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Lecture 30 Point-group symmetry III (c) So Hirata, Department of Chemistry, University of Illinois...
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![Page 1: Lecture 30 Point-group symmetry III (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022081514/56649d9c5503460f94a85394/html5/thumbnails/1.jpg)
Lecture 30Point-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.
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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 (D4h) has doubly degenerate wave functions.
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The D4h character table (h = 16)
D4h E 2C4 C2 2C2’ 2C2” i 2S4 σh 2σv 2σd
A1g 1 1 1 1 1 1 1 1 1 1
A2g 1 1 1 −1 −1 1 1 1 −1 −1
B1g 1 −1 1 1 −1 1 −1 1 1 −1
B2g 1 −1 1 −1 1 1 −1 1 −1 1
Eg 2 0 −2 0 0 2 0 −2 0 0
A1u 1 1 1 1 1 −1 −1 −1 −1 −1
A2u 1 1 1 −1 −1 −1 −1 −1 1 1
B1u 1 −1 1 1 −1 −1 1 −1 −1 1
B2u 1 −1 1 −1 1 −1 1 −1 1 −1
Eu 2 0 −2 0 0 −2 0 2 0 0
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C3v: another non-Abelian group
C3v, 3m E 2C3 3σv h = 6
A1 1 1 1 z, z2, x2+y2
A2 1 1 −1
E 2 −1 0 (x, y), (xy, x2−y2), (zx, yz)
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C3v: expanded character table
C3v, 3m E C3 C32 σv σv σv h = 6
A1 1 1 1 1 1 1 z, z2, x2+y2
A2 1 1 1 −1 −1 −1
E 2 −1 −1 0 0 0 (x, y), (xy, x2−y2), (zx, yz)
C3v, 3m E 2C3 3σv h = 6
A1 1 1 1 z, z2, x2+y2
A2 1 1 −1
E 2 −1 0 (x, y), (xy, x2−y2), (zx, yz)
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Integral of degenerate orbitals
C3v, 3m E C3 C32 σv σv σv h = 6
A1 1 1 1 1 1 1 z, z2, x2+y2
A2 1 1 1 −1 −1 −1
E 2 −1 −1 0 0 0 (x, y), (xy, x2−y2), (zx, yz)
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What is E ✕ E ?
What is the irrep for this set of characters?
C3v, 3m E C3 C32 σv σv σv h = 6
A1 1 1 1 1 1 1 z, z2, x2+y2
A2 1 1 1 −1 −1 −1
E 2 −1 −1 0 0 0 (x, y), (xy, x2−y2), (zx, yz)
E ✕ E 4 1 1 0 0 0
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 A1 is normalized.
The character vector of E is normalized.
The character vectors of A1 and E are orthogonal.
Orthonormal character vectorsC3v, 3m E C3 C3
2 σv σv σv h = 6
A1 1 1 1 1 1 1 z, z2, x2+y2
A2 1 1 1 −1 −1 −1
E 2 −1 −1 0 0 0 (x, y), (xy, x2−y2), (zx, yz)
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The contribution (cA1) of A1:
The contribution (cA2) of A2:
The contribution (cE) of E:
DecompositionC3v, 3m E C3 C3
2 σv σv σv h = 6
A1 1 1 1 1 1 1 z, z2, x2+y2
A2 1 1 1 −1 −1 −1
E 2 −1 −1 0 0 0 (x, y), (xy, x2−y2), (zx, yz)
E ✕ E 4 1 1 0 0 0
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 ruleThe photo and thermal pericyclic reactions yield different isomers of cyclobutene.
H
H
CH3
CH3
H
HCH3
CH3
CH3
HCH3
H
Reaction A
Reaction B
photochemical
thermal
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H
H
CH3
CH3
H
HCH3
CH3
CH3
HCH3
H
Reaction A
Reaction B
Woodward-Hoffmann ruleWhat are the symmetry groups to which these reactions A and B belong?
photochemical / disrotary / Cs
thermal / conrotary / C2
σ
C2
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Woodward-Hoffmann rule
higher energy
occupied
higher energy
dcba hgfe
occupied
Reactant Product
Process a b c d e f g h
Photochemical / Cs A” A’ A” A’ A” A” A’ A’
Thermal / C2 A B A B B A B A
Process a b c d e f g h
Photochemical / Cs A” A’ A” A’ A” A” A’ A’
Thermal / C2 A B A B B A B A
“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(NH3)6]2+, [Ni(en)3]2+, [NiCl4]2−, [Ni(H2O)6]2+
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Crystal field theory
d orbitalsdxy, dyz, dzx
dxy, dyz, dzx
dz2, dx2−y2
dz2, dx2−y2
Td Ohspherical
E
T2
Eg
T2g
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NiCl42− belongs to Td
d orbitals
dxy, dyz, dzx
dz2, dx2−y2
Td spherical
Td E 8C3 3C2 6S4 6σd h = 24
A1 1 1 1 1 1 x2+y2+z2
A2 1 1 1 −1 −1
E 2 −1 2 0 0 (z2, x2−y2)
T1 3 0 −1 1 −1
T2 3 0 −1 −1 1 (xy, yz, zx)
E
T2 dz2
+
dxy
CT transitionallowed
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Ni(OH2)62+ belongs to Oh
d orbitalsdxy, dyz, dzx
dz2, dx2−y2
Ohspherical
Oh E 8C2 6C2 6C4 … h = 48
A1g 1 1 1 1 x2+y2+z2
…
Eg 2 −1 0 0 (z2, x2−y2)
…
T2g 3 0 1 −1 (xy, yz, zx)
…
Eg
T2g
dz2dxy
+
d-d transitionforbidden
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Jahn-Teller distortion
Oh D4h
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Jahn-Teller distortion
dxy, dyz, dzx
dz2, dx2−y2
(3d)8
dxy, dyz, dzx
dz2, dx2−y2
(3d)9
Hunt’s rule no Hunt’s rule
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Cu(OH2)62+ belongs to D4h
dxy, dyz, dzx
dz2, dx2−y2
OhD4h
D4h E 2C4 C2 2C2’ … h = 48
A1g 1 1 1 1 x2+y2, z2
…
B1g 1 −1 1 1 x2−y2
B2g 1 −1 1 −1 xy
Eg 2 0 −2 0 xz, yz
…
Eg
T2g
dzxdxy
+
dx2−y2
dz2
dxy
dyz, dzx
Eg
B2g
B1g
A1g
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Jahn-Teller distortion
In Cu(OH2)62+, the distortion lowers the
energy of d electrons, but raises the energy of Cu-O bonds. The spontaneous distortion occurs.
In Ni(OH2)62+, 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.