THE ABSOLUTELY INEQUIVALENT PURE INTERCHANGE SYMMETRY FOR MOLECULES AND CRYSTALS

Abstract

The interchange and transport symmetries have been discussed $previously,^{1, 2}$ It turns out that while all special cases encountered before were treated correctly a refinement of terms is necessary for the most general case. An interchange operation is a geometrical covering operation that interchanges $nuclci.^{1}$ A pure interchange operation moves all nuclei within an orbit (set of equivalent molecules, or molecular fragments, or nuclei).$^{1}$ A set of inequivalent pure interchange operations is made of elements modulo pure fixing operations (a fixing operation fixes some nuclei, a pure fixing operation fixes all nuclci). An absolutely inequivalent pure interchange set is made of elements modulo any fixing operations. Two exact theorems were derived: a. An orbit of order n has at most (n-1) absolutely inequivalent pure interchange operations. b. Any pure interchange group (order n) is necessarily made of (n-1) absolutely inequivalent interchange operations and the identity. The utility of the above concepts will be discussed with respect to molecular crystals, with emphasis on exciton functions. $^{1}$ R. Kopelman, J. Chem. Phys. 47, 2631 (1967). $^{2}$ R. Kopelman, This Symposium, Paper R6 (1967).