.Moleculary Symmetry, Projection Diagrams, And Point Groups

5/28/2018 .Moleculary Symmetry, Projection Diagrams, And Point Groups

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Molecular Spectroscopy

January 29, 2008Molecular Groups, Projection Diagrams, and PointGroups


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Element Operation Example

E, the identity Does nothing All objects posses E

Cn, n-fold rotation

Rotate object by 2/n about the n-fold axis.The axis with the highest n is called the principleaxis.

For n>2 the sense is importantCn+ - clockwise

rotationCn

--counterclockwise rotation

H2O posses a C2 axisNH

3posses a C

3axis

Benzene posses a C2, C3, C6, and two sets ofthree C

2axes perpendicular to the C

6axis

, reflection plane Reflects through the plane Three types

v, vertical plane Plane containing the principle axis

H2O

h, horizontal plane Plane perpendicular to the principle axis Benzene

d, dihedral plane

Vertical plane which bisects two C2

axes perpendicular

to the principal axis

Benzene

i, inversion center

Sends each atom through the origin along a straight

line so that x-x, y-y, and z-z

H2

CO2

BenzeneSF

6

Sn, n-fold improper rotation axis Rotate object by 2/n, then reflects through plane

perpendicular to the axis of rotation

CH4


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Lets take the ECLIPSED conformation of ethane

One three-fold axis coincident with the C-C bond

Three two-fold axes perpendicular to the C-C bond and intersecting its midpoint

Three reflection planes, each containing the C-C bond and a pair of C-H bonds

One reflection plane perpendicular to the C-C bond and bisecting it

NO CENTER OF INVERSION IS PRESENT

One three-fold improperaxis coincident with the C-C bond


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Lets take the STAGGERED conformation of ethane

One three fold axis coincident with the C-C bond

Three two-fold axes perpendicular to the C-C bond and intersecting its midpoint

Three reflection planes, each containing the C-C bond and a pair of C-H bond

No reflection plane perpendicular to the C-C bondwe lost it!

One point of inversion at the midpoint of the C-C bondwe gained it!

One six-fold improper axis coincident with the C-C bondchanged its order!


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Stereographic Projections


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Lets consider the symmetry elements of the C2v point group and

the resulting symmetry operations


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Still considering the symmetry elements of the C2v point group

and the resulting symmetry operations


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Lets compare our results with the C2v character table

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


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Lets try this with another point group and build the character table

C2h E C2 h i

E

C2

h

i


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This gives us the following for the Character Table

C2h E C2 h i

E

i

C2

h

h C2

i

E

Complete the rest as we have done and hand it in on Tuesday


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Point Groups

Symmetry operations may be organized into groups that obey the four properties

of mathematical groupsnamely

1. There exists an identity operator that commutes with all other members

2. The product of any two members must also be a member3. Multiplication is associative

4. The existence of an inverse for each member

It will be possible to assign ANY molecule to one of these so-called Point Groups So named because the symmetry operation leaves one point

unchanged


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Point Groups

Point Group C1Trivial group with contains all molecules having no symmetry

There is no center of inversion, plane of reflection, improper axis of

rotation, etc.

The molecule HNClF (below) is such a molecule


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Point Groups

Point Group CsOnly molecules which have to them a plane of reflection as a symmetry

element

Formyl Chloride belongs to this point group


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Point Groups

Point Group Ci

Molecules whose sole symmetry element is an inversion center

As an example take the rotamer of 1,2-dichloro-1,2-diflouroethane in

which the fluorine atoms are off-set to one another

The only operations generated by this inversion are i and i2=Ethis is a

group of order 2 and is designated Ci

Important to note that i is the same as an S2 axis


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Point Groups

Point Group CnMolecules with only an n-fold axis of rotation

Boric acid with the hydrogen atoms lying above the BO3 plane is an

example


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Point Groups

Point Group CnvMolecules with an n-fold Cn axis of rotation and n vertical mirror planes

which are colinear with the Cn axis

Water belongs to such a point groupthe C2V

Note that if you find a Cn axis and one v plane then you are

gauranteed to find n v planes


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Point Groups

Point Group CnhThis point group is generated by a Cn axis of rotation AND a horizontal

mirror plane, h, which is perpendicularto the Cn axis

Trans-butadiene (trans-1,3-butene) is an example of the C2h point

group


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Point Groups

Point Group DnThis point group is generated by a Cn axis of rotation and a C2 axis which

is perpendicularto the Cn axisno mirror planes

Tris(ethylenediamine)cobalt(III) posses idealized D3 symmetry

Build this molecule and convince yourself that this in indeed the case!!


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Point Groups

Point Group DndThis point group is generated by a Cn axis of rotation and a C2 axis which

is perpendicularto the Cn axis and a dihedral mirror plane, dThe dihedral plane is colinear with the principal axis and bisects the

perpendicular C2 axisCrystals of Cs2CuCl4 have a squashed tetrahedral D2d structure


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Point Goups

Point Group DnhThis point group is generated by a Cn axis of rotation and n C2 axis which

is perpendicularto the Cn axis and a horizontal mirror plane which is by

definition perpendicularto the principal axis of rotation

Benzene is the classic example, possesing D6h symmetry


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Point Goups

Point Groups SnThese point groups are generated by an Sn axis.

For example, the spirononanetetraflourospirononane belongs to the

S4 point group and only elements generated by S4

S4S4

2 = C2S4

3

S44 = E

The S2 point group is just Ci since S2 = i

When n is odd the point groups are just the same as the Cnh point groups

and so are designated as such

As a direct result, only S4, S6, and S8, have a separate existence


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Special Point Goups

Tetrahedral molecules belong to the point group Td

Octahedral molecules belong to the point group Oh

Molecules with icosahedral (20 triangular faces) or dodecahedral (12 pentagonal

faces) belong to the point group Ih

Atomswhich have spherical symmetry, belong to the point group Kh

The point group Th, which may be obtained by adding to Td as set of h planes,

which contain pairs of C2 axisas opposed to the d planes which contain one C2axis and bisect another pair, giving Td)

.Moleculary Symmetry, Projection Diagrams, And Point Groups

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