For six atoms there is another possible configuration to be considered in which the atoms are arranged as follows. It has four characteristic distances which can be expressed in terms of parameters λ and ρ. The solution of this problem proved to be a little more difficult and a two dimension search done as a check showed that the two parameter version of Newton's method gave an erroneous result. The search again gave an equilibrium value of 0.956 for the two parameters and the same equilibrium potential as the previous configuration for the six atoms.
The potential is approximately parabolic.
One needs keen eyesight to determine what went wrong with Newton's method. A comparison of the actual potential with a surface fit produced this surface representing the difference between the two.
The light blue at the top is bounded by contours of zero potential difference. So the original surface appears to be slightly saddle shaped. Apparently there was a contradiction in the assumptions made for Newton's method.
Supplemental (Nov 4): This configuration is identical with the previous one. It's just seen from a different direction, for example, from along the line through the origin and a point in the direction θ=35.2644° and φ=0° or some equivalent.
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