3010 group symmetry

Cn

counterclockwise rotation about an axis by 2pi/n

sigma h

reflection through horizontal plane perpendicular to primary Cn

sigma v

reflection through a vertical plane that contains Cn

sigma d

vertical plane reflecting dihedral planes

i

inversion center. x,y,z → -x,-y,-z

Sn

Cn followed by sigma h

Mulliken symbols: first step

E=1 → A (symmetric to highest Cn) or B (anti)

E=2 → E

E= 3 → T

mulliken second step

does the group have inversion?

yes, symmetric (+) → subscript g

yes, anti (-) → subscript u

mulliken 3rd step

nonprincipal Cn or sigma v (first in character table)

symmetric (+) → subscript 1

anti (-) → subscript 2

last mulliken step (only if needed)

sigma h

symmetric → ‘

anti → ‘‘

num irreducible representations

num of classes

sum of squares of characters

group order

two different irreducible representations are..

orthogonal

hamiltonian for hydrogen atom

H = -(hbar²)/2mu * del + v(x)

mu

m1m2/(m1+m2)