Group theory and vibrational spectroscopy

Abstract

The energy of a molecule can be approximated as E = E rot + E vib + E elec + E tran + E spin + E nucl . If we make the approximation that the wavefunction is separable, then the problem can be reduced to several smaller tasks. This amounts to the assumption of the Born-Oppenheimer approximation, which permits the nuclear and electronic motions to be analyzed separately. In fact, this approximation breaks down, and leads to the Jahn-Teller phenomenon and other vibronic effects. Let us now address the pure vibrational problem. Exchange of energy between a molecule and the electromagnetic field occurs when hn = ∆E, where ∆E is the difference between initial and final quantized states. In terms of energies n = c l cm/sec cm Hz = n _ n cm -1 1 l = = = c Infrared absorption spectra usually cover the range 200-4000 cm -1 or 50-2.5 micrometers (microns). The conversion factor 1 e.v. = 8066 cm -1 or 23 kcal/mole is also useful to remember. Translational energies are about 200 cm -1 at room temperature and rotational energies are 1-100 cm -1 . Most infrared spectrometers provide the spectrum in the form of % transmittance vs. wavenumber. Transmittance spectra tend to emphasize weak absorptions in the spectrum. Spectra of the same sample recorded at different concentrations will have different relative peak heights, when displayed as % transmittance. Conversion to absorbance spectra (absorbance = -log 10 (transmittance) is an option available on most spectrometers. Absorbance spectra should be used whenever peak ratios or concentration information is desired (e.g., in kinetics, where the decrease or increase of concentration must be monitored). The other important aspect to IR and Raman spectroscopy is that the time scale of the measurement amounts to the time it takes for a vibration (~ 0.1 psec). Even rapidly isomerizing species show distinct vibrational spectra in contrast to slower techniques, such as NMR spectroscopy. Figure 5.1 shows the vibrational potential energy surface for a harmonic oscillator (A)