Symmetry and the point groups

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# Symmetry and the point groups - PowerPoint PPT Presentation

Symmetry and the point groups. 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 => E. Symmetry Elements and Symmetry Operations.

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## Symmetry and the point groups

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### Symmetry and the point groups

Symmetry Elements and Symmetry Operations

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

Symmetry Elements and Symmetry Operations

• Proper axis of rotation => 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

Symmetry Elements and Symmetry Operations

Mirror planes

sh => mirror plane perpendicular to a

principal axis of rotation

sv => mirror plane containing principal

axis of rotation

sd => mirror plane bisects dihedral angle made

by the principal axis of rotation and two

adjacent C2 axes perpendicular to principal

rotation axis

Mirrors

svsv

Cl Cl

sh

I sd

sd

Cl Cl

Symmetry Elements and Symmetry Operations

• Improper axis of rotation => Sn
• rotation about n axis (360° /n) followed by reflection through a plane perpendicular to the axis

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

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

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);

If all dihedral planes are removed but 3 sh remain, the point group is Th ( [Fe(py)6]2+ )

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 Ohgroup symmetry (E, i, 4S6, 4C3, 3S4, 3C4, 6C2, 3 C2, 3sh, 6sd )

by eliminating C4, S4 and some C2 axes andsd 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)

(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.

S1=s any molecule with a mirror plane is achiral.