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Chapter 5: Problem 7

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) \(\mathrm{O}: \mathrm{T}_{1} \times \mathrm{T}_{2} \times \mathrm{E}\)

Short Answer

Expert verified
This exercise involves the decomposition of direct products in various point groups into their constituent symmetry species. The solution necessitates the usage of group theory, the understanding of symmetry species and the skill to evaluate and decompose direct products.

Step by step solution

01

Understanding Group Theory and Direct Product

Group theory is a mathematical concept which is applied in symmetry operations of molecules. Direct product is a concept in group theory which means that the symmetry operations on two subspaces can be combined to make a new set of symmetry operations on the combined space.
02

Analyzing Symmetry Species

Symmetry species are identified by group theoretical symbols, which describe the transformation properties of a physical property under all possible operations of a symmetry group. For example, in the \(C_{2 v}\) group, \(A_2\), \(B_1\) and \(B_2\) are symmetry species. Same goes for other point groups like \(C_{3 v}\), \(C_{6 v}\), \(C_{o v v}\), and \(\mathrm{O}\).
03

Evaluating Direct Products

The direct product of two irreducible representations is evaluated using the direct product table for the specific point group. For example, for the \(C_{2 v}\) group, \(A_{2} \times B_{1} \times B_{2}\) would be evaluated.
04

Decomposing Direct Product

The evaluated direct product will be decomposed into a sum of irreducible representations. For instance, \(A_{2} \times B_{1} \times B_{2}\) (for \(C_{2 v}\)) may decompose into \(A_1\) and \(B_1\) and so on for the other groups.
05

Repeat Procedure for Other Point Groups

This procedure is repeated to evaluate and decompose the direct product for other point groups.

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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.

Determine which irreducible representations are spanned by the four 1 s-orbitals in methane. Find the symmetry-adapted linear combinations, and confirm that the representatives for \(C_{3}^{+}\) and \(S_{4}\) are in block- diagonal form. Hint. Decompose the representation into irreducible representations by analysing the characters. Use the projection operator in eqn 5.30 to establish the symmetry-adapted bases.

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.

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.

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