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Chapter 5: Problem 6

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.

Short Answer

Expert verified
The irreducible representations spanned by the four 1s orbitals in Methane are obtained by analyzing the characters. After forming symmetry-adapted linear combinations for each representation, the representatives for \(C_{3}^{+}\) and \(S_{4}\) are verified to be in block-diagonal form by checking the division into sub-matrices along the diagonal. Finally, the projection operator confirms the block form of the representatives.

Step by step solution

01

Character Table of Methane

Start with representing the four 1s-orbitals of Methane (CH4) in a character table. The operations possible are E, 8C3, 3C2, \(6S_{4}\), and \(6d\). The orbitals can be represented as a linear combination of each other, forming four representations.
02

Irreducible Representations

By applying all the symmetry operations possible and inspecting the characters, one can find the irreducible representations of methane.
03

Symmetry-Adapted Linear Combinations

Form symmetry-adapted linear combinations (SALCs) for each of the four representations. Each SALC would be associated with one irreducible representation, and thus span it.
04

Verify Representatives for \(C_{3}^{+}\) and \(S_{4}\)

Now check if the representatives for \(C_{3}^{+}\) and \(S_{4}\) are in block-diagonal form. For each representative matrix, check if it can be divided into blocks (sub-matrices) along its diagonal such that all elements outside these blocks are zero.
05

Project Operator

The projection operator in equation 5.30 may now be used to find how much of each irreducible representation can be found in the original representation (made up of the four 1s-orbitals). This involves multiplication of characters of the \(C_{3}^{+}\) and \(S_{4}\) operations from each irreducible representation with the characters of these operations from the original representation, and summing for all operations, then dividing by the order of the group. Finally, one should check if the obtained coefficients match the size of the corresponding irreducible representations. If they do, the representatives for \(C_{3}^{+}\) and \(S_{4}\) are in block form and the answer is confirmed.

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Most popular questions from this chapter

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.

We have seen that the angular momentum commutation rules are generated by considering consecutive infinitesimal rotations about perpendicular axes in three-dimensional space. Could it be that the fundamental quantum mechanical commutation rule \(\left[x, p_{x}\right]=\mathrm{i} \hbar\) is also just a manifestation of the geometry of three-dimensional space? Present an argument that makes use of the angular momentum result and the definition of the angular momentum operators in terms of position and linear momentum operators that could be used to justify this supposition.

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.

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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