In 3d with (|r|-c)2

mi=m (reduced mass of nuclei measured in electron mass units)

                Equation 2.7 of Variance Minimization becomes

             (1.1)

Ground State

                Try         

                             (2.1)

                                         (2.2)

                           (2.3)

                                            (2.4)

                                (2.5)

So that

                                                          (2.6)

(2.7)

Equation (1.1) becomes

(2.8)

Or

(2.9)

As in the one-d case a=Ö(m/2) cancels the variability of the variance near the maximum of the wave function at r=c.  The remaining E is

                                             (2.10)

If c were zero, this would be a 3d Harmonic Oscillator.  For a peak far from 0, r will be approximately equal to c.  This makes the oscillator approximately one d. 

Excited Rotational State

                Try         

                                       (3.1)

                       (3.2)

 

                     (3.3)

(3.4)

For c à 0, this is the sames as the first excited state in 3dHO.doc htm.  The value of  x has completely cancelled out of (3.4).   The largest variation of E is due to (r-c)2.  This is cancelled by making a=Ö(m/2).  Then the Energy is

 

                (3.5)

This state is orthogonal to the ground state by virtue of x being an odd function while any function of r is even.  Its energy differs from the ground state energy by

             (3.6)

There are three states, with x, y, and z in this position that are degenerate.  Other p states can be constructed by summing them in various combinations.