Question

Ammonia provides a particularly simple example of the dependence of vibrational frequencies on the atomic masses and of the use of vibrational frequencies to distinguish between a stable molecule and a transition state. First examine the vibrational spectrum of pyramidal ammonia ("ammonia" on the precalculated Spartan file). a. How many vibrational frequencies are there? How does this number relate to the number of atoms? Are all fre- quencies real numbers or are one or more imaginary numbers? Describe the motion associated with each frequency and characterize each as being primarily bond stretching, angle bending, or a combination of the two. Is bond stretching or angle bending easier? Do the stretching motions each involve a single \(\mathrm{NH}\) bond or do they involve combinations of two or three bonds? b. Next, consider changes to the vibrational frequencies of ammonia as a result of substituting deuteriums for hydrogens ("perdeuteroammonia" on the precalculated Spartan file \() .\) Are the frequencies in \(\mathrm{ND}_{3}\) larger, smaller, or unchanged from those in \(\mathrm{NH}_{3}\) ? Are any changes greater for motions that are primarily bond stretching or motions that are primarily angle bending? c. Finally, examine the vibrational spectrum of an ammonia molecule that has been constrained to a planar geometry ("planar ammonia"' on the Spartan download). Are all the frequencies real numbers? If not, describe the motions associated with any imaginary frequencies and relate them to the corresponding motion(s) in the pyramidal equilibrium form.

Short answer

In summary, pyramidal ammonia has 6 vibrational frequencies, all of which are real numbers, with some involving bond stretching and angle bending. Angle bending is generally easier in this molecule. When comparing perdeuteroammonia (ND3) with ammonia (NH3), the vibrational frequencies of ND3 are smaller due to the larger atomic mass of deuterium, especially in bond stretching motions. For planar ammonia, there is at least one imaginary frequency associated with the unstable planar geometry, corresponding to the nitrogen atom moving between planes of hydrogen atoms, and eventually resulting in the restoration of the pyramidal structure.

Step-by-step solution

Determine Number of Vibrational Frequencies

According to vibrational analysis, the number of vibrational frequencies for a nonlinear molecule is given by \(3N-6\), where N is the number of atoms. In the case of NH3, N = 4 (1 nitrogen atom and 3 hydrogen atoms), so the number of vibrational frequencies = \(3 \cdot 4 - 6\) = 6.

Analyzing Real/Imaginary Frequencies

All expected vibrational frequencies of pyramidal ammonia should be real numbers since it is a stable molecular configuration.

Describing and Characterizing Motions

The motions of these vibrational frequencies can be described and characterized as follows (this might differ slightly): 1. Frequency 1 - Bond bending: Two N-H bonds bending in and out. 2. Frequency 2 - Bond bending: Three N-H bonds bending in and out simultaneously. 3. Frequency 3 - Bond stretching: All three N-H bonds stretching together symmetrically. 4. Frequency 4 - Bond stretching: Two N-H bonds stretching and the third one contracting. 5. Frequency 5 - Bond stretching: One N-H bond stretching and the other two contracting. 6. Frequency 6 - Mixture of both (involving both bond stretching and angular bending). Angle bending in pyramidal ammonia is generally easier than bond stretching, as bending vibrations typically have lower energy. b.

Comparing Vibrational Frequencies

The vibrational frequencies of ND3 are smaller compared to those of NH3 since the masses of deuterium atoms are larger. This has a stronger effect on the frequencies related to bond stretching motions since these motions are more sensitive to changes in the atomic mass than angular bending motions. c.

Real/Imaginary Frequencies for Planar Ammonia

Planar ammonia is a vibrationally unstable (transition-state) structure, so it is expected to have at least one imaginary frequency. This indicates that the molecule tends to undergo motion in the direction associated with the imaginary frequency, leading to the more stable pyramidal geometry.

Motions Associated with the Imaginary Frequencies

The imaginary frequency in planar ammonia often corresponds to motion of the nitrogen atom switching between one plane of hydrogen atoms to another, which relates to the angular bending and restoring of the molecular structure to the pyramidal equilibrium form.

Key concepts

Vibrational Frequencies

Vibrational frequencies of a molecule help us understand its dynamic movements. For a nonlinear molecule like ammonia (NH_3), the number of vibrational frequencies is calculated using the formula \(3N-6\), where \(N\) is the number of atoms. In NH_3, \(N = 4\) (1 nitrogen and 3 hydrogen atoms), resulting in six vibrational frequencies. These frequencies are incredibly valuable as they demonstrate how different atoms in the molecule move, whether through stretching or bending.

• Frequencies involving bond bending often feature atoms moving in and out in a synchronized way, affecting the bond angles.
• Bond stretching frequencies involve changes in distance between bonded atoms, reflecting the bond length's slight expansion and contraction.

The NH_3 molecule is stable in its pyramidal form, thus all vibrational frequencies are real numbers. Any presence of imaginary frequencies would indicate a transition state, implying an unstable configuration.

Molecular Geometry

The molecular geometry of ammonia is essential when discussing its vibrational characteristics. In its equilibrium state, ammonia ( NH_3 ) has a pyramidal shape with a nitrogen atom sitting above the base formed by three hydrogen atoms. This structure leads to specific vibrational movements:

• Bond bending occurs as the nitrogen atom shifts slightly, altering the angles between the nitrogen and hydrogen atoms.
• Bond stretching happens when the bonds between nitrogen and each hydrogen atom stretch or compress.

When ammonia is forced into a planar geometry, it results in a less stable structure, evident by the occurrence of imaginary vibrational frequencies. This geometry has the nitrogen atom in the same plane as the hydrogen atoms, which is a transition state rather than the stable pyramidal form. Imaginary frequencies in this state denote instability, showing how the molecule naturally reverts to the pyramidal configuration.

Isotopic Substitution

Isotopic substitution involves replacing an atom in a molecule with another isotope, such as substituting hydrogen with deuterium in ammonia to form perdeuteroammonia ( ND_3 ). This substitution has a notable effect on vibrational frequencies:

• The larger mass of deuterium compared to hydrogen reduces vibrational frequencies, especially those related to bond stretching.
• Frequencies associated with angle bending are slightly less affected, as bending motions are generally less mass-dependent.

This change is crucial because it shows how vibrational spectra can be fine-tuned by the molecular mass, providing insights into molecular dynamics and stability. In essence, the heavier isotope adjusts the vibrational motion, typically slowing it down due to the increased mass.

Bond Stretching and Angle Bending

Vibrational motions in molecules involve both bond stretching and angle bending, each playing a key role in defining the vibrational spectrum. In ammonia, these motions represent:

• *Bond Stretching*: This occurs when the distance between nitrogen and hydrogen atoms in NH_3 alters. Stretching can involve one bond changing while the others contract or multiple bonds adjusting simultaneously.
• *Angle Bending*: This motion represents changes in the NH bond angles. The nitrogen atom may move closer or farther from the plane of hydrogen atoms, altering the bond angles and leading to angle bending vibrations.

Generally, angle bending requires less energy than bond stretching, making bending vibrations occur at lower frequencies. Understanding these motions helps to analyze the stability and dynamic behavior of the molecule, offering insights into molecular vibrations and their impact on molecular properties.