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Symmetry and the point groups

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Symmetry Elements and Symmetry Operations

- Identity
- Proper axis of rotation
- Mirror planes
- Center of symmetry
- Improper axis of rotation

Symmetry Elements and Symmetry Operations

- Identity gt E

Symmetry Elements and Symmetry Operations

- Proper axis of rotation gt Cn
- where n 2, 180o rotation
- n 3, 120o rotation
- n 4, 90o rotation
- n 6, 60o rotation
- n , (1/ )o rotation
- principal axis of rotation, Cn

2-Fold Axis of Rotation

3-Fold Axis of Rotation

Rotations for a Trigonal Planar Molecule

Symmetry Elements and Symmetry Operations

Mirror planes

sh gt mirror plane perpendicular to a

principal axis of rotation

sv gt mirror plane containing principal axis of

rotation

sd gt mirror plane bisects dihedral angle made

by the principal axis of rotation and two

adjacent C2 axes perpendicular to principal

rotation axis

Mirrors

sv

sv Cl Cl sh I

sd sd Cl Cl

Rotations and Mirrors in a Bent Molecule

Benzene Ring

Symmetry Elements and Symmetry Operations

- Center of symmetry gt i

Center of Inversion

Inversion vs. C2

Symmetry Elements and Symmetry Operations

- Improper axis of rotation gt Sn
- rotation about n axis (360 /n) followed by

reflection through a plane perpendicular to the

axis

Improper Rotation in a Tetrahedral Molecule

S1 and S2 Improper Rotations

Successive C3 Rotations onTrigonal Pyramidal

Molecule

Linear Molecules

Selection ofPoint Group from Shape

- first determine shape using Lewis Structure and

VSEPR Theory - next use models to determine which symmetry

operations are present - then use the flow chart to determine the point

group

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Decision Tree

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Selection ofPoint Group from Shape

1. determine the highest axis of

rotation 2. check for other non-coincident axis

of rotation 3. check for mirror planes

H2O and NH3

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Geometric Shapes

E, S4, C2

Point Groups with improper axes S2n (n 2)

1,3,5,7-tetrafluorocycloocta-1,3,5,7-tetraene

(S4)

2) Point Groups of high symmetry (cubic groups)

- In contrast to groups C, D, and S, cubic symmetry

groups are characterized by the presence of

several rotational axes of high order ( 3). - Cases of regular polyhedra
- Td (tetrahedral) BF4- , CH4
- Symmetry elements E, 4C3, 3C2, 3S4, 6sd
- Symmetry operations E, 8C3, 3C2, 6S4, 6sd
- If all planes of symmetry and i are missing, the

point group is T (pure rotational group, very

rare)

3) Point Groups of high symmetry

- Oh (octahedral) TiF62-, cubane C8H8
- Symmetry elements E, i, 4S6, 4C3, 3S4, 3C4,

6C2, 3 C2, 3sh, 6sd - Symmetry operations E, i, 8S6, 8C3, 6S4, 6C4,

6C2, 3 C2, 3sh, 6sd - Pure rotational analogue is the point group O (no

mirror planes and no Sn very rare)

4) Point Groups of high symmetry

Th group (symmetry elements E, i, 4S6, 4C3, 3C2,

3sh) can also be considered as a result of

reducing Oh group symmetry (E, i, 4S6, 4C3, 3S4,

3C4, 6C2, 3 C2, 3sh, 6sd ) by eliminating C4,

S4 and some C2 axes and sd planes

Point Groups of high symmetry

- Ih (icosahedral) B12H122-, C20
- Symmetry elements E, i, 6S10, 6C5, 10S6, 10C3,

15C2, 15s - Pure rotation analogue is the point group I (no

mirror planes and thus no Sn, very rare)

Enantiomer Pairs

Enantiomer Pairs

Polarimeter

(b) Chirality A chiral molecule not

superimposed on its mirror image. optically

active (rotate the plane of polarized)

A molecule may be chiral only if it does not

posses an axis of improper rotation Sn. All

molecules with center of inversion are achiral ?

optically inactive. S1s any molecule with a

mirror plane is achiral.

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