For four atoms there are six pairs of links which we can split into two groups in the following configuration represented by the relative distances ρ and λ.
By symmetry the first three positions are assumed to be points equidistant from the origin in the x,y plane and the fourth position is assumed to be on the z axis.
The total potential is just the sum of the number of links times the individual potential for each set of links. Using an extension of Newton's method for two independent parameters two arbitrary separations converge to a distance of κ. Comparing the equilibrium potential with one nearby confirms that there is no change in potential occurs if the parameters are separately changed.
Placing a fifth atom at position -z on the z-axis results in 10 pair of links but there is still only two independent parameters along with an additional potential term for the link between the two atoms on the z-axis.
The equilibrium distances again turn out to be κ.
Is this a pattern?
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