Electronic Structure ofOrganic Materials

- Periodic Table of Elements

- Rayleigh-Ritz Principle

- Atomic Orbitals (AO)

- Molecular Orbitals (MO - LCAO)

- Hybridization

- Example: Benzene

Periodic Table of Elements

Mendeleyev: Order by weight and chemical propertiesQuantum mechanics: Order by electron number and nature of orbitals!

Atomic Orbitals

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

Where is the Hamiltonian.

Thus the Schrödinger equation becomes:

Here the wavefunctions are the eigenfunctions to

the operator H, and the energy levels E are the corresponding eigenvalues of the solution.

Rayleigh-Ritz principle / method The best guess for the solution of the Schrödinger

equation is the trial wave function which

minimizes the energy expectation value

This is THE workhorse of all computational quantum chemistry and computational material sciences.

Analogy: Eigenvector problem of finite (symmetric) matrices Ax = x (xTAx) / (xTx) is extremum (minimal)

Atomic Orbitals

Atomic Orbitals

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

Atomic quantum numbers:n : mainl : 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 orbitalsAtoms that constitute typical organic compounds such asH, 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).

The ansatz for the Schrödinger equation is a linear combination of atomic single-electron wavefunctions.

Using the Rayleigh-Ritz method, is the initial

guess, from which the energy can be calculated as:

Now has to be

varied to minimize E

Molecular Orbitals

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

Atom A Atom B

HOMO

LUMO

Molecule AB

*

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

“band gap“

-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 a and b, and linear combination for the molecule:

is the ansatz for the Schrödinger equation / Ritz Principle: Approximate solution can be calculated according from

Molecular Orbitals

C – Coulomb integral (<0) <a| Vb| a>D – Resonance integral (<0) <a| Va| b>S – Overlap integral (0<S1) <a|b>

Molecular Orbitals

Solutions for :

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.

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 Rab into account, yields for H2

+ a binding energy of 1.7 eV:

E [eV]

Rab [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.

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

Esinglet

Etriplet

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 type bonds (“direkt 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

For the sp2-hybridization two p orbitals are mixed with one s orbital:

gives rise to three sp2-orbitals in the plane and

one singly occupied pz-orbital perpendicular to that plane

h1= s +21/2 py

h2= s + (3/2)1/2 px - (1/2)1/2 py

h3= s - (3/2)1/2 px - (1/2)1/2 py

Hybridization: sp2

Formation of -bonds from two pz 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:

1.) Electrons in the bonds are not considered as influence for the electrons2.) Ansatz for the wavefunction:

where i are the pz 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 -conjugated system, the smaller the optical band gap! Compare with particle in a box again...

Benzene

Particle in a box:

Most simple conjugated polymer: Polyacetylene (sp2-hybridized)

CC

CC

CC

CC

H H H H H H H H

H H H H H H H H

Compare with Polyethylene (sp3-hybridized):

CC

CC

CC

CC

H H H H

H H H H

Organic Semiconductors: Basics

Polyacetylene: dimerization unit cell a and 2a

The total energy (electronic plus lattice distortion) as a function of u. Note the double minimum associated with the spontaneous symmetry breaking and the twofold degenerate ground state.

Band structure (a) and density of states (b) for trans-(CH)x. The energy gap of 20 opens up at k=2ð/a due to the Peierls distortion

( )

( )

a 2a

metall?

semi- conductor!

Organic Semiconductors: Basics

Polyacetylene

(a) undimerized structure

(b) dimerized structure due to the Peierls instability.

(c) cis-polyacetylene

(d) degenerate A and B phases in trans-polyacetylene

(e) soliton in trans-polyacetylene

(f) ...again a bit more realistic.

Organic Semiconductors: Basics