Drawing the structure of polymer chains

1. 4. Intra- versus inter-molecular bands 4.1. From molecules to conjugated polymers: Evolution of the electronic structure 4.2. Electronic structure of systems with a degenerate ground state: Trans-polyacetylene These sections are based on notes prepared by Jean-Luc Brédas, Professor at the University of Georgia 4.3. π-π interactions in organic solids 4.4. Intermolecular bands in discotic liquid crystal Drawing the structure of polymer chains polyacetylene shorthand notation

2. Formation o f bands  Simple model for a solid: the one-dimensional solid, which consists of a single, infinitely long line of atoms, each one having one s orbital available for forming molecular orbitals (MOs). When the chain is extended:  The range of energies covered by the MOs is spread  This range of energies is filled in with more and more orbitals  For a chain of N→∞, the width of therange of energies of the MOs is finite, while the number of molecular orbitals is infinite: This is called a band . 4 “s” band

3.  For a chain of N atoms, i.e. for N atomic orbitals 1s, the chain has N molecular orbitals. The energy of these MOs can be found by solving the secular determinant:  Each molecular orbital is characterized by a energy labeled with k.  For N →∞, the Ek-Ek+1 → 0. But the “s” band still has finite width: EN -E1 → 4 a 4b

5. 4.1. From molecules to conjugated polymers: Evolution of the electronic structure 4.1.1. Electronic structure of dihydrogen H2 The zero in energy= e- and p+ are ∞ly separated In the H atom, e- is bound to p+ with 13.6 eV = 1 Rydberg (unit of energy) • When 2 hydrogen atoms approach one another, the ψ1s wavefunctions start overlapping: the 1s electrons start interacting. • To describe the molecular orbitals (MO’s), an easy way is to base the description on the atomic orbitals (AO’s) of the atoms forming the molecule • → Linear combination of atomic orbitals: LCAO • Note: from N AO’s, one gets N MO’s

7. 4.1.2. The polyene series • Methyl Radical • Planar Molecule • One unpair electron in a 2pz atomic arbital → π-OA

8. B. Methylene molecule • Planar molecule • Due to symmetry reason, the π-levels do not mix with the σ levels (requires planarity) • First optical transition: ≃ HOMO → LUMO ≃ 7 eV

9. C. Butadiene • From the point of view of the π-levels: the situation corresponds to the interaction between two ethylene subunits • First optical transition: ≃ 5.4 eV 3 nodes 2 nodes 1 node 0 nodes

10. Frontier molecular orbitals and structure: 1. The bonding-antibonding character of the HOMO wavefunction translates the double-bond/single-bond character of the geometry in the groundstate 2. The bonding-antibonding character is completly reversed in the LUMO The first optical transition (≃ HOMO to LUMO) will deeply change the structure of the molecule D. Hexatriene 3 interacting ”ethylene” subunits → 3 occupied π-levels and 3 unoccupied π*-levels

11. 5 nodes E 4 nodes * 3 nodes 4.7 eV 2 nodes 1 node  0 nodes Remarks: The energy of the π-molecular orbitals goes up as a function of the number of nodes → This is related to the kinetic energy term in the Schrödinger equation: this is related to the curvature of the wavefunction

12. In a bonding situation, the wavefunction evolves in a much smoother fashion than in an antibonding situation 2) Geometry wise: → In the absence of π-electrons (for alkanes): 1.52 Å All the C-C bond lengths would be nearly equal → When the π-electrons are throuwn in: the π-electron density distributes unevently over the π-bonds: Apparition of a bond-length alternation ≃ 1.34 Å ≃ 1.47 Å

20. 2tLUMO 2tHOMO By J.Cornil et al., Adv. Mater. 2001, 13, 1053 4.3. π-π intermolecular interactions π-π intermolecular interactionsdue to the overlap between π-orbitals of adjacent molecules  The strength of the interaction, i.e. the electronic coupling, is measured by the transfer integral: t = <Psi/H/Psi>

21. 4.4. Intermolecular band in organic solids π-π intermolecular interactionsdue to the overlap between π-orbitals of adjacent molecules  creation of a narrow π-band in the neutral ground state of the organic crystal. 2tLUMO W=4tLUMO  Electron mobility 2tHOMO W=4tHOMO  Hole mobility By J.Cornil et al., Adv. Mater. 2001, 13, 1053  The strength of the interaction, i.e. the electronic coupling, is measured by the transfer integral: t = <Psi/H/Psi> Band width in a solid can be estimated from from the splitting of the frontier levels“t” in a dimer

22. HOMO: 4 nodes HOMO-1: 3 nodes HOMO of HATNA-SH kcol= 0.31 Å-1 kcol= 0.15 Å-1 HOMO-4: 2 nodes HOMO-8: 1 node HOMO-11: 0 node kcol= 0.77 Å-1 kcol= 0.46 Å-1 kcol= 0.62 Å-1 Localized σ-orbitals Example with discotic liquid crystals Creation of quasi-bands for a stack of 5 molecules. For an infinite long stack, bands are formed and follow the expression: d=3.4 Å X. Crispin, et al, JACS, 126, 11889 (2004)