Diffusion of an asymmetric-top molecule in three dimensions. Langevin equations and new, one-particle cross-correlation functions

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Abstract
Langevin equations are developed for an asymmetric-top molecule rotating and translating as a Brownian particle in three dimensions. The translational Langevin equation is written in a rotating frame of reference (1, 2, 3)′ whose origin remains fixed at the origin of the laboratory frame (x, y, z). On the other hand, the rotational Langevin equation is written, as usual, in a moving frame (1, 2, 3), that of the molecular principal moments of inertia. There appears in both equations a common deterministic variable, ω, the molecular angular velocity, together with terms such as the molecular Coriolis and centripetal accelerations, which are shown to exist in the laboratory frame of reference (x, y, z). The structure of the two equations suggests the existence of numerous, hitherto unknown, single-molecule cross-correlation functions, both in frame (x, y, z) and frame (1, 2, 3). This is confirmed in this paper by computer simulation of two new types of cross-correlation function involving v, the centre-of-mass linear velocity of the molecule (i.e. a Brownian particle) and its own angular velocity, ω. By constructing vector and tensor products, the symmetry properties of these cross-correlation functions are computed and tabulated, both in the absence and presence of intense external electric field of force.

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