Symmetry

Abstract

Symmetry plays an important role in the description of molecular crystals since symmetry considerations permit the computation of bulk properties of the entire crystal from a knowledge of the behavior of a very small region. Group theoretical procedures can be used as is customary for isolated molecules, although there is a relevant difference between the collection of all symmetry elements which leave a given crystal invariant and those which leave a given molecule invariant. For isolated molecules, the set of all possible symmetry elements defines the molecular point group such that at least one point is left fixed in space. The corresponding coordinate transformation can be given in terms of orthogonal matrices representing rotations about some axis, reflections in some plane or combinations of the two. These transformations are usually referred to as (proper or improper) rotations. Rotational symmetry is also present in crystals which are, however, characterized by a structure made up of identical elementary cells, each containing a set of atoms which form the “basis” of the crystal structure and is repeated through space. Translations are thus symmetry operations for crystals. They transform every point into an equivalent one and the collection of all operations (translations, rotations and combinations of the two) which leave the crystal invariant forms a group, the “space group”. Obviously a given crystal cannot be reproduced by any translation, i.e. the elements of the translational subgroup of the space group are only those associated with translational vectors which interconnect points of a three-dimensional lattice.