Energies of the p molecular orbitals of benzene are obtained from:

Hij : matrix elements of Hückel operatorSij : overlap integrals between pz orbitals

Orbital energies of p-molecular orbitals of benzene

Most stable configuration:

Electronic ground state:

First excited configuration:

6 electronic states:(no restrictions based on Pauli principle)

Case of partially occupied degenerate orbitals: cyclopentadienyl cation (C5H5+)

It has 4p electrons; Organic chemists would call it «antiaromatic»

Matrix representation of Hückel operator:

Reduction of the 5-dimensional representation of 5 pz orbitals gives

Example:p-molecular orbitals of cyclopentadienyl cation Most stable configuration:

(a2’’)2(e1’’)2

Pauli principle imposesrestrictions in this case

A direct product has a symmetric part:

and an anti-symmetricpart:

For more information on the spectroscopy of C5H5+, see:

H. J. Wörner, F. Merkt, Angew. Chemie Int. Ed., 48, 6404- 6424 (2009)

General concept: The Frost-Musulin diagram:

For more information on the spectroscopy of C5H5+, see:

H. J. Wörner, F. Merkt, Angew. Chemie Int. Ed., 48, 6404- 6424 (2009)

The symmetric part of the direct product corresponds to spatially symmetric wave functions.These combine with a spatially anti-symmetric spin wave function (singlet): 1E2’, 1A1’ states

Similarly, the spatially anti-symmetric wave function combines with the spatially symmetricspin wave function (triplet): 3A2’ state

Conclusion: Only 3 states (instead of 6 in the case of benzene)

h: orbital energyJij: Coulomb integralKij: exchange integral

Hartree-Fock energies

Low-symmetry polyatomic molecules

Nomenclature of electronic states:• Capital letter for spin multiplicity: S for singlet, D for doublet, T for triplet, etc.• Numerical subscript indicates ground state (e.g. S0) and higher-lying states of

same multiplicity (e.g. S1, S2, etc.)• States of other multiplicities are also labelled in order of increasing energy, but

starting with the subscript «1», rather than «0» (e.g. T1, T2, etc.)

Nomenclature of electronic transitions:Electronic transitions are labeled in terms of the one-electron transition thatdominates the excitation:• Bonding orbitals (s, p, etc.)• Antibonding orbitals (s*, p*, etc.)• Non-bonding orbitals (n)

Propensity rules:• s s* and p p* are usually intense transitions• n s* or n p* are usually weak transitions

Example: DNA bases: They absorb strongly in the UV (200-300 nm), but fragment much lessthan comparable molecules.

Adenine has the following most stable configuration:

The first two excited states have configurations:

The energetic ordering was unclear for some time: were used

Example: Adenine

As in 3.4.3, we consider two-electron wave functionsBecause of Paul principle, the wave functions must either have

or vice versa

Two cases:1. Two electrons located in different molecular-orbital shells. No restrictions from

the Pauli principle. Electronic states are given by the direct product of therepresentations of the paritally occupied orbitals. All terms exist as both singletand triplet states (example: first excited configuration of benzene)

2. Two electrons located in the same molecular-orbital shell:a. Non-degenerate shell: Spatial part of wave function is necessarily

symmetric. Spin part must therefore by anti-symmetric.b. Degenerate shell: Spatial part has both symmetric and antisymmetric parts;

the symmetric part combines with singlet spin functions,the anti-symmetric part with triplet spin functions(example: the most stable configuration of the cyclopentadienyl cation)

Selection rule:

Rotational structure of vibrational transitions:

Selection rule:

Leads to a selection rule on the vibrational structure:

Franck-Condon factors:

This is the case, when the factorization

is not a good approximation. The selection rule for electronically forbidden, but vibronically allowed transitions is:

NH3

Treated in C3v symmetry:

Lowest-lying electronic transition is into 3s Rydberg state.NH3

+ is planar and belongs to D3h point group.

The electronic configuration of the excited state is:

C3v treatment:

Electronic transitionis allowed

Umbrella mode isTotally symmetric

Final states withall values of v2 areallowed

D3h treatment:

Electronic transitionis allowed

Umbrella mode isa2’’

Only final stateswith even valuesof v2 are allowed

Transitions fromthe uppertunneling level of GS to odd valuesof v2!

Excited configuration is (D6h point group)

is forbidden

is allowed

Vibrational modes of symmetry induce vibronic coupling between

the A and C states.

Access to vibronic levels of symmetry:

i.e. v1=n and v6=1