Mathematical Methods for Physics and Engineering - Matematica.NET

29.12 EXERCISESUse this to show that the character of the rotation in a representation based onthe orbital wavefunctions is given by1+2cosφ ′ +2cos2φ ′and hence that the characters of the representation, in the order of the symbolsgiven in exercise 29.5, is 5, −1, 1, −1, 1. Deduce that the five-fold degeneratelevel is split into two levels, a doublet and a triplet.29.8 Sulphur hexafluoride is a molecule with the same structure as the crystallinecompound in exercise 29.7, except that a sulphur atom is now the central atom.The following are the forms of some of the electronic orbitals of the sulphuratom, together with the irreps according to which they transform under thesymmetry group 432 (or O).Ψ s = f(r) A 1Ψ p1 = zf(r) T 1Ψ d1 =(3z 2 − r 2 )f(r) EΨ d2 =(x 2 − y 2 )f(r) EΨ d3 = xyf(r) T 2The function x transforms according to the irrep T 1 . Use the above data todetermine whether dipole matrix elements of the form J = ∫ φ 1 xφ 2 dτ can benon-zero for the following pairs of orbitals φ 1 ,φ 2 in a sulphur hexafluoridemolecule: (a) Ψ d1 , Ψ s ;(b)Ψ d1 , Ψ p1 ;(c)Ψ d2 , Ψ d1 ;(d)Ψ s , Ψ d3 ;(e)Ψ p1 , Ψ s .29.9 The hydrogen atoms in a methane molecule CH 4 form a perfect tetrahedronwith the carbon atom at its centre. The molecule is most conveniently describedmathematically by placing the hydrogen atoms at the points (1, 1, 1), (1, −1, −1),(−1, 1, −1) and (−1, −1, 1). The symmetry group to which it belongs, the tetrahedralgroup (¯43m or T d ),hasclassestypifiedbyI, 3,2 z , m d and ¯4 z , where the firstthree are as in exercise 29.5, m d is a reflection in the mirror plane x − y =0and¯4 z is a rotation of π/2 about the z-axis followed by an inversion in the origin. Areflection in a mirror plane can be considered as a rotation of π about an axisperpendicular to the plane, followed by an inversion in the origin.Thecharactertableforthegroup¯43m is very similar to that for the group432, and has the form shown in table 29.9.Typical element and class size Functions transformingIrreps I 3 2 z¯4z m d according to irrep1 8 3 6 6A 1 1 1 1 1 1 x 2 + y 2 + z 2A 2 1 1 1 −1 −1E 2 −1 2 0 0 (x 2 − y 2 , 3z 2 − r 2 )T 1 3 0 −1 1 −1 (R x ,R y ,R z )T 2 3 0 −1 −1 1 (x, y, z); (xy, yz, zx)Table 29.9The character table for group ¯43m.By following the steps given below, determine how many different internal vibrationfrequencies the CH 4 molecule has.(a) Consider a representation based on the twelve coordinates x i ,y i ,z i fori =1, 2, 3, 4. For those hydrogen atoms that transform into themselves, arotation through an angle θ about an axis parallel to one of the coordinateaxes gives rise in the natural representation to the diagonal elements 1 for1115