same equivalent class since their relative positions are different.