Center of Mass, Rotational Constants, and Moment of Intertia

About

In molecular physics and quantum chemistry, Center of Mass, Rotational Constants, and Moment of Inertia are key concepts used to describe the spatial arrangement, rotational behavior, and mass distribution of a molecule. These quantities are fundamental to understanding molecular rotation, vibrational spectra, and thermodynamic properties.

1. Center of Mass (COM)

The center of mass of a molecule is the point at which the total mass of the system can be considered to be concentrated for purposes of translational motion. It is the weighted average position of all the atoms in the molecule, with each atom's contribution being proportional to its mass.

The center of mass is important because:

• It simplifies the description of a molecule's motion. For translational motion, the entire molecule can be treated as if all its mass is concentrated at the COM.

• It helps in defining the moment of inertia, which is crucial for describing rotational behavior.

Formula for the Center of Mass:

For a molecule with $N$ atoms, the coordinates of the center of mass $\mathbf{R}_{\text{COM}}$ in three-dimensional space can be computed as:

$\mathbf{R}{\text{COM}} = \frac{1}{M{\text{total}}} \sum_{i=1}^{N} m_i \mathbf{r}_i$

Where:

• $M_{\text{total}} = \sum_{i=1}^{N} m_i$ is the total mass of the molecule.

• $m_i$ is the mass of the $i$-th atom.

• $\mathbf{r}_i$ is the position vector of the $i$-th atom.

This equation can be broken down into components along each axis (x, y, z):

$X_{\text{COM}} = \frac{1}{M_{\text{total}}} \sum_{i=1}^{N} m_i x_i \\ Y_{\text{COM}} = \frac{1}{M_{\text{total}}} \sum_{i=1}^{N} m_i y_i \\ Z_{\text{COM}} = \frac{1}{M_{\text{total}}} \sum_{i=1}^{N} m_i z_i$

2. Moment of Inertia (I)

The moment of inertia of a molecule is a measure of its resistance to rotational motion about a specific axis. It depends on both the mass distribution of the molecule and how far the masses are from the axis of rotation. In quantum mechanics and spectroscopy, the moment of inertia is crucial for understanding the rotational spectra of molecules.

Formula for Moment of Inertia:

For a molecule rotating about an axis, the moment of inertia $I$ is given by:

$I = \sum_{i=1}^{N} m_i r_i^2$

Where:

• $m_i$ is the mass of the $i$-th atom.

• $r_i$ is the perpendicular distance of the $i$-th atom from the axis of rotation.

For a molecule, the moment of inertia is often expressed relative to the three principal axes: principal moments of inertia $I_A$, $I_B$, and $I_C$, which correspond to rotation about the x, y, and z axes (or other defined axes, depending on the symmetry of the molecule).

Types of Molecules Based on Moment of Inertia:

Molecules can be classified based on the values of their moments of inertia about different axes:

1. Linear Molecules:

   • For linear molecules (e.g., CO₂, HCN), one of the moments of inertia is zero because the molecule is symmetric along its bond axis.

   • $I_A = 0$ , and $I_B = I_C$ (since rotation occurs around axes perpendicular to the molecular axis).

2. Symmetric Top Molecules:

   • Symmetric top molecules (e.g., CH₃Cl) have two moments of inertia that are equal.

   • For prolate symmetric tops (elongated along one axis), $I_A < I_B = I_C$.

   • For oblate symmetric tops (flattened along one axis), $I_A = I_B < I_C$.

3. Asymmetric Top Molecules:

   • Asymmetric top molecules (e.g., water, H₂O) have all three moments of inertia different.

   • $I_A \neq I_B \neq I_C$.

4. Spherical Top Molecules:

   • For molecules like methane (CH₄), all three moments of inertia are equal.

   • $I_A = I_B = I_C$.

3. Rotational Constants (B)

The rotational constant ( B ) describes the rotational energy levels of a molecule. It is inversely related to the moment of inertia and is an important quantity in molecular rotational spectroscopy.

Formula for Rotational Constant:

The rotational constant $B$ for a given axis is related to the moment of inertia $I$ by:

$B = \frac{h}{8 \pi^2 I c}$

Where:

• $h$ is Planck’s constant.

• $I$ is the moment of inertia about the axis of rotation.

• $c$ is the speed of light.

The rotational constant has units of inverse length (commonly expressed in cm⁻¹ in spectroscopy).

Rotational Energy Levels:

In quantum mechanics, the energy levels for rotational motion are quantized and given by:

$E_J = B J (J + 1)$

Where:

• $J$ is the rotational quantum number (an integer: $J = 0, 1, 2, \ldots$ ).

• $E_J$ is the energy associated with the $J$-th rotational level.

The rotational spectra of a molecule arise from transitions between these quantized rotational energy levels, typically in the microwave or far-infrared regions of the electromagnetic spectrum.

Summary of Key Relationships:

• Center of Mass (COM): The weighted average position of all atoms in a molecule, accounting for their masses.

• Moment of Inertia (I): The measure of a molecule’s resistance to rotational motion, depending on mass distribution and distance from the axis of rotation.

• Rotational Constant (B): A constant related to the moment of inertia, which defines the spacing of rotational energy levels.

These quantities are crucial for describing the rotational dynamics of molecules, predicting their rotational spectra, and providing insight into their structure and mass distribution.

Method

These properties are calculated from GFN2-xTB optimized structures in xTB

Find

The Center of Mass, Rotational Constants, and Moment of Intertia can be found in the Global property table.