# Solutions 13

- Page ID
- 52481

### Q1

By looking at the modes of water, we can visually inspect and see if the symmetry operations result in staying the same (designated by a 1) or a reflection (-1), and then assign a representation that matches these results.

Water Vibrational Mode |
\(E\) | \(C_2\) | \(\sigma_v(xz)\) | \(\sigma_v(yz)\) | Representation |
---|---|---|---|---|---|

Symmetric Stretch | 1 | 1 | 1 | 1 | \(A_1\) |

Antisymmetric Stretch | 1 | -1 | -1 | 1 | \(B_2\) |

Scissoring Bend | 1 | 1 | 1 | 1 | \(A_1\) |

### Q2

We use symmetry to evaluate integrals by comparing how symmetry operations affect

\[ M_{v \rightarrow v'} = \langle v' = 1 | \mu | v =0 \rangle\]

The left side is unaffected but each factor of the right side is either changed by a factor of 1 or -1, which is determined by the character of the symmetry operation on the representation. If we represent a symmetry operation by \(\hat{R}\) , remember that the ground state has the representation \(A_1\), we have:

\[\hat{R} M_{v \rightarrow v'} = M_{v \rightarrow v'} = \hat{R}(\langle v' = 1 |) \hat{R}( \mu) \hat{R} (| v =0 \rangle) = \chi_{A_1}(\hat{R}) \chi_{\mu}(\hat{R}) \chi_{v=1}(\hat{R}) M_{v \rightarrow v'} \]

We require

\[\chi_{\mu}(\hat{R}) \chi_{v=1}(\hat{R}) =1 \]

which occurs only if the representations of \(\mu\) and the excited state are the **same**. We can make a table to check how many combinations of \(\mu\) components and vibrational modes have the same representation, and therefore, an allowed transition:

Mode\ Dipole Moment Component |
\(\mu_x\) \((B_1)\) | \(\mu_y\)\((B_2)\) | \(\mu_z\)\((A_1)\) |
---|---|---|---|

Symmetric Stretch \((A_1)\) | same | ||

Antisymmetric Stretch \((B_2)\) | same | ||

Scissoring Bend \((A_1)\) | same |

Only three combinations result in allowed transitions which include the x and y dipole moment components with different modes.