Chapter 11

The Franck-Condon principle and the BornOppenheimer approximation have an important qualitative feature in common. What feature do they share that to a large extent justifies their usefulness?

Consider a case in which two electronic states have the same force constant but in which the equilibrium bond lengths differ by \(\Delta R\). Find an expression for the relative intensity of the \(0-1\) transition \((v=1\) is the upper vibrational state \()\) as a function of \(\Delta R .\) Hint: Follow Example \(11.1 ;\) use mathematical software to evaluate the integral numerically.

Show that in the carbonyl group the \(\pi^{*} \leftarrow \pi\) transition is allowed, its transition dipole moment lying along the bond. Hint. Consider the carbonyl group to be of \(C_{2 v}\) symmetry with the \(C=O\) bond along the \(z\) -axis.

Assess the polarization of the \(^{1} \mathrm{A}_{2} \leftarrow^{1} \mathrm{A}_{1}\) transition in \(\mathrm{H}_{2} \mathrm{CO}\) and of the \(^{1} \mathrm{B}_{2 \mathrm{u}} \leftarrow^{1} \mathrm{A}_{\mathrm{g}}\) transition in \(\mathrm{CH}_{2}=\mathrm{CH}_{2} .\) Hint. Use \(C_{2 v}\) and \(D_{2 \mathrm{h}}\) respectively; consider the role of vibrational coupling.

In a diamagnetic octahedral complex of \(\mathrm{Co}^{3+}\), two transitions can be assigned to \(^{1} \mathrm{T}_{1 \mathrm{g}} \leftarrow^{1} \mathrm{A}_{1 \mathrm{g}}\) and \(^{1} \mathrm{T}_{2 \mathrm{g}} \leftarrow^{1} \mathrm{A}_{1 \mathrm{g}}\) Are these transitions forbidden? If they are forbidden, what symmetries of vibrations would provide intensity? Can the intensities be ascribed to the admixture of configurations involving p-orbitals?

In an aromatic molecule of \(D_{2 \mathrm{h}}\) symmetry the lowest triplet term was identified as \(^{3} \mathrm{B}_{1 \mathrm{u}} .\) What is the polarization of its phosphorescence? Hint. Decide which singlet terms can mix with \(^{3} \mathrm{B}_{1 \mathrm{u}}\) and assess the polarization of the light involved in the return of that state to the \(^{1} \mathrm{A}_{\mathrm{g}}\) ground state.

The Bixon-Jortner approach to radiationless transitions was sketched in a very simplified form in Example \(11.4 .\) The following is a slightly more elaborate version. Let \(\psi,\) an eigenstate of the system hamiltonian \(H^{(\mathrm{sys})}\) with eigenvalue \(E,\) be the state populated initially, and let \(\varphi_{n}\) an eigenstate of the bath hamiltonian \(H^{( \text {bath) } }\) with eigenvalue \(E_{n},\) be a state of the bath. Let \(\Psi=a \psi+\Sigma_{n} b_{n} \varphi_{n}\) be an eigenstate of the true hamiltonian \(H\) with energy \(\mathcal{E}\). Let \(\left\langle\psi | \varphi_{n}\right\rangle=0\) and \(H^{\prime}=H-H^{(\mathrm{sys})}-H^{(\text {bath })}\) have constant matrix elements \(\left\langle\varphi_{n}\left|H^{\prime}\right| \psi\right\rangle=V\) for all \(n .\) Show that \(H \Psi=\mathcal{E} \Psi\) leads to \(V a+\left(E_{n}-\mathcal{E}\right) b_{n}=0\) and \((E-\mathcal{E}) a+V \Sigma_{n} b_{n}=0 .\) Hence find an expression relating \(a\) and \(b_{n^{*}}\) Letting \(\mathcal{E}-E_{n}=(\gamma-n) \varepsilon\) and using \(\Sigma_{n=-\infty}^{\infty} 1 /(\gamma-n)=-\pi \cot \pi \gamma\) and \(\rho=1 / \varepsilon,\) show that \(E-\mathcal{E}-\pi \rho V^{2} \cot \pi \gamma=0,\) an equation for \(\mathcal{E} .\) Go on to show on the basis that \(a^{2}+\Sigma_{n} b_{n}^{2}=1,\) that \(a^{2}=V^{2} /\left\\{(E-\mathcal{E})^{2}+V^{2}\right.\) \(\left.+\left(\pi V^{2} \rho\right)^{2}\right\\}\)