Molecular Reorientation in Liquids. I. Distribution Functions and Friction Constants

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Abstract
The orientational motion of rigid molecules in liquids obeying classical mechanics is considered. The equations of motion of these systems are Euler's equations. It is assumed that the torque on a molecule can be represented by a friction term proportional to angular velocity plus a randomly fluctuating torque. It is then shown that the equations of motion for a spherical top have solutions of the form of the solutions of the Langevin equation. The components of the diagonalized rotational friction constant tensor are given as time integrals of the autocorrelation functions of the components of the angular velocity and an approximate expression is obtained which gives the rotational friction constants in terms of ensemble averages of the angular derivatives of the intermolecular potential function. A differential equation is derived for the time‐dependent distribution function for the Eulerian angle displacements. The solutions of this equation can be written in terms of the solutions of the Schrödinger equation for the asymmetric top multiplied by exponentially decaying functions of time. Molecules possessing elements of symmetry in their interaction potential functions are considered, and it is shown that the components of the rotational friction tensor approach zero in these systems and that the rotational motion over relatively long time intervals is then determined by the inertial terms in the equation of motion. The application of these results to the theory of rotational nuclear magnetic and dielectric relaxation in liquids is briefly discussed.