Question

\(\quad \mathrm{NH}_{3}\) belongs to the \(C_{3 \mathrm{v}}\) group. The reducible representation for the vibrational modes is \(\Gamma_{\text {reducible}}=2 A_{1}+2 E\) a. How many vibrational modes does \(\mathrm{NH}_{3}\) have? b. How many of these modes are infrared active and to which representation do they belong? c. Are any of the infrared active modes degenerate in energy? d. How many of these modes are Raman active and to which representation do they belong? e. Are any of the Raman active modes degenerate in energy? f. How many modes are both infrared and Raman active?

Short answer

a. NH3 has 6 vibrational modes. b. There are 2 infrared-active modes, both belonging to the \(A_{1}\) representation. c. There is no degeneracy in the infrared active modes. d. There are 4 Raman active modes, 2 belonging to the \(A_{1}\) representation and 2 belonging to the \(E\) representation. e. The Raman active modes are degenerate; there are 2 degenerate Raman active modes. f. There are 2 modes that are both infrared and Raman active.

Step-by-step solution

a. Determining the number of vibrational modes

To find the number of vibrational modes of NH3, recall that a molecule with N atoms has 3N degrees of freedom. Since NH3 has 4 atoms, it has 3 * 4 = 12 degrees of freedom. Of these 12, there are 3 translational and 3 rotational degrees of freedom, leaving us with a total of 12 - 3 - 3 = 6 vibrational modes.

b. Infrared active modes and their representation

Infrared active modes correspond to changes in the dipole moment of the molecule. The reducible representation given is \(\Gamma_{\text {reducible}}=2 A_{1}+2 E\), which corresponds to the vibrational modes. Since molecules belonging to the \(C_{3v}\) point group have a dipole moment along the z-direction, we can determine which representations are infrared-active by looking at their transformation properties. As \(A_{1}\) representation is symmetric with respect to z, it is infrared-active. Hence, there are 2 infrared-active modes, both belonging to the \(A_{1}\) representation.

c. Degeneracy in infrared active modes

There is no degeneracy in the infrared active modes because they both belong to the \(A_{1}\) representation, which is not degenerate itself.

d. Raman active modes and their representation

Raman activity is determined by the polarizability tensor of the molecule. All representations present in the reducible representation for the vibrational modes are potentially Raman active. Therefore, there are 4 Raman active modes in NH3, with 2 of them belonging to the \(A_{1}\) representation and 2 belonging to the \(E\) representation.

e. Degeneracy in Raman active modes

The Raman active modes are degenerate, as there are 2 modes each belonging to the \(A_1\) and \(E\) representations. The \(E\) representation is doubly degenerate, so the total number of degenerate Raman active modes is 2.

f. Modes that are both infrared and Raman active

Since both of the infrared-active vibrational modes belong to the \(A_{1}\) representation and there are 2 Raman-active modes belonging to the \(A_{1}\) representation as well, we can deduce that there are 2 modes that are both infrared and Raman active.

Key concepts

Group Theory in Chemistry

Group Theory is an area of abstract mathematics used to study symmetry in molecules. In chemistry, it provides a system to categorize molecules according to their symmetry properties, described by operations like rotation and reflection. These operations can be applied to a molecule to generate a set of symmetrical configurations, which are collectively called a point group.

For NH3, Group Theory outlines that these configurations belong to the point group labeled as the C3v group. This classification helps chemists predict the symmetry of vibrational modes, such as whether they will be infrared or Raman active, based on the multiplication of these symmetry operations. Understanding these concepts is essential for analyzing vibrational spectra and for predicting the physical properties of molecules.

Infrared Spectroscopy

Infrared spectroscopy is an analytical technique used to identify and study chemicals through their interaction with infrared radiation. When a molecule absorbs infrared light, it undergoes molecular vibrations. These vibrations can be symmetrical or asymmetrical stretching and bending of the molecule's bonds.

The transition to excited vibrational states happens when the energy of the infrared light matches the energy required to change the vibrational quantum states. Only vibrational modes that result in a change in the dipole moment of the molecule are infrared active. In the case of NH3, the vibration modes that belong to the A1 representation are both infrared active because they involve such dipole moment changes.

Raman Spectroscopy

Raman spectroscopy is another vibrational spectroscopic technique like infrared, but it measures the scattering of monochromatic light, usually from a laser in the visible spectrum. During Raman scattering, photons are absorbed and immediately re-emitted. Most photons scatter without changing their energy, a process called Rayleigh scattering. However, a small fraction scatter with a change in energy, known as Raman scattering.

This energy change occurs due to the molecule's vibrational or rotational transitions. Unlike infrared spectroscopy, Raman activity does not require a change in the dipole moment; instead, it depends on the polarizability of the molecule. For NH3, the 2 A1 and 2 E modes are Raman active, with the E modes being degenerate and causing changes in the polarizability along two equivalent directions.

Molecular Vibrations

Molecular vibrations are the movements of atoms within a molecule that create different energy states. This concept is vital for understanding spectroscopic properties. There are different types of vibrational motions, including stretching (symmetrical and asymmetrical) and bending (in-plane and out-of-plane).

The NH3 molecule, being a triatomic species, demonstrates these motions. Each mode of vibration corresponds to a specific type of energy change that can be probed by spectroscopic techniques. Through infrared and Raman spectroscopy, we can understand these vibrations and determine the vibrational modes active in each spectral technique.

Symmetry Operations

Symmetry operations are movements that place molecules into a congruent position. They are fundamental to Group Theory and include rotations around an axis (referred to as Cn), reflections across a plane (σ), inversions through a point (i), and improper rotations (Sn).

Conducting a symmetry operation on NH3, such as a rotation around its trigonal axis, helps in understanding its molecular symmetry. These operations do not alter the molecule's position in space and form the basis of a molecule's point group classification, importantly affecting their spectroscopic behaviors.

Point Groups

Point groups are categories in Group Theory that represent sets of symmetry operations which define the overall symmetry of the molecule. They are useful for predicting physical and chemical properties and are imperative while analyzing spectroscopic data.

The NH3 molecule falls into the C3v point group, characterized by having a vertical axis of symmetry (C3) and three vertical planes of symmetry (σv). This point group classification can predict that there will be non-degenerate and doubly degenerate vibrational modes, which helps in understanding the vibrational spectra of the molecule.