Rotational motion in molecules is a key aspect of molecular physics, revealing insights into molecular structure and behavior. Diatomic molecules have simpler rotational motion due to a single rotational axis, while polyatomic molecules exhibit more complex rotational dynamics with multiple axes.

Understanding rotational motion helps explain molecular spectra and energy levels. The rigid rotor model approximates diatomic rotation, while symmetry and moments of inertia play crucial roles in polyatomic molecules. This knowledge connects to broader concepts of molecular vibration and rotation.

Rotational Motion of Molecules

Rotational Axes and Energy Levels

• Diatomic molecules have only one rotational axis perpendicular to the bond axis, while polyatomic molecules can have multiple rotational axes depending on their geometry and symmetry (linear, symmetric top, asymmetric top)
• The rotational energy levels and spectra of diatomic molecules are simpler compared to polyatomic molecules due to the presence of a single moment of inertia
• Polyatomic molecules exhibit more complex rotational motion, with different moments of inertia around each principal axis, leading to a greater number of rotational energy levels and more intricate spectra

Selection Rules and Transitions

• The selection rules for rotational transitions differ between diatomic and polyatomic molecules, with polyatomic molecules having additional allowed transitions based on their symmetry
  • Molecules with a permanent electric dipole moment (polar molecules) have allowed rotational transitions with ΔJ = ±1, while molecules without a permanent dipole moment (non-polar molecules) have forbidden rotational transitions
  • The parity of the rotational wavefunctions also affects the allowed transitions, with even-to-even and odd-to-odd transitions being allowed for symmetric molecules, while asymmetric molecules have more relaxed selection rules

Rotational Energy Levels for Diatomic Molecules

Rigid Rotor Approximation and Hamiltonian

• The rotational energy levels of a diatomic molecule can be derived using the rigid rotor approximation, which assumes the bond length is fixed and the molecule rotates as a single entity
• The rotational Hamiltonian for a diatomic molecule is given by $H = J^2 / (2I)$, where $J$ is the angular momentum operator and $I$ is the moment of inertia

Quantized Energy Levels and Wavefunctions

• The rotational energy levels are quantized and given by $E_J = J(J+1)ℏ^2 / (2I)$, where $J$ is the rotational quantum number ($J = 0, 1, 2, ...$) and $ℏ$ is the reduced Planck's constant
  • Example: For the CO molecule, with $I = 1.46 \times 10^{-46}$ kg m², the first few rotational energy levels are $E_0 = 0$, $E_1 = 7.63 \times 10^{-23}$ J, $E_2 = 3.05 \times 10^{-22}$ J
• The rotational wavefunctions for a diatomic molecule are the spherical harmonics, $Y_J^M(θ,φ)$, where $J$ is the rotational quantum number and $M$ is the magnetic quantum number ($M = -J, -J+1, ..., J-1, J$)
  • The rotational wavefunctions describe the angular distribution of the molecule and satisfy the Schrödinger equation for the rotational motion

Symmetry Effects on Rotational Motion

Degenerate Energy Levels and Symmetry

• Molecular symmetry plays a crucial role in determining the allowed rotational energy levels and transitions of a molecule
• Molecules with higher symmetry, such as linear and symmetric top molecules, have degenerate rotational energy levels due to the presence of multiple equivalent rotational axes
  • Example: The rotational energy levels of a linear molecule (CO₂) are doubly degenerate, while those of a spherical top molecule (CH₄) are (2J+1)-fold degenerate
• Asymmetric top molecules have distinct moments of inertia around each principal axis, leading to non-degenerate rotational energy levels and more complex spectra

Electric Dipole Moment and Parity

• The selection rules for rotational transitions are governed by the symmetry of the molecule and the electric dipole moment
• Molecules with a permanent electric dipole moment (polar molecules like HCl) have allowed rotational transitions with ΔJ = ±1, while molecules without a permanent dipole moment (non-polar molecules like O₂) have forbidden rotational transitions
• The parity of the rotational wavefunctions also affects the allowed transitions, with even-to-even and odd-to-odd transitions being allowed for symmetric molecules, while asymmetric molecules have more relaxed selection rules

Moments of Inertia for Different Geometries

Diatomic and Linear Molecules

• The moment of inertia is a measure of a molecule's resistance to rotational motion and depends on the mass distribution and geometry of the molecule
• For a diatomic molecule, the moment of inertia is given by $I = μr^2$, where $μ$ is the reduced mass and $r$ is the bond length
  • Example: For the HCl molecule, with $m_H = 1.67 \times 10^{-27}$ kg, $m_{Cl} = 5.89 \times 10^{-26}$ kg, and $r = 1.27 \times 10^{-10}$ m, the moment of inertia is $I = 2.65 \times 10^{-47}$ kg m²
• Linear polyatomic molecules have two equal moments of inertia perpendicular to the molecular axis ($I_x = I_y$) and a zero moment of inertia along the molecular axis ($I_z = 0$)

Symmetric and Asymmetric Top Molecules

• Symmetric top molecules, such as ammonia (NH₃) and methane (CH₄), have two equal moments of inertia ($I_x = I_y$) and a distinct moment of inertia along the symmetry axis ($I_z$)
• Asymmetric top molecules, such as water (H₂O) and hydrogen peroxide (H₂O₂), have three distinct moments of inertia ($I_x ≠ I_y ≠ I_z$)
• The moments of inertia can be calculated using the parallel axis theorem, $I = Σ(m_i r_i^2)$, where $m_i$ is the mass of each atom and $r_i$ is the distance of each atom from the rotational axis
  • Example: For the H₂O molecule, with $r_{OH} = 0.958$ Å and $∠HOH = 104.5°$, the moments of inertia are $I_x = 1.02 \times 10^{-47}$ kg m², $I_y = 1.92 \times 10^{-47}$ kg m², and $I_z = 2.94 \times 10^{-47}$ kg m²