1. Drawing the structure of polymer chains
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 the range 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 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. 4.1.1. Electronic structure of dihydrogen H2
4.1. From molecules to conjugated polymers: Evolution of the electronic structure
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. 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

10. → 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

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. 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

20. 4.3. π-π intermolecular interactions
 π-π intermolecular interactions due to the overlap between π-orbitals of adjacent molecules
2tLUMO
2tHOMO
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>

21. 4.4. Intermolecular band in organic solids
 π-π intermolecular interactions due 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. Example with discotic liquid crystals
Localized σ-orbitals
Creation of quasi-bands for a stack of 5 molecules.
For an infinite long stack, bands are formed and follow the expression:
HOMO of HATNA-SH
HOMO: 4 nodes
HOMO-1: 3 nodes
HOMO-4: 2 nodes
HOMO-8: 1 node
HOMO-11: 0 node
kcol= 0.77 Å-1
kcol= 0.62 Å-1
kcol= 0.46 Å-1
kcol= 0.31 Å-1
kcol= 0.15 Å-1
d=3.4 Å
X. Crispin, et al, JACS, 126, (2004)