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Introduction to molecular vibrations

Authors: Stéphane Nilsson and Giovanni Pizzi

Original notebook: Molecule_Vibration.ipynb

With this notebook, the concept of molecular vibration is explored and visualized in an interactive fashion.

Goals

• Learn about the nature of vibrational modes within molecules
• Investigate the different frequencies and oscillation patterns of these modes.
• Explore the variety of molecular vibrations and how they arise from the molecular topology.

Background theory

More on the background theory.

Tasks and exercises

1. How many vibrational modes are expected for O$_2$ ? How many for H$_2$O?

   Solution

   The total degrees of freedom in a molecule is given by $3N=N_{\text{translation}} + N_{\text{rotation}} + N_{\text{vibration}}$, where N is the number of atoms. $N_{\text{translation}}=3$ for the x,y,z directions. $N_{\text{rotation}}=3$ for a non-linear molecule and $N_{\text{rotation}}=2$ for a linear molecule, since the rotation along the molecule axis does not change the atoms positions. Therefore, $N_{\text{vibration}}=N-6$ for a non-linear molecule (H$_2$O) and $N_{\text{vibration}}=N-5$ for a linear one (O$_2$).

2. Compare O$_2$ and OH, what do you observe regarding oscillation amplitudes?

   Solution

   In a diatomic molecule, the relative amplitudes are given by $A_1=-\frac{M_2}{M_1}A_2$ (c.f. theory). Therefore, since hydrogen is a much lighter element than oxygen, its amplitude is much greater. This can be understood intuitively: as the hydrogen atom has a much smaller moment of inertia than oxygen, when given the same energy, the hydrogen atom will oscillate more easily.

3. Compare H$_2$O and CO$_2$, how many vibrational modes does each one have? Are all CO$_2$ vibrations distinct? Can you explain the difference in energy between vibrational modes?

   Solution

   Since CO2 is a linear molecule, it has one more vibrational mode. However, two vibrations are identical, as they are in two different planes, both containing the molecule.

   There exist different vibrational modes, such as : symmetric stretch, asymmetric stretch or bending. In bending, only the bond angle changes. In stretching, the bond distance is varied, and therefore, compared to bending, it takes more force/energy to move the atom. Bending modes are therefore the lowest in energy, whereas symmetrical stretching modes will have the highest energy.

4. Compute the conversion factor between energy in eV and frequency in cm$^{-1}$.

   Solution

   The relation between frequency and energy is given by $E=h\nu$, with $\nu$ the frequency. Here, $[h]=[m^2\cdot kg\cdot s^{-1}]$ and $\lambda=\frac{c}{\nu}$, with $c$ the speed of light and $\lambda$ the wavelength, we have :

   $$\begin{aligned} [eV]&=6.63\cdot10^{-34}[m^2\cdot kg\cdot s^{-1}][Hz]\\ 1.6\cdot10^{-19}[J]&=6.63\cdot10^{-34}[m^2\cdot kg\cdot s^{-1}]3\cdot10^8[m\cdot s^{-1}][m^{-1}]\\ 1.6\cdot10^{-19}[m^2\cdot kg\cdot s^{-2}]&=6.63\cdot10^{-34}\cdot3\cdot10^8[m^2\cdot kg\cdot s^{-1}][m\cdot s^{-1}]10^2[cm^{-1}] \end{aligned}$$

   which gives in the end the conversion factor of 1[eV]=8.056$\cdot10^3$ [cm$^{-1}$]

5. Compare N$_2$ and O$_2$. What does the energy/frequency tells you about bond strength?

   Solution

   The N$_2$ molecule has an higher energy mode than O$_2$. Since the vibrational mode energy is proportional to $\sqrt{k}$, it means that the N$_2$ molecule has a stronger bond. Indeed, N$_2$ has 3 bonds, where O$_2$ only has 2.

Interactive visualization

How to use the interactive visualization

Molecule viewer

The view shows an interactive animation of a molecular vibration.

• Use the Play / Pause button to start and stop the animation (it also stops while the view is off-screen).

• Rotate the surface by left clicking and dragging.

• Translate the surface by right clicking and dragging.

You can reset the view by clicking the corresponding ‘camera axis’ buttons.

Molecule

Above the view are parameters related to the molecular vibrations.

Select the molecule of interest by clicking on the dropdown. An extended version of the list can be accessed by checking “Extended list”.

Slide the temperature slider to give more energy to the vibrational modes.

Different vibrational modes can be selected using the “Vibrational mode” slider.

On the right, a summary of the energy and frequency of the selected vibration is shown.

Appearance

Under the view are parameters related to showing arrows along the oscillations or modifying the atoms appearance.

The “Aspect ratio” slider modifies the ratio between bond diameter and atomic radius.

The “Atoms label” check box toggles every atom label.