⚗️Theoretical Chemistry Unit 8 – Molecular Vibrations and Symmetry

Key Concepts and Fundamentals

• Molecular vibrations involve the periodic motion of atoms within a molecule around their equilibrium positions
• Vibrational energy levels are quantized and depend on the molecule's structure and the strength of its chemical bonds
• The number of vibrational modes in a molecule is determined by its degrees of freedom (3N-6 for non-linear molecules and 3N-5 for linear molecules, where N is the number of atoms)
• Fundamental vibrational frequencies correspond to the energy required to excite a molecule from its ground state to the first excited state of a particular vibrational mode
• Harmonic approximation assumes that the potential energy of a molecule varies quadratically with the displacement of its atoms from their equilibrium positions
  • Anharmonicity occurs when the potential energy deviates from the quadratic behavior, leading to non-evenly spaced energy levels
• The reduced mass ($\mu$) of a diatomic molecule is given by $\mu = \frac{m_1 m_2}{m_1 + m_2}$, where $m_1$ and $m_2$ are the masses of the individual atoms
• The force constant ($k$) is a measure of the strength of the chemical bond and determines the spacing between vibrational energy levels

Molecular Symmetry and Point Groups

• Molecular symmetry refers to the spatial arrangement of atoms in a molecule and the operations that leave the molecule unchanged
• Symmetry elements include rotation axes (C$_n$), reflection planes ($\sigma$), inversion centers (i), and improper rotation axes (S$_n$)
• Point groups are mathematical groups that describe the complete set of symmetry operations for a molecule
  • Examples of point groups include C$_{2v}$ (water), D$_{\infty h}$ (carbon dioxide), and T$_d$ (methane)
• The character table of a point group summarizes the symmetry operations and their effects on the molecule's properties
• Reducible representations can be decomposed into irreducible representations (IRs) using the great orthogonality theorem
• The symmetry of vibrational modes can be determined by the direct product of the IRs of the atomic displacements
• Degenerate vibrational modes occur when multiple modes have the same symmetry and energy

Vibrational Modes and Normal Coordinates

• Normal modes are independent, collective motions of atoms in a molecule that can be excited without affecting other modes
• Each normal mode has a specific frequency and symmetry, determined by the molecule's structure and force constants
• Normal coordinates ($Q_i$) are linear combinations of the Cartesian displacements of atoms that describe the motion of a normal mode
• The potential energy of a molecule can be expressed as a sum of quadratic terms in the normal coordinates: $V = \frac{1}{2} \sum_i k_i Q_i^2$
• The kinetic energy of a molecule can be expressed as a sum of quadratic terms in the time derivatives of the normal coordinates: $T = \frac{1}{2} \sum_i \dot{Q}_i^2$
• The Lagrangian formulation of classical mechanics can be used to derive the equations of motion for the normal coordinates
• The solutions to the equations of motion are harmonic oscillations with frequencies given by $\omega_i = \sqrt{\frac{k_i}{\mu_i}}$, where $\mu_i$ is the reduced mass associated with the i-th normal mode

Quantum Mechanical Treatment of Vibrations

• The quantum mechanical treatment of molecular vibrations involves solving the Schrödinger equation for the vibrational motion
• The vibrational Hamiltonian is given by $\hat{H} = -\frac{\hbar^2}{2\mu} \frac{\partial^2}{\partial Q^2} + \frac{1}{2} k Q^2$, where $Q$ is the normal coordinate and $\mu$ is the reduced mass
• The energy eigenvalues of the vibrational Hamiltonian are given by $E_n = \hbar \omega (n + \frac{1}{2})$, where $n$ is the vibrational quantum number (0, 1, 2, ...)
• The vibrational wave functions are given by the Hermite polynomials multiplied by a Gaussian function: $\psi_n(Q) = N_n H_n(\alpha Q) e^{-\alpha^2 Q^2/2}$, where $N_n$ is a normalization constant, $H_n$ is the n-th Hermite polynomial, and $\alpha = \sqrt{\frac{\mu \omega}{\hbar}}$
• The transition dipole moment between vibrational states determines the intensity of vibrational transitions in spectroscopy
• The Franck-Condon principle states that electronic transitions occur much faster than nuclear motion, leading to vertical transitions between vibrational states
• The Franck-Condon factors, which are the squares of the overlap integrals between vibrational wave functions, determine the relative intensities of vibrational transitions

Symmetry Selection Rules

• Selection rules determine which vibrational transitions are allowed or forbidden based on the symmetry of the initial and final states
• The transition dipole moment integral, $\int \psi_f^* \hat{\mu} \psi_i d\tau$, must be non-zero for a transition to be allowed
• For a transition to be allowed, the direct product of the IRs of the initial state, the transition dipole moment operator, and the final state must contain the totally symmetric IR
• The symmetry of the transition dipole moment operator depends on the direction of the electric field (x, y, or z) and the corresponding IR in the character table
• Overtone transitions ($\Delta v = 2, 3, ...$) are generally weaker than fundamental transitions ($\Delta v = 1$) due to the smaller overlap of the vibrational wave functions
• Combination bands arise from the simultaneous excitation of two or more vibrational modes and can be observed if the direct product of their IRs contains the totally symmetric IR
• Fermi resonance occurs when two vibrational states with the same symmetry and similar energies interact, leading to a mixing of their wave functions and a splitting of their energy levels

Spectroscopic Applications

• Infrared (IR) spectroscopy probes the vibrational transitions of molecules by absorbing infrared light
  • IR active modes must have a change in the dipole moment during the vibration
• Raman spectroscopy probes the vibrational transitions by inelastic scattering of monochromatic light
  • Raman active modes must have a change in the polarizability during the vibration
• The complementary nature of IR and Raman spectroscopy allows for a comprehensive analysis of a molecule's vibrational modes
• Vibrational spectroscopy can be used to identify functional groups, determine molecular structure, and study intermolecular interactions
• The position, intensity, and shape of vibrational bands provide information about the molecule's environment, such as hydrogen bonding, solvation, and phase transitions
• Time-resolved vibrational spectroscopy can be used to study the dynamics of chemical reactions and energy transfer processes
• Surface-enhanced Raman spectroscopy (SERS) exploits the enhanced electric fields near metal nanostructures to increase the Raman scattering intensity of adsorbed molecules

Computational Methods and Tools

• Ab initio methods, such as Hartree-Fock (HF) and density functional theory (DFT), can be used to calculate the vibrational frequencies and normal modes of molecules
• The harmonic frequencies are obtained by diagonalizing the mass-weighted Hessian matrix, which contains the second derivatives of the energy with respect to the atomic coordinates
• Anharmonic corrections to the vibrational frequencies can be obtained using perturbation theory or variational methods
• Normal mode analysis can be performed to visualize the atomic displacements and to assign the vibrational modes to specific molecular motions
• Potential energy distribution (PED) analysis decomposes the normal modes into contributions from individual internal coordinates (bond stretches, angle bends, etc.)
• Vibrational spectral simulation can be used to predict the IR and Raman spectra of molecules based on their calculated frequencies and intensities
• Molecular dynamics simulations can be used to study the time-dependent vibrational motion of molecules and to calculate vibrational spectra from the Fourier transform of the dipole moment or polarizability autocorrelation functions

Advanced Topics and Current Research

• Vibrational anharmonicity and coupling between modes can lead to complex spectral features, such as Fermi resonances and combination bands
• Vibrational energy redistribution (VER) describes the flow of vibrational energy among the modes of a molecule following excitation
• Intramolecular vibrational energy redistribution (IVR) can influence the rates and pathways of chemical reactions
• Coherent multidimensional vibrational spectroscopy (2D IR, 2D Raman) can be used to study the coupling and dynamics of vibrational modes
• Vibrational strong coupling occurs when the interaction between a molecular vibration and an optical cavity mode becomes stronger than the dissipation rates, leading to the formation of hybrid light-matter states (polaritons)
• Vibrational polaritons have potential applications in chemical reactivity control, energy transfer, and quantum information processing
• Tip-enhanced Raman spectroscopy (TERS) combines the high spatial resolution of scanning probe microscopy with the chemical specificity of Raman spectroscopy to study the vibrational properties of individual molecules and nanostructures
• Quantum cascade lasers (QCLs) and difference frequency generation (DFG) sources have enabled the development of compact, tunable, and high-power infrared light sources for vibrational spectroscopy