Excited States

As a first test we compute the energy of an excited 1S2S state of Helium with one electron in a 1st shell and one in a 2nd shell in spherical symmetry, to get an energy of -2.19 to be compared with the -2.90 for the 1S^2 ground state,  which fits with observation.  The about 5 eV for orthohelium in the Helium energy level plot below represents -5/27.2 =  -0.18 to be added to the 1S state of energy -2, which gives -2.18  to be compared with the -2.19 of realQM.

In stdQM excited states of Helium appear as parahelium and orthohelium with different spin characteristics and exchange-correlation energies with different sign, as part of a complex picture seeking  with observation.

In realQM spin plays no role and the picture is clear for 1st excitation in spherical symmetry with one electron in a 1st shell and one excited electron in a 2nd shell, which agrees with observation.

In realQM the excited states of Helium with one electron in a first shell and one excited  electron in a 2nd shell naturally comes out as analogous to the excited states of Hydrogen with the kernel of Hydrogen replaced by an effective pseudo-kernel consisting of the kernel of Helium surrounded by the (spherically symmetric) electron in the 1st shell. The same argument applies to all atoms with electrons in the outermost shell subject to excitation. A key question then is the effective boundary condition at the free boundary of outermost shell: homogeneous Neumann or continuity (or both?).

In a next test we consider excitations of the electron in the 2nd shell from the 1S considered above to a form of 2S and 3S states and again obtain fair agreement with the 3S and 4S states according to the Helium energy level plot. The excited 2S and 3S states in the 2nd shell are computed by minimisation and orthogonalisation over the 2nd shell with homogeneous Neumann condition on a free boundary determined by 1S states in both shells. This results in excited 2S and 3S states in the 2nd shell with a jump discontinuity at the free boundary. Alternatively, we may keep continuity and relax the homogeneous Neumann condition, which shows similar but less accurate results, see below. It is thinkable that excited states give up one of the two conditions at the free boundary and the computations then suggest that the homogeneous condition is preferred.