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Chapter 5: Problem 3

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 \(\mathrm{O}_{\mathrm{h}}\)

Short Answer

Expert verified
In an octahedral array with a total of six 1s hydrogen orbitals, we have three types of representations, namely A1g, Eg, and T1u, under the Oh point group. A1g representation is formed by the symmetric combination of all six hydrogen orbitals. Two degenerate Eg states are formed by combining orbitals in such a way that these resultants are represented under rotation about the z-axis. Finally, T1u is the triply degenerate set where combinations of orbitals are taken in such a manner that they transform as x, y, and z under symmetry operations.

Step by step solution

01

- Understanding the Octahedral Array

An octahedral array means six atoms are attached to a central atom in such a way as to form an octahedron. Consider six hydrogen 1s orbitals, centered on the corners of an octahedron, with the origin at the center of the octahedron. The orientation of the axes is traditional: z-axis up and down, y-axis out of the plane of the paper, x-axis right and left.
02

- A1g Representation

A1g representation is known as the symmetric identity. If we take a linear combination of all six hydrogen orbitals, positive or negative, we see the formation of A1g representation. It remains mostly unchanged under the operations of Oh symmetry point group. Mathematically this is represented as \( \phi_{A_1g} = \phi_1+\phi_2+\phi_3+\phi_4+\phi_5+\phi_6 \).
03

- Eg Representation

Eg representation generates doubly degenerate states. Two combinations of orbitals can be formed which remain unchanged when we rotate around z-axis by 120 degrees. Mathematically, these states can be represented as, \( \phi_{E_{g1}} = \phi_1 + \phi_3 - \phi_2 - \phi_4 \) and \( \phi_{E_{g2}} = 2\phi_5 - \phi_6 \). The subscript represents the different combinations in the same Eg representation.
04

- T1u Representation

Eg representation generates triply degenerate states. Three combinations of 1s orbitals are taken in such a way that they transform as x, y and z. That is, these combinations are the basis under rotations and reflections in 3D. Mathematically, this can be represented as, \( \phi_{T_{1u1}} = \phi_1 - \phi_2 \), \( \phi_{T_{1u2}} = \phi_3 - \phi_4 \) and \( \phi_{T_{1u3}} = \phi_5 - \phi_6 \). The subscript refers to these three different combinations in the same T1u representation.

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Most popular questions from this chapter

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.

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.

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.

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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