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I'm generally interested in numerical chemistry. In our lecture, we have discussed the $E \otimes e$ Jahn–Teller molecule with the following Hamiltonian \begin{align} H = \epsilon n_d + \omega ( a^{\dagger}_x a_x + a^{\dagger}_y a_y) + \lambda (Q_x \sigma_x + Q_y \sigma_y )n_d. \end{align} So we have two degenerate electronic levels and two degenerate vibrational modes. The operators $Q_{x/y}$ are the position operators and $n_d$ is one when the molecule is in the charged state. I'm interested in both the eigenenergies and eigenstates of this Hamiltonian, I initially thought, that this is not complicated, as we have just the operators given in second quantuzation.

I found in some publications that it is convenient to write down this Hamiltonian in the polar basis of the harmonic oscillator. So I transformed the Hamiltonian in this basis. There the basis states look like $|n,m\rangle$, with $n$ the radial excitation and $m$ the angular momentum. So, as a first try, I expressed the Hamiltonian for the uncharged molecule, in order to be sure that I understood the Hamiltonian in a correct way and I receive the correct energies according to $H|n,m\rangle = \omega n |n,m\rangle$.

Now I'm trying to express the charged states in this basis. Again, $m$ is the angular momentum of the state in the 2D isotropic harmonic oscillator and ranges in $n-1,n-3,\ldots,-n+1$. Now in the lecture, we claimed that the total angular momentum is conserved, i.e. I also need to consider the spin of the electron. However, I'm not sure how to adopt this in the basis. For sure, I need to introduce an additional parameter in my basis states, namely the spin.

My only try was to introduce states $|n,j\rangle$ with the total angular momentum $j=m\pm \sigma$, since I really have no idea to do so. However, the $j$ would be a half-integer, what is impossible for the 2D harmonic oscillator states. So there is something that I'm not getting correct.

Has someone a hint that could bring me back to the track? Do you have any references that discuss the eigenstates of this Hamiltonian?

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