Normal Modes

About

Normal modes in the context of molecular vibrations refer to the specific, independent patterns in which the atoms of a molecule move during vibrational motion. Each normal mode corresponds to a particular type of collective motion involving the atoms in the molecule. Understanding normal modes is crucial for analyzing molecular vibrations, infrared (IR) spectroscopy, and Raman spectroscopy.

1. Molecular Vibrations and Normal Modes

In a molecule, the atoms are constantly in motion, even at absolute zero, due to quantum mechanical effects. These vibrations can be complex because all the atoms in a molecule are connected and can influence each other's movements. However, these complex vibrations can be broken down into simpler, independent motions known as normal modes.

• Normal modes are the simplest possible vibrational motions of a molecule, where all parts of the molecule move with the same frequency and in a coordinated fashion.

• Each normal mode is characterized by a specific frequency (or energy) and a pattern of atomic displacements.

2. Number of Normal Modes

The number of normal modes a molecule has depends on the number of atoms $N$ in the molecule:

• Nonlinear molecules: A molecule with $N$ atoms has $3N - 6$ normal modes. These modes correspond to the possible independent vibrational movements of the molecule, excluding translational and rotational motions.

• Linear molecules: A linear molecule has $3N - 5$ normal modes, as one less vibrational mode is needed due to the linear geometry of the molecule.

For example:

• A water molecule (H₂O), which is nonlinear and has 3 atoms, has $3(3) - 6 = 3$ normal modes.

• A carbon dioxide molecule (CO₂), which is linear and has 3 atoms, has $3(3) - 5 = 4$ normal modes.

3. Types of Normal Modes

Normal modes can be classified into different types based on the nature of the atomic displacements:

• Stretching modes: These involve changes in the bond lengths between atoms.

  • Symmetric stretching: All bonds stretch or compress in unison.

  • Asymmetric stretching: Bonds stretch or compress in different directions or to different extents.

• Bending modes: These involve changes in the bond angles between atoms.

  • In-plane bending: The angle changes within the plane of the molecule.

  • Out-of-plane bending: The angle changes out of the plane of the molecule.

4. Visualization of Normal Modes

Normal modes can be visualized as patterns of motion:

• Stretching: Atoms move towards or away from each other along the bond axis.

• Bending: Atoms move towards or away from each other, changing the bond angle but not the bond length.

For example, in the water molecule (H₂O):

1. Symmetric stretching: Both O-H bonds stretch or compress in the same direction.

2. Asymmetric stretching: One O-H bond stretches while the other compresses.

3. Bending mode: The H-O-H bond angle changes, with hydrogen atoms moving in opposite directions relative to the oxygen atom.

5. Vibrational Frequency and Energy

Each normal mode has an associated vibrational frequency, corresponding to the energy required for that particular vibration. The frequency of a normal mode depends on the mass of the atoms involved and the strength of the bonds (force constants) between them. Heavier atoms and weaker bonds generally result in lower frequencies, while lighter atoms and stronger bonds result in higher frequencies.

The energy of a vibrational mode is quantized and can be expressed as:

$E = \left( n + \frac{1}{2} \right) h \nu$

Where:

• $n$ is the vibrational quantum number (0, 1, 2, ...),

• $h$ is Planck’s constant,

• $\nu$ is the vibrational frequency of the normal mode.

6. Normal Modes in Spectroscopy

Normal modes are crucial for understanding and interpreting vibrational spectroscopy techniques like Infrared (IR) spectroscopy and Raman spectroscopy:

• IR Spectroscopy: Measures the absorption of infrared light by a molecule, which occurs when the frequency of the light matches the frequency of a vibrational mode. Only normal modes that involve a change in the dipole moment of the molecule are IR-active.

• Raman Spectroscopy: Measures the scattering of light by a molecule, which can also occur when light interacts with vibrational modes. Normal modes that involve a change in polarizability are Raman-active.

Each normal mode gives rise to a characteristic peak in the IR or Raman spectrum, allowing for the identification of functional groups and structural information about the molecule.

7. Example: CO₂ and H₂O Normal Modes

Carbon Dioxide (CO₂):

• Symmetric stretching (no change in dipole moment, not IR-active, but Raman-active).

• Asymmetric stretching (changes the dipole moment, IR-active).

• Bending mode (degenerate, meaning it has two equivalent bending vibrations, IR-active).

Water (H₂O):

• Symmetric stretching (IR-active).

• Asymmetric stretching (IR-active).

• Bending mode (IR-active).

Summary:

• Normal modes describe the independent vibrational movements of a molecule.

• The number of normal modes depends on the number of atoms in the molecule and whether it is linear or nonlinear.

• Normal modes can be classified into stretching and bending modes.

• Each normal mode has a specific vibrational frequency and energy.

• Normal modes are important for interpreting vibrational spectra in IR and Raman spectroscopy.

Method

The Normal Modes are calculated with the --ohess option in xTB 6.6.0

Find

The Normal Modes can be found under the Spectra category in the property viewer. To visualize a normal mode, select the mode in the right menu or select the peak in the bottom plot.