Solutions 13
- Page ID
- 204096
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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.