Helium atom: excited states

We do consider the first excited state, that is . Since we have to dispone two electrons on two different orbital levels we can get four global wf:

For the mean value of the energy we have:

Where and are called direct and switch integrals, and take on the values:

Excited states allow us to introduce new energy levels, called orbitals. For the first excited state we have the configuration ; means that , and so . We thus have three possible wf for this state: . Actually only is a real wf. Since we want to have real functions in order to describe orbitals[1], we take linear combinations of them:

  1. To be precise, here we should have taken only the radial part of the w.f. Since orbitals do describe the probability of occupation of a particular angular space region. Anyway, if you're given a state in terms of orbitals you can rewrite your state in terms of a w.f. using the combinations given below.