Extending Newton's method to higher dimensions is proving to be a little more difficult but I've had some success. A configuration of seven atoms has 21 pair of links and can be arranged according to this scheme. The three independent parameters chosen are q,z,w with dimensionless counterparts θ,ζ,χ.
This is the executive summary of the procedure used. To check the formulas I derived for Newton's method for a higher dimensions I did a search for the minimum total potential energy. The value for χ was 1.53580.
A 3D plot shows the surface looks like this:
The surface can be can be approximated by a quadratic form derived from 1st and 2nd order derivatives and can be written like this.
The independent parameters converge fairly quickly to a point slightly displaced from the search value.
It's a little puzzling why this doesn't work out exactly but the formulas derived may be somewhat biased estimators of the minimum.
Supplemental (Nov 9): The quadratic form isn't a perfect match for the potential surface. A least squares fit of the potential surface gave the following set of coefficients.
A plot shows the difference between the two surfaces.
The difference between the quadratic form and the configuration potential may account for the deviation of the computed minimum.
A 3D plot shows the surface looks like this:
The surface can be can be approximated by a quadratic form derived from 1st and 2nd order derivatives and can be written like this.
The independent parameters converge fairly quickly to a point slightly displaced from the search value.
It's a little puzzling why this doesn't work out exactly but the formulas derived may be somewhat biased estimators of the minimum.
Supplemental (Nov 9): The quadratic form isn't a perfect match for the potential surface. A least squares fit of the potential surface gave the following set of coefficients.
A plot shows the difference between the two surfaces.
The difference between the quadratic form and the configuration potential may account for the deviation of the computed minimum.
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