Research: Science & Education
On the Character Tables of Finite Point Groups Ivan Baraldi and Davide Vanossi Dipartimento di Chimica, Universita di Modena, Via Campi 183, 41100 Modena, Italy The table of characters of a finite group G = {Kj j = 1, 2, … , r} of order g separated into r classes Kj is presented in the form
Γ(2)
K 1;E 1 f2
:
G Γ(1)
K2 … 1 … ( 2 ) χ ( K 2) …
:
:
Γ(i )
fi
χ(i )(K
:
:
:
fr
χ(r )(K
Γ(r )
2)
Kj 1 ( 2 χ )(Kj )
…
…
:
…
:
…
χ(i )(Kj )
…
χ(i )(Kr )
:
…
:
χ(r )(Kj )
…
χ(r )(Kr )
… 2) …
Kr 1 ( 2 … χ )(Kr )
…
ters of the 1-dimensional irreps (A, B), while for multidimensional ones the situation is much more complex. In the latter case, it is necessary to resolve some relations between unitary and periodic matrices of sizes n × n with n = 2, 3, 4, 5. Such matrices have the property that |det(D)| = 1; that is, det(D) = exp(iΘ), where D indicates a generic unitary matrix and Θ the argument (a real number). In particular, det(D) = ±1 for the orthogonal matrices: +1 corresponds to a proper rotation and {1 to an improper one. Dealing with the proper groups, the matrices of the 2-dimensional (E) and 3-dimensional (T) irreps coincide, in general, with rotation matrices in the 2-dimensional E(2) and 3-dimensional E(3) Euclidean space, respectively. The traces of these matrices are χ_C(ω)+ = 2 cosω
Γ(i)
the where K j indicates the jth class of rj elements Gj, ith irreducible representation (irrep) of dimension f i (Γ(1) is the totally symmetric representation), and χ(i) (Kj) the character of the ith irrep corresponding to the jth class. In the following the dimension of irrep is also indicated as f. The elements of this table can be considered as forming the r × r square matrix Χ(G) = [χ(i) (K j)]. In the case of point groups, such tables are of fundamental importance in the applications of group theory to quantum chemistry and to molecular spectroscopy. Think, for example, of the reduction of one representation into its irreducible components and the determination of projection operators. For this reason, ample collections of these tables have been made (1–3). One of the more recent and extensive (Altmann and Herzig [4]) also contains other useful tables for the point groups. The general properties used to determine the character tables of the finite groups are widely discussed in several books (5–7). See, in particular, that of Lomont (7). In the case of finite point groups, it is well known that it is sufficient to determine the character tables of the five nonisomorphic rotation groups Cn, Dn, T, O, and I,1 the finite subgroups of the group of the rotations in 3-dimensional space (K). Those of the remaining finite point groups (S2n , Cnh, Cnv, Dnh, Dnd, Th, Td, Oh , Ih) are obtained through isomorphisms (S2n ≅ C2n, Cnv ≅ Dn, Dnd ≅ D2n, Td ≅ O) and direct products (Cnh = Ci × Cn [n even], Cnh = Cs × Cn [n odd], Dnh = Ci × Dn [n even], Dnh = Cs × Dn [n odd], Th = Ci × T, Oh = Ci × O, Ih = Ci × I). In fact, the character tables of the isomorphic groups are the same, and those obtained from the direct product have the form
Χ G′ Χ G = 1 1 × Χ G′ = Χ G′ 1 {1
Χ G′ {Χ G′
(1)
with G′ = Cn, Dn, T, O, I. In this paper we pursue the study of the theory of symmetry point groups based, where possible, in terms of the generator theory (8, 9). In more detail, we will obtain the character tables of the proper groups and, as a consequence, of all finite point groups. This topic is usually taught in an incomplete way. The defining relations of the group are easily applicable to the determination of the number and charac-
806
(2)
in E(2) for a rotation about an axis perpendicular to the plane, and χ_C(ω)+ = 1 + 2 cosω
(3)
in E(3) for any rotation about any axis passing through the center of the Cartesian coordinate, where in both expressions ω indicates the rotation angle and C(ω) a rotation of ω. For ω = 2π/n , the rotation is indicated as C n; and in what follows, the identity is represented by E. As regards the characters of the 5-dimensional irrep (H) of the icosahedral group I, their values can be calculated with the formula χ_C(ω)+ = 1+ 2 cosω + 2 cos2ω
(4)
Such a formula gives the trace of the rotation matrix of the five d orbitals (dz2 , dx2{y 2, dxy, dxz, d yz), which form a basis for the H representation. The missing characters of the multidimensional irreps can be determined by using the general properties. Some of these characters, in particular those of the second 3-dimensional irrep of O and those of the 4dimensional irrep (G) of I, can be determined by exploiting the following isomorphisms: O ≅ S4 and I ≅ A5, where Sn is the symmetric group and An its alternating subgroup. For Sn groups, with n > 2, and An, with n > 3, there exists an irrep of degree n – 1 whose characters are given by the formula χ(Ki) = N(Ki) – 1
(5)
where N(K i) is the number of basis vectors of the natural representation that remain fixed under the action of one of the symmetry operations belonging to the class Ki and that in turn is equal to the number of objects unaffected by one of the permutations corresponding to the symmetry operations of K i. Such a formula is a consequence of the reduction of the natural representation of a 2-transitive permutation group in the sum of the unit representation and an irrep. On this subject see the books by Kostrikin (10) and Burrow (11). Let us now determine the character tables of the proper rotation groups, remembering that the nomenclature of the irreps is the one developed by R. S. Mulliken (Newburyport 1896–Arlington 1986), who received the Nobel prize for chemistry in 1966.
Journal of Chemical Education • Vol. 74 No. 7 July 1997
Research: Science & Education 1. Cn = = {E, Cn , Cn 2, … , Cnn{1} The cyclic group Cn is the set of the n distinct proper rotations about an axis of n order. It is abelian with n 1dimensional irreps Γ (0), Γ (1), Γ (2), ... , Γ (n{1) (f = 1). In this case the irrep matrices coincide with the characters χ, and from the relation Cnn = E it follows that χ(Cnn ) = χ(Cn )n = 1
(6)
χ Cn = n 1
(7)
From the general relation that the dimension of the irreps must satisfy (∑i f i2 = g), one obtains that the remaining irreps, (p – 1) for D2p and p for D 2p+1 , are 2dimensional (E k k = 1, 2, … , (p – 1) or p; f = 2). The matrices of the 2-dimensional irreps corresponding to the generator of D n must satisfy the relations n
2
2
D Cn = D C2 = D C2 D Cn
from which
The n distinct roots of unity in the complex plane are, l 2πl i2πl cos 2πl n + i × sin n = exp n = ε
(8)
cos ω ± sin ω
:
χ C nm = 2 cos 2πm n ; m = 1, 2, … , p – 1 or p
:
…
:
εn -2
(εn -2)2
(εn -2)3
…
(εn -2)n -2
(εn -2)n -1
εn -1
(εn -1)2
(εn -1)3
…
(εn -1)n -2
(εn -1)n -1
:
:
:
Γ(n -2)
1
Γ(n -1)
1
… … …
In the construction of the table the relation χ(Cnm) = χ(Cn) m
(13)
But for C nm, the only valid solution is that in which det_D(C nm)+ = +1; thus the character of D(C nm) is obtained from eq 2, giving at ω the appropriate values, ω = (2π m/n). We have
:
Γ(2)
…
{ sin ω ± cos ω
Cnn -1 1 εn -1 (ε2)n -1
Cn 3 1 ε3 (ε2)3
Cn 1 ε ε2
Γ(1)
Cnn -2 1 εn -2 (ε2)n -2
Cn 2 1 ε2 (ε2)2
E 1 1 1
Γ(0)
(12)
where D(G) (G∈{Cn, C2}) is a 2 × 2 unitary matrix. In particular, since the determinants are numbers, we get det_D(Cn )+ = ± 1 for n even and det_D(Cn)+ = +1 for n odd, with det_D(C2)+ = ± 1; that is, the matrices D(Cn ) and D(C2) are orthogonal matrices of the general form
where l = 0, 1, … , (n – 1), and ε = exp (i2π/n). The table of characters of Cn is immediate and is Cn
=D E = 1 0 01
(9)
has been used.
(14)
As regards D(C2), the only acceptable solution is that with det_D(C2)+ = {1, and consequently χ(C2) = 0. (They are obtained from eq 13 using minus signs. This is a consequence of the fact that the C2 rotation about an axis perpendicular to the Cn axis is equivalent to a rotoreflection σCnm, where σ is the reflection about the plane containing the C2 axes.) Finally, from eq 11 we necessarily have χ(C2′) = 0. The character tables obtained are
2. Dn = The dihedral group D n is the set of disD2p E 2Cn 2Cn 2 … 2Cnm … 2Cnp -1 C 2 = C 2pp pC 2 tinct proper rotations of a molecule that has A1 1 1 1 … 1 … 1 1 1 an n-fold principal axis and n perpendicuA2 1 1 1 … 1 … 1 1 {1 lar 2-fold axes, which intersect at an angle (π/n) if they are adjacent. The 2n elements B1 1 {1 ({1)2 … ({1)m … ({1)p -1 ({1)p 1 of Dn are subdivided into p + 3 classes for 2 m p -1 p B 1 {1 ({ 1 ) … ({ 1 ) … ({ 1 ) ({ 1 ) {1 2 n even (n = 2p), D2p = {E, 2C n, 2C n2, … , E1 2 2c 2c 2 … 2cm … 2cp -1 2cp 0 2Cnp{1, C2 = C2pp, pC 2, pC2′}, and p + 2 classes 2 2 2 m 2 p 1 2 p 2 E2 2 2(c ) 2(c ) … 2(c ) … 2(c ) 2(c ) 0 for n odd (n = 2p + 1), D 2p+1 = {E, 2Cn , 2Cn 2, … , 2Cn p, pC2}. Since the number of … … : : : : : : : : the inequivalent irreps is equal to the numEp -1 2 2(c )p -1 2(c 2)p -1 … 2(cm )p -1 … 2(cp -1)p -1 2(cp )p -1 0 ber of the classes, the groups with n even will have p + 3 irreps (Γ(i) i = 1, 2, … , p + 3) and those with n odd p + 2 irreps (Γ(i) i = c = cos(π/p ); (cm )k = cos(πmk /p ) 1, 2, … , p + 2). The characters of the 1-dimensional irreps of Dn must satisfy relations analogous to those for the definD2p +1 E 2Cn 2Cn 2 … 2Cnm … 2Cnp ing group Dn. It follows that χ(Cn )n = χ(C2) 2 = _χ(C2)χ(Cn)+2 = χ(C2) 2χ(Cn)2 = 1 (10) For n even the solution is χ(Cn ) = ±1, χ(C 2) = ±1, while for n odd it is χ(Cn) = 1, χ(C2) = ±1. The pair combinations of these characters are four for n even (+1 +1, +1 {1, {1 +1, {1 {1) and two for n odd (+1 +1, +1 {1). There D2p has four 1-dimensional irreps (A1, A2, B1, B 2; f = 1) and D2p+1 two (A1, A2; f = 1) The characters that remain to be defined for these irreps are found from eq 9 and from χ(C′2) = χ(C 2) χ(Cn)
0
: 0
…
1
{1
2c p
0
…
2(c p )2
0
:
…
:
:
2(c m )p
…
2(c p )p
0
1
1
1
…
1
E1 E2
2
2c
2c 2
…
2c m
2
2(c )2
2(c 2)2
…
2(c m )2
:
:
:
:
…
2
2(c )p
2(c 2)p
…
Ep
0
…
A2
…
1
pC 2 1
1
1
{1 {1
1
A1
1
pC 2′ 1
1
…
c = cos(2π/n ); (cm )k = cos(2πmk /n )
(11)
Vol. 74 No. 7 July 1997 • Journal of Chemical Education
807
Research: Science & Education 3. T = The tetrahedral group T is the group of the 12 proper rotations of the tetrahedron. These symmetry operations are combined in four classes, T = {E, 3C2, 4C3, 4C32}. The character table of T will then have four inequivalent irreps. The characters of the 1-dimensional irreps must satisfy the relations χ(C3)3 = χ(C 2) 2 = χ(C3)3χ(C2)3 = 1
3
3
2
100 =D E = 010 001
E
4C 3 4C 32 3C 2
A
1
1
1
1
E
{ 11
ε ε*
ε* ε
1 } 1
T
3
0
0
{1
This table shows that the characters of the second and third 1-dimensional irrep of T are complex conjugate and indicated with E, the symbol of a 2-dimensional irrep. The reason for this must be sought in the properties of quantum mechanics connected with the time-reversal symmetry. In the absence of magnetic fields, the Hamiltonian operator is invariant under time reversal, and for stationary states for a time-reversal invariant potential the complex conjugate of an eigenfunction is an eigenfunction for the same eigenvalue of the energy. The basis functions for the present pair of 1-dimensional irreps are complex conjugate and degenerate, so that for quantum-mechanical applications, it is convenient to classify such pairs as 2-dimensional reps. In the C n groups with n ≥ 3, one had p of these pairs for n = 2p + 1 and (p – 1) for n = 2p; and because (ε1)* = εn{1 these pairs are formed by Γ(k) and Γ(n{k) (k = 1, 2, … , p for n = 2p + 1; k = 1, 2, … , p – 1 for n = 2p). Evidently, such pairs are present also in the S 2n groups (n ≥ 2) because of the isomorphism with C2n, and in the Cnh (n ≥ 3) and Th groups because they are obtained from direct products involving C n and T. To be more exact, the direct product doubles the number of such pairs. 4. O = The octahedral group O is composed of the 24 proper rotations of the octahedron or of the cube (dual solids). Such transformations are separated into five classes, O = {E, 8C3, 3C42, 6C4, 6C2}, and thus the total number of
808
χ(C4) χ(C3)2 χ(C4) = χ(C 3)
(17)
χ(C4) χ(C3) χ(C4) = χ(C 3)χ(C4) 2χ(C3) give χ(C3) = 1, χ(C4) = ±1. The pair combinations of these characters are two, {χ(C3), χ(C4)} = {+1, +1} and {+1, {1}. The O group then has two 1-dimensional irreps (A1, A2; f = 1). Moreover, χ(C42) = χ(C4)2 = 1 and χ(C2) = χ(C3)χ(C4) = ±1. The remaining three irreps of O give one 2-dimensional irrep (E; f = 2). and two 3-dimensional irreps (T1, T2; f = 3). The matrices of these multidimensional irreps satisfy the relations D(C4)4 = D(C 3) 3 = D(E)
(16)
where D(G) (G ∈{C3, C2}) is a 3 × 3 unitary matrix. Moreover, det_D(C3)+ = 1, ε, ε* and det_D(C2)+ = 1, but, as mentioned shortly before eq 2, the opportune solution for det_D(C3)+ is +1. Thus the characters of D(C 3) and D(C2) are given by eq 3 with ω = (2π/3) and ω = π, respectively. One has χ(C3) = 0 χ(C 2) = {1, and in consequence χ(C32) = 0. The resulting character table is T
χ(C4)4 = χ(C3)3 = 1
(15)
which are valid only for χ(C3) = 1, ε, ε* χ(C2) = 1, where ε = exp(i2π/3). The pair combinations of these characters are three (1 1, ε 1, ε* 1), and give three 1-dimensional irreps (f = 1). Moreover, χ(C32) = χ(C3)2 = 1, ε*, ε. The remaining fourth irrep must inevitably be 3-dimensional (T; f = 3). The matrices of the T irrep satisfy the relations
D C3 = D C2 = D C2 D C3
inequivalent irreps is five. The character table of O is a 5 × 5 square matrix. For the 1-dimensional irreps, the solution of the equations
D(C4) D(C3)2D(C4) = D(C3)
(18)
2
D(C4)D(C3)D(C4) = D(C3)D(C4) D(C3) where the matrices have 2 × 2 or 3 × 3 dimension according to whether the irrep is of E or T type, respectively. Moreover, det_D(C3)+ = 1 and the det_D(C4)+ = ± 1 but, as mentioned at the beginning, the correct value of det_D(C4)+ is only +1. As regards T1, the characters are obtained using eq 3 with ω = π/2, (2/3)π, π. One has χ(C4) = 1, χ(C3) = 0, χ(C42) = χ(C2) = {1. For the characters of T2, one can use eq 5, which gives χ(C4) = {1, χ(C3) = 0, χ(C 42) = {1, χ(C2) = 1 because N(C4) = 0, N(C3) =1, N(C42) = 0, and N(C2) = 2. The group T being a subgroup of O, for the common characters of the 2-dimensional irrep E one has χ(C3) = ε + ε* = {1 and χ(C42) = 1 + 1 = 2. The unknown characters of E can be obtained from the orthogonality relations. Thus O has the following character table. O
E
8C 3
3C 42
6C 4
6C 2
A1
1
1
1
1
1
A2
1
1
1
{1
{1
E
2
{1
2
0
0
T1
3
0
{1
1
{1
T2
3
0
{1
{1
1
5. I = The icosahedral group I is composed of the 60 distinctive proper rotations of the icosahedron or of the dodecahedron (dual solids). They are separated into five classes, I = {E, 12C5, 12C52, 20C3, 15C2}. I then has five inequivalent irreps. The relations χ(C5)5 = χ(C 2) 2 = χ(C2)3χ(C5)3 = 1
(19)
are only satisfied by the value χ(C5) = χ(C2) = 1, and thus there exists only one 1-dimensional irrep (A; f = 1). where all the characters are equal to one, the totally symmetric representation. From the general relation of fi, it results that the remaining four irreps are composed of two 3-dimensional irreps (T1, T 2; f = 3), one 4-dimensional irrep (G; f = 4) and one 5-dimensional one (H; f = 5). For
Journal of Chemical Education • Vol. 74 No. 7 July 1997
Research: Science & Education these multidimensional irreps the following matrix relations D(C5)5 = D(C2)2 = _D(C2)D(C5)+3 = D(E)
(20)
must be satisfied, and thus det_D(C5)+ = det_D(C 2)+ = 1. The characters of T1 are deduced via eq 3 with ω = 2/ π, π, 4/ π, 2/ π. They are: χ(C ) = 1/ (1 + √5), χ(C ) = {1, 5 5 3 5 2 2 χ(C52) = 1/2(1 – √5), χ(C 3) = 0. For the G irrep, the characters are obtained by eq 5 giving χ(C5) = χ(C52) = {1, χ(C2) = 0 and χ(C3) = 1, because N(C5) = N(C52) = 0, N(C2) = 1, N(C3) = 2. As regards the characters of the H irrep, they are deduced via eq 4 giving ω the values previously listed. It follows that χ(C5) = χ(C52) = 0, χ(C2) = {1, and χ(C3) = {1. Finally, the characters of T2 are obtained by using the orthogonality relations. The character table of the icosahedral group is then
of the structure and the calculation of the character tables of finite point groups as a function of the proper rotation groups, and to the development of the generator theory in the field of irreps of these groups. Acknowledgment Financial support from the Ministero dell’Università e della Ricerca Scientifica e Tecnologica (Roma) is gratefully acknowledged. Note 1. These are also known as (+)-groups or proper-rotation point groups.
Literature Cited 2
I A
E 1
12C 5
12C 5
1
1
T1
3
(1/2)(1+√5)
T2
3
(1/2)(1{ √5)
G
4
{1
H
5
0
20C 3 15C 2 1
1
(1/2)(1{ √5)
0
{1
(1/2)(1+√5)
0
{1
{1
1
0
0
{1
1
In conclusion, this paper presents a general discussion and a “complete” determination of character tables for the finite point groups. The approach is that of the defining relations and proper rotation groups. The characters of the multidimensional irreps are obtained using more intuitive arguments. Generally, we think we have given a useful contribution both to the arrangement
1. Cotton, F. A. Chemical Application of Group Theory, 3rd ed.; Wiley: New York, 1990. 2. Atkins, P. W.; Child, M. S.; Phillips, C. S. G. Tables for Group Theory; Oxford University: Salisbury, 1970. 3. Harris, D. C.; Bertolucci, M. D. Symmetry and Spectroscopy. An Introduction to Vibrational and Electronic Spectroscopy; Dover: New York, 1989. 4. Altmann, S. L; Herzig, P. Point-Group Theory Tables; Clarendon: Oxford, 1994. 5. Hamermesh, M. Group Theory and Its Application to Physical Problems; Dover: New York, 1989. 6. Elliott, J. P.; Dawber, P. G. Symmetry in Physics; Macmillan: London, 1979; Vol. 1. 7. Lomont, J. S. Application of Finite Groups; Academic: New York, 1959. 8. Baraldi, I.; Carnevali, A. J. Chem. Educ. 1993, 70, 964–967. 9. Baraldi, I. La Chimica nella Scuola 1995, 17, 71–73. 10. Kostrinikin. A. Introduction à l’Algèbre; Editions MIR: Moscou, 1981. 11. Burrow, M. Representation Theory of Finite Groups; Dover: New York, 1993.
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