Doing a post mortem on the failed Newton's method for six atoms seems to indicate that a term was missing due to the assumption of ceteris paribus. This appears to be the correct procedure.
The values for ρ and ζ converge to the values found by doing a search for the minimum of the sum of the potentials.
Supplemental (Nov 5): This is a problem in the calculus of variations since we want the change in the potential for arbitrary displacements from the initial position. The displacements are the variables here. The changes along the differential axes are directional derivatives and don't provide enough information about the quadratic surface to determine the minimum value. The treatment of differentials in texts can be rather intuitive at times. For example see Price, A Treatise on the Differential Calculus.
Edit (Nov 9): Correct sequence number in the title.
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