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Chapter 5: Problem 16

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

Short Answer

Expert verified
The symmetry species of the f-orbitals under \(C_{3 v}\) and \(T_{d}\) symmetries can be identified by constructing transformation matrices according to the symmetry operations in each point group, and then using these matrices to determine the irreducible representation for each orbital, which labels the symmetry species.

Step by step solution

01

Remember the shapes of the f-orbitals

The five 'd' orbitals split into two groups in a ligand field, three 't' orbitals with one orientation, and two 'e' with other orientation. A similar concept applies to 'f' orbitals, which splits into three groups under octahedral field, a group of 'a' orbitals, a group of 't' orbitals and another 't' orbital. Analyzing their shapes, we will have: \[yz(x^2-y^2), zx(y^2-z^2), xy(z^2-x^2)\] for the 't' orbital and \[z(x^2-3y^2), y(3x^2-y^2)\] for 'e' orbital.
02

Compose Transformation Matrices for \(C_{3 v}\) Symmetry

For the \(C_{3 v}\) point group, orbital transformation matrices can be defined according to the symmetry elements of the group. These are: identity (E), two \(C_{3}^{2}\) operations and three \(v\) planes. Analyzing these symmetries and applying them to X, Y and Z will let us know how the orbitals transform. In the end, the character table under \(C_{3 v}\) symmetry for each f-orbital can be written out in order to determine the irreducible representation (or symmetry species).
03

Compose Transformation Matrices for \(T_{d}\) Symmetry

Same as before, but now considering the \(T_{d}\) point group symmetry. The main operations (beyond identity) to consider fora \(T_{d}\) point group symmetry are: 8 \(C_{3}\) rotations (with different axes), three \(C_{2}\) rotations and 6 \(S_{4}\) operations. Similar to step 2, for each f-orbital, we construct the transformation matrix according to these symmetry operations, which are then used to compute the character of the orbital under each operation. The resulting characters will lead us to the irreducible representations of the f-orbitals in the \(T_{d}\) symmetry point group.
04

Symmetry Species of the f-orbitals

After determining the irreducible representation for each f-orbital, for \(C_{3 v}\) and \(T_{d}\) symmetries, we can identify the symmetry species for them. The symmetry species is just the label of the irreducible representation in terms of the point group.

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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.

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.

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.

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.
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