1. Drawing the structure of polymer chains
Electronic structure of conjugated polymers
This chapter is based on notes prepared by Jean-Luc Brédas, Professor at the University of Georgia.
1. From molecules to conjugated polymers: Evolution of the electronic structure
2. Electronic structure of systems with a degenerate ground state: Trans-polyacetylene
Drawing the structure of polymer chains
polyacetylene
shorthand notation

2. The Born interpretation of the wavefunction
Example of a 1-dimensional system
 Physical meaning of the wavefunction:
If the wavefunction of a particle has the value (r) at some point r of the space, the probability of finding the particle in an infinitesimal volume d=dxdydz at that point is proportional to |(r)|2d
 |(r)|2 = (r)*(r) is a probability density. It is always positive! Hence, if the wavefunction has a negative or complex value, it does not mean that it has no physical meaning… because what is important is the value of |(r)|2 ≥ 0; for all r. But, the change in sign of (r) in space (presence of a node) is interesting to observe in chemistry: antibonding orbital overlap.
Node

3. 1.1. Electronic structure of dihydrogen H2
1. From molecules to conjugated polymers: Evolution of the electronic structure
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

5. 1.2. The polyene series Methyl Radical Planar Molecule
One unpair electron in a 2pz atomic arbital → π-OA

6. 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

7. First optical transition: ≃ 5.4 eV
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

8. → 3 occupied π-levels and 3 unoccupied π*-levels
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

9. 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

10. Apparition of a bond-length alternation
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