🌈Spectroscopy Unit 3 – Molecular Structure and Vibrational Modes

Key Concepts and Definitions

• Molecular structure refers to the arrangement of atoms within a molecule, including bond lengths, bond angles, and dihedral angles
• Vibrational modes are the different ways a molecule can vibrate, determined by its structure and symmetry
  • Stretching modes involve changes in bond lengths (symmetric and asymmetric stretching)
  • Bending modes involve changes in bond angles (in-plane and out-of-plane bending)
• Normal modes are independent, collective vibrations of atoms in a molecule, each with a specific frequency and symmetry
• Selection rules determine which vibrational transitions are allowed based on the symmetry of the molecule and the vibrational modes
• Spectroscopy is the study of the interaction between matter and electromagnetic radiation, used to probe molecular structure and dynamics
• Resonance occurs when the frequency of the incident radiation matches the natural frequency of a vibrational mode, leading to enhanced absorption or emission

Molecular Geometry and Bonding

• Molecular geometry is determined by the arrangement of atoms and the types of bonds between them (covalent, ionic, metallic)
• VSEPR (Valence Shell Electron Pair Repulsion) theory predicts molecular geometries based on the number of electron pairs around a central atom
  • Common geometries include linear (CO2), trigonal planar (BF3), tetrahedral (CH4), and octahedral (SF6)
• Hybridization describes the mixing of atomic orbitals to form new hybrid orbitals, which influence the geometry and bonding of molecules
  • sp hybridization results in linear geometry (C2H2)
  • sp2 hybridization results in trigonal planar geometry (C6H6)
  • sp3 hybridization results in tetrahedral geometry (CH4)
• Molecular symmetry is determined by the presence of symmetry elements such as rotation axes, reflection planes, and inversion centers
• Point groups are used to classify molecules based on their symmetry elements, which have implications for their vibrational modes and spectroscopic properties

Vibrational Modes and Symmetry

• The number of vibrational modes in a molecule depends on its number of atoms (N) and its linearity
  • Non-linear molecules have 3N-6 vibrational modes
  • Linear molecules have 3N-5 vibrational modes
• Vibrational modes can be classified as Raman-active or IR-active based on their symmetry and the selection rules
  • Raman-active modes involve changes in polarizability during the vibration
  • IR-active modes involve changes in the dipole moment during the vibration
• Degenerate vibrational modes have the same frequency and symmetry but different orientations (e.g., doubly degenerate bending modes in CO2)
• Group theory is used to determine the symmetry of vibrational modes and predict their spectroscopic activity
  • Irreducible representations describe the symmetry of vibrational modes and their behavior under symmetry operations
• Vibrational coupling can occur between modes with the same symmetry, leading to mixing and shifts in their frequencies

Energy Levels and Transitions

• Vibrational energy levels are quantized, with each level characterized by a vibrational quantum number (v = 0, 1, 2, ...)
• The energy spacing between vibrational levels is determined by the force constant and reduced mass of the vibrating atoms
  • Harmonic oscillator model assumes equally spaced energy levels, but real molecules exhibit anharmonicity
• Vibrational transitions occur when a molecule absorbs or emits a photon with energy matching the difference between two vibrational levels
  • Fundamental transitions involve a change in the vibrational quantum number by one unit (Δv = ±1)
  • Overtone transitions involve a change in the vibrational quantum number by more than one unit (Δv = ±2, ±3, ...)
  • Combination bands result from the simultaneous excitation of two or more vibrational modes
• Fermi resonance occurs when an overtone or combination band has nearly the same energy as a fundamental transition, leading to mixing and intensity borrowing
• Hot bands arise from transitions originating from excited vibrational states, which are populated at higher temperatures

Spectroscopic Techniques

• Infrared (IR) spectroscopy probes the absorption of IR radiation by molecules, which can excite vibrational transitions
  • Fourier-transform infrared (FTIR) spectroscopy uses an interferometer to obtain high-resolution spectra
  • Attenuated total reflectance (ATR) is a sampling technique that allows the analysis of solid and liquid samples without extensive preparation
• Raman spectroscopy measures the inelastic scattering of monochromatic light by molecules, which can excite or de-excite vibrational modes
  • Stokes scattering occurs when the scattered photon has lower energy than the incident photon (vibrational excitation)
  • Anti-Stokes scattering occurs when the scattered photon has higher energy than the incident photon (vibrational de-excitation)
• Vibrational sum-frequency generation (VSFG) is a nonlinear spectroscopic technique that probes the vibrational structure of interfaces and surfaces
• Cavity ring-down spectroscopy (CRDS) is a highly sensitive technique that measures the decay of light in an optical cavity to determine the absorption spectrum of a sample

Mathematical Models and Calculations

• The harmonic oscillator model treats a vibrating molecule as a system of masses connected by springs, with a potential energy given by:

$V(x) = \frac{1}{2}kx^2$

where $k$ is the force constant and $x$ is the displacement from equilibrium

• The vibrational frequency of a harmonic oscillator is given by:

$\nu = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}}$

where $\mu$ is the reduced mass of the vibrating atoms

• The anharmonic oscillator model includes higher-order terms in the potential energy expression to account for the non-ideal behavior of real molecules:

$V(x) = \frac{1}{2}kx^2 + \frac{1}{3!}k'x^3 + \frac{1}{4!}k''x^4 + ...$

• Normal mode analysis involves solving the secular equation to determine the frequencies and eigenvectors of the normal modes:

$\det(F - \lambda G) = 0$

where $F$ is the force constant matrix, $G$ is the kinetic energy matrix, and $\lambda$ is the eigenvalue (squared frequency)

• Density functional theory (DFT) is a computational method that calculates the electronic structure of molecules and can predict their vibrational spectra
  • Functionals such as B3LYP and M06-2X are commonly used for vibrational frequency calculations
  • Basis sets (e.g., 6-31G*, cc-pVTZ) describe the atomic orbitals used in the calculations

Applications in Research and Industry

• Vibrational spectroscopy is used to identify and characterize molecules in various fields, including chemistry, biology, and materials science
  • Functional group identification is based on characteristic vibrational frequencies (e.g., C=O stretching in ketones at ~1700 cm-1)
  • Conformational analysis can be performed by comparing experimental and calculated vibrational spectra
• Reaction monitoring and kinetics studies use vibrational spectroscopy to follow the progress of chemical reactions in real-time
  • Disappearance of reactant peaks and appearance of product peaks can be used to determine reaction rates and mechanisms
• Quality control and process analytical technology (PAT) employ vibrational spectroscopy for the rapid, non-destructive analysis of products and intermediates
  • Quantitative analysis can be performed using calibration models that relate spectral features to analyte concentrations
• Environmental monitoring and remote sensing rely on vibrational spectroscopy to detect and quantify pollutants, greenhouse gases, and other atmospheric species
  • Satellite-based instruments (e.g., MOPITT, TROPOMI) measure the absorption of infrared radiation by molecules in the Earth's atmosphere

Common Challenges and Misconceptions

• Spectral interpretation can be challenging due to the complexity of molecular vibrations and the presence of overlapping bands
  • Deconvolution techniques (e.g., curve fitting, second derivatives) can help resolve overlapping features
  • Isotopic substitution (e.g., deuteration) can simplify spectra by shifting the frequencies of certain vibrational modes
• Selection rules are not always strictly followed, and forbidden transitions may be weakly allowed due to vibronic coupling or symmetry breaking
• Anharmonicity can lead to deviations from the ideal harmonic oscillator behavior, resulting in non-equally spaced energy levels and the appearance of overtones and combination bands
• Fermi resonance can complicate the assignment of vibrational modes by causing intensity borrowing and frequency shifts
• Environmental effects (e.g., solvent, pH, temperature) can influence the vibrational spectra of molecules, leading to changes in band positions, intensities, and shapes
  • Hydrogen bonding can cause significant shifts in the vibrational frequencies of certain functional groups (e.g., O-H, N-H)
  • Conformational changes can alter the vibrational spectra of flexible molecules (e.g., proteins, polymers)
• Computational methods have limitations in accurately predicting vibrational spectra, particularly for large and complex molecules
  • Anharmonic corrections and scaling factors are often applied to improve the agreement between calculated and experimental frequencies
  • The choice of functional and basis set can significantly impact the accuracy of the calculated spectra