Of course the in the Hamiltonian doesn't change.

So our ground state trial function is

Minimize the energy.

We can recycle our previous work to do these integrals. First, replace the in with a and put in a correction term. This makes the part just a hydrogen energy. The correction term is just a constant over so we can also write that in terms of the hydrogen ground state energy.

Then we reuse the perturbation theory calculation to get the term.

Use the variational principle to determine the best
.

Putting these together we get our estimate of the ground state energy.

Now we are within a few percent. We could use more parameters for better results.

Jim Branson 2013-04-22