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.

Short Answer

Expert verified
The group \(T_d\) is represented by 24 matrices obtained by permuting the vertices of a tetrahedron corresponding to the symmetry operations of the group (one identity matrix, eight 120º rotation matrices, fifteen 180º rotation matrices).

Step by step solution

01

Define the basis

Choose 1s-orbitals at the vertices of a regular tetrahedron as a basis. Label them as \(|1\rangle\), \(|2\rangle\), \(|3\rangle\), \(|4\rangle\) corresponding to different vertices of the tetrahedron.
02

Find the Transformation

Each symmetry operation of the \(T_d\) group will map the tetrahedron onto itself, resulting in some permutation of the four vertices. The permutation of the basis vectors is represented by 4x4 permutation matrices.
03

Identify Elements of Group

There are three kinds of operations that can be performed on the tetrahedron: identity, permutation of three vertices, permutation of two pairs of vertices. The identity operation leaves the vertices unchanged. Permutation of three vertices corresponds to rotating the tetrahedron by \(120^\circ\) about an axis passing through one vertex and the center of the opposite face, while permutation of two pairs of vertices corresponds to rotating the tetrahedron by \(180^\circ\) about an axis passing through the mid-points of two opposite edges.
04

Find Identity Matrix

The identity operation leaves all vertices unchanged, represented by the matrix: \[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]\].
05

Find Matrices for 120º Rotation

Performing a clockwise 120º rotation of vertices 2, 3, 4 while keeping 1 stationary is represented by the matrix: \[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]\]. There are similar matrices for clockwise and anticlockwise rotations about the axes through the other vertices. Altogether, there are eight 120º rotation matrices.
06

Find Matrices for 180º Rotation

A 180º rotation of the tetrahedron about an axis between vertex 1 and 2 swaps vertices 1 and 2, and 3 and 4, represented by the matrix: \[[0,1,0,0], [1,0,0,0], [0,0,0,1], [0,0,1,0]\]. Similar matrices can be found for the other axes. Altogether, there are fifteen 180º rotation matrices.
07

Combined 24 Matrices

Altogether, the group \(T_d\) is represented by the 24 matrices: one identity matrix, eight 120º rotation matrices, and fifteen 180º rotation matrices.

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Most popular questions from this chapter

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.

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

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.

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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