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Lecture 30 Point-group symmetry III

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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.

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

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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

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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)

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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)

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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)

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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

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

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

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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)

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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

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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

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

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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

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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”

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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+

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

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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

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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

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Jahn-Teller distortion OhOh D4hD4h

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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

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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

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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.

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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.

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