Mathematical Methods for Physics and Engineering - Matematica.NET

29.10 PRODUCT REPRESENTATIONSgive a large selection of character tables; our aim is to demonstrate and justifythe use of those found in the literature specifically dedicated to crystal physics ormolecular 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 henceall members, of a conjugacy class in a group. In physics these are usually of thetypes n z , ¯n z or m x . The first of these denotes a rotation of 2π/n about the z-axis,and the second the same thing followed by parity inversion (all vectors r go to−r), whilst the third indicates a mirror reflection in a plane, in this case the planex =0.Typical chemistry symbols for classes are NC n , NC 2 n, NC x n , NS n , σ v , σ xy .Herethe first symbol N, where it appears, shows that there are N elements in theclass (a useful feature). The subscript n has the same meaning as in the physicsnotation, but σ rather than m is used for a mirror reflection, subscripts v, d or h orsuperscripts xy, xz or yz denoting the various orientations of the relevant mirrorplanes. Symmetries involving parity inversions are denoted by S; thus S n is thechemistry analogue of ¯n. None of what is said in this and the previous paragraphshould be taken as definitive, but merely as a warning of common variations innomenclature and as an initial guide to corresponding entities. Before using anyset of group character tables, the reader should ensure that he or she understandsthe precise notation being employed.29.10 Product representationsIn quantum mechanical investigations we are often faced with the calculation ofwhat are called matrix elements. These normally take the form of integrals over allspace of the product of two or more functions whose analytic forms depend on themicroscopic properties (usually angular momentum and its components) of theelectrons or nuclei involved. For ‘bonding’ calculations involving ‘overlap integrals’there are usually two functions involved, whilst for transition probabilities a thirdfunction, giving the spatial variation of the interaction Hamiltonian, also appearsunder the integral sign.If the environment of the microscopic system under investigation has somesymmetry properties, then sometimes these can be used to establish, withoutdetailed evaluation, that the multiple integral must have zero value. We nowexpress the essential content of these ideas in group theoretical language.Suppose we are given an integral of the form∫∫J = Ψφdτ or J = Ψξφdτto be evaluated over all space in a situation in which the physical system isinvariant under a particular group G of symmetry operations. For the integral to1103