Electronic Configurations:

What is the symmetry of the molecular electronic state?

Electrons occupy the molecular orbitals, the overall wavefunction of the state is a product of electrons in orbitals.

Molecular orbitals of molecules transform as one of the representations of the molecular point group.

The symmetry of the state is the product of the symmetry of occupied molecular orbitals.

Most orbitals are fully occupied (2, 4 or 6 electrons), which results in symmetry Γ1 .

The symmetry of the state is due to the unpaired electrons. Multiply the symmetries of the unpaired electrons.

e.g. H2O

ground state configuration

a12 b22a12 b12

Symmetry of ground state wavefunction A1

1st excited state

configuration

a12 b22a12 b11b21

Symmetry of excited state wavefunction

b1xb2= A2

Is the spectroscopic transition A1 A2 allowed? (See Later!)

Formaldehyde:

ground state: … (n , b2) 2 : A1

1st excited state: …(n , b2) 1(π , b1) 1 : A2

What about molecules with degenerate orbitals?

benzene π  π* transition in the UV.

benzene π m.o.s, D6h symmetry in energy order:

ππ*

a2u < e1ge2u < b2g

ground state: … a2u 2 e1g 4 : A1g

1st excited state:… a2u 2 e1g 3 e2u 1

1 unpaired electron in e1g and 1 in e2u

e1g x e2u = B2u + B1u + E1u

(check by multiplying to get reducible representation and then reducing it)

States of these three symmetries are observed in the absorption spectrum of the molecule.

A1g B2u at ~ 220 nm (uv) very weak

A1g B1u at ~ 180 nm (vuv) weak

A1g E1uat ~ 160 nm (vuv) very strong

[Spin adds an extra complexity to this analysis.]

Spin Quantum Numbers

The unpaired electrons determine the total spin quantum number S.

0 unpaired electrons S = 0

1 unpaired electrons S = ½

2 unpaired electrons S = 1 and S = 0

(for molecules very unusual to have more than 2 unpaired electrons)

Why 2 values for 2 unpaired electrons? The spins of the electrons can either add ½ + ½ = 1, or subtract ½ - ½ = 0. (Really the spin angular momenta can add or cancel.)

To describe the molecular state need to give both S and symmetry (Γ).

Molecular term symbol (2S+1) Γ

(2S+1) is the spin degeneracy – called multiplicity.

S = 0(2S+1) = 1singlet

S = ½ (2S+1) = 2doublet

S = 1 (2S+1) = 3triplet

Transition Intensity, Selection Rules and Symmetry

Transition from molecular State 1 to State 2

The intensity, Iab, of a transition from state a, energy Ea, wavefunction a to a state b, Eb, b can be shown to obey:

where  is the dipole moment of the system

d is integration over all coordinates

All selection rules can be derived from this equation.

For an electronic transition:

ground state

excited state

As  only involves spatial coordinates and cannot change the spin of the electrons, the spin selection rule ΔS = 0 applies to the molecular states. (Sa = Sb)

I12 is a number (scalar) and had symmetry Γ1

means that

Γ1 = Γax Γx Γb

is required for an allowed transition (Iab≠ 0)

Γ are the representations of the coordinates (x, y, z) because is a vector.

For a closed shell ground state molecule, Γa = Γ1 and therefore Γb = Γ

Molecular states of H2 CO:

For an allowed transition ΔS = 0, therefore

1A11 Γ

C2v: Γ = A1 (z) , B1 (x) ,B2 (y)

For an allowed transition: Γ 1 = Γax Γx Γb

for the nπ* transition: does Γ1 = A1x Γx A2?

no – forbidden.

for the ππ* transition: does Γ1 = A1x Γx A1?

yes – allowed.