Solutions 13

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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	\(\mu_y\)\((B_2)\)	\(\mu_z\)\((A_1)\)
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.