Molecular rotations and energy levels are key to understanding how molecules move and interact with light. This topic explores the rigid rotor model, which simplifies molecular rotation, and introduces rotational constants that describe energy spacing between levels.

Selection rules govern allowed transitions between rotational states, leading to distinct spectral patterns. We'll see how different molecular geometries, from linear to asymmetric top molecules, affect rotational behavior and spectral characteristics.

Rigid Rotor Model and Rotational Constants

Rigid Rotor Model and Moment of Inertia

• Rigid rotor model approximates molecules as rigid objects rotating around their center of mass
• Assumes fixed bond lengths and angles during rotation
• Applies primarily to diatomic molecules and linear polyatomic molecules
• Moment of inertia measures resistance to rotational acceleration
• Calculated using formula $I = \mu r^2$, where μ represents reduced mass and r equals bond length
• Larger moment of inertia results in slower rotation for a given angular momentum

Rotational Constants and Angular Momentum

• Rotational constants describe energy spacing between rotational levels
• Expressed as $B = \frac{h}{8\pi^2I}$, where h represents Planck's constant and I equals moment of inertia
• Units of rotational constants typically given in cm^-1 or MHz
• Angular momentum quantized in rotational motion
• Magnitude of angular momentum given by $\sqrt{J(J+1)}\hbar$, where J represents rotational quantum number
• Angular momentum vector precesses around molecular axis (space quantization)

Selection Rules and Rotational Spectra

Selection Rules for Rotational Transitions

• Selection rules determine allowed transitions between rotational energy levels
• For pure rotational spectroscopy, ΔJ = ±1
• Transitions must involve a change in dipole moment
• Homonuclear diatomic molecules (O₂, N₂) lack permanent dipole moment, no pure rotational spectrum
• Heteronuclear diatomic molecules (CO, HCl) exhibit rotational spectra due to permanent dipole moment

Rotational Spectra and Energy Levels

• Rotational spectra consist of equally spaced lines in frequency domain
• Energy levels given by $E_J = BJ(J+1)$, where B represents rotational constant and J equals rotational quantum number
• Spacing between adjacent energy levels increases with increasing J
• Intensity of spectral lines depends on population of rotational levels (Boltzmann distribution)
• Temperature affects relative intensities of rotational lines
• Rotational spectra typically observed in microwave region of electromagnetic spectrum

J Quantum Number and Spectral Analysis

• J quantum number describes rotational state of molecule
• J = 0, 1, 2, ... (non-negative integers)
• Each J value corresponds to specific rotational energy level
• Spectral analysis involves measuring line positions and intensities
• Determination of rotational constants from spectral data allows calculation of bond lengths and molecular geometry

Molecular Geometries

Linear Molecules and Their Rotational Behavior

• Linear molecules possess one principal axis of rotation
• Examples include CO₂, HCN, and acetylene (C₂H₂)
• Single moment of inertia describes rotational motion
• Rotational energy levels depend on single rotational constant
• Spectral pattern consists of equally spaced lines (rigid rotor approximation)
• Centrifugal distortion effects become important at higher J values

Symmetric Top Molecules

• Symmetric top molecules have two equal moments of inertia
• Examples include NH₃, CH₃Cl, and benzene (C₆H₆)
• Classified as prolate (cigar-shaped) or oblate (disk-shaped) symmetric tops
• Rotational energy levels depend on two rotational constants
• Additional quantum number K describes projection of angular momentum along symmetry axis
• Spectral patterns more complex than linear molecules due to K-structure

Asymmetric Top Molecules

• Asymmetric top molecules have three distinct moments of inertia
• Most polyatomic molecules fall into this category (H₂O, CH₂O)
• Rotational energy levels depend on three rotational constants
• No closed-form expression for energy levels, numerical methods required
• Spectral patterns highly complex, often requiring computer simulations for analysis
• Asymmetry parameters κ used to characterize degree of asymmetry
• Rotational spectra provide valuable information about molecular structure and dynamics