Chapter 15, Problem 15.1E

Linearly polarized light can be assigned a specific irreducible representation of a symmetry point group. If the electronic ground state of benzene has $A_{1\text{g}}$ symmetry and $y\text{-}$ polarized light has the label $E_{1\text{u}}$, what are the symmetry label(s) of allowed excited electronic states? Use the $D_{\text{6h}}$ character table in Appendix 3.

Interpretation Introduction

Interpretation:

The symmetry label(s) of allowed excited electronic states for the electronic ground state of benzene as $A_{1\text{g}}$ symmetry and $y\text{-}$ polarized light is to be predicted.

Concept introduction:

The electronic transitions should have a nonzero transition moment, given by the expression shown below.

$M={\displaystyle \int {\Psi }_{\text{final}}^{*}}\widehat{\mu }{\Psi }_{\text{initial}}d\tau$

Answer to Problem 15.1E

The symmetry label of allowed excited state for the $y-\text{polarized}$ light is $E_{\text{1u}}$.

Explanation of Solution

The electronic ground state of benzene has $A_{\text{1g}}$ symmetry and $y\text{-}$ polarized light has $E_{\text{1u}}$ symmetry.

The character table for point group $D_{\text{6h}}$ is shown below.

$\begin{array}{ccccccccccccc}{D}_{\text{6h}}& E& 2C_6& 2C_3& C_2& 3C_2^{\text{'}}& 3C_2^{\text{'}\text{'}}& i& 2S_3& 2S_6& {\sigma }_{\text{h}}& 3{\sigma }_{\text{d}}& 3{\sigma }_{\text{v}}\\ A_{\text{1g}}& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ E_{\text{1u}}& 2& 1& -1& -2& 0& 0& -2& -1& 1& 2& 0& 0\end{array}$

The representations of $A_{\text{1g}}\otimes E_{\text{1u}}$ is calculated as follows:

$\begin{array}{ccccccccccccc}& E& 2C_6& 2C_3& C_2& 3C_2^{\text{'}}& 3C_2^{\text{'}\text{'}}& i& 2S_3& 2S_6& {\sigma }_{\text{h}}& 3{\sigma }_{\text{d}}& 3{\sigma }_{\text{v}}\\ A_{\text{1g}}\otimes E_{\text{1u}}& 2& 1& -1& -2& 0& 0& -2& -1& 1& 2& 0& 0\end{array}$

The representations of $A_{\text{1g}}\otimes E_{\text{1u}}$ is the same as the representations of $E_{\text{1u}}$. Therefore, the symmetry label of allowed excited state for the $y-\text{polarized}$ light is $E_{\text{1u}}$.

Conclusion

The symmetry label of allowed excited state for the $y-\text{polarized}$ light is $E_{\text{1u}}$.