Solutions 13

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.