Chapter 12

Evaluate the polarizability \(\alpha_{z z}\) and polarizability volume of a hydrogen atom; for simplicity, confine the perturbation sum to the 2 p-orbitals. Use the following results for matrix elements: \\[ \begin{array}{l} \left\langle n^{\prime}, l+1, m_{l}|z| n l m_{l}\right\rangle=\left\\{\frac{(l+1)^{2}-m_{l}^{2}}{(2 l+3)(2 l+1)}\right\\}^{1 / 2}\left\langle n^{\prime}, l+1|r| n l\right\rangle \\ \langle n p|r| 1 \mathrm{s}\rangle=\left\\{\frac{2^{8} n^{7}(n-1)^{2 n-5}}{(n+1)^{2 n+5}}\right\\}^{1 / 2} a_{0} \end{array} \\]

Show group theoretically that in a tetrahedral molecule (a) the mean hyperpolarizability is zero, (b) the only non-zero components are \(\beta_{x y z}\) and the permutations of its indices. Hint. The mean is defined as \(\frac{3}{3}\left(\beta_{x x z}+\beta_{y z z}+\beta_{z z z}\right)\) and so (b) implies (a). For (b) consider the symmetry characteristics of \(E=-(1 / 3 !) \Sigma_{a, b, c} \beta_{a b c} E_{a} E_{b} E_{c}\) the generalization of eqn 12.11.

Prove the sum rule \\[ \Sigma_{p} x_{m f} x_{f m} \omega_{f n}=\left(\hbar / 2 m_{e}\right) \delta_{m n}+\frac{1}{2} \omega_{m n}\left(x^{2}\right)_{m n^{+}} \\] Hint. Consider the matrix elements of the commutator \\[ \left[H, x^{2}\right] \\].

Evaluate the rotational strength of a transition of an electron from a \(2 \mathrm{p}_{x}\) -orbital to a \(2 \mathrm{p}_{z}, 3 \mathrm{d}_{x y}\) -hybrid orbital. Assume the orbitals are on a carbon atom. Estimate the optical rotation angle for 590 nm light. Hint. Follow Example \(12.4,\) with changes of detail. For carbon, take \(\zeta_{p}=1.57 / a_{0}\) and \(\zeta_{d}=0.33 / a_{0}\) and use \(\lambda_{k 0}=200 \mathrm{nm}\).