Quantum Theory of PolymersII.a2 Electron transfer in polymers:

Marcus semi­classical theory

Jean­Marie André

EC ­ Socrates ­ Erasmusprogramme

FUNDP, NamurUniversity of Warsaw

Time-Dependent Perturbation Theory: "Exact" TDPT

H=H 0V t

Calculate the probability of transition to a final state that is not the same as the initial one under the effect of a given perturbation

k=2 ph∑α

∣V 0α∣2 d E 0−E

α

δ(E0-Ea) = energy conservation law for the transition; only those transitions from 0> to α> are possible for which the energy of the initial state E0 matches some energy Eα of the final states.

Time-Dependent Perturbation Theory: From First-Order TDPT to Fermi's Golden Rule

Calculate the probability of transition to a final state that is not the same as the initial one under the perturbation of a constant potential turned on at time t = 0 and turned off at time t = τ.

Example of a radiationless transition: light absorption initiates the coupling between the ground and the excited electronic state (which are coupled by nonadiabatic or spin-orbit interactions). The coupling itself is a function of the coordinates of the nuclei but is not an explicit function of time and thus can be thought to be constant in TDPT.

Time-Dependent Perturbation Theory: From First-Order TDPT to Fermi's Golden Rule

Calculate the probability of transition to a final state that is not the same as the initial one under the perturbation of a constant potential turned on at time t = 0 and turned off at time t = τ.

A time τ after the initial electronic excitation, transitions to the second state have generated a distribution of those states according to the sin k/k distribution.

Assume:1° The final states are so closely spaced in energy that they form a near continuum of energy levels density ρ(Ek), 2° We are only interested in the long-time behavior of the system, 3° Coupling Vkm and ρ(Ek) have a weak dependence on k,4° First-order PT is still valid under these assumptions.

Vkm

In practice, the assumptions (1)-(4) are generally satisfied when light absorption and emission is concerned.

Rate of the transition = Probability /time = wT=PT

t=

2 phρ Em ∣V km∣

2

Fermi's Golden RuleSecond Alternative Formulations (Delta Dirac Function)

wkm= lim t ¥

Pkm

t=

2 ph

d E k−Em ∣V km∣2

wT=PT

t=

2 phρ Em ∣V km∣

2

Physically, Fermi's golden rule describes how molecules evolve under a time-dependent perturbation.

two conditions :

1° the delta function expresses the result that in the limit, only transitions which obey energy conservation can be caused by a time-independent interaction.

2° the transition will be permitted only if the external perturbation V has the correct form to mix state φm with state φn (non-zero transition matrix element).

Fermi's Golden Rule: Periodic InteractionThird Alternative Formulation (Electronic Absorption or Emission)

V t =U  exp ±iwt

U is independent of time but can be a function of coordinate and momentum operator.

V kmUkm

wkmwkm±w

E k−EmE k−Em±hw

Peak in occurs at E k=Em m hw

wT=2 ph

∣Ukm∣2ρ Em mh w

wkm=2 ph

∣Ukm∣2 d E k−Em±hw

exp(iωt) causes transitions with final energy Ek = Em-hν "emission”exp(-iωt) causes transitions with final energy Ek = Em+hν ”absorption"

Nonadiabatic ET in a Donor-Acceptor Complex,High-Temperature Case: Semi-classical Marcus equation

as a particular case of Fermi's Golden Rule

Semi-classical Marcus theory adopts Marcus' harmonic curves in order to describe the donor (D) and acceptor (A) states:

with the conditions that the frequencies are the same and are considered as classical:

U A q =U 0 A hwA

2 q−q A 2

U D q =U 0 D hwD

2 q−qD 2

hwA=hwD << k B T

FGR is generalized as:

in order to take into the ensemble average:

Solving the various Gaussian integrals is straightforward and results into Marcus semi-classical formula:

The general structure of the formula is similar to Marcus classical formula:

but defines the preexponential factor and introduces, a third parameter besides the free energy of the reaction, and the reorganization energy, λ, i.e., the coupling between the initial and final states, VDA

kET=2 ph∫ dq f q ∣V DA∣

2 d U D q −U A q

f q = 1Z

exp −U D q k B T

kET=∣V DA∣2 p

h2 k BTλ exp {− DG°+ λ 2

4 kλ B T }

kET µ  exp {− DG°+ λ 2

4 kλ B T }

(1) Saturated alkane chains, oligomethylene (OM) (2) Polyene chains, oligovinylene (OV)(3) alternating p­phenylene and acetylene chains, oligo p­phenylene­ethynylene (OPE)

M.D. NewtonInternational Journal of Quantum Chemistry, Vol. 77, 255–263 (2000)

HDA=HDA0  exp −β2 r−R0

kET=2 ph

HDA

2ρ FC

µ k 0  exp −β r−R0

Rate constantdecreases exponentially with distanceconsistent with the exponential radial

dependence of the WF, if ET between D and A not enhanced by electronic mixing with the intervening space

i.e., unless transported by localization and hoppingor by resonance

in general, no more than 20 Å

Results of Marcus Approach as cited in Marcus' Nobel Lecture

Adiabatic ? Diabatic ? Non adiabatic ?

Results of Marcus Approach as cited in Marcus' Nobel Lecture

When the splitting caused by the electronic coupling between the electron donor and acceptor is large enough at the intersection, a system crossing S from the lower surface on the reactants' side of S continues onto the lower surface on the products' side, and so an electron transfer in the dark has then occurred (adiabatic ET).

Results of Marcus Approach as cited in Marcus' Nobel Lecture

When the coupling is, instead, very weak the probability of successfully reaching the lower surface on the products' side is small and can be calculated using quantum mechanical perturbation theory, for example, using Fermi's "Golden Rule" (non-adiabatic ET)

Adiabatic ⇔ Nonadiabatic Process

Adiabatic αδιαβατοσ ≡ impassable, occuring without loss or gain of heatimpassable ≡ incapable of being passed, traveled, crossed or surmounted

Adiabatic thermodynamics: process occuring without exchange of heat with environment (isoentropic)quantum mechanics: a change is occuring so that the system makes no transition to other states

Adiabatic An adiabatic process is one in which the system under investigation is thermally isolated so that there is no exchange of heat with the surroundings

Webster

Atkins

Alberty ,  Silbey

Adiabatic ⇔ Nonadiabatic Process Thermodynamics ⇔ Quantum mechanics

U=∑ini ei

dU=∑i

ei dni∑i

ni dei

dU=        dQ      −      PdV

εi can only be modified by volume changes of the perfect gas:

ei=enx , ny , nz=h2

8  mL2nx 2ny 2nz 2

microscopic

macroscopic

dQ=0 =∑i

ei dni

−PdV=0 =∑i

ni dei

dU=0  dQ=PdV ∑i

ni dei=∑i

ei dni

Adiabatic

Constant volume

Constant temperature

Adiabatic Constant volumecompression cooling

Characteristic times

Electronic time, proportional to the time the electron needs to move from the donor site to the acceptor site.

Vibrational time, characteristic of the vibrational motion

tel=h∣H AB∣

=hJ

t vib=2 pwvib

Adiabatic ⇔ Nonadiabatic Process

P.F. Barbara, T.J. Meyer, M.A. Ratner, Contemporary issues in Electron Transfer Research, J. Phys. Chem., 1996, 100, 13148-13168, p. 13165

The most important usage of the words adiabatic and nonadiabatic involves the nature of the transfer process itself:

Adiabatic ET's are envisioned as taking place on curves like those of the figure, in which the upper state is ignored; in this case, actual electronic coupling between minima decreases the barrier height, but does not affect the dynamic barrier top crossing.

The difference between adiabatic and nonadiabatic has to do with the relative magnitude of the HRP matrix element compared to other energy quantities in the system such as the frequency, the inverse relaxation time, or the gap.

Precise analysis of the relative adiabaticity or nonadiabaticity of ET reactions, the role of multidimensionality, and developing a general formalism that smoothly bridges adiabatic and non adiabatic limits remain a major challenge in theoretical approaches to ET reactions.

Marcus TheoryVanishing kET at 0 K not observedExperimental low T kET temperature independent

Jortner:Introduction of quantum vibrationsinstead of classical vibrations

classicalhwvib << kT

ßhwvib » kT     or     hwvibkT

quantum

But this will be another story (R. Kipling)