6.2 Rotational motion of diatomic and polyatomic molecules
4 min read•july 30, 2024
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 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
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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 and spectra of diatomic molecules are simpler compared to polyatomic molecules due to the presence of a single
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=J2/(2I), where J is the operator and I is the moment of inertia
Quantized Energy Levels and Wavefunctions
The rotational energy levels are quantized and given by EJ=J(J+1)ℏ2/(2I), where J is the rotational quantum number (J=0,1,2,...) and ℏ is the reduced
Example: For the CO molecule, with I=1.46×10−46 kg m², the first few rotational energy levels are E0=0, E1=7.63×10−23 J, E2=3.05×10−22 J
The rotational wavefunctions for a diatomic molecule are the spherical harmonics, YJM(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=μr2, where μ is the reduced mass and r is the bond length
Example: For the HCl molecule, with mH=1.67×10−27 kg, mCl=5.89×10−26 kg, and r=1.27×10−10 m, the moment of inertia is I=2.65×10−47 kg m²
Linear polyatomic molecules have two equal moments of inertia perpendicular to the molecular axis (Ix=Iy) and a zero moment of inertia along the molecular axis (Iz=0)
Symmetric and Asymmetric Top Molecules
Symmetric top molecules, such as ammonia (NH₃) and methane (CH₄), have two equal moments of inertia (Ix=Iy) and a distinct moment of inertia along the symmetry axis (Iz)
Asymmetric top molecules, such as water (H₂O) and hydrogen peroxide (H₂O₂), have three distinct moments of inertia (Ix=Iy=Iz)
The moments of inertia can be calculated using the parallel axis theorem, I=Σ(miri2), where mi is the mass of each atom and ri is the distance of each atom from the rotational axis
Example: For the H₂O molecule, with rOH=0.958 Å and ∠HOH=104.5°, the moments of inertia are Ix=1.02×10−47 kg m², Iy=1.92×10−47 kg m², and Iz=2.94×10−47 kg m²