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Chapter 5: Problem 10

Classify the terms that may arise from the following configurations: (a) \(C_{2 v}: a_{1}^{2} b_{1}^{1} b_{2}^{1} ;\) (b) \(C_{3 v}: a_{2}^{1} e^{1}, e^{2} ;(c) T_{d}: a_{2}^{1} e^{1}\) \(\mathrm{e}^{1} \mathrm{t}_{1}^{1}, \mathrm{t}_{1}^{1} \mathrm{t}_{2}^{1}, \mathrm{t}_{1}^{2}, \mathrm{t}_{2}^{2} ;\) (d) \(\mathrm{O}: \mathrm{e}^{2}, \mathrm{e}^{1} \mathrm{t}_{1}^{1}, \mathrm{t}_{2}^{2} .\) Hint. Use the direct- product tables; triplet terms have antisymmetric spatial functions.

Short Answer

Expert verified
The terms are derived by direct multiplication of the given configurations under different groups, and reducing the result to a sum of normal modes (irreducible representations). These character tables along with the antisymmetric spatial function hint guide the triplet term classifications.

Step by step solution

01

Classification of Terms for \(C_{2v}\)

Starting with the configuration \(a_{1}^{2} b_{1}^{1} b_{2}^{1}\) under the point group \(C_{2v}\), we can use the direct-product tables for \(C_{2v}\) and multiply the representations for \(a_{1}\), \(b_{1}\), and \(b_{2}\) together. Do this by multiplying the characters under each class, and then reduce the product to a sum of irreducible representations.
02

Classification of Terms for \(C_{3v}\)

For the configuration \(a_{2}^{1} e^{1}, e^{2}\) under the point group \(C_{3v}\), we will multiply the representations for \(a_{2}\) and \(e\) together using the direct-product tables for \(C_{3v}\), and reduce the product to a sum of irreducible representations. This might result in multiple terms in the results, involving symmetric and antisymmetric combinations.
03

Classification of Terms for \(Td\)

For configurations \(a_{2}^{1} e^{1}\), \(e^{1} t_{1}^{1}\), \(t_{1}^{1} t_{2}^{1}\), \(t_{1}^{2}\), and \(t_{2}^{2}\) under point group \(Td\), use the same procedure i.e., use the direct-product tables for \(T_d\) to get the possible terms from these configurations by considering both direct multiplication and the combinations of the representations.
04

Classification of Terms for \(O\)

For configurations \(e^{2}\), \(e^{1} t_{1}^{1}\), and \(t_{2}^{2}\) of the point group \(O\), use the direct-product tables for \(O\) to find the possible terms. Taking the products of the irreducible representations will give terms in symmetry species of \(O\). Antisymmetrize the product if it belongs to a triplet term, following the provided hint.

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Most popular questions from this chapter

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.

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.

Show that in an octahedral array, hydrogen 1 s-orbitals \(\operatorname{span} \mathrm{A}_{1 \mathrm{g}}+\mathrm{E}_{\mathrm{g}}+\mathrm{T}_{1 \mathrm{u}}\) of the group \(\mathrm{O}_{\mathrm{h}}\)

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