Elliptical Coordinates

Figure: The Elliptical Coordinate System for Diatomic Molecules. The coordinate is not depicted.
Each problem in mathematical chemistry and physics either has an appropriate coordinate system, in which the mathematics simplifies termendously, or it doesn't, whereupon no coordinate transformations will result in helping the analysis. For diatomic molecules, fortuitously, there is a coordinate system, Confocal Elliptical Coordinates, which fits the bill, just as for the Hydrogen Atom, Spherical Polar Coordinates were the most appropriate.

If rA is the distance from nucleus A to a point P(x,y,z) (where the electron is located, in H2+, presumably), and rB is the distance from nucleus B to the same point(!), then Elliptical Coordinates are defined as:

\begin{displaymath}\lambda = { {r_A+r_B} \over {R}}


\begin{displaymath}\mu = { {r_A-r_B} \over {R}}

(where is the same as the coordinate used in Spherical Polar Coordinates), which means that


This also means that


What is K?

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Last edit: 13hrs:30min, Tuesday, January 21, 20103