Ensemble averages and molecular symmetry

Abstract

It is shown that a simple assumption enables ensemble average tensor components, including time correlation functions, on molecular or laboratory axes, to be identified as zero or else as potentially non-zero. The tensors may refer to any properties and are illustrated predominantly by linear and angular velocities and their products. The assumption is that the properties of a molecule subject to intermolecular forces in the liquid, solution or dense gas states, have ensemble averages based on molecular axes which reduce as fully symmetric representations of the molecular point group. These quantities can be related to laboratory axes by projections as indicated, but the tensors are then of higher order. The zero elements are not of great importance, save as a guide to the avoidance of computer simulations which lead to zero, whereas the potentially non-zero features have a more general interest. There are normally few such terms with molecules of high symmetry. Tables are given relating to molecular symmetries C₂, C_2h, C_2v, C_3v and T_d as examples and extend to fourth order cartesian tensors. The understanding of a molecular level of some non-zero terms is discussed. The number of independent auto-correlation functions derived from two vectors is discussed in an appendix and it is shown that, in some point groups, such functions are not independent of other correlation functions.