Mathematical Methods for Physics and Engineering - Matematica.NET

REPRESENTATION THEORY(i) Under I all eight basis functions are unchanged, and χ(I) =8.(ii) The rotations R, R ′ and Q change the value of N in every case and soall diagonal elements of the natural representation are zero and χ(R) =χ(Q) =0.(iii) m x takes x into −x and y into y and, for N = 1 and 3, leaves N unchanged,with the consequences (remember the forms of Ψ x (N) andΨ y (N)) thatΨ x (1) →−Ψ x (1), Ψ x (3) →−Ψ x (3),Ψ y (1) → Ψ y (1), Ψ y (3) → Ψ y (3).Thus χ(m x ) has four non-zero contributions, −1, −1, 1 and 1, togetherwith four zero contributions. The total is thus zero.(iv) m d and m d ′ leave no atom unchanged and so χ(m d )=0.The character set of the natural representation is thus 8, 0, 0, 0, 0, which, eitherby inspection or by applying formula (29.18), shows thatD nat =A 1 ⊕ A 2 ⊕ B 1 ⊕ B 2 ⊕ 2E,i.e. that all possible irreps are present. We have constructed previously thecombinations of Ψ x (N) and Ψ y (N) that transform according to A 1 and B 1 .The others can be found in the same way.29.11.2 Matrix elements in quantum mechanicsIn section 29.10 we outlined the procedure for determining whether a matrixelement that involves the product of three factors as an integrand is necessarilyzero. We now illustrate this with a specific worked example.◮Determine whether a ‘dipole’ matrix element of the form∫J = Ψ d1 xΨ d2 dτ,where Ψ d1 and Ψ d2 are d-state wavefunctions of the forms xyf(r) and (x 2 − y 2 )g(r) respectively,can be non-zero (i) inamoleculewithsymmetryC 3v (or 3m), such as ammonia, and(ii) in a molecule with symmetry C 4v (or 4mm), such as the MnI 4 molecule considered inthe previous example.We will need to make reference to the character tables of the two groups. The table forC 3v is table 29.1 (section 29.6); that for C 4v is reproduced as table 29.5 from table 29.4 butwith the addition of another column showing how some common functions transform.We make use of (29.23), extended to the product of three functions. No attention needbe paid to f(r) andg(r) as they are unaffected by the group operations.Case (i). From the character table 29.1 for C 3v , we see that each of xy, x and x 2 − y 2forms part of a basis set transforming according to the two-dimensional irrep E. Thus wemay fill in the array of characters (using chemical notation for the classes, except thatwe continue to use I rather than E) as shown in table 29.6. The last line is obtained by1108