Configurations 07

I got the nD version of Newton's method for finding the zeros of a function to improve on a search for the minimum potential energy of a configuration of 7 atoms. With 7 "atoms" there are 21 links to optimize which can be split up into 6 types for the chosen configuration of 2 sets of 3 coplanar atoms and another on the z-axis. The configuration is as follows.

This figure will help to keep track of the arrangement of the atoms and the links.

I completed the post mortem and found an error in the formula used to compute change in position needed for an improved estimate for the zero. Typos are easy to make when entering complex formulas and difficult to find in Excel. I went back and rechecked the calculations for the earlier configurations the new formula. For the configuration of 7 atoms the estimates converged quite nicely to the minimum.

The larger number of links pulls the atoms together more so the sides of the horizontal coplanar sets of atoms have different lengths for ρ and r. The value of κ was chosen to make the equilibrium distance for two atom equal to one. The center plane of three atoms is pushed out from the center somewhat and all the other lengths for nearest neighbor atoms are reduced with the length u of the single atom on the z-axis being pulled towards the center the most.