# electronic transition for polyenic molecules

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?

• 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. Feb 8, 2021 at 14:16
• 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. Feb 8, 2021 at 16:59