1. Lecture 30 Point-group symmetry III (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.

2. Non-Abelian groups and chemical applications of symmetry In this lecture, we learn non-Abelian point groups and the decomposition of a product of irreps. We also apply the symmetry theory to chemistry problems.

3. Degeneracy The particle in a square well (D 4h ) has doubly degenerate wave functions.

4. The D 4h character table (h = 16) D 4h E2C42C4 C2C2 2C2’2C2’2C2”2C2”i2S42S4 σhσh 2σv2σv 2σd2σd A 1g 1111111111 A 2g 111−1 111 B 1g 1−111 1 11 B 2g 1−11 11 1 1 EgEg 20−20020 00 A 1u 11111−1 A 2u 111−1 11 B 1u 1−111 1 1 B 2u 1−11 1 1 1 EuEu 20−200 0200

5. C 3v : another non-Abelian group C 3v, 3mE2C32C3 3σv3σv h = 6 A1A1 111z, z 2, x 2 +y 2 A2A2 11−1 E2 0(x, y), (xy, x 2 −y 2 ), (zx, yz)

6. C 3v : expanded character table C 3v, 3mEC3C3 C32C32 σvσv σvσv σvσv h = 6 A1A1 111111z, z 2, x 2 +y 2 A2A2 111−1 E2 000(x, y), (xy, x 2 −y 2 ), (zx, yz) C 3v, 3mE2C32C3 3σv3σv h = 6 A1A1 111z, z 2, x 2 +y 2 A2A2 11−1 E2 0(x, y), (xy, x 2 −y 2 ), (zx, yz)

7. Integral of degenerate orbitals C 3v, 3mEC3C3 C32C32 σvσv σvσv σvσv h = 6 A1A1 111111z, z 2, x 2 +y 2 A2A2 111−1 E2 000(x, y), (xy, x 2 −y 2 ), (zx, yz)

8. What is E ✕ E ? What is the irrep for this set of characters? C 3v, 3mEC3C3 C32C32 σvσv σvσv σvσv h = 6 A1A1 111111z, z 2, x 2 +y 2 A2A2 111−1 E2 000(x, y), (xy, x 2 −y 2 ), (zx, yz) E ✕ E 411000 It is not a single irrep. It is a linear combination of irreps

9. Superposition principle (review) Eigenfunctions of a Hermitian operator are complete. Eigenfunctions of a Hermitian operator are orthogonal.

10. Decomposition An irrep is a simultaneous eigenfunction of all symmetry operations.

11. The character vector of A 1 is normalized. The character vector of E is normalized. The character vectors of A 1 and E are orthogonal. Orthonormal character vectors C 3v, 3mEC3C3 C32C32 σvσv σvσv σvσv h = 6 A1A1 111111z, z 2, x 2 +y 2 A2A2 111−1 E2 000(x, y), (xy, x 2 −y 2 ), (zx, yz)

12. The contribution (c A1 ) of A 1 : The contribution (c A2 ) of A 2 : The contribution (c E ) of E: Decomposition C 3v, 3mEC3C3 C32C32 σvσv σvσv σvσv h = 6 A1A1 111111z, z 2, x 2 +y 2 A2A2 111−1 E2 000(x, y), (xy, x 2 −y 2 ), (zx, yz) E ✕ E 411000 Degeneracy = 2 × 2 = 1 + 1 + 2

13. Chemical applications While the primary benefit of point-group symmetry lies in our ability to know whether some integrals are zero by symmetry, there are other chemical concepts derived from symmetry. We discuss the following three: Woodward-Hoffmann rule Crystal field theory Jahn-Teller distortion

14. Woodward-Hoffmann rule The photo and thermal pericyclic reactions yield different isomers of cyclobutene. photochemical thermal

15. Woodward-Hoffmann rule What are the symmetry groups to which these reactions A and B belong? photochemical / disrotary / C s thermal / conrotary / C 2 σ C2C2

16. Woodward-Hoffmann rule higher energy occupied higher energy dcbahgfe occupied ReactantProduct Processabcdefgh Photochemical / C s A”A’A”A’A” A’ Thermal / C 2 ABABBABA Processabcdefgh Photochemical / C s A”A’A”A’A” A’ Thermal / C 2 ABABBABA “Conservation of orbital symmetry”

17. Crystal field theory Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. [Ni(NH 3 ) 6 ] 2+, [Ni(en) 3 ] 2+, [NiCl 4 ] 2−, [Ni(H 2 O) 6 ] 2+

18. Crystal field theory d orbitals d xy, d yz, d zx d z2, d x2−y2 TdTd OhOh spherical E T2T2 EgEg T 2g

19. NiCl 4 2− belongs to T d d orbitals d xy, d yz, d zx d z2, d x2−y2 TdTd spherical TdTd E8C38C3 3C23C2 6S46S4 6σd6σd h = 24 A1A1 11111x2+y2+z2x2+y2+z2 A2A2 111−1 E2 200(z 2, x 2 −y 2 ) T1T1 30−11 T2T2 30 1(xy, yz, zx) E T2T2 dz2dz2 + + d xy CT transition allowed

20. Ni(OH 2 ) 6 2+ belongs to O h d orbitals d xy, d yz, d zx d z2, d x2−y2 OhOh spherical OhOh E8C28C2 6C 2 6C46C4 …h = 48 A 1g 1111x2+y2+z2x2+y2+z2 … EgEg 2−100(z 2, x 2 −y 2 ) … T 2g 301−1(xy, yz, zx) … EgEg T 2g dz2dz2 d xy + + d-d transition forbidden

21. Jahn-Teller distortion OhOh D4hD4h

22. d xy, d yz, d zx d z2, d x2−y2 (3d) 8 d xy, d yz, d zx d z2, d x2−y2 (3d) 9 Hunt’s ruleno Hunt’s rule

23. Cu(OH 2 ) 6 2+ belongs to D 4h d xy, d yz, d zx d z2, d x2−y2 OhOh D4hD4h D4hD4h E2C42C4 C2C2 2C2’2C2’…h = 48 A 1g 1111x 2 +y 2, z 2 … B 1g 1−111x2−y2x2−y2 B 2g 1−11 xy EgEg 20−20xz, yz … EgEg T 2g d zx d xy + + d x2−y2 dz2dz2 d xy d yz, d zx EgEg B 2g B 1g A 1g

24. Jahn-Teller distortion In Cu(OH 2 ) 6 2+, the distortion lowers the energy of d electrons, but raises the energy of Cu-O bonds. The spontaneous distortion occurs. In Ni(OH 2 ) 6 2+, the distortion lowers the energy of d electrons, but loses the spin correlation as well as raises the energy of Ni-O bonds. The distortion does not occur.

25. Summary We have learned how to apply the symmetry theory in the case of molecules with non- Abelian symmetry. We have learned the decomposition of characters into irreps. We have discussed three chemical concepts derived from symmetry, which are Woodward-Hoffmann rule, crystal field theory, and Jahn-Teller distortion.