Chem 104A, UC, Berkeley Groups with very high symmetry:...

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Chem 104A, UC, Berkeley

Groups with very high symmetry: multiple high-fold rotation axes

The platonic solids: polyhedra constructed from regular polygons with all vertices and edges equivalent: 5 possibilities only.

icosahedron


None

dodecahedron

2

Chem 104A, UC, BerkeleyTetrahedron: Td

}6,3,3,3,4,4,{ 3442

233 dSSCCCE

CH4

group order: 24

Chem 104A, UC, BerkeleyRotational subgroup: T

}3,4,4,{ 2233 CCCE

1264 ][ LFe

C2 symmetry of the ligand

3


T + i Th

}3,4,4,,3,4,4,{ 5662

233 hSSiCCCE

2655 ])([ NHCFe


Octahedron: Oh

}6,3,4,4,3,3,,6,4,4,3,3,3,{ 566

3442

233

3424 vhSSSSiCCCCCCE

26 ][PtCl

group order: 48

4


Rotational subgroup: O

}6,4,4,3,3,3,{ 2233

3424 CCCCCCE

Cube : Oh

Example: ferritin protein

}6,3,4,4,3,3,,6,4,4,3,3,3,{ 566

3442

233

3424 vhSSSSiCCCCCCE


5


Chem 104A, UC, BerkeleyIcosahedron: Ih

}15,10,10,6,6,6,6,,15,10,10,6,6,6,6,{ 566

910

710

310102

233

45

35

255 SSSSSSiCCCCCCCE

21212 ][ HB

group order: 120

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Chem 104A, UC, BerkeleyRotational subgroup: I

}15,10,10,6,6,6,6,{ 2233

45

35

255 CCCCCCCE

The rhinoviruses are single stranded positive sense RNA viruses. They are the most common viral infective agents in humans. The most well known disease caused by rhinoviruses is the common cold.


Dodecahedra: Ih

}15,10,10,6,6,6,6,,15,10,10,6,6,6,6,{ 566

910

710

310102

233

45

35

255 SSSSSSiCCCCCCCE

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Symmetry of a perfect sphere: Kh

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},,,,,{ '''233 vvvCCE C3v

},,,,,{ 533

233 SSCCE hhC3

{E, Cn, Cn2, Cn

3….Cn n-1}Cn

{E}=C1, Schönflies Symbol/notation

{E,} =Cs

{E, i} =Ci

4

342

244 },,,{

S

SCSSE S4


}15,10,10,6,6,6,6,,15,10,10,6,6,6,6,{ 566

910

710

310102

233

45

35

255 SSSSSSiCCCCCCCEIh

}15,10,10,6,6,6,6,{ 2233

45

35

255 CCCCCCCEI

}6,4,4,3,3,3,{ 2233

3424 CCCCCCEO

}6,3,4,4,3,3,,6,4,4,3,3,3,{ 566

3442

233

3424 vhSSSSiCCCCCCE Oh

}3,4,4,,3,4,4,{ 5662

233 hSSiCCCE Th

}3,4,4,{ 2233 CCCET

}6,3,3,3,4,4,{ 3442

233 dSSCCCE Td

}'',',,,,,'',',,,,{ 566222

2333 dddd SSiCCCCCED

}'',',,,,,'',',,,,{ 533222

2333 vvvhh SSCCCCCED

},,,,,{ ''2

'22

2333 CCCCCED

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Roadmap to determine the point group


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Cv

YesNo

Linear molecules

C ? i ? Dh

Yes

Cn ?

? Cs

i ? Ci

C1

No

No

No

No

Yes

Yes

Molecules ofLow symmetry

Yes

Molecules ofHigh symmetry

Chem 104A, UC, BerkeleyYes Yes

No No

I

3C4 ? i ? Oh

No

Yes Yes

No

O

Cn ? 6C5 ? i ? IhYes

4C3? i? Th

6 ? Td

T

Yes Yes

No

No

Yes

Molecules ofHigh symmetry

No

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nC2Cn ? h ? Dnh

S2n? S2n n v? Dnd

h? Cnh Dn

n v? Cnv

Cn

Yes Yes

YesYes

Yes

Yes

No

No

No

No

No

No


Oh

12



D2d

13



14



15


O=C=O

O=C=S

Dh

Cv


hD

CNNi

4

24 ])([

1. Ion is completely planar2. Ni-C-N bond angles are 180 degree3. C-Ni-C bond angles are 90 degree4. Ni-C bond lengths are all equal5. C-N bond lengths are all equal.

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Symmetry & Group Theory

Predict vibrational spectra

Determine optical activity

Construct bonding based on atomic orbitals

Access reaction pathway


100

010

001

100

010

001

100

010

001

100

010

001

Any symmetry operation can be carried out using a 3x3 transformation matrix

Collectively, these matrices form a representation of the group

17



‘

18


'"'23

"'3

vv

vv

C

C

x

z

y


E C2 v v’

1 1 1 1 z

1 1 -1 -1 Rz xy

1 -1 1 -1 x, Ry xz

1 -1 -1 1 y,Rx yz

C2v

A1

A2

B1

B2

222 ;; zyx

Character Table

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C3v E 2C3 3v

A1 1 1 1

A2 1 1 -1

E 2 -1 0

z

Rz

(x,y),(Rx,Ry)

222 ; zyx

),(

),( 22

yzxz

xyyx

Chem 104A, UC, BerkeleyCharacter Table

Two elements (A and B) of a group (G) are conjugate if there exists X G, such that B=X-1AX; B is called the similarity transform of A by X

Symmetry operations that are conjugate serve to transform regions similarly.

A complete set of elements that are conjugate to one another is called a class of the group.

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'3

'

3

CC

C

vv

v

Example:

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C3v E 2C3 3v

A1 1 1 1

A2 1 1 -1

E 2 -1 0

z

Rz

(x,y),(Rx,Ry)

222 ; zyx

),(

),( 22

yzxz

xyyx

100

010

001

100

03/2cos3/2sin

03/2sin3/2cos

100

010

001


Character of a square matrix A is the sum of its diagonal elements

(A)= aii

Matrices for symmetry elements belonging to the same class have identical character.

(converse not true).

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If the matrices in a representation are all block-factored in the same way, then the representation is reducible, and may be decomposed into new representations consisting of sets of the isolated blocks.

Irreducible representation

The number of irreducible representations of a group is equal to the number of classes.

Chem 104A, UC, Berkeley Groups with very high symmetry:...

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