Symmetry Operations

Identity (E)

No change in molecule, every molecule has it

Rotation (Cn)

Rotation through 360/n. Rotate at Cn angle and it’s symmetrical

Reflection (σ)

Mirror plane

σh

When plane is perpendicular to principal axis of rotation

σv

mirror plane aligns with C2 axis

σd

mirror plane bisects C2

Inversion (i)

Each point moves through the center of the molecule to a position opposite of the original

Improper rotation (s)

Rotation by 360/n, followed by reflection through a plane perpendicular of rotation axis

point group

set of symmetry operations that describes molecules overall symmetry

C1

low symmetry; no symmetry operation besides identity operation

Cs

low symmetry; only one mirror plane

Ci

low symmetry; Only an inversion center

C∞

molecules are linear with an infinite number of rotations and reflections; no inversion center

D∞

linear, infinite number of rotations and reflection, perpendicular C2, and contains an inversion center

Td

4C3s, 3C2 axes, 3S4, 6σd, no C4; mostly tetrahedral molecules

Oh

Octahedral structure; 48 symmetry operations

Ih

Icosahedral structures; six C5, 120 symmetry operations