Point Groups and Character Tables

point group

describes a set of symmetry operations that share at least one common point and can be performed on a molecule or object

how to assign point group

1. what is the highest fold rotation axis?
2. are there n C2 axes perpendicular to the principle Cn axis? if yes, D, if no, C
3. is there a σh plane? if yes, h
4. Are there n σv planes containing Cn?

what is the point group for carbon monoxide?

C∞v

what is the point group for O2?

D∞h

special cases: high symmetry

tetrahedral, Td
octahedral, Oh

special cases: low symmetry

Ci if it contains an inversion center but C1 axis, no C2 axes, no mirror planes
Cs if C1 and a σh plane
C2 if the C2 and no mirror planes

character table

a complete set of irreducible representations for a point group, contains info about the structural and spectroscopic behavior for members of the point group

order of a point group

the total number of symmetry operations in a group, add all coefficients of operations

class of a point group

number of columns of operations

how to determine the character for irreducible representation

apply the symmetry operations to the orbital(s) in question, if the phase changes, contributes -1, if not, 1

what is special about s orbitals in character tables?

always transform as the fully symmetric representation

reducible representation

the sum of the irreducible representations in the character table

irreducible representation

the row of characters for the given symmetry species

how to reduce a reducible representation

with the equation, do that for each row to determine number of reducible representations

how to generate a reducible representation for a vibration

assign x,y and z vector to each atom, apply symmetry operation, if the atom moves: 0, if the atom stays put, any vector that does not change is 1 and those that change is -1

IR active mode

results in a change in molecular dipole, symmetry labels with x y or z

Raman active mode

results in a change in polarizability, symmetry labels associated with a product term like x^2, xy