Point groups

What should you first ask yourself for deciding point groups?

If the molecule is linear, has high symmetry, or low symmetry

Linear point groups

C∞V and D∞h

C∞V

linear

No inversion center (i)

D∞h

Linear

Has inversion center (i)

High symmetry

Must have two or more Cn that are n > 2

Td, Oh, and Ih

Td

Has multiple C3 rotation axes

No inversion center (i)

Oh

Has multiple C4 rotation axes

Has an inversion center (i)

Does not have a C5

Ih

Has multiple C5 rotation axes

Has an inversion center (i)

Low symmetry

No Cn

Cs, Ci, and C1

Cs

No Cn

Yes mirror planes

only identity and mirror planes

Ci

No Cn

No mirror planes

Yes inversion center (i)

only identity and inversion center

C1

No Cn

No mirror planes

No inversion center (i)

only identity

If linear, high, and low symmetry query is no then …….

the highest order Cn will be principal → where we define “n” in point group naming

Dihedral groups

Have nC2 axes that are perpendicular to the principal

Dnh, Dnd, and Dn

Dnh

Has nC2 axes that are perpendicular to the principal

Has horizontal mirror planes

Dnd

Has nC2 axes that are perpendicular to the principal

Has no horizontal mirror planes

Has dihedral mirror planes

Dn

Has nC2 axes that are perpendicular to the principal

Has no horizontal mirror planes

Has no dihedral mirror planes

Cyclic / C groups

Has no nC2 axes that are perpendicular to the principal

Cn, S2n, Cnv, and Cnh

Cn

No nC2 axes that are perpendicular to the principal

No horizontal mirror planes

No vertical mirror planes

No S2n (improper rotation axis)

S2n

No nC2 axes that are perpendicular to the principal

No horizontal mirror planes

No vertical mirror planes

Yes S2n (improper rotation axis)

Cnv

No nC2 axes that are perpendicular to the principal

No horizontal mirror planes

Yes vertical mirror planes

Cnh

No nC2 axes that are perpendicular to the principal

Yes horizontal mirror planes