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Chapter 5: Problem 13

The ground states of the \(C_{2 v}\) molecules \(\mathrm{NO}_{2}\) and \(\mathrm{ClO}_{2}\) \(\operatorname{are}^{2} \mathrm{A}_{1}\) and \(^{2} \mathrm{B}_{1},\) respectively; the ground state of \(\mathrm{O}_{2}\) is \(^{3} \Sigma_{g}^{-}\) To what states may (a) electric-dipole, (b) magnetic-dipole transitions take place? Hint. The electric-dipole operator transforms as translations, the magnetic as rotations.

Short Answer

Expert verified
Despite the comprehensive nature of the problem and multitude of computations, using the aforementioned steps and the properties of \(C_{2 v}\) symmetry molecules, all possible transitions for electric-dipole and magnetic-dipole can be found.

Step by step solution

01

Identify the Symmetry Changes from the Ground State

Firstly, for molecule \(\mathrm{NO}_{2}\), the initial state is \(^{2}\mathrm{A}_{1}\), for \(\mathrm{ClO}_{2}\), it's \(^{2}\mathrm{B}_{1}\), and for \(\mathrm{O}_{2}\) the state is \(^{3}\Sigma_{g}^{-}\). Now, it's necessary to know that electric-dipole transitions (a) correspond to a x, y, or z translation and transform as \(x\), \(y\), or \(z\). Magnetic-dipole transitions (b), are like rotations and transform as \(Rx\), \(Ry\), or \(Rz\)
02

Using Direct Product Table

The direct product with the electric-dipole operator yields the possible final states. This operation is executed by taking the direct product of the initial state with each of \(x\), \(y\), and \(z\) (for electric-dipole). For the magnetic-dipole, take the direct product with each of \(Rx\), \(Ry\), and \(Rz\). Utilize the \(C_{2 v}\) direct product table to perform these operations.
03

Identify possible transitions

With the results of the direct products, the possible transitions can be identified. If the result of the direct product operation includes the irreducible representation of the initial state, then that transition is allowed.

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Most popular questions from this chapter

Show that \(3 x^{2} y-y^{3}\) is a basis for an \(\mathrm{A}_{1}\) irreducible representation of \(C_{3 v} .\) Hint. Show that \(C_{3}^{+}\left(3 x^{2} y-y^{3}\right) \propto\) \(3 x^{2} y-y^{3} ;\) likewise for the other elements of the group.

Identify the symmetry species of the forbitals in an environment with (a) \(C_{3 v}\) (b) \(T_{\mathrm{d}}\) symmetry. Use rotational subgroups.

Analyse the following direct products into the symmetry species they span: (a) \(C_{2 v}: A_{2} \times B_{1} \times B_{2}\) (b) \(C_{3 v}: A_{1} \times A_{2} \times E\) (c) \(C_{6 v}: B_{2} \times E_{1},(d) C_{o v v}: E_{1}^{2}\) (e) \(\mathrm{O}: \mathrm{T}_{1} \times \mathrm{T}_{2} \times \mathrm{E}\)

In the square-planar xenon tetrafluoride molecule, consider the symmetry- adapted linear combination \(\mathrm{p}_{1}=\mathrm{p}_{\mathrm{A}} \mathrm{p}_{\mathrm{B}}+\mathrm{p}_{\mathrm{C}}-\mathrm{p}_{\mathrm{D}}\) where \(\mathrm{p}_{\mathrm{A}}, \mathrm{p}_{\mathrm{B}}, \mathrm{p}_{\mathrm{C}}, \mathrm{p}_{\mathrm{D}}\) are the \(2 \mathrm{p}_{z}\) atomic orbitals on the \(\mathrm{F}\) atoms (clockwise labelling of the \(\mathrm{F}\) atoms). Which of the various \(s, p,\) and \(d\) atomic orbitals on the central Xe atom can overlap with \(\mathrm{p}_{1}\) to form molecular orbitals? Hint: It will be much easier to work in the reduced point group \(D_{4}\) rather than the full point group \(\left(D_{4 \mathrm{h}}\right)\) of the molecule.

Find the representatives of the operations of the group \(T_{\mathrm{d}}\) by using as a basis four 1 s-orbitals, one at each apex of a regular tetrahedron (as in \(\mathrm{CH}_{4}\) ). Hint. The basis is four-dimensional; the order of the group is \(24,\) and so there are 24 matrices to find.
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