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Chapter 5: Problem 15

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.

Short Answer

Expert verified
The atomic orbitals on the central Xe atom that can overlap with \(p_{1}\) to form molecular orbitals are \(p_x\), \(p_y\), \(d_{xy}\), and \(d_{x^2-y^2}\).

Step by step solution

01

Identify the Symmetry Operations

Start by identifying the symmetry operations that leave \(p_{1}\) unchanged under the point group \(D_{4}\). This molecule is four-fold rotationally symmetric (360/4=90 degrees rotation) along the z-axis and includes mirror planes sigma_v and sigma_d (vertical and diagonal).
02

Determine the Symmetry of \(p_{1}\)

To find which orbitals on the central atom \(p_{1}\) can overlap with, determine the symmetry of \(p_{1}\) under the point group \(D_{4}\). Considering the given molecular construct, \(p_{1}\) behaves as a \(d_{xz}\) atomic orbital. This means it will transform the same way as a \(d_{xz}\) orbital under \(D_{4}\) operations. Hence, \(p_{1}\) belongs to the \(E\) symmetry species in \(D_{4}\) point group.
03

Identify the Overlapping Orbitals

The central Xe atom s, p, and d atomic orbitals are going to have \(A_1\) (for s and \(d_{z^2}\)), \(E\) (for \(p_x, p_y, d_{xy}, d_{x^2-y^2}\)), and \(B_1\) (for \(d_{xz}, d_{yz}\)) symmetries. The orbital of the Xe atom overlapping with \(p_{1}\) should belong to the same symmetry species \(E\). So, \(p_x, p_y, d_{xy},\) and \(d_{x^2-y^2}\) atomic orbitals on the central Xe atom can overlap with \(p_{1}\) to form molecular orbitals.

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Most popular questions from this chapter

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

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.

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.

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

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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