6.1 Molecular vibrations and normal modes

Molecular vibrations are the rhythmic dance of atoms within molecules. These movements, like stretching and bending, shape a molecule's behavior. Understanding vibrations helps us grasp how molecules interact with light and each other, key to many chemical processes.

Normal modes are the unique vibration patterns of molecules. They're like fingerprints, revealing a molecule's structure and properties. By studying these modes, we can predict how molecules will behave in different environments and reactions, crucial for many applications in chemistry and physics.

Molecular vibrations and their characteristics

Types of molecular vibrations

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• Molecular vibrations involve the periodic motion of atoms in a molecule relative to each other, resulting in changes in bond lengths (stretching) and angles (bending)
• Stretching vibrations involve changes in the interatomic distances along bond axes
  • Can be symmetric (in-phase motion) or asymmetric (out-of-phase motion), depending on the symmetry of the molecule (CO2, H2O)
• Bending vibrations involve changes in the angles between bonds
  • Include rocking, scissoring, wagging, and twisting motions, which are distinguished by the relative motion of the atoms in the molecular plane (CH4, NH3)

Factors influencing vibrational modes

• The number and types of vibrational modes depend on the molecular structure and symmetry
  • More complex molecules have a greater number of vibrational modes (benzene, glucose)
• The frequency and energy of a vibrational mode are determined by the masses of the atoms involved and the strength of the bonds
  • Heavier atoms and weaker bonds lead to lower frequencies and energies (C-H vs. C-Cl stretching)
• The reduced mass (μ) of the vibrating atoms affects the frequency, given by μ = (m1 × m2) / (m1 + m2), where m1 and m2 are the masses of the vibrating atoms
  • A higher reduced mass results in a lower vibrational frequency (H-Cl vs. H-Br stretching)

Normal modes of molecular vibration

Characteristics of normal modes

• Normal modes are the independent, collective motions of atoms in a molecule that occur at specific frequencies
• Each normal mode involves all atoms in the molecule vibrating with the same frequency and phase, but with different amplitudes
• Normal modes are orthogonal, meaning they are independent of each other and can be excited separately (CO2 symmetric and asymmetric stretching)

Symmetry and normal modes

• Symmetry plays a crucial role in determining the normal modes of a molecule
  • Modes that belong to the same symmetry species are allowed to couple and mix (benzene)
• Group theory can be used to predict the number and symmetry of normal modes based on the molecular point group
  • The character table of a point group provides information on the symmetry species and their vibrational activity (H2O, NH3)

Degrees of freedom and normal modes

• The number of normal modes in a molecule is determined by the degrees of freedom
  • 3N - 6 for non-linear molecules and 3N - 5 for linear molecules, where N is the number of atoms
• The 3N term represents the total degrees of freedom for N atoms, each having three translational degrees of freedom (x, y, z)
• In a non-linear molecule, 6 degrees of freedom are subtracted to account for the three translational and three rotational modes of the entire molecule, which do not contribute to vibrations
• In a linear molecule, only 5 degrees of freedom are subtracted because there are only two rotational modes (rotation about the molecular axis does not change the molecule's orientation)

Frequencies and energies of vibrations

Calculating vibrational frequencies

• The frequency of a molecular vibration depends on the reduced mass of the vibrating atoms and the force constant of the bond
• The force constant (k) is a measure of the strength of the bond and can be determined experimentally or calculated using computational methods
• The vibrational frequency (ν) is related to the reduced mass and force constant by the equation: $ν = (1 / 2π) × √(k / μ)$
  • Higher force constants and lower reduced masses lead to higher vibrational frequencies (C≡C vs. C=C stretching)

Quantization of vibrational energy

• The energy of a vibrational mode is quantized, with each quantum level separated by hν, where h is Planck's constant
• The vibrational energy levels can be described by the harmonic oscillator model, with the energy given by $E = (n + 1/2) × hν$, where n is the vibrational quantum number (n = 0, 1, 2, ...)
  • The lowest energy level (n = 0) is called the ground state or zero-point energy
• Anharmonicity in real molecules leads to deviations from the harmonic oscillator model, resulting in non-equidistant energy levels and the possibility of vibrational overtones and combination bands
  • Overtones occur at approximately integer multiples of the fundamental frequency (2ν, 3ν, etc.)
  • Combination bands result from the simultaneous excitation of two or more normal modes (ν1 + ν2, ν1 + ν3, etc.)

Vibrational degrees of freedom

Definition and calculation

• Vibrational degrees of freedom refer to the number of independent ways a molecule can vibrate, determined by the number of atoms and the molecular structure
• For a non-linear molecule with N atoms, there are 3N - 6 vibrational degrees of freedom, while for a linear molecule, there are 3N - 5 vibrational degrees of freedom
  • Example: H2O (non-linear, N = 3) has 3 × 3 - 6 = 3 vibrational degrees of freedom
  • Example: CO2 (linear, N = 3) has 3 × 3 - 5 = 4 vibrational degrees of freedom

Relationship to normal modes and vibrational spectra

• The number of vibrational degrees of freedom determines the number of normal modes and the complexity of the vibrational spectrum of a molecule
  • Each normal mode corresponds to a specific vibrational degree of freedom
• The vibrational spectrum of a molecule, typically obtained through infrared or Raman spectroscopy, reflects the number and characteristics of the normal modes
  • The number of peaks in the spectrum is related to the number of vibrational degrees of freedom, although some modes may be inactive or degenerate
  • The position, intensity, and shape of the peaks provide information about the frequencies, energies, and nature of the vibrational modes