Diatomic and polyatomic molecules

Electronic transitions

Diatomic and polyatomic molecules

X

A

B

A

a

b

Potential curves and the electronic states

e.g., singlet

e.g., triplet

Vib-rot

Vib-rot

Orbitals and states

Diatomic molecule

elelelel EH ψψ =ˆThe approximate solution of equation is accomplihed by assuming that ψelis made up of molecular orbitals MO:

elelelel EH ψψ =ˆ

)(

)...3()2()1( 211

BjiB

AjiA

ji

el

jjCC φφφ

φφφψ

+=

=

∑are atomic orbitals localized on atoms A and B, respectively.

)(

)...2()1(

iiMO

MOMO

c φφ

φφψ

∑=

=

Electronic wavefunctions are constructed insuch way that they are eigenfunctions ofthe symmmetry operators for a specificmolecular point group

(Within Born –Oppenheimer Approximation)

Polyatomic molecule

The electronic wavefunctions are simultaneously eigenfunctions of the Hamiltonian because the symmetry operator commute with the electronic Hamiltonian

[Hel ,OR ]=0

They wf are classified by the irreducible representations of the appropriatemolecular point group and the electronicwavefunction belongs to a particular irreducible representation.

The atomic orbitals are in an environment of reduced symmetry from spherical (Kh )to axial (D~h , C~v ). Each electron with orbital angular momentum Lwill process about the internuclear axis and direction of the circulation can be right or left.

zλ

l

Only projection of L onto internuclear axis Λ

remains useful. The circulationdoes not affect the energy but generate +/- vlues of Λ (double degeneracy)

H2 O C2v A1 , A2 , B1 , B2

-Predictions of the geometry and electronic structure is of importance to build themolecular orbitals!!

e.g., Walsh‘s rules:

for predicting a bent or linear geometryof triatomic molecules

-Symmetry-adapted linear combinations (SALCs) of atomic orbitals are formed byinspection or by use the projection operators

Molecular orbitals:

Atomic and molecular orbitals

Li2 to N2

Diatomic molecules

s, px , py , pza1 , b1 , b2 , a1

H H HH

1SA 1SB

Polyatomic molecule

HOMO

LUMO

X1A1

Atomic and molecular orbitals

Electronic states of diatomic molecule

2S+1ΛJ Λ=Σ λi

Λ=0−−−−−−−−−−−−−ΣΛ=1−−−−−−−−−−−−−ΠΛ=2−−−−−−−−−−−−−Δ

Selection rules for the electronic transitions (more in attached appendix)

ΔΛ=0, +/-1Σ−Σ, Π−Σ, Δ−Π, and so forth

ΔS=0transitions which change multiplicity are very weak for

molecules from light atoms; for haevy atoms, transitions with ΔS/=0 become more strongly allowed

ΔΣ=0transition for Hunds case a

DΩ=0, +/-1

Σ+−Σ+, Σ−

−Σ−

no transition for Σ+

−Σ−

(trnsition dipole moment have

Σ+ symmetry)

Σ+, Σ−

−Π g<->u The transitions e.g., 1Πg

-1Πu are allowed for centrosymmetric molecules

Transitions among the electronic states of O2

Only the B3Σ-u – X3Σ-

g transition is allowedSchumann-Runge system is responsible

for the absorption of UV light for wavelengths λ<200nm in the earth‘s atmosphere

Appendix: Determining selection rules with group theory