Chemistry:Finite subgroups of SU(2)

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Short description: Use of mathematical groups in magnetochemistry

[[File:Frobenius_character_tables_binary_subgroups_1899.pdf|thumb|upright=1|The character tables of the binary tetrahedral, octahedral and icosahedral groups following Frobenius (1899)Cite error: Closing </ref> missing for <ref> tag[1] Instances when a double group is commonly used include 6-coordinate complexes of copper(II), titanium(III) and cerium(III). In these double groups rotation by 360° is treated as a symmetry operation separate from the identity operation; the double group is formed by combining these two symmetry operations with a point group such as a dihedral group or the full octahedral group.

Definition and theory

Let Γ be a finite subgroup of SO(3), the three-dimensional rotation group. There is a natural homomorphism f of SU(2) onto SO(3) which has kernel {±I}.[2] This double cover can be realised using the adjoint action of SU(2) on the Lie algebra of traceless 2-by-2 skew-adjoint matrices or using the action by conjugation of unit quaternions. The double group Γ' is defined as f−1 (Γ). By construction {±I} is a central subgroup of Γ' and the quotient is isomorphic to Γ. Thus Γ' is a central extension of the group Γ by {±1}, the cyclic group of order 2. Ordinary representations of Γ' are just mappings of Γ into the general linear group that are homomorphisms up to a sign; equivalently, they are projective representations of Γ with a factor system or Schur multiplier in {±1}. Two projective representations of Γ are closed under the tensor product operation, with their corresponding factor systems in {±1} multiplying. The central extensions of Γ by {±1} also have a natural product.[3]

The finite subgroups of SU(2) and SO(3) were determined in 1876 by Felix Klein in an article in Mathematische Annalen, later incorporated in his celebrated 1884 "Lectures on the Icosahedron": for SU(2), the subgroups correspond to the cyclic groups, the binary dihedral groups, the binary tetrahedral group, the binary octahedral group, and the binary icosahedral group; and for SO(3), they correspond to the cyclic groups, the dihedral groups, the tetrahedral group, the octahedral group and the icosahedral group. The correspondence can be found in numerous text books, and goes back to the classification of platonic solids. From Klein's classifications of binary subgroups, it follows that, if Γ a finite subgroup of SO(3), then, up to equivalence, there are exactly two central extensions of Γ by {±1}: the one obtained by lifting the double cover Γ' = f−1 (Γ); and the trivial extension Γ x {±1}.[3][4][5][6][7]

The character tables of the finite subgroups of SU(2) and SO(3) were determined and tabulated by F. G. Frobenius in 1898,[8] with alternative derivations by I. Schur and H. E. Jordan in 1907 independently. Branching rules and tensor product formulas were also determined. For each binary subgroup, i.e. finite subgroup of SU(2), the irreducible representations of Γ are labelled by extended Dynkin diagrams of type A, D and E; the rules for tensoring with the two-dimensional vector representation are given graphically by an undirected graph.[4][5][6] By Schur's lemma, irreducible representations of Γ x {±1} are just irreducible representations of Γ multiplied by either the trivial or the sign character of {±1}. Likewise, irreducible representations of Γ' which send –1 to I are just ordinary representations of Γ; while those which send –1 to –I are genuinely double-valued or spinor representations.[3]

Example. For the double icosahedral group, if [math]\displaystyle{ \varphi }[/math] is the golden ratio [math]\displaystyle{ {1\over 2} (1 +\sqrt{5}) }[/math] with inverse [math]\displaystyle{ \tilde{\varphi}={1\over 2}(-1 + \sqrt{5}) }[/math], the character table is given below: spinor characters are denoted by asterisks. The character table of the icosahedral group is also given.[9][10]

Character table: double icosahedral group
[math]\displaystyle{ I_h^\prime }[/math] 1 12C2[5] 12C3[5] 1C4[2] 12C5[10] 12C6[10] 20C7[3] 20C8[6] 30C9[4]
[math]\displaystyle{ \chi_{1} }[/math] 1 1 1 1 1 1 1 1 1
[math]\displaystyle{ \chi_{3} }[/math] 3 [math]\displaystyle{ \varphi }[/math] [math]\displaystyle{ -\tilde{\varphi} }[/math] 3 [math]\displaystyle{ -\tilde{\varphi} }[/math] [math]\displaystyle{ \varphi }[/math] 0 0 –1
[math]\displaystyle{ \chi_{3^\prime} }[/math] 3 [math]\displaystyle{ -\tilde{\varphi} }[/math] [math]\displaystyle{ \varphi }[/math] 3 [math]\displaystyle{ \varphi }[/math] [math]\displaystyle{ -\tilde{\varphi} }[/math] 0 0 –1
[math]\displaystyle{ \chi_{4} }[/math] 4 –1 –1 4 –1 –1 1 1 0
[math]\displaystyle{ \chi_{5} }[/math] 5 0 0 5 0 0 –1 –1 0
[math]\displaystyle{ \chi_2^* }[/math] 2 [math]\displaystyle{ -\varphi }[/math] [math]\displaystyle{ \tilde{\varphi} }[/math] –2 [math]\displaystyle{ -\tilde{\varphi} }[/math] [math]\displaystyle{ \varphi }[/math] –1 1 0
[math]\displaystyle{ \chi_{2^\prime}^* }[/math] 2 [math]\displaystyle{ \tilde{\varphi} }[/math] [math]\displaystyle{ -\varphi }[/math] –2 [math]\displaystyle{ \varphi }[/math] [math]\displaystyle{ -\tilde{\varphi} }[/math] –1 1 0
[math]\displaystyle{ \chi_4^* }[/math] 4 –1 –1 -4 –1 –1 1 0 –1
[math]\displaystyle{ \chi_6^* }[/math] 6 1 1 –6 –1 –1 0 0 0
Character table: icosahedral group
[math]\displaystyle{ I_h }[/math] 1 20C2[3] 15C3[2] 12C4[5] 12C5[5]
[math]\displaystyle{ \chi_1 }[/math] 1 1 1 1 1
[math]\displaystyle{ \chi_3 }[/math] 3 0 –1 [math]\displaystyle{ \varphi }[/math] [math]\displaystyle{ -\tilde{\varphi} }[/math]
[math]\displaystyle{ \chi_{3^\prime} }[/math] 3 0 –1 [math]\displaystyle{ -\tilde{\varphi} }[/math] [math]\displaystyle{ \varphi }[/math]
[math]\displaystyle{ \chi_4 }[/math] 4 1 0 –1 –1
[math]\displaystyle{ \chi_5 }[/math] 5 –1 1 0 0

The tensor product rules for tensoring with the two-dimensional representation are encoded diagrammatically below:

Affine E8 diagram.svg

The numbering has at the top [math]\displaystyle{ \chi_{3^\prime} }[/math] and then below, from left to right, [math]\displaystyle{ \chi_{2^\prime}^* }[/math], [math]\displaystyle{ \chi_{4} }[/math], [math]\displaystyle{ \chi_6^* }[/math], [math]\displaystyle{ \chi_{5} }[/math], [math]\displaystyle{ \chi_4^* }[/math], [math]\displaystyle{ \chi_{3} }[/math], [math]\displaystyle{ \chi_2^* }[/math], and [math]\displaystyle{ \chi_{1} }[/math]. Thus, on labelling the vertices by irreducible characters, the result of multiplying [math]\displaystyle{ \chi_2^* }[/math] by a given irreducible character equals the sum of all irreducible characters labelled by an adjacent vertex.[11]

The representation theory of SU(2) goes back to the nineteenth century and the theory of invariants of binary forms, with the figures of Alfred Clebsch and Paul Gordan prominent.[12][13][14][15][16]Cite error: Closing </ref> missing for <ref> tag[5][6]

In a 1929 article on splitting of atoms in crystals, the physicist H. Bethe first coined the term "double group" (Doppelgruppe),[17][18] a concept that allowed double-valued or spinor representations of finite subgroups of the rotation group to be regarded as ordinary linear representations of their double covers.[lower-alpha 1][lower-alpha 2] In particular, Bethe applied his theory to relativistic quantum mechanics and crystallographic point groups, where a natural physical restriction to 32 point groups occurs. Subsequently, the non-crystallographic icosahedral case has also been investigated more extensively, resulting most recently in groundbreaking advances on carbon 60 and fullerenes in the 1980s and 90s.[20][21][22] In 1982–1984, there was another breakthrough involving the icosahedral group, this time through materials scientist Dan Shechtman's remarkable work on quasicrystals, for which he was awarded a Nobel Prize in Chemistry in 2011.[23][24][25][lower-alpha 3]

Applications

Magnetochemistry

In magnetochemistry, the need for a double group arises in a very particular circumstance, namely, in the treatment of the magnetic properties of complexes of a metal ion in whose electronic structure there is a single unpaired electron (or its equivalent, a single vacancy) in a metal ion's d- or f- shell. This occurs, for example, with the elements copper, silver and gold in the +2 oxidation state, where there is a single vacancy in the d-electron shell, with titanium(III) which has a single electron in the 3d shell and with cerium(III) which has a single electron in the 4f shell.

In group theory, the character [math]\displaystyle{ \chi }[/math], for rotation, by an angle α, of a wavefunction for half-integer angular momentum is given by

[math]\displaystyle{ \chi^J (\alpha) = \frac{\sin [J+1/2] \alpha } {\sin (1/2) \alpha } }[/math]

where angular momentum is the vector sum of spin and orbital momentum, [math]\displaystyle{ J= L + S }[/math]. This formula applies with angular momentum in general.

In atoms with a single unpaired electron the character for a rotation through an angle of [math]\displaystyle{ 2\pi+\alpha }[/math] is equal to [math]\displaystyle{ -\chi^J (\alpha) }[/math]. The change of sign cannot be true for an identity operation in any point group. Therefore, a double group, in which rotation by [math]\displaystyle{ 2\pi }[/math] is classified as being distinct from the identity operation, is used. A character table for the double group D'4 is as follows. The new operation is labelled R in this example. The character table for the point group D4 is shown for comparison.

Character table: double group D'4
D'4 C4 C43 C2 2C'2 2C''2
E R C4R C43R C2R 2C'2R 2C''2R
A'1 1 1 1 1 1 1 1
A'2 1 1 1 1 1 -1 -1
B'1 1 1 -1 -1 1 1 -1
B'2 1 1 -1 -1 1 -1 1
E'1 2 -2 0 0 -2 0 0
E'2 2 -2 √2 -√2 0 0 0
E'3 2 -2 -√2 √2 0 0 0
Character table: point group D4
D4 E 2 C4  C2  2 C2'  2 C2
A1 1 1 1 1 1 +
A2 1 1 1 −1 −1
B1 1 −1 1 1 −1
B2 1 −1 1 −1 1
E 2 0 −2 0 0

In the table for the double group, the symmetry operations such as C4 and C4R belong to the same class but the header is shown, for convenience, in two rows, rather than C4, C4R in a single row.

Character tables for the double groups T', O', Td', D3h', C6v', D6', D2d', C4v', D4', C3v', D3', C2v', D2' and R(3)' are given in (Koster Dimmock), (Salthouse Ware) and (Cornwell 1984).[27][28][29][lower-alpha 4] thumb|150px|left|Sub-structure at the center of an octahedral complex thumb|170px|Structure of a square-planar complex ion such as [AgF4]2-

An atom or ion (red) held in a C60 fullerene cage

The need for a double group occurs, for example, in the treatment of magnetic properties of 6-coordinate complexes of copper(II). The electronic configuration of the central Cu2+ ion can be written as [Ar]3d9. It can be said that there is a single vacancy, or hole, in the copper 3d-electron shell, which can contain up to 10 electrons. The ion [Cu(H2O)6]2+ is a typical example of a compound with this characteristic.

(1) Six-coordinate complexes of the Cu(II) ion, with the generic formula [CuL6]2+, are subject to the Jahn-Teller effect so that the symmetry is reduced from octahedral (point group Oh) to tetragonal (point group D4h). Since d orbitals are centrosymmetric the related atomic term symbols can be classified in the subgroup D4 .
(2) To a first approximation spin-orbit coupling can be ignored and the magnetic moment is then predicted to be 1.73 Bohr magnetons, the so-called spin-only value. However, for a more accurate prediction spin-orbit coupling must be taken into consideration. This means that the relevant quantum number is J, where J = L + S.
(3) When J is half-integer, the character for a rotation by an angle of α + 2π radians is equal to minus the character for rotation by an angle α. This cannot be true for an identity in a point group. Consequently, a group must be used in which rotations by α + 2π are classed as symmetry operations distinct from rotations by an angle α. This group is known as the double group, D4'.

With species such as the square-planar complex of the silver(II) ion [AgF4]2- the relevant double group is also D4'; deviations from the spin-only value are greater as the magnitude of spin-orbit coupling is greater for silver(II) than for copper(II).[30]

A double group is also used for some compounds of titanium in the +3 oxidation state. Compounds of titanium(III) have a single electron in the 3d shell. The magnetic moments of octahedral complexes with the generic formula [TiL6]n+ have been found to lie in the range 1.63 - 1.81 B.M. at room temperature.[31] The double group O' is used to classify their electronic states.

The cerium(III) ion, Ce3+, has a single electron in the 4f shell. The magnetic properties of octahedral complexes of this ion are treated using the double group O'.

When a cerium(III) ion is encapsulated in a C60 cage, the formula of the endohedral fullerene is written as {Ce3+@C603-}.[32]

Free radicals

Double groups may be used in connection with free radicals. This has been illustrated for the species CH3F+ and CH3BF2+ which both contain a single unpaired electron.[33]

See also

Notes

  1. ↑ In his 1931 book Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren,[19][13] Eugene Wigner describes in detail how SU(2) arises as a double cover of SO(3), following Hermann Weyl. Referring to a 1928 article of von Neumann and Wigner in Zeitschrift für Physik (vol. 49), (Bethe 1929) explains why double-valued representations of finite groups are involved.
  2. ↑ In mathematical language, a double group Γ' is defined to be a central extension of the group Γ by {±1}, the cyclic group of order 2: thus ordinary representations of Γ' are just mappings of Γ into the general linear group that are homomorphisms up to a sign; equivalently, they are projective representations of Γ with a factor system or Schur multiplier in {±1}.
  3. ↑ Ted Janssen has outlined how the characters of the double icosahedral group appear to play a role.[26]
  4. ↑ R(3)' refers to SU(2), the double cover of the three-dimensional rotation group SO(3)

References

  1. ↑ Tsukerblat, Boris S. (2006). Group Theory in Chemistry and Spectroscopy. Mineola, New York: Dover Publications Inc.. pp. 245–253. ISBN 0-486-45035-X. 
  2. ↑ Wigner 1959.
  3. ↑ 3.0 3.1 3.2 Bethe 1929.
  4. ↑ 4.0 4.1 Cite error: Invalid <ref> tag; no text was provided for refs named Griffith
  5. ↑ 5.0 5.1 5.2 Ramond, Pierre (2010). Group theory. A physicist's survey. Cambridge University Press. ISBN 978-0-521-89603-0. 
  6. ↑ 6.0 6.1 6.2 Jacobs, Patrick (2005). Group Theory with Applications in Chemical Physics. Cambridge University Press. doi:10.1017/CBO9780511535390. ISBN 9780511535390. 
  7. ↑ Wolf, Joseph A. (1967). Spaces of constant curvature. New York-London-Sydney: McGraw-Hill. pp. 83–88. 
  8. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Frobenius
  9. ↑ Griffith 1961, pp. 383,385
  10. ↑ Ramond 2010, p. 351
  11. ↑ Ramond 2010.
  12. ↑ Weyl, Hermann (1946). The Classical Groups. Their Invariants and Representations. Princeton Landmarks in Mathematics (2nd ed.). Princeton University Press. ISBN 0-691-05756-7.  Archive, Institut Fourier
  13. ↑ 13.0 13.1 Wigner, Eugene (1959). Group theory and its application to the quantum mechanics of atomic spectra. Pure and Applied Physics. 5 (Expanded and improved ed.). New York–London: Academic Press. p. 157. 
  14. ↑ Weyl, Hermann (1950). The theory of groups and quantum mechanics [Gruppentheorie und Quantenmechanik] (Second revised ed.). New York: Dover Books.  First edition in 1928 from notes of von Neumann; second edition (in German), expanded and simplified in 1931.
  15. ↑ van der Waerden, B. L. (1974). Group theory and quantum mechanics (translated from the 1932 German original). Die Grundlehren der mathematischen Wissenschaften. 214. New York-Heidelberg: Springer-Verlag. 
  16. ↑ Tinkham, Michael (1964). Group Theory and Quantum Mechanics. McGraw-Hill. 
  17. ↑ Bethe, Hans (1929). "Termaufspaltung in Kristallen" (in de). Ann. Physik 395 (3): 133–206. doi:10.1002/andp.19293950202. Bibcode: 1929AnP...395..133B. https://gallica.bnf.fr/ark:/12148/bpt6k15392p/f141.item. 
  18. ↑ English translation in Bethe, Hans (1996). Selected Works of Hans A. Bethe with commentary. World Scientific. pp. 1–72. ISBN 9789810228767.  Bethe's commentary: "If an atom is placed in a crystal, its energy levels are split. The splitting depends on the symmetry of the location of the atom in the crystal. The splitting is derived here from group theory. This paper has been widely used, especially by physical chemists."
  19. ↑ Wigner, Eugen (1931) (in de). Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Vieweg+Teubner Verlag. ISBN 9783663025559. https://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/33355459/33355459.pdf. 
  20. ↑ Chung, Fan R. K.; Kostant, Bertram; Sternberg, Shlomo (1994). "Groups and the buckyball". Lie theory and geometry. Progress in Mathematics. 123. Birkhäuser. pp. 97–126. doi:10.1007/978-1-4612-0261-5_4. ISBN 978-1-4612-6685-3. https://link.springer.com/chapter/10.1007/978-1-4612-0261-5_4.  (Subscription content?)
  21. ↑ Yang, C. N. (1994). "Fullerenes and carbon 60". Perspectives in mathematical physics. Conf. Proc. Lecture Notes Math. Phys., III. Int. Press. pp. 303–307. https://academic.hep.com.cn/fib/EN/chapter/9781571460097/chapter15. 
  22. ↑ Chancey, C. C.; O'Brien, M. C. M. (1998). The Jahn-Teller Effect in C60 and Other Icosahedral Complexes. Princeton University Press. doi:10.1515/9780691225340. ISBN 9780691225340. 
  23. ↑ See:
  24. ↑ Penrose, Roger (1978). "Pentaplexity: a class of non-periodic tilings of the plane". Eureka (University of Cambridge) 39: 16–22. 
  25. ↑ Au-Yang, Helen; Perk, Jacques (2013). "Quasicrystals—the impact of N. G. de Bruijn". Indag. Math. 24 (4): 996–1017. doi:10.1016/j.indag.2013.07.003. 
  26. ↑ Janssen, Ted (2014). "Development of Symmetry Concepts for Aperiodic Crystals". Symmetry 6 (2): 171–188. doi:10.3390/sym6020171. Bibcode: 2014Symm....6..171J. 
  27. ↑ Salthouse, J.A.; Ware, M.J. (1972). Point group character tables and related data. Cambridge: Cambridge University Press. pp. 55–57. ISBN 0-521-081394. 
  28. ↑ Koster, George F.; Dimmock, John O.; Wheeler, Robert G.; Statz, Hermann (1963). Properties of the thirty-two point groups. Cambridge, Mass.: The M.I.T. Press. 
  29. ↑ Cornwell, J. F. (1984). "The double crystallographic point groups". Group theory in physics. Vol. I. Academic Press. pp. 342–355. ISBN 0-12-189801-6. 
  30. ↑ Foëx, D.; Gorter, C. J.; Smits, L.J. (1957). Constantes Sélectionées Diamagnetism et Paramagnetism. Paris: Masson et Cie. 
  31. ↑ Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 971. ISBN 978-0-08-037941-8. 
  32. ↑ Chai, Yan; Guo, Ting; Jin, Changming; Haufler, Robert E.; Chibante, L. P. Felipe; Fure, Jan; Wang, Lihong; Alford, J. Michael et al. (1991). "Fullerenes with metals inside". The Journal of Physical Chemistry 95 (20): 7564–7568. doi:10.1021/j100173a002. 
  33. ↑ Bunker, P.R. (1979), "The Spin Double Groups of Molecular Symmetry Groups", in Hinze, J., The Permutation Group in Physics and Chemistry, Lecture Notes in Chemistry, 12, Springer, pp. 38–56, doi:10.1007/978-3-642-93124-6_4, ISBN 978-3-540-09707-5 

Further reading



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