UNIVERSITA’ DEGLI STUDI DI PADOVA
Laurea specialistica in Scienza e Ingegneria dei Materiali
Curriculum Scienza dei Materiali
Chimica Fisica dei Materiali Avanzati
Part 7b – Photophysics and photochemistry of molecular
materials
(excitons, energy transfer and electron transfer)
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Exciton
 Exciton: In some applications it is useful to consider an
electronic excitation as a quasi-particles capable of migrating.
This is termed an exciton. In organic materials two models are
used: the band or wave model (low temperature, high crystalline
order) and the hopping model (higher temperature, low
crystalline order or amorphous state). Energy transfer in the
hopping limit is identical with energy migration.
The following consideration follows M. Kasha et al [Pure Appl. Chem.
11 (1965) 371] model of dimer exciton interaction (the band or wave
model).
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Exciton model: ground state
 We consider a weak interaction between 2 molecules (chromophores),
so that perturbation theory can be used.
 The ground state wave-function is
G   u v
 u and  v are ground state wave-functions of molecules u and v.
 The Hamiltonian operator is
H  Hu  H v  Vuv
Hu and Hv are the Hamiltonians of the isolated molecules,
Vuv is the intermolecular perturbation potential approximated by (point)
dipole-dipole interaction term.
 The energy of the ground state is
Eg  Eu  Ev   u v Vuv  u v
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Exciton model: excited state wavefunctions
 If molecules u and v are the same and have identical environments and
*
*
the excited state wave-functions of the molecules are denoted  u and  v ,
*
*
the excited dimer wavefunctions  u v and  u v are degenerate.
 Thus, the excited state wave-function of the dimer must be written
E  r u* v  s u v*
with r and s coefficients to be determined.
 The Schrödinger equation is
H r u* v  s u v*   EE r u* v  s u v* 
 Solution for wave-functions (with Hamiltonian matrix elements Huu = Hvv
and Hvu = Hvu)
1

E 
 u* v   u v* 
2
1

E 
 u* v   u v* 
2
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Exciton model: excited state energies
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Parallel transition dipoles
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In-line dipoles
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Exciton in asymmetric dimer
If the molecules in the dimer are not equivalent because of, say,
local perturbations, disorder, etc., the diagonal elements of the
Hamiltonian are different. The excitation energies then become:
 D
2
1 *
D1  D2*  
 L* 
2
 4

D*  D1*  D2*
E*   * 



*
 
1
2

and the wavefunctions:
*  cos   1* 2  sin   1 2*
*  sin   1* 2  cos   1 2*
 2 H12  1
 2 L* 
1
  arctan 

  arctan 
* 
2
 D 
 H11  H 22  2
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L* and D compete
for the exciton
delocalization: the
first favors, the
second opposes it.
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Main mechanisms of energy transfer
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Radiative electronic energy transfer
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Radiative energy transfer: distance dependence
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Radiative energy transfer: efficiency
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Förster mechanism from Fermi “Golden rule”
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Coulombic or dipole-dipole interaction
(Förster energy transfer)
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Förster formulation of the dipole-dipole
energy transfer
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Level diagrams for energy stransfer
 Förster mechanism
 Dexter mechanism
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Electronic exchange energy transfer:
Dexter mechanism
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Applications of energy transfer
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Orientation factor
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How to measure energy transfer
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Energetics of the electron transfer process
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Classical theory of electron transfer
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Marcus formulation
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Normal and inverted regions
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Marcus rate plot
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Reorganization energy
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Solvent reorganization energy (examples)
Quite a substantial energy!
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Adiabatic and non-adiabatic processes
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Quantum mechanical theory
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Quantum theory - 2
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Comparison of quantum and classical theories
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Electronic coupling and maximum ET rate
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Distance dependence of the ET rate
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Effect of solvent dynamics
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Photoinduced charge separation and
charge recombination
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