29.12 EXERCISES
Use this to show that the character of the rotation in a representation based on
the orbital wavefunctions is given by1+2cosφ′+2cos2φ′
and hence that the characters of the representation, in the order of the symbols
given in exercise 29.5, is 5,−1, 1,−1, 1. Deduce that the five-fold degenerate
level is split into two levels, a doublet and a triplet.
29.8 Sulphur hexafluoride is a molecule with the same structure as the crystalline
compound 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 sulphur
atom, together with the irreps according to which they transform under the
symmetry group 432 (orO).
Ψs=f(r)A 1
Ψp 1 =zf(r)T 1
Ψd 1 =(3z^2 −r^2 )f(r)E
Ψd 2 =(x^2 −y^2 )f(r)E
Ψd 3 =xy f(r)T 2
The functionxtransforms according to the irrep T 1. Use the above data to
determine whether dipole matrix elements of the formJ=∫
φ 1 xφ 2 dτcan be
non-zero for the following pairs of orbitalsφ 1 ,φ 2 in a sulphur hexafluoride
molecule: (a) Ψd 1 ,Ψs;(b)Ψd 1 ,Ψp 1 ;(c)Ψd 2 ,Ψd 1 ;(d)Ψs,Ψd 3 ;(e)Ψp 1 ,Ψs.
29.9 The hydrogen atoms in a methane molecule CH 4 form a perfect tetrahedron
with the carbon atom at its centre. The molecule is most conveniently described
mathematically 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 tetrahe-
dral group ( ̄ 43 morTd),hasclassestypifiedbyI,3,2z,mdand 4 ̄z, where the first
three are as in exercise 29.5,mdis a reflection in the mirror planex−y=0and
4 ̄zis a rotation ofπ/2 about thez-axis followed by an inversion in the origin. A
reflection in a mirror plane can be considered as a rotation ofπabout an axis
perpendicular to the plane, followed by an inversion in the origin.
Thecharactertableforthegroup ̄ 43 mis very similar to that for the group
432, and has the form shown in table 29.9.
Typical element and class size Functions transformingIrreps I (^32) z 4 ̄z md according to irrep
1836 6
A 1 1111 1x^2 +y^2 +z^2
A 2 111 − 1 − 1
E 2 −120 0(x^2 −y^2 , 3 z^2 −r^2 )
T 1 30 − 11 − 1 (Rx,Ry,Rz)
T 2 30 − 1 − 11 (x, y, z); (xy , y z , zx)
Table 29.9 The character table for group ̄ 43 m.
By following the steps given below, determine how many different internal vibra-
tion frequencies the CH 4 molecule has.
(a) Consider a representation based on the twelve coordinatesxi,yi,zi for
i=1, 2 , 3 ,4. For those hydrogen atoms that transform into themselves, a
rotation through an angleθabout an axis parallel to one of the coordinate
axes gives rise in the natural representation to the diagonal elements 1 for