Exercises

29.12 EXERCISES
as the sum of two one-dimensional irreps and, using the reasoning given in the previous
example, are therefore split in frequency by the perturbation. For other values of n the
representation is irreducible and so the degeneracy cannot be split. 29.12 Exercises
29.1
29.2
29.3
A group G has four elements I, X, Y and Z, which satisfy X 2 = Y 2 = Z 2 =
XY Z = I. Show that G is Abelian and hence deduce the form of its character
table.
Show that the matrices
1 0
−1
0
D(I) =
,
D(X) =
,
0 1
0
−1
−1 −p
1
p
D(Y ) =
,
D(Z) =
,
0
1
0 −1
where p is a real number, form a representation D of G. Find its characters and
decompose it into irreps.
Using a square whose corners lie at coordinates (±1, ±1), form a natural representation of the dihedral group D4 . Find the characters of the representation,
and, using the information (and class order) in table 29.4 (p. 1102), express the
representation in terms of irreps.
Now form a representation in terms of eight 2 × 2 orthogonal matrices, by
considering the effect of each of the elements of D4 on a general vector (x, y).
Confirm that this representation is one of the irreps found using the natural
representation.
The quaternion group Q (see exercise 28.20) has eight elements {±1, ±i, ±j, ±k}
obeying the relations
i2 = j 2 = k 2 = −1,
29.4
ij = k = −ji.
Determine the conjugacy classes of Q and deduce the dimensions of its irreps.
Show that Q is homomorphic to the four-element group V, which is generated by
two distinct elements a and b with a2 = b2 = (ab)2 = I. Find the one-dimensional
irreps of V and use these to help determine the full character table for Q.
Construct the character table for the irreps of the permutation group S4 as
follows.
(a) By considering the possible forms of its cycle notation, determine the number
of elements in each conjugacy class of the permutation group S4 , and show
that S4 has five irreps. Give the logical reasoning that shows they must consist
of two three-dimensional, one two-dimensional, and two one-dimensional
irreps.
(b) By considering the odd and even permutations in the group S4 , establish the
characters for one of the one-dimensional irreps.
(c) Form a natural matrix representation of 4 × 4 matrices based on a set of
objects {a, b, c, d}, which may or may not be equal to each other, and, by
selecting one example from each conjugacy class, show that this natural
representation has characters 4, 2, 1, 0, 0. In the four-dimensional vector
space in which each of the four coordinates takes on one of the four values
a, b, c or d, the one-dimensional subspace consisting of the four points with
coordinates of the form {a, a, a, a} is invariant under the permutation group
and hence transforms according to the invariant irrep A1 . The remaining
three-dimensional subspace is irreducible; use this and the characters deduced
above to establish the characters for one of the three-dimensional irreps, T1 .
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REPRESENTATION THEORY
(d) Complete the character table using orthogonality properties, and check the
summation rule for each irrep. You should obtain table 29.8.
Irrep
A1
A2
E
T1
T2
(1)
1
1
1
2
3
3
Typical element and class size
(12) (123) (1234) (12)(34)
6
8
6
3
1
1
1
1
−1
1
−1
1
0
−1
0
2
1
0
−1
−1
−1
0
1
−1
Table 29.8 The character table for the permutation group S4 .
29.5
In exercise 28.10, the group of pure rotations taking a cube into itself was found
to have 24 elements. The group is isomorphic to the permutation group S4 ,
considered in the previous question, and hence has the same character table, once
corresponding classes have been established. By counting the number of elements
in each class, make the correspondences below (the final two cannot be decided
purely by counting, and should be taken as given).
Permutation
class type
(1)
(123)
(12)(34)
(1234)
(12)
29.6
29.7
Symbol
(physics)
I
3
2z
4z
2d
Action
none
rotations about a body diagonal
rotation of π about the normal to a face
rotations of ±π/2 about the normal to a face
rotation of π about an axis through the
centres of opposite edges
Reformulate the character table 29.8 in terms of the elements of the rotation
symmetry group (432 or O) of a cube and use it when answering exercises 29.7
and 29.8.
Consider a regular hexagon orientated so that two of its vertices lie on the x-axis.
Find matrix representations of a rotation R through 2π/6 and a reflection my in
the y-axis by determining their effects on vectors lying in the xy-plane . Show
that a reflection mx in the x-axis can be written as mx = my R 3 , and that the 12
elements of the symmetry group of the hexagon are given by R n or R n my .
Using the representations of R and my as generators, find a two-dimensional
representation of the symmetry group, C6 , of the regular hexagon. Is it a faithful
representation?
In a certain crystalline compound, a thorium atom lies at the centre of a regular
octahedron of six sulphur atoms at positions (±a, 0, 0), (0, ±a, 0), (0, 0, ±a). These
can be considered as being positioned at the centres of the faces of a cube of
side 2a. The sulphur atoms produce at the site of the thorium atom an electric
field that has the same symmetry group as a cube (432 or O).
The five degenerate d-electron orbitals of the thorium atom can be expressed,
relative to any arbitrary polar axis, as
(3 cos2 θ − 1)f(r),
e±iφ sin θ cos θf(r),
e±2iφ sin2 θf(r).
A rotation about that polar axis by an angle φ effectively changes φ to φ − φ .
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29.12 EXERCISES
Use this to show that the character of the rotation in a representation based on
the orbital wavefunctions is given by
1 + 2 cos φ + 2 cos 2φ
29.8
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.
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 (or O).
Ψs = f(r)
Ψp1 = zf(r)
Ψd1 = (3z 2 − r2 )f(r)
Ψd2 = (x2 − y 2 )f(r)
Ψd3 = xyf(r)
29.9
A1
T1
E
E
T2
The function x transforms according to the irrep T1 . Use the
above data to
determine whether dipole matrix elements of the form J = φ1 xφ2 dτ can be
non-zero for the following pairs of orbitals φ1 , φ2 in a sulphur hexafluoride
molecule: (a) Ψd1 , Ψs ; (b) Ψd1 , Ψp1 ; (c) Ψd2 , Ψd1 ; (d) Ψs , Ψd3 ; (e) Ψp1 , Ψs .
The hydrogen atoms in a methane molecule CH4 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 tetrahedral group (4̄3m or Td ), has classes typified by I, 3, 2z , md and 4¯z , where the first
three are as in exercise 29.5, md is a reflection in the mirror plane x − y = 0 and
4¯z is a rotation of π/2 about the z-axis followed by an inversion in the origin. A
reflection in a mirror plane can be considered as a rotation of π about an axis
perpendicular to the plane, followed by an inversion in the origin.
The character table for the group 4̄3m is very similar to that for the group
432, and has the form shown in table 29.9.
Irreps
Typical element and class size
I
3
2z
md
4¯z
1
8
3
6
6
Functions transforming
according to irrep
A1
A2
E
T1
T2
1
1
2
3
3
x2 + y 2 + z 2
1
1
−1
0
0
1
1
2
−1
−1
1
−1
0
1
−1
1
−1
0
−1
1
(x2 − y 2 , 3z 2 − r2 )
(Rx , Ry , Rz )
(x, y, z); (xy, yz, zx)
Table 29.9 The character table for group 4̄3m.
By following the steps given below, determine how many different internal vibration frequencies the CH4 molecule has.
(a) Consider a representation based on the twelve coordinates xi , yi , zi for
i = 1, 2, 3, 4. For those hydrogen atoms that transform into themselves, a
rotation through an angle θ about an axis parallel to one of the coordinate
axes gives rise in the natural representation to the diagonal elements 1 for
1115
REPRESENTATION THEORY
the corresponding coordinate and 2 cos θ for the two orthogonal coordinates.
If the rotation is followed by an inversion then these entries are multiplied
by −1. Atoms not transforming into themselves give a zero diagonal contribution. Show that the characters of the natural representation are 12, 0, 0,
0, 2 and hence that its expression in terms of irreps is
A1 ⊕ E ⊕ T1 ⊕ 2T2 .
(b) The irreps of the bodily translational and rotational motions are included in
this expression and need to be identified and removed. Show that when this
is done it can be concluded that there are three different internal vibration
frequencies in the CH4 molecule. State their degeneracies and check that
they are consistent with the expected number of normal coordinates needed
to describe the internal motions of the molecule.
29.10
Investigate the properties of an alternating group and construct its character
table as follows.
(a) The set of even permutations of four objects (a proper subgroup of S4 )
is known as the alternating group A4 . List its twelve members using cycle
notation.
(b) Assume that all permutations with the same cycle structure belong to the
same conjugacy class. Show that this leads to a contradiction, and hence
demonstrates that, even if two permutations have the same cycle structure,
they do not necessarily belong to the same class.
(c) By evaluating the products
p1 = (123)(4) • (12)(34) • (132)(4) and p2 = (132)(4) • (12)(34) • (123)(4)
deduce that the three elements of A4 with structure of the form (12)(34)
belong to the same class.
(d) By evaluating products of the form (1α)(βγ) • (123)(4) • (1α)(βγ), where α, β, γ
are various combinations of 2, 3, 4, show that the class to which (123)(4)
belongs contains at least four members. Show the same for (124)(3).
(e) By combining results (b), (c) and (d) deduce that A4 has exactly four classes,
and determine the dimensions of its irreps.
(f) Using the orthogonality properties of characters and noting that elements of
the form (124)(3) have order 3, find the character table for A4 .
29.11
29.12
Use the results of exercise 28.23 to find the character table for the dihedral group
D5 , the symmetry group of a regular pentagon.
Demonstrate that equation (29.24) does, indeed, generate a set of vectors transforming according to an irrep λ, by sketching and superposing drawings of an
equilateral triangle of springs and masses, based on that shown in figure 29.5.
C
A
30◦
C
B
(a)
C
B 30◦
A
(b)
A
B
(c)
Figure 29.7 The three normal vibration modes of the equilateral array. Mode
(a) is known as the ‘breathing mode’. Modes (b) and (c) transform according
to irrep E and have equal vibrational frequencies.
1116