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For E1 selection rules (embodied in the electronic transition moment), to have an allowed transition the direct product of irreducible representations (IRs) must contain the totally symmetric IR. So, for transitions between two $\Sigma$ states, the following transitions are symmetry allowed: $\Sigma^- \leftrightarrow \Sigma^-$ and $\Sigma^+ \leftrightarrow \Sigma^+$.

But I read in Herzberg's book "Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules" that in Hund's case b when we have a rotational distortion of the electronic motion or spin-orbit interaction, the $\Sigma^- \leftrightarrow \Sigma^+$ transition can appear.

Although for rotational distortion I understood that this violation is due to the $- \leftrightarrow +$ rule. I am wondering whether the direct product rule does not apply anymore?

Could you please tell me if there are other cases to consider where the $\Sigma^- \leftrightarrow \Sigma^+$ transition is allowed? And did I misunderstood the selection rule governing the transitions between two $\Sigma$ states?

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    $\begingroup$ The z transition dipole belongs to $A_1\equiv \sum^+$ symmetry species so product $\sum^+\times \sum^+\times \sum^+\equiv A_1$ and similarly $\sum^-\times \sum^+\times \sum^-\equiv A_1$. If there is spin orbit 'distortion' which has symmetry $A_2\equiv \sum^-$ then the case is $(\sum^+\times \sum^- )\times \sum^+\times \sum^-\equiv A_1$ so is allowed $\endgroup$
    – porphyrin
    Feb 13, 2021 at 9:26
  • $\begingroup$ Thank you very much porphyrin for your answer, it is clearer now. $\endgroup$
    – sarah bnm
    Feb 13, 2021 at 15:15

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