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$$\gamma \begin{pmatrix}

O_{11} & O_{22} \\

O_{21} & O_{22}

\end{pmatrix}$$

where ##O_{ij} = <i|\hat{O}|j>##, where ##|i>## and ##|j>## and the 2 Hund cases basis. I think that up to here I understand it well. However, I am not sure how we account for off diagonal terms in this hamiltonian. When we do a fit to the data (which in this case would be a measurement of the energy difference between ##|i>## and ##|j>##) in order to extract ##\gamma##, do we just ignore the off diagonal terms, or do we diagonalize this hamiltonian (which in practice can have hundreds of rows, depending on how many lines were measured)? Usually when the energy levels are labeled in a diagram, they have the quantum numbers of the hund case chosen, which would imply that we ignore the off diagonal entries. Are they so small that we can ignore them? Or are they actually zero? They shouldn't be zero, as in an actual hamiltonian there are terms which break the perfect coupling picture of a perfect hund case. Can someone help me understand how do we connect hund energy levels to real energy levels? Thank you!