The First Excited State(s)

Now we will look at the energies of the excited states. The Pauli principle will cause big energy differences between the different spin states, even though we neglect all spin contribution in $H_1$ This effect is called the exchange interaction. In the equation below, the $s$ stands for singlet corresponding to the plus sign.

$$\begin{eqnarray*} E^{(s,t)}_{1st} &=& {e^2\over 2} \left< \phi_{100}\phi_{2\ell... ...\phi_{100} \right> \right\} \\ &\equiv& J_{2\ell}\pm K_{2\ell} \end{eqnarray*}$$

It's easy to show that $K_{2\ell}>0$. Therefore, the spin triplet energy is lower. We can write the energy in terms of the Pauli matrices:

$$\begin{eqnarray*} \vec{S}_1\cdot\vec{S}_2 &=& {1\over 2}(S^2-S^2_1-S^2_2) ={1\o... ... 2}\left( 1 + \vec{\sigma}_1\cdot\vec{\sigma}_2\right) K_{n\ell} \end{eqnarray*}$$

Thus we have a large effective spin-spin interaction entirely due to electron repulsion. There is a large difference in energy between the singlet and triplet states. This is due to the exchange antisymmetry and the effect of the spin state on the spatial state (as in ferromagnetism).

The first diagram below shows the result of our calculation. All states increase in energy due to the Coulomb repulsion of the electrons. Before the perturbation, the first excited state is degenerate. After the perturbation, the singlet and triplet spin states split significantly due to the symmetry of the spatial part of the wavefunction. We designate the states with the usual spectroscopic notation.

In addition to the large energy shift between the singlet and triplet states, Electric Dipole decay selection rules

$$\begin{eqnarray*} \Delta\ell&=&\pm 1 \\ \Delta s&=&0 \end{eqnarray*}$$

cause decays from triplet to singlet states (or vice-versa) to be suppressed by a large factor (compared to decays from singlet to singlet or from triplet to triplet). This caused early researchers to think that there were two separate kinds of Helium. The diagrams below shows the levels for ParaHelium (singlet) and for OtrhoHelium (triplet). The second diagrams shows the dominant decay modes.