Research: Science & Education
More than One Character Table? Yes! Ian J. McNaught School of Chemistry, University of Sydney, Sydney, NSW 2006, Australia Recently Contreras-Ortega et al. (1) gave a method for unambiguously determining the correct character table for the point groups Dnh and Cnh from a set of several that obey the great orthogonality theorem (2). They show that the correct character table has the structure n even
D∞ h ≠ C ∞ v⊗ Ci. The character table for D∞h cannot be generated using the great orthogonality theorem because the theorem does not apply to infinite groups despite claims that it applies to any group (5). The character table for D∞h is not unique; there are at least four different tables in the literature and it is instructive to examine them. The following can be noted from an examination of the tables:
Dnh
Cn
Sn
Cnh
Cn
Sn
Γg
Dn
Dn
Γg
Cn
Cn
Γu
Dn
{ Dn
Γu
Cn
{ Cn
2. Tables 3 and 4 have the same operations but different characters for the operations i and 2iC∞φ for the degenerate representations. 3. The orthogonality theorems do not apply to any of these tables. For example, applying the little orthogonality theorem (11) and letting (1) represent ∑ g+ and (2) represent ∏g, then summing over the classes shows that ∑g(c)χ(1)(c)χ(2) (c) has the value 4 in Tables 1 and 3 and the value 2 in Tables 2 and 4, rather than zero. For Table 3 each of the g representations is orthogonal to each of the u representations.
1. Tables 1, 2, and 3 all have different operations.
n odd
Dnh
Cn
Sn
Cnh
Cn
Sn
Γ′
Dn
Dn
Γ′
Cn
Cn
Γ′′
Dn
{ Dn
Γ′′
Cn
{ Cn
where Cn and Sn are sets of proper and improper rotations, and D n and C n are the characters for the point groups Dn and C n. They note that this is a direct result of the fact that for n even, Dnh = D n⊗Ci and Cnh = C n⊗Ci; while for n odd, Dnh = Dn ⊗Cs and Cnh = Cn ⊗Cs. It may be additionally noted that for n odd Dnd
Cn
Sn
S2n
Cn
Sn
Γg
Dn
Dn
Γg
Cn
Cn
Γu
Dn
{ Dn
Γu
Cn
{ Cn
D∞h
C∞ v C∞v
A
Γu
C∞v
{A
Dnh
D∞
Γg
iD∞
D∞
D∞
Γu
D∞
{ D∞
where D∞ is the set of operations and D ∞ the set of characters of the infinite dihedral group D ∞. This illustrates the fact that D∞h = D∞⊗Ci.
as a consequence of the relations D nd = Dn ⊗ Ci and S2n = Cn ⊗ Ci . This means that, for example, the character tables for D3h and D3d are the same as are those of C3h and S6, although the labels change (′ is replaced by g, ′′ is replaced by u). Application of the great orthogonality theorem led to several sets of possible character tables (4) because satisfying the great orthogonality theorem is a necessary but not sufficient condition for a character table for a finite group. In contrast to the above there is a case where this structure apparently is not observed, namely the D∞h point group. Here the structure in the commonly used character table (3) is Γg
4. Table 3 does have the structure
5. All four different character tables are valid in the solution of any physical problem. 6. Although Table 2 contains an additional class and symmetry operation compared with Table 1, this does not invalidate the result that the number of classes equals the number of irreducible representations, infinity in all four tables. This extra operation is intrinsically contained in Table 1 as σh = S∞0. 7. Despite the differences in the operations and characters of the irreducible representations, translations and rotations transform as the same irreducible representations in all four character tables, (x, y) as E 1u(∏u), z as A2u(∑ u+), (Rx, Ry) as E 1g(∏ g) and Rz as A2g(∑ g{). This was not true of all six character tables in ref 4; four of the character tables did not have irreducible representations transforming as z or Rz.
Literature Cited where C∞v is the set of operations and C∞v the set of characters of the C ∞ v point group, A being a set of characters independent of C∞ v. The difference arises because
1. Contreras-Ortega, C.; Vera, L.; Quiroz-Reyes, E. J. Chem. Educ. 1995, 72, 821–822. 2. Tinkham, M. Group Theory and Quantum Mechanics; McGraw-
Vol. 74 No. 7 July 1997 • Journal of Chemical Education
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Research: Science & Education Table 3 (9 )
Table 1 (3, 6 )
E
D ∞h
φ
2C∞
∞σv
∞C 2′
φ
2S ∞
i
2 C ∞φ
∞C 2′
i
2i C ∞φ
∞iC 2′
+)
1
1
1
1
1
1
{1
A 2g (∑g { )
1
1
{1
1
1
{1
0
E 1g (∏g )
2
2 cos φ
0
2
2 cos φ
0
2
2 cos2 φ
0
2
2 cos2 φ
0
+)
1
1
1
1
1
1
A 2g (∑g { )
1
1
{1
1
1
E 1g (∏g )
2
2 cos φ
0
2
{2 cos φ
2 cos2 φ
2 cos2 φ
A 1g (∑g
E
D ∞h A 1g (∑g
E 2g (∆g )
2
0
2
0
E 2g (∆g )
A 1u (∑u { )
1
1
{1
{1
{1
1
A 1u (∑u { )
1
1
1
{1
{1
{1
+)
1
1
1
{1
{1
{1
A 2u (∑u +)
1
1
{1
{1
{1
1
0
{2
2 cos φ
0
E 1u (∏u )
2
2 cos φ
0
{2
{2 cos φ
0
0
{2
{2 cos2 φ
0
E 2u (∆u )
2
2 cos2 φ
0
{2
{2 cos2 φ
0
σh
D ∞h
E
2 C ∞φ
A 2u (∑u
2
2 cos φ
E 2u (∆u )
2
2 cos2 φ
D ∞h
E
2 C ∞φ
E 1u (∏u )
Table 2 (7, 8 ) ∞σv
i
Table 4 (10 ) 2 S ∞φ
∞C 2′
+)
1
1
1
1
1
1
A 2g (∑g { )
1
1
{1
1
1
{1
A 1g (∑g
∞C 2′
i
2iC ∞φ
∞iC 2′
1
A 1g (∑g
+)
1
1
1
1
1
1
1
A 2g (∑g { )
1
1
{1
1
1
{1
E 1g (∏g )
2
2 cos φ
0
2
{2 cos φ
0
{2
E 1g (∏g )
2
2 cos φ
0
0
{2 cos φ
0
E 2g (∆g )
2
2 cos2 φ
0
2
2 cos2 φ
0
2
E 2g (∆g )
2
2 cos2 φ
0
0
2 cos2 φ
0
A 1u (∑u { )
1
1
{1
{1
{1
1
{1
A 1u (∑u { )
1
1
1
{1
{1
+)
1
1
1
{1
{1
{1
{1
A 2u (∑u +)
1
1
{1
{1
{1
1
A 2u (∑u
{1
E 1u (∏u )
2
2 cos φ
0
{2
2 cos φ
0
2
E 1u (∏u )
2
2 cos φ
0
0
2 cos φ
0
E 2u (∆u )
2
2 cos2 φ
0
{2
{2 cos2 φ
0
{2
E 2u (∆u )
2
2 cos2 φ
0
0
{2 cos2 φ
0
Hill: New York, 1964; p 23. 3. Cotton, A. F. Chemical Applications of Group Theory, 3rd ed; Wiley: New York, 1990; p 435. 4. Contreras-Ortega, C.; Quiroz-Reyes, E; Vera, L. J. Chem. Educ. 1991, 68, 200–202. 5. Bunker P. R. Molecular Symmetry; Academic: New York, 1979; p 49. 6. Wilson, E. B., Jr.; Decius, J. C.; Cross, P. C. Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra; McGrawHill: New York, 1955; p 330. 7. Herzberg, G. Molecular Spectra and Molecular Structure Part II.
812
8. 9. 10. 11.
Journal of Chemical Education • Vol. 74 No. 7 July 1997
Infrared and Raman Spectra of Polyatomic Molecules; Van Nostrand: New York, 1945; p 119. King, G. W. Spectroscopy and Molecular Structure; Holt, Rinehart and Winston: New York, 1964; p 464. Eyring, H; Walter, J; Kimball, G. E. Quantum Chemistry; Wiley: New York, 1944; p 388. Atkins, P. W. Molecular Quantum Mechanics, 2nd ed; Oxford University: Oxford, 1983; p 440. Atkins, P. W. Physical Chemistry, 5th ed; Oxford University: Oxford, 1994; p 528.
