Mathematical Methods for Physics and Engineering - Matematica.NET

REPRESENTATION THEORY4mm I Q R, R ′ m x , m y m d , m d ′A 1 1 1 1 1 1A 2 1 1 1 −1 −1B 1 1 1 −1 1 −1B 2 1 1 −1 −1 1E 2 −2 0 0 0Table 29.4 The character table deduced for the group 4mm. For an explanationof the entries in bold see the text.with the characters of A 1 requires that1(1)(1) + 1(1)(1) + 2(1)(p) + 2(1)(q) + 2(1)(r) =0.The only possibility is that two of p, q, andr equal −1 and the other equals +1. Thiscan be achieved in three different ways, corresponding to the need to find three furtherdifferent one-dimensional irreps. Thus the first four lines of entries in character table 29.4can be completed. The final line can be completed by requiring it to be orthogonal to theother four. Property (v) has not been used here though it could have replaced part of theargument given. ◭29.9 Group nomenclatureThe nomenclature of published character tables, as we have said before, is erraticand sometimes unfortunate; for example, often E is used to represent, not onlya two-dimensional irrep, but also the identity operation, where we have used I.Thus the symbol E might appear in both the column and row headings of atable, though with quite different meanings in the two cases. In this book we useroman capitals to denote irreps.One-dimensional irreps are regularly denoted by A and B, B being used if arotation about the principal axis of 2π/n has character −1. Here n is the highestinteger such that a rotation of 2π/n is a symmetry operation of the system, andthe principal axis is the one about which this occurs. For the group of operationson a square, n = 4, the axis is the perpendicular to the square and the rotationin question is R. The names for the group, 4mm and C 4v , derive from the factthat here n is equal to 4. Similarly, for the operations on an equilateral triangle,n = 3 and the group names are 3m and C 3v , but because the rotation by 2π/3 hascharacter +1 in all its one-dimensional irreps (see table 29.1), only A appears inthe irrep list.Two-dimensional irreps are denoted by E, as we have already noted, and threedimensionalirreps by T, although in many cases the symbols are modified byprimes and other alphabetic labels to denote variations in behaviour from oneirrep to another in respect of mirror reflections and parity inversions. In the studyof molecules, alternative names based on molecular angular momentum propertiesare common. It is beyond the scope of this book to list all these variations, or to1102