Vibrational Hot Bands
In the harmonic approximation, the normal modes of a molecule are not coupled, and all vibrational quantum levels are equally spaced, so hot bands would not be distinguishable from so-called "fundamental" transitions arising from the overall vibrational ground state. However, vibrations of real molecules always have some anharmonicity, which causes coupling between different vibrational modes that in turn shifts the observed frequencies of hot bands in vibrational spectra. Because anharmonicity decreases the spacing between adjacent vibrational levels, hot bands exhibit red shifts (appear at lower frequencies) than the corresponding fundamental transitions. The magnitude of the observed shift is correlated to the degree of anharmonicity in the corresponding normal modes.
Both the lower and upper states involved in the transition are excited states. Therefore, the lower excited state must be populated for a hot band to be observed. The most common form of excitation is by thermal energy. The population of the lower excited state is then given by the Boltzmann distribution. In general the population can be expressed as
where kB is the Boltzmann constant and E is the energy difference between the two states. In simplified form this can be expressed as
where ν is the wavenumber of the hot band and T is the temperature . Thus, the intensity of a hot band, which is proportional to the population of the lower excited state, increases as the temperature increases.
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