⚛Molecular Physics Unit 8 – Rotational and Vibrational Spectroscopy

Key Concepts and Terminology

• Rotational spectroscopy involves the study of molecular rotations and their interactions with electromagnetic radiation
• Vibrational spectroscopy focuses on the vibrational motions of molecules and how they absorb and emit energy
• Molecular energy levels refer to the discrete energy states that molecules can occupy based on their rotational and vibrational motions
  • Transitions between these energy levels give rise to the observed spectroscopic features
• Selection rules determine which transitions between energy levels are allowed and can be observed in rotational and vibrational spectra
• Rotational constants ($B$) characterize the rotational energy levels and depend on the molecule's moment of inertia
• Vibrational frequencies ($\nu$) are determined by the molecule's force constants and reduced masses
• Spectroscopic techniques used in rotational and vibrational spectroscopy include absorption, emission, and Raman spectroscopy

Rotational Spectroscopy Basics

• Rotational spectroscopy probes the rotational motion of molecules in the gas phase
• Molecules with permanent dipole moments can absorb and emit electromagnetic radiation in the microwave and far-infrared regions
• The rotational energy levels of a diatomic molecule are given by $E_J = BJ(J+1)$, where $J$ is the rotational quantum number and $B$ is the rotational constant
  • The rotational constant $B$ is inversely proportional to the molecule's moment of inertia
• Selection rules for rotational transitions require $\Delta J = \pm 1$, resulting in a series of equally spaced lines in the rotational spectrum
• The intensity of rotational transitions depends on the population of the initial rotational state and the transition dipole moment
• Rotational spectra can provide information about the molecule's structure, bond lengths, and dipole moments
• Centrifugal distortion effects can cause deviations from the rigid rotor approximation and lead to more complex rotational spectra

Vibrational Spectroscopy Fundamentals

• Vibrational spectroscopy probes the vibrational motions of molecules, typically in the infrared region
• Molecules can vibrate in various modes, such as stretching and bending, depending on their geometry and number of atoms
• The vibrational energy levels of a diatomic molecule can be approximated by the harmonic oscillator model, with energy levels given by $E_v = (v + \frac{1}{2})h\nu$, where $v$ is the vibrational quantum number and $\nu$ is the vibrational frequency
• Selection rules for vibrational transitions require $\Delta v = \pm 1$ for harmonic oscillators, resulting in a fundamental vibrational band
• Anharmonicity in real molecules leads to the presence of overtones and combination bands in the vibrational spectrum
  • Overtones occur when $\Delta v = \pm 2, \pm 3, ...$, while combination bands involve the simultaneous excitation of multiple vibrational modes
• The intensity of vibrational transitions depends on the change in the molecule's dipole moment during the vibration
• Vibrational spectra can provide information about the molecule's functional groups, bond strengths, and intermolecular interactions

Molecular Energy Levels and Transitions

• Molecules possess both rotational and vibrational energy levels, which can be coupled to form ro-vibrational energy levels
• The total energy of a molecule is the sum of its electronic, vibrational, and rotational energies
• Born-Oppenheimer approximation allows the separation of electronic and nuclear motions, simplifying the treatment of molecular energy levels
• Franck-Condon principle governs the intensity of vibrational transitions based on the overlap of the initial and final vibrational wavefunctions
• Rotational-vibrational coupling can lead to the appearance of fine structure in the vibrational spectra, such as P, Q, and R branches
  • P branch corresponds to transitions with $\Delta J = -1$, Q branch to $\Delta J = 0$, and R branch to $\Delta J = +1$
• Fermi resonance occurs when two vibrational energy levels with the same symmetry are close in energy, leading to the mixing of their wavefunctions and intensity borrowing
• Anharmonic corrections to the vibrational energy levels can be described by the Morse potential, which accounts for the dissociation of the molecule at high vibrational excitations

Spectroscopic Techniques and Instrumentation

• Fourier-transform infrared (FTIR) spectroscopy is widely used for vibrational spectroscopy, offering high resolution and fast data acquisition
  • FTIR spectrometers use an interferometer to generate an interferogram, which is then Fourier-transformed to obtain the spectrum
• Microwave spectroscopy is the primary technique for studying pure rotational spectra, using microwave radiation to excite rotational transitions
• Cavity ring-down spectroscopy (CRDS) is a sensitive technique for measuring absorption spectra, based on the decay of light intensity in a high-finesse optical cavity
• Raman spectroscopy probes vibrational transitions through inelastic scattering of monochromatic light, providing complementary information to IR spectroscopy
  • Raman spectroscopy is particularly useful for studying symmetric molecules and vibrational modes that are inactive in IR
• Supersonic jet expansion techniques can be used to cool molecules to their lowest rotational and vibrational states, simplifying the spectra and improving resolution
• Laser-based techniques, such as laser-induced fluorescence (LIF) and resonance-enhanced multiphoton ionization (REMPI), offer high sensitivity and selectivity for studying molecular spectra

Applications in Molecular Analysis

• Rotational and vibrational spectroscopy can be used to identify and characterize molecules in various fields, such as atmospheric chemistry, astrochemistry, and pharmaceutical analysis
• Molecular structure determination can be achieved by analyzing rotational constants and vibrational frequencies, providing information about bond lengths, angles, and functional groups
• Conformational analysis can be performed by studying the rotational spectra of molecules with multiple stable conformations, such as sugars and peptides
• Isotope effects on rotational and vibrational spectra can be used to study reaction mechanisms, kinetics, and isotope fractionation processes
  • Isotopic substitution can also help in the assignment of spectral features and the determination of molecular symmetry
• Quantitative analysis can be performed using the intensity of rotational and vibrational transitions, enabling the determination of molecular concentrations and partial pressures
• Environmental monitoring and trace gas detection can be achieved using high-resolution rotational and vibrational spectroscopy, such as in the monitoring of greenhouse gases and pollutants
• Astrochemical applications include the identification of molecules in interstellar space and planetary atmospheres, providing insights into the chemical evolution of the universe

Problem-Solving Strategies

• Identify the type of molecule (diatomic, linear, symmetric top, asymmetric top) and its spectroscopic properties (dipole moment, polarizability) to determine the appropriate spectroscopic technique
• Use selection rules to predict the allowed transitions and the appearance of the spectrum (P, Q, R branches, overtones, combination bands)
• Calculate rotational constants and vibrational frequencies using the molecule's moment of inertia and force constants, respectively
  • For diatomic molecules, use the formulas $B = \frac{h}{8\pi^2cI}$ and $\nu = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}$, where $I$ is the moment of inertia, $k$ is the force constant, and $\mu$ is the reduced mass
• Assign spectral features to specific transitions using the calculated energy levels and selection rules, considering the effects of anharmonicity and centrifugal distortion
• Interpret the spectrum in terms of molecular structure, functional groups, and intermolecular interactions, comparing with reference spectra and theoretical predictions
• Use spectral simulations and fitting techniques to extract molecular parameters and refine the analysis, considering the instrumental resolution and line broadening mechanisms
• Apply error analysis and statistical methods to assess the uncertainty of the derived molecular parameters and the quality of the spectral fit

Real-World Examples and Case Studies

• Atmospheric chemistry: Rotational and vibrational spectroscopy are used to monitor the concentrations of greenhouse gases (CO2, CH4, N2O) and ozone-depleting substances (CFCs) in the Earth's atmosphere
  • High-resolution FTIR spectroscopy from satellites and ground-based stations provides global coverage and long-term trends of these atmospheric constituents
• Astrochemistry: Rotational spectroscopy has been instrumental in the discovery of complex organic molecules in interstellar space, such as amino acids (glycine) and sugars (glycolaldehyde)
  • The Atacama Large Millimeter/submillimeter Array (ALMA) and the Green Bank Telescope (GBT) are powerful facilities for studying the rotational spectra of interstellar molecules
• Pharmaceutical analysis: Vibrational spectroscopy (IR and Raman) is routinely used for the identification and quality control of pharmaceutical compounds, polymorphs, and formulations
  • Raman spectroscopy can be used for non-invasive, in-line monitoring of drug manufacturing processes, ensuring the consistency and purity of the final product
• Breath analysis: Rotational-vibrational spectroscopy can be used to detect trace amounts of volatile organic compounds (VOCs) in human breath, which can serve as biomarkers for various diseases (lung cancer, diabetes)
  • Cavity ring-down spectroscopy (CRDS) and photoacoustic spectroscopy (PAS) are sensitive techniques for breath analysis, capable of detecting VOCs at parts-per-billion (ppb) levels
• Art conservation: Vibrational spectroscopy (FTIR and Raman) is used to analyze the composition and degradation of pigments, binders, and varnishes in paintings and other artwork
  • Non-invasive and portable spectroscopic techniques allow the in-situ analysis of valuable art objects, guiding conservation and restoration efforts