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Chapter 25: Problem 9

The hydrogen atoms in a methane molecule \(\mathrm{CH}_{4}\) form a perfect tetrahedron with the carbon atom at its centre. The molecule is most conveniently described mathematically by placing the hydrogen atoms at the points \((1,1,1),(1,-1,-1)\), \((-1,1,-1)\) and \((-1,-1,1) .\) The symmetry group to which it belongs, the tetrahedral group \(\left(\overline{4} 3 m\right.\) or \(T_{d}\) ) has classes typified by \(I, 3,2_{z}, m_{d}\) and \(\overline{4}_{z}\), where the first three are as in exercise \(25.5, m_{d}\) is a reflection in the mirror plane \(x-y=0\) and \(\overline{4}_{2}\) is a rotation of \(\pi / 2\) about the \(z\)-axis followed by an inversion in the origin. \(A\) reflection in a mirror plane can be considered as a rotation of \(\pi\) about an axis perpendicular to the plane, followed by an inversion in the origin. The character table for the group \(43 m\) is very similar to that for the group 432, and has the form shown in table \(25.9 .\) By following the steps given below, determine how many different internal vibration frequencies the \(\mathrm{CH}_{4}\) molecule has. (a) Consider a representation based on the 12 coordinates \(x_{i}, y_{i}, z_{l}\) for \(i=\) \(1,2,3,4 .\) For those hydrogen atoms that transform into themselves, a rotation through an angle \(\theta\) about an axis parallel to one of the coordinate axes gives rise in the natural representation to the diagonal elements 1 for the corresponding coordinate and \(2 \cos \theta\) for the two orthogonal coordinates. If the rotation is followed by an inversion then these entries are multiplied by -1. Atoms not transforming into themselves give a zero diagonal contribution. Show that the characters of the natural representation are \(12,0,0,0,2\) and hence that its expression in terms of irreps is $$ \mathrm{A}_{1} \oplus \mathrm{E} \oplus \mathrm{T}_{1} \oplus 2 \mathrm{~T}_{2} $$ (b) The irreps of the bodily translational and rotational motions are included in this expression and need to be identified and removed. Show that when this is done it can be concluded that there are three different internal vibration frequencies in the \(\mathrm{CH}_{4}\) molecule. State their degeneracies and check that they are consistent with the expected number of normal coordinates needed to describe the internal motions of the molecule.

Short Answer

Expert verified
Methane \(CH_4\) has three distinct internal vibration frequencies: one \(\text{A}_1\) mode, one \(\text{E}\) mode (doubly degenerate), and one \(\text{T}_2\) mode (triply degenerate).

Step by step solution

01

Define the representation based on the 12 coordinates

Consider a representation based on the 12 coordinates given by the atoms' positions: \ (\(x_{i}, y_{i}, z_{i}\)) for \(i=1,2,3,4\). Each hydrogen atom has coordinates corresponding to \(x, y, z\) axes. For example, hydrogen at point \((1,1,1)\) has coordinates (1,1,1).
02

Identify transformation effects

Identify how the hydrogen atoms and their coordinates change under the group's symmetry transformations. For rotations about an axis parallel to one coordinate axis, the diagonal elements transform as 1 for the axis of rotation and \(2 \cos \theta\) for the other two coordinates. For example, a \(\pi/2\) rotation about the z-axis followed by inversion multiplies these entries by -1, resulting in -1 and -2.
03

Determine characters for natural representation

For each symmetry operation, sum up the contributions to the diagonal positions to get the characters of the natural representation. Atoms transforming into themselves would have their coordinates altered while others will contribute zeros. Calculating this provides characters: \(12,0,0,0,2\).
04

Express in terms of irreducible representations (irreps)

From the character table for group \(T_d\), decompose the natural representation's characters to find its irreducible components. Using the table, we get the decomposition: \ \( \text{A}_1 \oplus \text{E} \oplus \text{T}_1 \oplus 2\text{T}_2 \)
05

Identify and remove bodily motions

Bodily translations and rotations are included in the representation. Translational motions correspond to the irreducible representation \( \text{T}_2 \), and rotational motions correspond to \(\text{T}_1\). Removing these components leaves the internal vibrations as: \ \( \text{A}_1 \oplus \text{E} \oplus \text{T}_2 \ \).
06

Conclude internal vibration frequencies and validate

The remaining irreducible representations correspond to three different internal vibration frequencies. \( \text{A}_1 \) is non-degenerate (1 frequency), \( \text{E} \) is doubly degenerate (2 frequencies), \( \text{T}_2 \) is triply degenerate (3 frequencies). Hence, there are three different modes of vibration consistent with the (3N-6) vibrational modes expected for a methane molecule.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tetrahedral Group
The tetrahedral group, denoted as \( T_d \) or \( \bar{4} 3m \), describes the symmetries of a tetrahedron. In the context of a methane molecule (\(CH_4\)), this group captures all the symmetry operations that leave the tetrahedral shape of the molecule unchanged. These operations include:
  • Identity (\(I\))
  • 3-fold rotations (\( C_3 \))
  • 2-fold rotations (\( C_2 \))
  • Reflection planes (\( m \))
  • Rotations with inversion (\( \overline{4}_z \))

Understanding these symmetries is key to analyzing the molecular vibrations and other properties. The tetrahedral symmetry helps in simplifying the complexity of the molecule's geometry by categorizing movements that preserve its shape.
Irreducible Representations
Irreducible representations (irreps) are core to understanding molecular symmetry. Essentially, they are the fundamental building blocks that cannot be decomposed further. For the \( T_d \) group, irreps describe the symmetry of the molecule's normal modes of vibration or other properties. These normal modes form patterns of transformation under the geometric operations of the group.
The natural representation for \( CH_4 \) can be decomposed into the sum of these irreps. From our exercise, the character table method provided:
  • \( \text{A}_1 \)
  • \( \text{E} \)
  • \( \text{T}_1 \)
  • 2\( \text{T}_2 \)

We remove the irreps associated with translational and rotational motions, which simplifies the analysis and isolation of vibrational modes.
Vibrational Modes
Vibrational modes in molecules are specific patterns of atomic movements that occur when a molecule vibrates. For methane (\(CH_4\)), we look at internal movements of the atoms that lead to distinct vibrational frequencies.
By removing the contributions from translational and rotational movements, we focus on:
  • \( \text{A}_1 \) - Symmetric stretch (1 frequency, non-degenerate)
  • \( \text{E} \) - Twofold degenerate bending (2 frequencies)
  • \( \text{T}_2 \) - Triply degenerate asymmetric stretch (3 frequencies)

These modes correspond to different ways the hydrogen atoms move relative to the central carbon atom, whether in symmetrical stretching or more complex bending movements.
Character Table
A character table is a tabular representation of the characters for the irreducible representations of a group under each symmetry operation. Characters are traces of the matrices representing the symmetry operations in the group.
For the tetrahedral group \( T_d \), the character table helps in breaking down the natural representation of methane's vibrational modes into its irreducible components. Understanding the character table allows us to:
  • Identify the irreps present in various motions (vibrational, rotational, translational)
  • Associate these irreps with specific symmetry operations within the group

This decomposition is crucial to determine how different vibrations manifest and how certain frequencies arise due to the molecular structure's symmetry.
Symmetry Operations
Symmetry operations are transformations that leave the structure of a molecule unchanged. For \( CH_4 \), these operations include:
  • Identity (\( E \)): Does nothing to the molecule.
  • 3-fold rotations (\( C_3 \)): Rotate by 120° around an axis.
  • 2-fold rotations (\( C_2 \)): Rotate by 180° around an axis.
  • Mirror reflections (\( \text{σ} \)): Reflect through a mirror plane.
  • Improper rotations (\( \overline{4}_z \)): Rotation followed by inversion.

Each operation places constraints on the position and movement of the atoms. By understanding these operations, we can better explain how various modes of vibration relate to the overall molecular symmetry.

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Most popular questions from this chapter

A group \(\mathcal{G}\) has four elements \(I, X, Y\) and \(Z\), which satisfy \(X^{2}=Y^{2}=Z^{2}=\) \(X Y Z=I\). Show that \(\mathcal{G}\) is Abelian and hence deduce the form of its character table. Show that the matrices $$ \begin{aligned} &\mathrm{D}(I)=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right), & \mathrm{D}(X)=\left(\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right) \\ &\mathrm{D}(\boldsymbol{Y})=\left(\begin{array}{cc} -1 & -p \\ 0 & 1 \end{array}\right), & \mathrm{D}(Z)=\left(\begin{array}{cc} 1 & p \\ 0 & -1 \end{array}\right) \end{aligned} $$ where \(p\) is a real number, form a representation \(\mathrm{D}\) of \(\mathcal{G} .\) Find its characters and decompose it into irreps.

(a) By considering the possible forms of its cycle notation, determine the number of elements in each conjugacy class of the permutation group \(S_{4}\) and show that \(S_{4}\) has five irreps. Give the logical reasoning that shows they must consist of two three-dimensional, one two-dimensional, and two one- dimensional irreps. (b) By considering the odd and even permutations in the group \(S_{4}\) establish the characters for one of the one-dimensional irreps. (c) Form a natural matrix representation of \(4 \times 4\) matrices based on a set of objects \(\\{a, b, c, d\\}\), which may or may not be equal to each other, and, by selecting one example from each conjugacy class, show that this natural representation has characters \(4,2,1,0,0 .\) The one-dimensional vector subspace spanned by sets of the form \(\\{a, a, a, a\\}\) is invariant under the permutation group and hence transforms according to the invariant irrep \(\mathrm{A}_{1}\). The remaining three-dimensional subspace is irreducible; use this and the characters deduced above to establish the characters for one of the three-dimensional irreps, \(T_{1}\) (d) Complete the character table using orthogonality properties, and check the summation rule for each irrep. You should obtain table \(25.8 .\)

(a) The set of even permutations of four objects (a proper subgroup of \(S_{4}\) ) is known as the alternating group \(A_{4-}\) List its twelve members using cycle notation. (b) Assume that all permutations with the same cycle structure belong to the same conjugacy class. Show that this leads to a contradiction and hence demonstrates that even if two permutations have the same cycle structure they do not necessarily belong to the same class. (c) By evaluating the products \(p_{1}=(123)(4) \cdot(12)(34) \cdot(132)(4)\) and \(p_{2}=\) \((132)(4) \cdot(12)(34) \cdot(123)(4)\) deduce that the three elements of \(A_{4}\) with structure of the form (12)(34) belong to the same class. (d) By evaluating products of the form \((1 \alpha)(\beta \gamma) \bullet(123)(4) \bullet(1 \alpha)(\beta \gamma)\), where \(\alpha, \beta, \gamma\) are various combinations of \(2,3,4\), show that the class to which \((123)(4)\) belongs contains at least four members. Show the same for \((124)(3)\). (e) By combining results (b), (c) and (d) deduce that \(A_{4}\) has exactly four classes, and determine the dimensions of its irreps. (f) Using the orthogonality properties of characters and noting that elements of the form \((124)\) (3) have order 3, find the character table for \(A_{4}\)

Consider a regular hexagon orientated so that two of its vertices lie on the \(x\)-axis. Find matrix representations of a rotation \(R\) through \(\pi / 6\) and a reflection \(m_{y}\) in the \(y\)-axis by determining their effects on vectors lying in the \(x y\)-plane. Show that a reflection \(m_{x}\) in the \(x\)-axis can be written as \(m_{x}=m_{y} R^{3}\) and that the \((12)\) elements of the symmetry group of the hexagon are given by \(R^{n}\) or \(R^{n} m_{y} .\) Using the representations of \(R\) and \(m_{y}\) as generators, find a two- dimensional representation of the symmetry group, \(C_{6}\), of the regular hexagon. Is it a faithful representation?
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