8.1 Molecular vibrations and normal modes

Molecular vibrations and normal modes are key concepts in understanding how atoms move within molecules. They help explain vibrational spectra and molecular behavior, connecting atomic-level motion to observable properties.

Normal modes represent independent vibrations in molecules, with specific frequencies and patterns. By studying these modes, we can predict and interpret vibrational spectra, providing insights into molecular structure, bonding, and reactivity.

Vibrational Motion and Normal Modes

Fundamentals of Molecular Vibrations

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• Normal modes represent the independent, collective motions of atoms in a molecule where all atoms oscillate with the same frequency and phase
• Degrees of freedom determine the number of independent ways a molecule can move, with 3N-6 vibrational degrees of freedom for nonlinear molecules and 3N-5 for linear molecules, where N is the number of atoms
• Vibrational frequency, denoted by $\nu$, is the number of vibrations per unit time for a particular normal mode, typically expressed in units of inverse centimeters ($cm^{-1}$) or Hertz (Hz)
• Harmonic oscillator model approximates the vibrational motion of a diatomic molecule as two masses connected by a spring, obeying Hooke's law with a restoring force proportional to the displacement from equilibrium
• Anharmonicity describes the deviation of real molecular vibrations from the ideal harmonic oscillator model due to the presence of higher-order terms in the potential energy function, leading to non-evenly spaced energy levels and overtones in vibrational spectra (H2O, CO2)

Types and Characteristics of Normal Modes

• Stretching modes involve changes in bond lengths, with symmetric stretching preserving the symmetry of the molecule and asymmetric stretching breaking the symmetry (CO2, NH3)
• Bending modes involve changes in bond angles, including in-plane bending (scissoring) and out-of-plane bending (wagging, twisting, rocking) (H2O, CH4)
• Degenerate modes occur when multiple normal modes have the same vibrational frequency due to molecular symmetry, such as the two bending modes in CO2
• Combination bands arise from the simultaneous excitation of two or more normal modes, resulting in new bands in the vibrational spectrum at frequencies equal to the sum of the individual mode frequencies (CH4, C6H6)

Potential Energy and Force Constants

Potential Energy Surfaces

• Potential energy surface is a multidimensional plot of the potential energy of a molecule as a function of its atomic coordinates, providing insight into the relative stability of different molecular geometries
• Equilibrium geometry corresponds to the minimum point on the potential energy surface, representing the most stable configuration of the molecule
• Saddle points on the potential energy surface represent transition states, which are high-energy configurations that connect different equilibrium geometries and play a crucial role in chemical reactions (H2 + H → H3 transition state)
• Normal mode displacements can be visualized as orthogonal vectors on the potential energy surface, indicating the direction and magnitude of atomic motion for each vibrational mode

Force Constants and Reduced Mass

• Force constants, denoted by $k$, measure the strength of the chemical bond and the resistance to deformation, with higher values indicating stiffer bonds and higher vibrational frequencies
• Reduced mass, denoted by $\mu$, is a combination of the masses of the atoms involved in a vibration, simplifying the two-body problem into a one-body problem with a single effective mass
  • For a diatomic molecule with masses $m_1$ and $m_2$, the reduced mass is given by $\mu = \frac{m_1m_2}{m_1+m_2}$
  • Reduced mass is used in the calculation of vibrational frequencies and the vibrational Schrödinger equation (H2, CO)

Vibrational Quantum Mechanics

Quantum Description of Molecular Vibrations

• Vibrational quantum number, denoted by $v$, is a non-negative integer that specifies the vibrational energy level of a molecule, with $v=0$ corresponding to the ground state and $v=1, 2, 3, ...$ representing excited states
• Vibrational wave functions, denoted by $\psi_v(x)$, are solutions to the vibrational Schrödinger equation and describe the probability distribution of the atomic positions in a given vibrational state
• Selection rules govern the allowed transitions between vibrational energy levels, with the fundamental transition ($\Delta v = \pm 1$) being the most intense and overtones ($\Delta v = \pm 2, \pm 3, ...$) having diminishing intensity (HCl, CO)

Zero-Point Energy and Anharmonicity Effects

• Zero-point energy is the lowest possible vibrational energy of a molecule, corresponding to the $v=0$ state, and arises from the Heisenberg uncertainty principle
  • Even at absolute zero temperature, molecules possess a non-zero vibrational energy, given by $E_0 = \frac{1}{2}h\nu$, where $h$ is Planck's constant
• Anharmonicity causes the spacing between vibrational energy levels to decrease with increasing $v$, leading to the convergence of energy levels and the dissociation of the molecule at high vibrational excitations
• Morse potential is a more accurate model for the potential energy of a diatomic molecule, accounting for anharmonicity by including higher-order terms in the potential energy function (H2, I2)
  • The Morse potential is given by $V(r) = D_e[1-e^{-\alpha(r-r_e)}]^2$, where $D_e$ is the dissociation energy, $\alpha$ is a parameter related to the force constant, and $r_e$ is the equilibrium bond length