1. Electronic Structure of Organic Materials - Periodic Table of Elements - Rayleigh-Ritz Principle - Atomic Orbitals (AO) - Molecular Orbitals (MO - LCAO) - Hybridization - Example: Benzene

2. Periodic Table of Elements Mendeleyev: Order by weight and chemical properties Quantum mechanics: Order by electron number and nature of orbitals!

3. 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.

4. 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  (x T  A  x) / (x T  x) is extremum (minimal) Atomic Orbitals

5. For the most simple atom, the H-atom, the orbitals with n > 1 are energetically (almost) degenerated. Atomic quantum numbers: n : main l : angular m : magnetic

6. 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.

7. 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  

8. 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“

9.  -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

10. Molecular Orbitals Example: The most simple molecule: H 2 + Good approximation, as the nuclei have a much higher mass than the electron.

11. 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

12. Molecular Orbitals C – Coulomb integral ( D – Resonance integral ( S – Overlap integral(0

13. 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.

14. Molecular Orbitals Taking the Coulomb repulsion between the two nuclei at distance R ab into account, yields for H 2 + a binding energy of 1.7 eV: E [eV] R ab [10 -10 m] anti-bonding bonding

15. 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): E singlet E triplet

16. Hybridization: sp 3 Hybridization of atomic orbitals allow optimized geometries for bonds: p-orbital in carbon

17. Hybridization: sp 3 4 3

18. The sp 3 hybridization leads to  type bonds (“direkt bonds“).

19. Hybridization: sp 3 Nitrogen in ammonia shows sp 3 -hybridization as well (note: free electron pair!)

20. Hybridization: sp 2 Carbon-carbon double bonds are described by sp 2 -hybridization

21. Hybridization: sp 2 For the sp 2 -hybridization two p orbitals are mixed with one s orbital: gives rise to three sp 2 -orbitals in the plane and one singly occupied p z -orbital perpendicular to that plane h 1 = s +2 1/2 p y h 2 = s + (3/2) 1/2 p x - (1/2) 1/2 p y h 3 = s - (3/2) 1/2 p x - (1/2) 1/2 p y

22. Hybridization: sp 2 Formation of  -bonds from two p z orbitals ethylene

23. 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

24. Benzene To find a solution, the Hückel method is applied: 1.) Electrons in the  bonds are not considered as influence for the  electrons 2.) Ansatz for the wavefunction: where  i are the p z wavefunctions of individual C-atoms at pos. i

25. Benzene has degenerate molecular orbitals! Benzene

27. 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...

28. Benzene Particle in a box:

29. Most simple conjugated polymer: Polyacetylene (sp 2 -hybridized) C C C C C C C C H HH HH HH H H H H H H H H H Compare with Polyethylene (sp 3 -hybridized): C C C C C C C C H H H H HH H H Organic Semiconductors: Basics

30. 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 2  0 opens up at k=2ð/a due to the Peierls distortion ( ) a 2a metall? semi- conductor! Organic Semiconductors: Basics

31. 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