Electronic Structure of

Organic Materials

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- Periodic Table of Elements

- Atomic Orbitals (AO)

- Molecular Orbitals (MO - LCAO)

- Hybridization

- Example: Benzene

Periodic Table of Elements

Mendeleyev: Order by weight and chemical properties

Quantum mechanics: Order by electron number and nature of orbitals!

Atomic Orbitals

Atomic Orbitals represent a solution of the time-independent Schrödinger equation:

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Where is the Hamiltonian.

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Thus the Schrödinger equation becomes:

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Here the wavefunctions Ψ are the eigenfunctions to the operator H, and the energy levels E are the corresponding eigenvalues of the solution.

Atomic Orbitals

For the most simple atom, the H-atom, the orbitals with n > 1 are energetically (almost) degenerated.

Atomic quantum numbers:

n : principle

l : angular

m : magnetic

Atomic Orbitals

The orbitals are ordered according to their angular momentum: s=0, p=1 & d=2.

(The coloration differentiates between positive and negative parts of the wave functions.)

3D visualization of atomic orbitals

Atoms that constitute typical organic compounds such as

H, C, N, O, F, P, S, Cl have outermost (valence) electrons in s and p orbitals.

Molecular Orbitals

Commonly molecular orbitals are derived by the method of

Linear Combination of Atomic Orbitals (LCAO).

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The ansatz for the Schrödinger equation is a linear combination of atomic single-electron wavefunctions. Using the Rayleigh-Ritz method, Ψ_α is the initial guess, from which the energy can be calculated as:

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Now Ψ_α has to be

varied to minimize E_α.

Molecular Orbitals

Formation of bonding and anti-bonding(*) levels:

Atom A

HOMO

LUMO

If more atoms are used to construct the molecule, HOMO and LUMO consist of the corresponding number of levels. (overlap)

-bands in conjugated polymers

Molecular Orbitals

Isolated atomic orbital

Isolated molecular

orbital

2 “inter-acting” molecular

orbitals

many “inter-acting” molecular

orbitals

Very many “inter-

acting” molecular

orbitals

Molecular Orbitals

Example: The most simple molecule: H₂⁺

Good approximation, as the nuclei have a much higher mass than the electron.

Molecular Orbitals

Single atom wavefunctions ϕ_a and ϕ_b, and linear combination for the molecule:

Molecular Orbitals

C – Coulomb integral (<0) <Ψ_a| V_b| Ψ_a>

D – Resonance integral (<0) <Ψ_a| V_a| Ψ_b>

S – Overlap integral (0<S≤1) <Ψ_a|Ψ_b>

Molecular Orbitals

Solutions for Ψ:

A bond is formed as the electron has an increased probability between the two nuclei (top):

The kinetic energy is lowered as the electron is spread over a larger spatial region.

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In the anti-bonding state, the kinetic energy is increased as the probability is going to zero between the two nuclei.

Molecular Orbitals

Taking the Coulomb repulsion between the two nuclei at distance R_ab into account, yields for H₂⁺ a binding energy of 1.7 eV:

E [eV]

R_ab [10^-10m]

anti-bonding

bonding

Molecular Orbitals

If more than one electron are to be considered, the Pauli principle has to be obeyed, i.e. one orbital can be populated by maximal two electrons with opposite spin.

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The energetic degeneracy is lifted by the exchange interaction (Spin orbit coupling):

E_singlet

E_triplet

Hybridization: sp³

Hybridization of atomic orbitals allow optimized geometries for bonds:

p-orbital in carbon

Hybridization: sp³

4

3

Hybridization: sp³

The sp³ hybridization leads to σ−type bonds .

Hybridization: sp³

Nitrogen in ammonia shows sp³-hybridization as well (note: free electron pair!)

Hybridization: sp²

Carbon-carbon double bonds are described by sp²-hybridization

Hybridization: sp²

Formation of π-bonds from two p_z orbitals

ethylene

Bond Length

The following generalizations can be made about bond length:

1. Bond lengths between atoms of a given type decrease with the amount of multiple bonding. Thus, bond lengths for carbon-carbon bonds are in the order C-C > C =C > C=C

2. Bond lengths tend to increase with the size of the bonded atoms. This effect is most dramatic as we proceed down the periodic table. Thus, a CH bond is shorter then a C-F bond, which is shorter then a C-Cl bond. Since bond length is the distance between the center of bonded atoms, it is reasonable that larger atoms should form longer bonds.

3. When we make comparisons within a given row of the periodic table, bonds of a certain type (single, double, or triple) between a given atom and a series of other atoms become shorter with increasing electro negativity.

Thus, the C-F bond in H3C-F is shorter then the C-C bond in H3C-CH3. This effect occurs because a more electronegative atoms has a greater attraction for the electrons of the bonding partner, and therefore ‘pulls it closer,’ than a less electronegative atom.

Quoted from Organic Chemistry by G.M. Loudon

Benzene

To find a solution, the Hückel method is applied:

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1.) Electrons in the σ−bonds are not considered as influence for the π−electrons

2.) Ansatz for the wavefunction:

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where ϕ_i are the p_z wavefunctions of individual C-atoms at pos. i

Benzene has degenerate molecular orbitals!

Benzene

Benzene

Benzene

From benzene to polyacenes: red-shift in absorption

Thus the larger the π-conjugated system, the smaller the optical band gap! Compare with particle in a box again...

Benzene

Particle in a box:

Thank you 😌