Time-Correlation Function of Molecular Random Motion and Shape of Spectral Bands

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Abstract
It is shown that there is a close relation between the spectral density of Brownian motion and the band shape of a spectrum related to a general excitation‐response process, and that time‐correlation functions of a molecular system are obtained from the observed band shape of spectra. The correlation function for rotational Brownian motion of a molecular system is considered theoretically. Two limits of Brownian motion—the inertial limit and the Debye limit—are conveniently assumed. In the inertial limit, the correlation function is composed from two different parts, owing to the discrete nature of fluid in which the molecules are immersed. The first part of the correlation function gives the broadening of spectrum and the second part the narrowing. The relation between molecular motion and infrared spectrum is chosen as an illustration, because it provides simple correlation functions. Consequently, a theory of the band shape of infrared spectrum is presented in terms of the correlation function. The general features of the correlation function obtained from theoretical consideration are consistent with those obtained from the infrared spectrum of trans‐dichloroethylene. In the Debye limit the effect of motion of neighboring molecules is considered. This leads a narrowing in the distribution of correlation times. The result is compared with the distribution function obtained experimentally through the Cole—Cole diagram. The theory of nuclear spin relaxation is examined carefully. It is shown that Schwinger and Bloembergen's theory cannot be applicable to large molecules and that Bloembergen's treatment for the relaxation in the liquid state is not applicable to nonviscous liquid. A more general description is proposed.