23

Rudolph A. Marcus (1923- )

Nobel prize 1992

Rudolph A. Marcus developed an original theory to explain the rate of ET reactions, linking the thermodynamics of the process to its kinetics. It was originally formulated to address outer sphere electron transfer reactions.

His original article (1956) referred to the self exchange reaction in aqueous solution Fe2+ + Fe3+ → Fe3+ + Fe2+, where the reorganization energy is solely due to reorganization of water dipoles around solvated ions.

Based on statistical mechanics considerations, Marcus theory postulates that :

a) Both precursor and successor states can be described by parabolic potential curves of similar shape and widths

b) Nuclear reorganization proceeds classically through low frequency vibrational and rotational modes

c) Nuclear reorganization is kinetically "symetrical" with kn = k–n .

Marcus classical theory

Andrei Tokmakoff, MIT Department of Chemistry, 3/25/08 12-6

12.2 MARCUS THEORY1

The displaced harmonic oscillator (DHO) formalism and the Energy Gap Hamiltonian have been

used extensively in describing charge transport reaction, such as electron and proton transfer.

Here we describe the rates of electron transfer between weakly coupled donor and acceptor states

when the potential energy depends on a nuclear coordinate, i.e., non-adiabatic electron transfer.

These results reflect the findings of Marcus’ theory of electron transfer.

We can represent the problem as calculating the transfer or reaction rate for the transfer

of an electron from a donor to an acceptor

D A D A� �� � � (12.22)

This reaction is mediated by a nuclear coordinate q. This need

not be, and generally isn’t, a simple local coordinate. For

electron transfer in solution, we most commonly consider

electron transfer to progress along a solvent rearrangement

coordinate in which solvent reorganizes its configuration so that

dipoles or charges help to stabilize the extra negative charge at

the acceptor site. This type of collective coordinate is illustrated

in the figure to the right. The external response of the medium

along the electron transfer coordinate is referred to as “outer

shell” electron transfer, whereas the influence of internal

vibrational modes that promote ET is called “inner shell”.

The influence of collective solvent rearrangements or

intramolecular vibrations can be captured with the use of an

electronic transition coupled to a harmonic bath.

Normally we associate the rates of electron transfer

with the free-energy along the electron transfer coordinate

q. The electronic coupling that leads to transfer mixes the

diabatic states that define the potential with the electron on the donor or on the acceptor in the

region of their crossing. From this adiabatic surface, we would obtain the rate of transfer from

� �†expk G kT� �� . If the coupling is weak, and the splitting is small at the crossing point, we

24

Geometrical derivation of Marcus equation

x

y

i

f

yi = ax2

yf = a(x + b)2 + c

Parabolas :

Intersection (x†, y†) :

b

c

y† = a(x†)2 = a(x† + b)2 + c

a(x†)2 = a(x†)2 + 2 abx† + (ab2 + c)

2 abx† = – ab2 – c

– (ab2 + c )x† =

2 ab

y†

Λ

x†

y† = a(x†)2 =(ab2 + c )2

4 ab2

Λ = ab2

ΔrG0 = c

ΔG† = y†⎬

( Λ + ΔrG0 )2

4 ΛΔG† =

⎫

⎭

25

ΔG† = (Λ + ΔrG0)2

4Λ

ΔG†

ΔrG0–

Λ

0

Marcus-Hush theory

Marcus theory was further extended to inner sphere electron transfer reactions by Noel Hush. Parabolic potential energy curves were derived in this case from harmonic oscillator potentials. The final formulation is practically identical to Marcus' (Hush's formulation is known as Marcus-Hush theory) and reads:

Noel S. Hush (1924- )

For an excellent review of the theory, see for instance:

Marcus, R. A. & Sutin, N. Biochimica et Biophysica Acta, 1985, 811, 265.

26

Nuclear coordinate

i f

ΔG†

ΔrG0

i

f

ΔG†=0

ΔrG0

i

f

ΔG†

ΔrG0

Λ

−ΔrG0 = Λ−ΔrG0 < Λ

ΔrG0=Gf–Gi

−ΔrG0 > Λ

Activation energy (ΔG†) dependence upon exoergicity (ΔrG0) of ET

27

Avoided crossing of potential energy surfaces

[D/A]

[D+...A–]

Λ

2 |HDA|

ΔrG0

HDA

HDA

In order for electron transfer to occur, an overlap between the populated orbital of the donor and the empty orbital of the acceptor is necessary in the activated complex. This electronic interaction involves a split of electronic energy levels.

If the electronic states are of the same symmetry, breakage of the Frank-Condon principle at the saddle point causes in turn a split of potential energy levels and avoided crossing of the two curves.

As a result two potential surfaces are created that are separated at the configuration of the activated complex by an energy gap 2|HDA|. |HDA| is a matrix element for electronic coupling between the donor and the acceptor.

[D/A] [D+...A–]

28

Adiabatic vs. diabatic electron transfer

adiabatic

non-adiabatic

2 |HDA|

In an adiabatic electron transfer process (|HDA| >> kBT ) the reacting system remains on the lowest potential energy s u r f a ce i n p a s s i n g f rom reactants to products. The act ivat ion-control led rate constant ket is related to the activation energy by Eyring's equation :

νn is the nuclear reorganization frequency of the precursor complex. In the simplest model, νn = kBT / h.

ket = νn· exp ( – )ΔG†

kBT

When the electronic coupling is sufficiently small ( |HDA| ≤ 3 kBT ), the precursor complex can borough some thermal energy from the environment and jump from one potential energy curve to the other. The rate of the diabatic (or non-adiabatic) electron transfer process in this case is reduced considerably and is not govern by the sole nuclear reorganization rate νn anymore. A treatment of the problem will be proposed later in the course (p. 33).

29

Arrhenius/ Eyring Marcus

3.02.01.00.0-1.0-4

0

4

8

12

30001500

500ϖ = 50 cm-1

log

( ket

[s-1

] )

- ΔetG [ eV ]

Λ = 0.75 eV

‘inverted’ region‘normal’ region

Λ

Marcus' quadratic equation predicts some counter-intuitive kinetic behavior as the driving force becomes larger than the reorgani-zation energy (– ΔrG0 > Λ) : The theoretical value of the activation energy ΔG† starts to raise again, hence reducing the overall ET process rate constant ket.

The predicted existence of such an "inverted region" was controversial from the time the theory was proposed in 1956 until experimental evidence of it was found for a s e t o f p h o t o r e d o x reactions (Closs & Miller, 1986).

Marcus normal- and inverted regions

In electrochemistry, Butler-Volmer's equation relates the current density j through an electrode to the overpotential η , defined as the difference between the electrode potential and the standard potential of the redox couple in solution. If the process is controlled by activation, the current density j is proportional to the rate of electron transfer reaction at the interface and, therefore, to its first order rate constant ket. Butler-Volmer's equation simplifies for η << 0, yielding Tafel's linear equation:

where ket0 represents the rate constant of

electron transfer in reversible conditions (η  = 0), α is a symmetry- or configuration coefficient and F is the Faraday constant.

!30

Marcus vs. Tafel equation

ln ket = ln ket

0 −α  FRT

ηc !

Provided the value of the symmetry coefficient is set to α = 0.5, Tafel's linear equation is compatible with Marcus' parabolic relationship for – ΔetG0 << Λ.

· ( + + )· ( – ) dD dA rm

31

Commonly the total reorganization energy Λ is expressed as the sum of an inner-shell contribution λin (also sometime written λv ) accounting for vibrational reorganization of the redox partners and an outer-shell contribution λout (also sometime written λs ) attributed to the reorganization of the solvent dipoles :

Λ = λin + λout (always > 0)

Approximations have been proposed to calculate the value of λout taking into account the dielectric constant ε (relative permittivity) of the solvent :

λout = (Δe)2

4π· ε0 1 1 1

n2 ε 1 1

· ( – )· ( – ) d 2 rm

λout = (Δe)2

8π· ε0 2 1

n2 ε 1 1

(homogeneous system)

(heterogeneous system)

e : elementary charged : diameter of the reactant speciesrm : optimum reaction distanceε : static dielectric constant of the solventn : solvent refractive index. n2 represents the optical (high frequency) dielectric constant of the medium.

Nuclear reorganization energy Λ

32

Exercise 7.3

A strongly emissive neutral perylene dye (S) dissolved in ethanol has a fluorescence quantum yield Φf = 0.90. The lifetime of its excited state is τ = 50 ns.

In the presence of tri-ethylamine (TEA) electron donor in large concentration (~ 1 M), the fluorescence of the dye is quenched down to Φf ≈ 0.01.

Estimate the inner-sphere reorganization energy λin [eV] associated to the light-induced electron transfer taking place between S* and TEA.

Numerical data

φ0 (S/ S– ) = – 1.0 V / SCE

φ0 (TEA+/ TEA ) = + 0.7 V / SCE

ΔE0,0 (S) = 2.5 eV

n (EtOH) = 1.3611ε (EtOH) = 24.3

ε0 = 8.86· 10–12 A· s· V–1· m–1 (dielectric permittivity of vacuum)

33

Fermi's golden rule for diabatic ET

The general expression describing the rate of non-adiabatic ET processes is given by Fermi's golden rule:

This relation applies to all non-adiabatic processes and is based on the Born-Oppenheimer approximation decoupling electronic and nuclear contributions. It includes the electronic coupling matrix element squared |HDA|2 (with the dimensions of an energy). |HDA| depends on the overlap of the electronic wave functions of the donor and the acceptor and thus upon the distance r, on which electron tunneling takes place:

β is a damping factor, whose value is typically <1.2 Å–1 for through-space ET.

ρ(Ef) is a Franck-Condon factor-weighted density of states. It expresses the contribution of nuclear reorganization to the electron transfer rate. The ρ(Ef) term is proportional to the matrix element describing the overlap of nuclear wave functions of vibronic states of the precursor and successor complexes.

wv

i

f

δE

34

Nuclear factor of Fermi's golden rule

ket =2πh

H 2FC

Fermi golden rule :

classical limit :

FC =1

4π ⋅ Λ⋅ kBT⋅exp −

ΔG†kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

ΔG† =Λ + ΔG°( )2

4Λdensity of states

(S+/S*)

(S+/S)

ΛD

E donor states

acceptor states

hν

(A/A-)

e-

ΛA

ΔGo

nuclear wave functionvibronic levelspopulation density of the vibrational state v(i)energy difference betweenlevels v and w

χ :v, w :ρv :

δE :⎬⎫

⎭Classical limit

If nuclear reorganization proceeds only through low frequency rotational and vibrational modes, quantum levels are so close that they form a quasi-continuum of states.

(equivalent to Marcus' equation)

35

Marcus-Jortner semi-classical theory

When nuclear reorganization proceeds through high frequency vibrational modes, a nuclear tunneling effect cannot be neglected. This quantum treatment is applied to high frequency modes, while a classical treatment is still applied to low frequency modes, such as those associated to the solvent dipoles rotational reorganization (Marcus-Jortner semi-classical theory).

In the simplified case, where only a single vibrational frequency ν is considered (single-phonon model), the Fermi golden rule can be expressed by the Marcus-Dogonadze equation:

When the values of λin or ν are small, the equation is reduced back to the Marcus classical result. As well, when the overlap between vibrational wave functions of initial and final states is weak, the nuclear tunneling effect becomes negligible, even if high frequency reorganization modes associated to the electron transfer do exist.

Joshua Jortner (1933 - )

ket = HDA2 · 1

2π · λout · kBT · exp −λin

hν⎛⎝⎜

⎞⎠⎟ ·

λin

hν⎛⎝⎜

⎞⎠⎟

n!n=0

∞

∑ · exp− ΔGr

0 + nhν+ λout( )2

4 λout · kBT

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Semi-classical treatment of electron transfer

ΔG

ΔG

f

i i

f

ΔG

nuclear configuration nuclear configuration 36

Simple considerations, such as illustrated by the scheme here below, show that efficient vibrational coupling and nuclear tunneling take place more specifically in the Marcus inverted situation (right). The semi-classical treatment is therefore necessary in the inverted region, while the classical approach is sufficient in the normal region.

'normal' region 'inverted' region

Semi-classical treatment of electron transfer

37

3.02.01.00.0-1.0-4

0

4

8

12

30001500

500ϖ = 50 cm-1

log

( ket

[s-1

] )

- ΔetG [ eV ] -ΔrG0 [eV]

log

(ket [

s–1])

ν = 50 cm–1

1'5003'000

500

Λ = 0.5 eV

ket = HDA2 · 1

2π · λout · kBT · exp −λin

hν⎛⎝⎜

⎞⎠⎟ ·

λin

hν⎛⎝⎜

⎞⎠⎟

n!n=0

∞

∑ · exp− ΔGr

0 + nhν+ λout( )2

4 λout · kBT

⎡

⎣⎢⎢

⎤

⎦⎥⎥

38

Temperature-dependence of ET kinetics

100 150 200 250 300T [ K ]

2

6

4

8

10

-2.0-1.0

0

Δ etG [ eV ]

log

k et

ΔrG0 [eV]

log

ket

T [K]

Nuclear tunneling in the inverted region means that electron transfer does not proceed over the activation energy barrier ΔG†. The Marcus-Dogonadze equation thus predicts ET does not follow Arrhenius' law and that ket does not depend upon temperature in the inverted region.

E

ΔGEx

e–cb (SC)

S*

v = 0

S+ (v > 0)

hω

hν

cbΛ

Nuclear coordinate

f

i

Charge injection into a continuum of acceptor states

Photoinduced electron transfer from a dye excited state S* into the conduction band of a semiconductor S* + SC → S+ + e–cb (SC) is called charge injection. In this case the acceptor is not represented by a discrete electronic level but by a continuum of empty states. ΔrG0 of the reaction is not well defined and an activation-less ET process is usually possible.

39

Charge injection into a continuum of acceptor states

density of states

(S+/S*)

(S+/S)

Λ

E donor states

acceptor states

hν

e-

Because overlap of nuclear wave functions in the precursor and successor complexes is optimum in the case of a continuum of acceptor levels, the nuclear factor ρ(Ef) is always very large. Fermi's golden rule reduces then to:

40

ρ(Ef) ≈ na · 1hω

na (0≤  na ≤1) is the reduced density of electronic acceptor states in the semiconductor's conduction band. 1/hω is the density of vibrational product states.

Note that, in such a situation, the electron transfer rate does not depend upon T nor ∆rG0 anymore. Two parameters are now essentially controlling the injection rate ket : the density of acceptor states in the conduction band and the electronic coupling factor |HDA|2.

Dynamics of light-induced electron transfer

nuclear configuration

ΔG

Δ-etG

ΔetG

S + SC

S* + SC S+ + e-cb (SC)

ΔG‡

hν

D/A

D*/A D+/ A–

ΔG†

Δ–etG0

ΔetG0

In practical applications of photo-induced ET react ions , charge separation has to be maintained during a period of time sufficient for further redox reactions to take place. Ideally, forward ET involving an excited state has to be as fast as possible, while back electron transfer during which charges recombine has to be slow.

Such an ideal situation is achieved when the energetics of the system implies activation-less forward ET and very exoergic back ET process situated in the inverted Marcus region.

41