29.10 PRODUCT REPRESENTATIONS
give a large selection of character tables; our aim is to demonstrate and justify
the use of those found in the literature specifically dedicated to crystal physics or
molecular chemistry.
Variations in notation are not restricted to the naming of groups and theirirreps, but extend to the symbols used to identify a typical element, and hence
all members, of a conjugacy class in a group. In physics these are usually of the
typesnz, ̄nzormx. The first of these denotes a rotation of 2π/nabout thez-axis,
and the second the same thing followed by parity inversion (all vectorsrgo to
−r), whilst the third indicates a mirror reflection in a plane, in this case the plane
x=0.
Typical chemistry symbols for classes areNCn,NC^2 n,NCnx,NSn,σv,σxy.Herethe first symbolN, where it appears, shows that there areNelements in the
class (a useful feature). The subscriptnhas the same meaning as in the physics
notation, butσrather thanmis used for a mirror reflection, subscriptsv,dorhor
superscriptsxy,xzoryzdenoting the various orientations of the relevant mirror
planes. Symmetries involving parity inversions are denoted byS; thusSnis the
chemistry analogue of ̄n. None of what is said in this and the previous paragraph
should be taken as definitive, but merely as a warning of common variations in
nomenclature and as an initial guide to corresponding entities. Before using any
set of group character tables, the reader should ensure that he or she understands
the precise notation being employed.
29.10 Product representationsIn quantum mechanical investigations we are often faced with the calculation of
what are called matrix elements. These normally take the form of integrals over all
space of the product of two or more functions whose analytic forms depend on the
microscopic properties (usually angular momentum and its components) of the
electrons or nuclei involved. For ‘bonding’ calculations involving ‘overlap integrals’
there are usually two functions involved, whilst for transition probabilities a third
function, giving the spatial variation of the interaction Hamiltonian, also appears
under the integral sign.
If the environment of the microscopic system under investigation has somesymmetry properties, then sometimes these can be used to establish, without
detailed evaluation, that the multiple integral must have zero value. We now
express the essential content of these ideas in group theoretical language.
Suppose we are given an integral of the formJ=∫
Ψφdτ or J=∫
Ψξφ dτto be evaluated over all space in a situation in which the physical system is
invariant under a particular groupGof symmetry operations. For the integral to