Infrared

The energies of photons in the infrared region of the spectrum correspond to the separation of vibrational energy levels in molecules. Absorption in the infrared can occur if two criteria are met:

• The product of Planck's constant and frequency must correspond to the difference in energy of the ground vibrational state and the first excited state.
• The vibration in the molecule is accompanied by an oscillating dipole moment.
  • This criteria is met in heteronuclear diatomics (such as HCl) but not in homonuclear diatomics (such as N₂ or O₂). Therefore atmospheric nitrogen and oxygen do not interfere with the measurement.
  • This selection rule changes if the vibrational transitions are induced by scattering, as in a Raman measurement. In the scattering experiment, we require an oscillating polarizability. Some vibrational transitions are invisible in the absorption measurement, but can be observed in the Raman measurement.

Theory for Simple Vibrations

The spacing of the energy levels for a vibration can be extracted from the Schroedinger Equation for a few cases in which the potential energy takes a simple form. Fortunately, two of these simple forms are sufficiently close to the real situation encountered in molecules so that they are of great use in interpreting the ir spectra of polymers.

• Harmonic oscillator, with potential energy given by Hooke's Law
• Anharmonic oscillator, with a more complicated (and more realistic, for covalent bonds) Morse potential

Normal Modes

In three-dimensional space, the N atoms in a molecule have 3N degrees of freedom. Three degrees of freedom define the center of mass. The orientation of the molecule in space is defined by an additional 2 or 3 degrees of freedom, depending on whether the molecule is linear or nonlinear. The remaining degrees of freedom are internal. Thus there are 3N - 5 or 3N - 6 internal modes, depending of whether the molecule is linear or nonlinear. Some of these modes may have the same frequency. Examples:

• Diatomic: one vibrational mode (bond stretching). For diatomic hydrogen, the time for 1 vibration is 8 fs.
• Nonlinear triatomic, such as water. Three vibrational modes, all of different energy. The times for one vibration are in the range 9-21 fs
  • Bond angle bending
  • Asymmetric stretch
  • Symmetric stretch
• Linear triatomic, such as carbon dioxide (four vibrational modes, two of which have the same energy). The times for one vibration are in the range 14-50 fs.
  • Bond angle bending (two modes, of the same energy, for bending in two planes oriented at 90 deg to one another)
  • Asymmetric stretch
  • Symmetric stretch

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