Quantum Theory of PolymersII.b Energy transfer in polymers:Förster and Dexter mechanisms

Jean­Marie André

EC ­ Socrates ­ Erasmusprogramme

FUNDP, NamurUniversity of Warsaw

Light emitting devices:

1° charge injection2° charge transport to a recombination region3° formation of excited states from recombination of radical-ion (polaron) states4° light emission.

Energy harvesting / light transducing devices such as photovoltaic cells:

1° optical absorption2° photoinduced charge separation3° charge transport4° charge collection at the electrodes

Photovoltaïc Cell

Separation of holes and electrons

OLED

Fluorescence K ≈ 106 → 109 s-1 Phosphorescence K ≈ 10-2 → 106 s-1

Franck-Condon Shift

The rate of non-radiative phonon emission a2*→ a0

*- c0

* is much larger than the radiative transition rate. Therefore, radiative transitions are most likely to occur from the lowest vibrational level of an electronic state → KASHA’s rule

Franck-Condon energy = red shift= Stokes shift → 0.5 eV

Non-radiative Electronic Energy TransferFörster & Dexter Mechanisms

During energy transfer between molecules, an excited donor molecule D* transfers its energy to an acceptor A which in turn is promoted into an excited state:

D* + A → D + A*

In some doped organic thin films, the exciton energy can be transferred from the host to the guest molecule, quenching luminescence of the host while increasing that of the guest. If D and A are the same = energy migration

Electrostatic interaction

Exchange interaction

Photoinduced Electron Transfer

Nonradiative Energy Transfer (followed by emission)

From Fermi's Golden Ruleto Dexter & Förster Non-Radiative Energy Transfer Equations

D* + A → D + A*

k m n    =2 ph

⟨ ym∣V∣yn⟩2ρ E n =

2 phβ 2ρ E

β =⟨ y f∣H '∣yi⟩

D∗¿ 2  y A 1  D∗¿ 1  y A 2  −y¿

y¿

¿

D∗¿ yA=

1

2¿

yi=åy ¿

¿

A∗¿ 1  A∗¿ 2  −yD 2  y¿

yD 1  y¿

¿

A∗¿=1

2¿

yf=åy

Dy¿

¿

yD∗¿ 1  y

A2  H '

yD 1  y A∗¿ 2  dt

1dt

2

β =βC

   =∫ ¿−βE

   =∫ yD∗¿ 1  y

A2  H '

yD 2  y A∗¿ 1  dt

1dt

2

¿¿

¿

H’ = electrostatic interaction of all electrons and nuclei

Scheme of the Active Orbitals in Energy Transfer

General Scheme:

“Active” Orbitals:

Digression: Indiscernability of Electrons and Pauli Principle

"Energetic space"

"Physical direct" space

Y 1,2  =yD∗¿ 1  yA

2 

¿

then a correct antisymmetric wave function would be:

D∗¿ 1  y A 2  −yD∗¿ 2  yA

1 

D∗¿ 1  y A 2  D∗¿ 2  y A 1 

D∗¿ 1  y A 2  −y¿

y ¿=−Y 1,2  ¿

D∗¿ 2  y A 1  −y ¿=−¿

Y 1,2  µ y ¿Y 2,1  µ y ¿

¿¿

¿

if:

ψ(2,1)2 = ψ(1,2)2

ψ(2,1) = -ψ(1,2) antisymmetry Pauli principle for fermionsψ(2,1) = +ψ(1,2) symmetry Pauli principle for bosons

Förster Equation

Electronic energy transfer process = one-step transfer of electronic excitation from an excited DONOR molecule (D*) to an ACCEPTOR molecule (A) in separate molecules (INTERMOLECULAR ENERGY TRANSFER) or in different parts of the same molecule (INTRAMOLECULAR ENERGY TRANSFER)D + hν → D*D* + A → D + A*

Förster = Non-radiative resonance excitation (dipole-dipole) energy transferoccurs when an excited molecule (D*) can transfer its excitation energy to an acceptor (A) molecule over distances much greater than collisional diameters (e.g. 50-100 Å).

β Coulomb dipole−dipole »   M D M A

RAD3

The Coulomb term represents the classical interaction of the charge distributions:Q1(1) = eφD*(1) φD(1) and Q2(2) = eφA(2) φA*(2) that can be expanded into multipole terms : dipole-dipole, dipole-quadrupole, …

At not too small distances RAD between the donor and the acceptor, the dominant dipole term is:

Förster Equation

Electronic energy transfer process = one-step transfer of electronic excitation from an excited DONOR molecule (D*) to an ACCEPTOR molecule (A) in separate molecules (INTERMOLECULAR ENERGY TRANSFER) or in different parts of the same molecule (INTRAMOLECULAR ENERGY TRANSFER)D + hν → D*D* + A → D + A*

Förster = Non-radiative resonance excitation (dipole-dipole) energy transferoccurs when an excited molecule (D*) can transfer its excitation energy to an acceptor (A) molecule over distances much greater than collisional diameters (e.g. 50-100 Å).

Thus, ET can be approximated to occur via dipole (donor)-dipole (acceptor) interaction (Coulombic interaction). The oscillating dipole of the excited donor (D*) causes electrostatic forces which can be exerted on the electronic system of the acceptor:

kET=1t D

R0

R 6

R06=1  . 25 ´ 101 7 f E

n4∫0

¥F D ν α A ν dν

ν 4

k m n    =2 ph

⟨ ym∣V∣yn⟩2ρ E n =

2 phβ 2ρ E

Förster Equation

τD = average donor exciton lifetime for recombination in the absence of energy transferR0 = Förster critical distance given by the “overlap” integral over all energies hνφE = quantum efficiency of donor emissionn = refractive index of the hostFD = normalized emission spectrum of the donorαA = molar extinction coefficient (absorption spectrum) of the acceptor

equation valid if D and A are well separated (20 Å)D and A exhibit broadened relatively unstructured spectraspectral overlap is insignificantno important medium or solvent interactionssolvent excited states lying much higher than those of D and A

kET=1t D

R0

R 6

R06=1  . 25 ´ 101 7 f E

n4∫0

¥F D ν α A ν dν

ν 4

«  overlap  » integral:

Digression: 4-coordinates-spinorbitals

yD∗¿ 1  yA

2  H 'yD 1  y A∗¿ 2  dt

1dt

2

A∗¿ 2  y A 2  dt 1 dt 2

¿β C=∫ ¿=∫ yD∗¿ 1  y

D1  H '

y ¿¹ 0 

¿

y 1  = f x1 , y1 , z1 s w1

y 1  = f x1 , y1 , z1{α 1  β 1 

∫α w α w dw=∫β w β w dw=1

∫α w β w dw=∫β w α w dw=0Orthogonality of spin functions:

Thus after integration, ψD*(1)ψD(1) will give a result different of zero only if the spins of ψD* and ψD are the same.

and after integration, ψA*(2)ψA(2) will give a result different of zero only if

the spins of ψA* and ψA are the same.

Förster Mechanisms

1D*(singlet) + 1A → 1D + 1A*(singlet)Allowed

yD∗¿ 1  yA

2  H 'yD 1  y A∗¿ 2  dt

1dt

2

A∗¿ 2  y A 2  dt 1 dt 2

¿β

C=∫ ¿=∫ y

D∗¿ 1  yD

1  H 'y ¿¹ 0 

¿

yD∗¿ 1  yD

1  H 'y A∗¿ 2  y

A2  dt

1dt

2

β C=∫¿=∫ f D∗¿ 1  fD

1  H 'f A∗¿ 2  f

A2  dr

1dr

2∫β w

1β w1dw

1∫β w

2β w2dw

2¹ 0 

¿¿

¿

Spins of D and D*and of A and A*

are the same

Förster Mechanisms

1D*(singlet) + 3A → 1D + 3A*(triplet)Allowed

yD∗¿ 1  yA

2  H 'yD 1  y A∗¿ 2  dt

1dt

2

A∗¿ 2  y A 2  dt 1 dt 2

¿β

C=∫ ¿=∫ y

D∗¿ 1  yD

1  H 'y ¿¹ 0 

¿

yD∗¿ 1  yD

1  H 'y A∗¿ 2  y

A2  dt

1dt

2

β C=∫ ¿=∫ f D∗¿ 1  fD

1  H 'f

A∗¿ 2  fA

2  dr1

dr2∫β w

1β w1dw

1∫α w

2α w2dw

2¹ 0 

¿¿

¿

Förster Mechanisms

3D*(triplet) + 1A → 1D + 3A*(triplet)Not allowed

yD∗¿ 1  yA

2  H 'yD 1  y A∗¿ 2  dt

1dt

2

A∗¿ 2  y A 2  dt 1 dt 2

¿β

C=∫ ¿=∫ y

D∗¿ 1  yD

1  H 'y ¿¹ 0 

¿

yD∗¿ 1  yD

1  H 'y A∗¿ 2  y

A2  dt

1dt

2

β C=∫ ¿=∫ f D∗¿ 1  fD

1  H 'f

A∗¿ 2  fA

2  dr1

dr2∫α w

1β w1dw

1∫β w

2α w2dw

2=0 

¿¿

¿

Förster Equation

spin-allowed transitions:

1D* (singlet)+ 1A → 1D + 1A* (singlet)1D* (singlet) + 3A → 1D + 3A* (triplet)

yD∗¿ 1  yA

2  H 'yD 1  y A∗¿ 2  dt

1dt

2

A∗¿ 2  y A 2  dt 1 dt 2

¿β C=∫ ¿=∫ yD∗¿ 1  y

D1  H '

y ¿¹ 0 

¿

Spin conservation of both D and A =Allowed transitions on D and AForbidden transitions (triplet-triplet, singlet-triplet)

3D* (triplet) + 1A → 1D + 3A* (triplet)1D* (singlet) + 1A → 1D + 3A* (triplet)

The triplet-singlet transition

3D* (triplet) + 1A → 1D + 1A* (singlet)

is forbidden, but sometimes observed since 3D* has a long lifetime and although slow kET can be larger than the 3D* → 1D transition rate.

Dexter Equation

Electronic energy transfer process = one-step transfer of electronic excitation from an excited DONOR molecule (D*) to an ACCEPTOR molecule (A) in separate molecules (INTERMOLECULAR ENERGY TRANSFER) or in different parts of the same molecule (INTRAMOLECULAR ENERGY TRANSFER)D + hν → D*D* + A → D + A*

Dexter = Non-radiative electron exchange energy transfer (EEET)

occurs when an excited donor molecule (D*) and an acceptor molecule (A) are close enough (10-15 Å) to be in molecular contact. If their electron clouds sufficiently overlap each other, an exciton could diffusively hop from one molecule to the next with no change of spin. Also called overlap or collision mechanism.

The exchange term represents the interaction of the exchange charge distributions:Q1(1) = eφD*(1) φA*(1) and Q2(2) = eφD(2) φA(2). They vanish if the spin-orbitals φD*(1) φA*(1) or φD(2) φA(2) contain different spin functions and exponentially decrease with distance.

Dexter Equation

Electronic energy transfer process = one-step transfer of electronic excitation from an excited DONOR molecule (D*) to an ACCEPTOR molecule (A) in separate molecules (INTERMOLECULAR ENERGY TRANSFER) or in different parts of the same molecule (INTRAMOLECULAR ENERGY TRANSFER)D + hν → D*D* + A → D + A*

Dexter = Non-radiative electron exchange energy transfer (EEET)

occurs when an excited donor molecule (D*) and an acceptor molecule (A) are close enough (10-15 Å) to be in molecular contact. If their electron clouds sufficiently overlap each other, an exciton could diffusively hop from one molecule to the next with no change of spin. .Also called overlap or collision mechanism.

For transfers between states with allowed transitions, Dexter transfer is typically overwhelmed by long-range Förster dipole-dipole processes. But since triplet-to-triplet energy transfer is forbidden by spin conservation in Förster mechanism, Dexter is the only mechanism permitting triplet energy transfer.

kET=2 pZ 2

h   ∫0

¥F D ν F A ν dν

FD = normalized emission spectrum of the donorFA = normalized absorption spectrum of the acceptorZ, determined by the molecular overlap and related to the intermolecular spacing R

Z 2  µ  exp −2 RL

L = effective average Bohr radiusR = distance between the centers of D and A

J=   ∫0

¥F D ν F A ν dν

spectroscopic overlap integral, measure of the donor-emission and acceptor absorption

Dexter Mechanisms

1D*(singlet) + 1A → 1D + 1A*(singlet)Allowed

yD∗¿ 1  yA

2  H 'yD 2  y A∗¿ 1  dt

1dt

2

D∗¿ 1  y A∗¿ 1  H 'yD 2  y A 2  dt 1 dt 2

¿β E=∫¿=∫ y ¿¹ 0 

¿

yD∗¿ 1  yA∗¿1   H '

yD

2  yA

2  dt1

dt2

β E=∫ ¿=∫ f D∗¿ 1  fA∗¿1   H '

fD

2  fA

2  dr1

dr2∫β w

1β w1dw

1∫β w

2β w2dw

2¹ 0 

¿¿

¿

Dexter Mechanisms

3D*(triplet) + 1A → 1D + 3A*(triplet)Allowed

yD∗¿ 1  yA

2  H 'yD 2  y A∗¿ 1  dt

1dt

2

D∗¿ 1  y A∗¿ 1  H 'yD 2  y A 2  dt 1 dt 2

¿β E=∫¿=∫ y ¿¹ 0 

¿

yD∗¿ 1  yA∗¿1   H '

yD

2  yA

2  dt1

dt2

β E=∫ ¿=∫ fD∗¿ 1  f

A∗¿1   H 'fD

2  fA

2  dr1

dr2∫α w

1α w1dw

1∫β w

2β w2dw

2¹ 0 

¿¿

¿

Another model: Dexter transport = simultaneous transfer of an electron and a hole between D* and A.

K ET=K e K h Thermally activated !

Mobility of the the triplet exciton = product of electron (Ke) and hole (Kh) transfer rates:

Dexter Equation

spin-allowed EEET:

1D* (singlet) + 1A → 1D + 1A* (singlet)(also allowed under dominating long-range Förster mechanism )

3D* (triplet) + 1A → 1D + 3A* (triplet) →→ triplet-triplet energy transfer(forbidden by resonance-excitation energy transfer)

yD∗¿ 1  yA

2  H 'yD 2  y A∗¿ 1  dt

1dt

2

D∗¿ 1  y A∗¿ 1  H 'yD 2  y A 2  dt 1 dt 2

¿β E=∫ ¿=∫ y ¿¹ 0 

¿

Spin conservation of total spin of D*A system

Summary:

TDPT:

ρE = density of states, related to spectral overlap J

β →

Förster (1951):

Dexter (1953):

k D∗ A=2 phβ 2ρ E

kET Coulomb »f D f A

RDA6 ν2

J

kET Exchange »e−2 R

DA/L

J

Förster: Coulomb term = interaction of charge distributions : and may be expanded in multipole expansion, limited to dipole-dipole

β C µM D M A

RAD3

k D∗ ACb µ

f D f A

RDA6 ν2

J

Dexter: Exchange term = interaction of exchange charge distributions : and vanishes if ΨD* and ΨA* or ΨD and ΨA correspond to different

spin functionsdepend on the spatial overlap of orbitals of D and Adecreases exponentially with increasing internuclear distance

β E µ  exp −RDA

L k D∗ AExch µ  exp −RDA

L J

Useful Relations

1 eV = 8,066 cm-1

DE eV =1241.5  1

λ nm

References

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P.F. Barbara, T.J. Meyer, M.A. Ratner, Contemporary issues in Electron Transfer Research, J. Phys. Chem., 1996, 100, 13148-13168

R. Farchioni, G. Grosso (Eds.), Organic Electronic Materials, Conjugated Polymers and Low Molecular Weight Organic Solids, Springer, Berlin (2001)

M. Klessinger, J. Michl, Excited States and Photochemistry of Organic Molecules, VCH Publishers, New York (1995)

R.A. Marcus, Nobel lecture (1992), Rev. Modern Phys., 1993, 65, 599

J.F. Rabek, Mechanisms of Photophysical Processes and Photochemical Recations in Polymers, Theory and Applications, J. Wiley, New York (1987)

G.C. Schatz, M.A. ratner, Quantum mechanics in Chemistry, Prentice Hall, Englewood Cliffs (1993)