I know that analytic forms of the Helium wavefunctions are not known. However, are there empirical expressions for the low-lying states of Helium? I'd like to use them to calculate some transition matrix elements for fun.
If you want a quick and dirty expressions in terms of hydrogen atom wave functions, you can try this:
For ground state you can get a quick approximation by applying effective nuclear charge. Then you get a hydrogen-like wave function with a modified nuclear charge between 1 and 2; solve that and get orbital. It will be doubly occupied, spin singlet.
The lowest excited states of He are all Rydberg states. The roughest approximation would be to consider it He+ plus a Rydberg electron, and you can even use hydrogen-like atom formula to get the Rydberg orbital. The total wave function would be antisymmetrized product of these two orbitals, coupled to be either singlet or triplet.
I am not sure you want transition density matrix or transition dipole matrix. Either way, its just quite straightforward integration from there. I think there might be ways to do this analytically, but frankly I do not remember and you can probably try that on Mathematica.
Yes, wavefunctions for helium can be calculated numerically with various levels of approximation with quantum chemistry methods. Be aware, though, that wavefunctions depend on a specific representation of the system (the use of molecular orbitals) but are not physical observables of the system (only the electronic density is an observable).