Calculating the first order correction to the ground state is simple in principle.
of the energy shift in first order
involves an integral over the coordinates of both electrons.
So the ground state energy to first order is
compared to -78.975 eV from experiment.
A 10% error is not bad considering the size of the perturbation.
First order perturbation theory neglects
the change in the electron's wavefunction
due to screening of the nuclear charge by the other electron.
Higher order perturbation theory would correct this, however, it is
hard work doing that infinite sum.
We will find a better way to improve the calculation a bit.