Mathematical Methods for Physics and Engineering - Matematica.NET

29.11 PHYSICAL APPLICATIONS OF GROUP THEORY4mm I Q R, R ′ m x , m y m d , m d ′A 1 1 1 1 1 1 z; z 2 ; x 2 + y 2A 2 1 1 1 −1 −1 R zB 1 1 1 −1 1 −1 x 2 − y 2B 2 1 1 −1 −1 1 xyE 2 −2 0 0 0 (x, y); (xz, yz); (R x ,R y )Table 29.5 The character table for the irreps of group 4mm (or C 4v ). Theright-hand column lists some common functions, or, for the two-dimensionalirrep E, pairs of functions, that transform according to the irrep against whichthey are shown.Function Irrep ClassesI 2C 3 3σ vxy E 2 −1 0x E 2 −1 0x 2 − y 2 E 2 −1 0product 8 −1 0Table 29.6 The character sets, for the group C 3v (or 3mm), of three functionsand of their product x 2 y(x 2 − y 2 ).Function Irrep ClassesI C 2 2C 6 2σ v 2σ dxy B 2 1 1 −1 −1 1x E 2 −2 0 0 0x 2 − y 2 B 1 1 1 −1 1 −1product 2 −2 0 0 0Table 29.7 The character sets, for the group C 4v (or 4mm), of three functions,and of their product x 2 y(x 2 − y 2 ).multiplying together the corresponding characters for each of the three elements. Now, byinspection, or by applying (29.18), i.e.m A1 = 1 [1(1)(8) + 2(1)(−1) + 3(1)(0)] = 1,6we see that irrep A 1 does appear in the reduced representation of the product, and so Jis not necessarily zero.Case (ii). From table 29.5 we find that, under the group C 4v , xy and x 2 − y 2 transformas irreps B 2 and B 1 respectively and that x is part of a basis set transforming as E. Thusthe calculation table takes the form of table 29.7 (again, chemical notation for the classeshas been used).Here inspection is sufficient, as the product is exactly that of irrep E and irrep A 1 iscertainly not present. Thus J is necessarily zero and the dipole matrix element vanishes. ◭1109