Lecture 6 Molecular Symmetry B 092210

8/13/2019 Lecture 6 Molecular Symmetry B 092210

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II.PointGroupsPoint Group

A complete set of symmetry operations that can be carried out on a single object

(reflecting the overall symmetry of the object).

These symmetry operations are called point group symmetry operations because

they leave at least one point of the object unmoved.

E nothing moved

a whole plane not moved

Cn a line not moved

i (S2) a point not moved

Objects (including molecules) that possess all symmetry operations of a given point

group are said to belong to that (point) group.


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Crystallographic Point Groups

A set of point groups used to classify molecules (based on crystallography).

There are 32 crystallographic point groups that classify most known molecules.

(See pictures shown on next page)

About 40point groups classify all molecules (32 crystallographic groups listedon next page, plus a couple of others, including dodecahedron and icosahedron.

1. Assigning Molecules to Point Groups(a) Point Group C1

Consisting of a single symmetry operationE ( = C1).

Any molecules that do not have any real symmetry belong to this point group.Ex/. CHBrClF

H

Br

F

Cl


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32 Point Groups


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(b) Point Group Cs

Consisting of two and only two symmetry operations:E and .Any molecules that contain a reflection symmetry plane (only) belong to this point

group.

Ex/. CH(Cl2)F

(c) Point Group Ci

Consisting of two and only two symmetry operations:E and i.

Any molecules that contain only a center of symmetry belong to this point group.

Ex/. (CHBrCl)2 (staggered)

H

F

Cl

Cl

H

Br

Cl

Br

HCl


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(d) Point Group Cn

Consisting ofn symmetry operations: E, Cn , Cn2

Cnn-1

Any molecules that contain a n-fold symmetry rotational axis Cnbelong to this

point group.

Ex/. H2O2(Hydrogen Peroxide)

(e) Point Group Cnv (Cn as a subgroup)

Consisting of2n symmetry operations: E, Cn , Cn2 Cn

n-1 , n vMolecules that have an n-fold rotational axis Cn andn v vertical reflection planes

(passing the n-fold rotational axis Cn) belong to this point group (no C2

Cn).

Ex/. H2O (2n = 4) E, C2(z), v(xz), v(yz)

NH3 (2n = 6) E, C3(z), C32(z), v, v, v

H

O

H

C2or

N

H

H

Hy

x

z

O

H

OH


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(f) Point group Cnh (Cn as a subgroup)

Consisting of2n symmetry operations

Molecules that have a Cn symmetry axis with a h symmetry reflection plane

perpendicular to the Cn axis belong to this point group (no C2

Cn).

Ex/. C2h (2n = 4): E, C2(z), h(xy), i

C3h (2n = 6): E (= C33), C3(z), h(xy), C3

2, S3, S35

(S32 = C3

2, S33 = h, S3

4 = C3)

Also, C1h = Cs

For n = even, Cnh has an inversion center (i) C2h, C4h, C6h

B

O O

O

H

H

H

x

y

H

CC

F

H F


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(g) Point GroupDn (Cn as a subgroup)

Consisting of2n symmetry operations.

Molecules that have a Cn rotational axis andn C2 rotational axis perpendicular to

the Cn axis belong to this point group.

Ex/. Ethane (C2H

6) (configuration that is neither a staggered nor eclipsed)

E, C3, C32, 3C2

(h) Point GroupDnh (Dn as a subgroup)


Molecules that have a Cn axis,n C2 axes perpendicular to Cn, a horizontal

symmetry plane ( h) andn v vertical symmetry planes (passing through Cn)belong to this symmetry point group.

C C

H

H

H

H

H

HC2

C2C2

The two equilateral H3

triangles are not

symmetrically placed

w.r.t. each other


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Ex/. BF3 (D3h), 4n = 12

symmetry elements: C3, h(xy), v, v, v, C2, C2,

C2, S3

symmetry operations: E, C3, C32, S3, S3

2, v, v, v, h,

C2, C2, C2 (Note: S33 = h)

For even n,Dnh has an inversion center i.

(i) Point Group Dnd (Dn as a subgroup)


Molecules that have a Cn axis,n C2 axes perpendicular to Cn, andn d dihedral

planes (passing through Cn) belong to this symmetry point group.

Ex/. staggered ethane (D3d)

E, C3, C32, 2S6, d, d, d, C2, C2, C2 , i

For odd n,Dnd has an inversion center i.

B

F

FF

C2

H

H

H

H

HH

C2

C2


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(j) Point Group Sn (n = even)

Consisting ofn symmetry operations (Sn, Sn2 Sn

n =E)

Molecules that have an Sn axis only without mirror planes belonging to this

group.

n = odd Sn Cnh group [since (Sn)n h and (Sn

)2n

E]

Note:

a) Dnd also contains Sn axis, but it also containsn d mirror planes

b) S1

Cs

C1h group (recall S1 operation h operation)

c) A Sn axis may coincide with a Cn/2 axis

Ex/. S4point group: S4, S42 = C2 , S4

3 ,E

(k) Linear Point Groups

Cv: Containing

symmetry operations:E, C

,

v

Molecules that have a C

axis,

v vertical reflection planes, but no C2 axis

perpendicular to the C

axis belong to this group.

All heteronuclear diatomic molecules, and all asymmetric linear polyatomicmolecules belong to this point group (e.g. C-O, H-F, S-C-N-)


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Dh:All symmetry operations in Cvplus

C2 rotations perpendicular to the C

axis, and a hplane.

All homonuclear diatomic molecules and symmetric linear polyatomicmolecules belong to this group.

Ex/. O=C=O, NCC N, HCC H, NN

(l) Point Groups Arised from Platonic Solids

Platonic Solid polyhedra with all vertices, edges, and faces equivalent.

In 3D space, five exist:Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron (See next page).

Conjugate of a Platonic Solid solid figure obtained by connecting the centers of

adjacent faces in a platonic solid with lines

Cubic Octahedron

Tetrahedron Tetrahedron

Icosahedron Dodecahedron

No. Symmetry Operations = 2(No. Faces) (No. Edges Per Face)


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Five Platonic Solids TETRAHEDRONFour triangular faces, four

vertices, and six edges.

(2

4

3 = 24)

CUBE

Six square faces, eight vertices,

and twelve edges.

(2

6

4 = 48)

OCTAHEDRON

Eight triangular faces, six

vertices, and twelve edges.

(2

8

3 = 48)

DODECAHEDRON

Twelve pentagonal faces,

twenty vertices, and thirtyedges.

ICOSAHEDRON

Twenty triangular faces,

twelve vertices, and thirty

edges.


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Tetrahedron (Td)

24 symmetry operations: E, 8C3 , 3C2 , 6S4 , 6d

Any molecules with a tetrahedral geometry belong to this group

Ex/. CH4 , CCl4

Octahedron (Oh)

48 symmetry operations

Any molecules with an octahedral symmetry belong to this group.

Ex/. SF6 , (PtCl6)2-

Cube (Th)


Any molecules with a cubic symmetry belong to this point group.

Icosahedron and Dodecahedron (Ih)


Molecules that have one of these two symmetries (geometries) belongto this group.


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(m) A systematic way to assign molecule to a point group

Use Figure 4-7 (text p. 83) as a guide while assigning a molecule to a point group. First determine if the molecule is linear.

Decide if there are two or more Cn with n > 2.

Find the principal rotational axis Cn

for molecules with only one Cn

(n > 2).

Work with each of the subgroups to determine the actual point group.

Ex/. (a) PF5 (40 valence e-, trigonal bipyramidal)

C3 and no other Cn with n > 2, 3C2

C3 , h

D3h (no i)

(b) (AgC6H6)+

C6 , no other Cn with n > 2,

No n C2

C6 , no h

Yes 6 v

C6v

H H

H

H

H

H

Ag

P

F

F

F

F

F


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Point Group Assignment

N


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III.

SymmetryApplicationsThe symmetry of a molecule is crucial for certain properties to exist. Here wediscuss two such properties.

1. Polarity

A molecule is polar if it has a permanent Electric Dipole Moment (), whensummation of all individual Bond Moment Vector (b) is nonzero.Dipole Moment:

bb

bb

dq

dq

=

==

)(r

rr

Where, q = charge

db = distance between the two charge centers

Since a symmetry operation produces a configuration physically indistinguishablefrom the original one, thedirection and themagnitude of the dipole momentvector must remain unchanged after a symmetry operation. Therefore, the dipolevector must be contained in all symmetry elements of the molecule that has apermanent dipole moment.

+q -q

db


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Important Notes

For a proper rotation axis Cn must lie in the axis

For two or more non-coincident symmetry axes no dipole

(This excludes all

point groups containingDn as a subgroup)

Ex/. CH4 (Td), 4C3 axes, no dipole

For symmetry plane must lie in the plane.

For several symmetry planes must lie along the line of intersection of theseplanes.

Ex/. H2O,

in C2 axis, which is also the intersection line of the two vplanes

For center of symmetry i no dipole

(direction change)

Ex/. Any atoms containing a center of symmetry do not have dipole

moments, e.g. (CHBrCl)2

(Ci)

Conclusion

Only molecules that belong to C1 , Cs , Cn or Cnv point groups possess dipolemoments and they are polar.

H

Br

Cl

Br

HCl


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Examples

a) CHCl2F

Cs dipole

b) H2O2 (hydrogen peroxide)

C2 dipole

2. Chirality and Optical Activity

Chiral Molecule

Molecule/object that is not superimposable on its mirror image (ex/ hands)

Chiral Greek hand

Enantiomer

A chiral molecule and its mirror image together form a pair of enantiomers.

Enantimer Greek both

Chiral molecules are (potentially) optically active. They can rotate the plane ofpolarized light that passes through them.

C2or

H

F

Cl

Cl

O

H

O

H


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How can we make use of symmetry to identify a molecule in terms of its optical

activity?

Consider a Sn symmetry operation on a molecule A:

Since:S1

, S2

i

Molecules having any plane of symmetry or center of symmetry can not be chiral,

and they are optically inactive.

ACn Rotated

A

Mirror Imageof (Rotated) A

Superimposable

since Sn is a

symmetry operation

As shown above, an improper symmetry rotation Sn always converts a right-handedobject into a left-handed one and they are superimposable. Thus, molecules

containing any Sn axiscan notbe chiral, and theycan not be optically active.


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Conclusion

Only molecules that belong to C1 and Cn point groups are potentially opticallyactive.

Examples:

Lecture 6 Molecular Symmetry B 092210

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