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I am having troubles with polyenes states symmetry.

I found an exercise on the pentadien, where they ask us to analyze the symmetry of the singlet excited states corresponding to: State S1: π+ →π+ State S2: π+ →π- State S3: π- →π+ State S4: π- →π-

1- I would like to understand what the signs + AND - refer to in this problem. Is there any forbidden transition? I searched in plenty of books and I have not encountered these signs, I thought that it corresponds to the wave function sign but I do not see the relation between the transition rules and the signs of the π MO, I thought +/- signs matter only for transitions between Σ diatomic states .

2-In the case of a polyene molecule, in order to find the symmetry of the ground and excited states, do we need to find the symmetry of the π orbitals in relation to m and C2 operations only? or do we have to find the group symmetry of the molecule and apply all the operations of the groupe?

3-For an excited singlet state having two unpaired electrons, do we multiply the symmetry of the π molecular orbitals in order to get the symmetry of the state? Like m(S)*m(A)=m(A)? Can we do this?

4-Could you please help me and suggest books that are related to my questions?

Thank you all.

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  • $\begingroup$ Could you please provide a source for this exercise? I'm not sure why they use capital Pi letter as an orbital symbol, and why an underscore is used for what is logically supposed to be minus. $\endgroup$
    – andselisk
    Feb 8, 2021 at 10:25
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    $\begingroup$ Thank you for replaying It is a test given by my University. I also don't understand why they used the capitol letter. For the underscore it is supposed to represent a minus. $\endgroup$
    – sarah bnm
    Feb 8, 2021 at 10:41
  • $\begingroup$ Capital $\Pi$ can be used as a label for molecular orbitals, but only in linear species ($D_{\infty\mathrm{h}}$ or $C_{\infty\mathrm{v}}$). For polyenes it appears that small $\pi$ is the appropriate description. However, this doesn't mean that they are doubly degenerate orbitals; it indicates that they are part of the $\pi$ system of the molecule (i.e. conjugated p orbitals). At least to me, the $+$ and $-$ labels don't seem to have an easy explanation: usually these denote symmetry with respect to some mirror plane, but it's not obvious how that applies here. $\endgroup$ Feb 8, 2021 at 14:16
  • $\begingroup$ Thank you very much for your explanations, I corrected the post. $\endgroup$
    – sarah bnm
    Feb 8, 2021 at 15:45
  • $\begingroup$ It is hard to know but if +/- are replaced with u/g the $S_0 \to S_1$ transition is forbidden as the symmetry of both states is the same, and $S_0\to S_2$ allowed. However, $\pi-$ in $S_3 , S_4$ are then difficult to understand. $\endgroup$
    – porphyrin
    Feb 8, 2021 at 16:59

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