Electronic Structure of Organic Materials

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

4. Atomic Orbitals Rayleigh-Ritz principle / method The best guess for the solution of the Schrödinger equation is the trial wave function Ya 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 = lx  (xTAx) / (xTx) is extremum (minimal)

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

6. Atomic Orbitals 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 sand porbitals. 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.)

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, Ya is the initial guess, from which the energy can be calculated as: Now Ya has to be varied to minimize Ea.

8. Molecule AB Atom B * “band gap“ Molecular Orbitals Formation of bonding and anti-bonding(*) levels: Atom A LUMO HOMO If more atoms are used to construct the molecule, HOMO and LUMO consist of the corresponding number of levels. (overlap)

9. Molecular Orbitals -bands in conjugated polymers many “inter-acting” molecular orbitals Very many “inter- acting” molecular orbitals 2 “inter-acting” molecular orbitals Isolated atomic orbital Isolated molecular orbital

10. Molecular Orbitals Example: The most simple molecule: H2+ Good approximation, as the nuclei have a much higher mass than the electron.

11. Molecular Orbitals Single atom wavefunctions ja and jb, 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 (<0) <Ya| Vb| Ya> D – Resonance integral (<0) <Ya| Va| Yb> S – Overlap integral (0<S1) <Ya|Yb>

13. Molecular Orbitals Solutions for Y: 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 Rab into account, yields for H2+a binding energy of 1.7 eV: E [eV] anti-bonding Rab [10-10m] 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): Esinglet Etriplet

16. Hybridization: sp3 Hybridization of atomic orbitals allow optimized geometries for bonds: p-orbital in carbon

17. Hybridization: sp3 3 4

18. Hybridization: sp3 The sp3 hybridization leads to s-type bonds (“direkt bonds“).

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

20. Hybridization: sp2 Carbon-carbon double bonds are described by sp2-hybridization

21. 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 onesingly 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

22. Hybridization: sp2 ethylene Formation of p-bonds from two pz orbitals

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 ofmultiple bonding. Thus, bond lengths for carbon-carbon bonds are in theorder C-C > C =C > C=C 2. Bond lengths tend to increase with the size of the bonded atoms. Thiseffect is most dramatic as we proceed down the periodic table. Thus, a CHbond is shorter then a C-F bond, which is shorter then a C-Cl bond. Sincebond length is the distance between the center of bonded atoms, it isreasonable that larger atoms should form longer bonds. 3. When we make comparisons within a given row of the periodic table, bondsof a certain type (single, double, or triple) between a given atom and aseries 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. Thiseffect occurs because a more electronegative atoms has a greaterattraction for the electrons of the bonding partner, and therefore ‘pulls itcloser,’ 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 s-bonds are not considered as influence for the p-electrons 2.) Ansatz for the wavefunction: where ji are the pz wavefunctions of individual C-atoms at pos. i

25. Benzene Benzene has degenerate molecular orbitals!

26. Benzene

27. Benzene From benzene to polyacenes: red-shift in absorption Thus the larger the p-conjugated system, the smaller the optical band gap! Compare with particle in a box again...

28. Benzene Particle in a box:

29. H H H H C C C C C C C C C C C C C C C C H H H H H H H H H H H H H H H H H H H H Organic Semiconductors: Basics Most simple conjugated polymer: Polyacetylene (sp2-hybridized) Compare with Polyethylene (sp3-hybridized):

30. Organic Semiconductors: Basics Polyacetylene: dimerization unit cell a and 2a ( ) metall? semi- conductor! a 2a ( ) The total energy (electronic plus lattice distortion) as a function of u. Note thedouble 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

31. Organic Semiconductors: Basics • Polyacetylene • 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.