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Chapter 11: Problem 8

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?

Short Answer

Expert verified
Both transitions, \(^1 T_{1g} \leftarrow ^1 A_{1g}\) and \(^1 T_{2g} \leftarrow ^1 A_{1g}\), in the \(\mathrm{Co}^{3+}\) complex are forbidden due to the Laporte rule. Intensity to these forbidden transitions can be provided through vibronic coupling, where vibrations have the same symmetry as the tensor product of the initial and final state representations, specifically, vibrations with the symmetry of \(^1 A_{1g}\). However, an admixture of configurations involving p-orbitals will not contribute to the intensities.

Step by step solution

01

Checking for Transition Permissibility

The permissibility of electronic transitions in coordination compounds is determined by the Laporte selection rule, which states that only g-->u or u-->g transitions are allowed. However, since the given complex is centrosymmetric (octahedral), all Laporte-allowed transitions are forbidden. For the transitions \(^1 T_{1g} \leftarrow ^1 A_{1g}\) and \(^1 T_{2g} \leftarrow ^1 A_{1g}\), both originate from the \(^1 A_{1g}\) ground state, which is a g state. As the excited states \((^1 T_{1g}\) and \(^1 T_{2g})\) are also g states, this violates the Laporte rule, causing both transitions to be forbidden.
02

Symmetry of Vibrations Influencing the Intensity

Forbidden transitions can gain intensity through vibronic coupling, where a vibrational motion of the same symmetry as the electronic transition can provide pathway for the transition. The type of vibrations which would provide intensity are those having the same symmetry as the tensor product of initial and final state representations. Hence, in this context, the initial and final states are both g, so the vibrational motion required for intensity would be with the symmetry of \(^1 A_{1g}\). This coincides with the three translational motions and three rotational motions.
03

Role of p-Orbital Configurations

In octahedral complexes, it is unlikely that an admixture of configurations involving p-orbitals would lead to magnitude intensities in these transitions. The electron would likely move from t2g to eg orbital, not p-orbitals, because Δoct is significantly larger than Δtet. This is because the Laporte rule allows transitions with a change in l=±1, but these are not considered here as the provided complex is octahedral in nature, which does not include p-orbitals.

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Most popular questions from this chapter

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

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.

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.

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.

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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