Local Energy in small r limit

The Schroedinger equation in Rydberg units is

(1.1)

This is frequently “solved” by minimizing

(1.2)

 

with respect to the components of a vector a that represents a parameterization of the wave function.

An S electron – the cusp condition

For an electron near a nucleus at the origin with charge Z, the integrand in equation (1.2) is

            (2.1)

The 1/r part of this can become very large for small r.  In the small r limit the del² can be taken in spherical coordinates for which

          (2.2)

For a trial wave function of the form , this becomes

(2.3)

Thus (2.1) becomes

 

(2.4)

The choice a₁ = Z, removes the singularity in the integrand of (1.2) and also gives a reasonable “guess” at the eigenvalue E.

Lennard-Jones  ..\..\definitions\Lennard Jones Potential.htm .doc

 

            For two nuclei interacting with a Lennard-Jones potential as rà 0, the integrand in (1.2) becomes

   (3.1)

Consider a trial wave function of the form

           (3.2)

Again in spherical coordinates the del² is given by

(3.3)

So that

(3.4)

Note that in this generalization of (2.3) that the second term which was constant has the capability of becoming the most singular.  Substituting (3.4) into (3.1) yields

(3.5)

For n=-5, (3.5) becomes

(3.6)

Then solve

(3.7)

To reduce (3.6) to

(3.8)

The remaining singularity is unfortunately not exactly canceled by the attractive part of the Lennard Jones potential, but is a lot easier to deal with than the much higher one just removed.