1. Closure: A, B G AB G

2. Associativity: A, B, C A(BC)= (AB)C

3. Identity: There exists E G such that AE=EA=A for all A G

4. Inverse: A G there exists A-1 G such that AA-1 =A-1A=E

Mathematical Group

Order of a group: the number of elements it contains

Symmetry of an object point group (symmetry about a point){E, C2,v,v'} = point group C2v

Binary operation: one operation followed by another

C2v E C2 v v’

E E C2 v v’

C2 C2 E v’ v

v v v’ E C2

v’ v’ v C2 E

Closure: Associativity: Identity: Inverse:

C2v E C2 v v’

E E C2 v v’

C2 C2 E v’ v

v v v’ E C2

v’ v’ v C2 E

Rearrangement Theorem: each row and each column in a group multiplication table lists each of the elements once and only once.

Proof: suppose AB=AC, i.e. two column entries are identicalthen:

EB=ECB=C

ACAABA 11

C2v E C2 v v’

E E C2 v v’

C2 C2 E v’ v

v v v’ E C2

v’ v’ v C2 E

A group is Abelian if AB=BA ( the multiplication is completely commutative).

Not all groups are abelian.

vv CC 33

Any object (or molecule) may be classified into a point group uniquely determined by its symmetry.

Groups with low symmetry:{E}=C1, Schönflies Symbol/notation

{E,} =Cs

{E, i} =Ci

ONCl, Cs

H

HCl

Cl

F

F

Ci

Groups with a single Cn axis

{E, Cn, Cn2, Cn

3….Cn n-1} =Cn

H2O2

Groups with a single Cn axis plus

a perpendicular h plane: Cnh

222ON },,,{ 2 iCE hhC2

},,,{ 2 iCE hhC2

C = C

Cl

ClH

H

},,,,,{ 533

233 SSCCE hhC3

B

O

O O

H

H

H

Groups with a single Cn axis plus n vertical v planes: Cnv

vC2 OH 2

},,,{ '2 vvCE

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

3

3

NH

C v

5

4

BrF

C v },,,,,,{ ','3424 ddvvCCCE

d: dihedral reflection planes (bisects v )

Br

F

FF

F F

Square pyramidal

n-gonal pyramidal shape: Cnv

vCHF

...},...,{ vCE

Groups generated by an Sn axis: Sn

n odd Cnh

n even: Sn

B

O

O O

H

H

H

hC3

iCS 2

H

HCl

Cl

F

F

4

342

244 },,,{

S

SCSSE

1,3,5,7-tetrafluorocyclooctatetraene

F

F

F

F

Dihedral groups: Groups with a Cn axis plus n perpendicular C2 axes: Dn

C2H6 partly rotated (not staggered, not eclipsed).

},,,,,{ ''2

'22

2333 CCCCCED

Other example:

32422 )( NHHCNHRu

22422 )( NHHCNHPt

Groups with a Cn axis, n C2 axes and a h plane: Dnh

Example: Eclipsed Conformation of Hydrazine, N2H4

},,,),(),(),(,{ 2222 xzyzxyh ixCyCzCED

D3h: BF3

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

2333 vvvhh SSCCCCCED

D4h: PtCl4

,....},,,.........,.....,,{

,

2

22:

SiCCE

BeFHD

hv

h

n-gonal prism

Groups with a Cn axis, n C2 axes, and n d planes: Dnd

Symmetry Elements for D3d Point Group (A2X6)

C2H6 staggered.

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

2333 dddd SSiCCCCCED

}',,'',',,,,{ 2234422 ddd CCSSCED

Allene

C

C

C

staggered regular polygons (n-gonal antiprism)

B2Cl4

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