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Chapter 11: Problem 4

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.

Short Answer

Expert verified
The relative intensity of the 0-1 transition as a function of \(\Delta R\) is given by \(I_{01(\Delta R)} = | S_{01(\Delta R)} |^2\), where \(S_{01(\Delta R)}\) is the overlap integral which needs to be evaluated numerically using mathematical software.

Step by step solution

01

Understanding the Problem

In this case, there are two electronic states with the same force constant but different equilibrium bond lengths. The problem is asking for an expression for the relative intensity of the 0-1 transition as a function of \(\Delta R\), where \(v = 1\) is the upper vibrational state.
02

Applying the Franck-Condon Principle

According to Franck-Condon principle, during a vertical transition, there is no change in the nuclear position. So, it can be inferred that the intensity of a transition is proportional to the Franck-Condon factor, which is the square of the overlap integral between the vibrational wavefunctions of the initial and final electronic states.
03

Define the Wavefunctions

The wavefunction for the vibrational ground state of the initial electronic state can be defined as \(\Psi_0 (R)\). Similarly, the wavefunction for the vibrational excited state of the final electronic state can be defined as \(\Psi_1 (R - \Delta R)\), where \(\Delta R\) denotes the change in equilibrium bond lengths of the two states. These equations play a fundamental role in solving this problem.
04

Calculate Overlap Integral

The overlap integral can be calculated by multiplying the two wave functions and integrating over all possible space. Thus, the overlap integral \(S_{01(\Delta R)}\) is: \[ S_{01(\Delta R)} = \int \Psi_0(R) \Psi_1(R - \Delta R) dR \]
05

Find the Relative Intensity

The relative intensity of the 0-1 transition can be obtained by squaring the overlap integral. Thus, the relative intensity \(I_{01(\Delta R)}\) is: \[ I_{01(\Delta R)} = | S_{01(\Delta R)} |^2 \] This module generally involves numerical computation, which could be done using any mathematical software.

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Most popular questions from this chapter

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.

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 \(a^{2}=V^{2} /\left\\{(E-\mathcal{E})^{2}+V^{2}\right.\) \(\left.+\left(\pi V^{2} \rho\right)^{2}\right\\}\)

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?

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?
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