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Marcus theory
Marcus theory is a theory originally developed by Rudolph A. Marcus, starting in 1956, to explain the rates of electron
transfer reactions – the rate at which an electron can move or jump from one chemical species (called the electron donor)
to another (called the electron acceptor).[1] It was originally formulated to address outer sphere electron transfer
reactions, in which the two chemical species only change in their charge with an electron jumping (e.g. the oxidation of an
ion like Fe2+/Fe3+), but do not undergo large structural changes. It was extended to include inner sphere electron transfer
contributions, in which a change of distances or geometry in the solvation or coordination shells of the two chemical
species is taken into account (the Fe-O distances in Fe(H2O)2+ and Fe(H2O)3+ are different).[2][3]
For electron transfer reactions without making or breaking bonds Marcus theory takes the place of Eyring's transition
state theory[4][5] which has been derived for reactions with structural changes. Both theories lead to rate equations of the
same exponential form. However, whereas in Eyring theory the reaction partners become strongly coupled in the course of
the reaction to form a structurally defined activated complex, in Marcus theory they are weakly coupled and retain their
individuality. It is the thermally induced reorganization of the surroundings, the solvent (outer sphere) and the solvent
sheath or the ligands (inner sphere) which create the geometrically favourable situation prior to and independent of the
electron jump.
The original classical Marcus theory for outer sphere electron transfer reactions demonstrates the importance of the
solvent and leads the way to the calculation of the Gibbs free energy of activation, using the polarization properties of the
solvent, the size of the reactants, the transfer distance and the Gibbs free energy
G0 of the redox reaction. The most
startling result of Marcus' theory was the "inverted region": whereas the reaction rates usually become higher with
increasing exergonicity of the reaction, electron transfer should, according to Marcus theory, become slower in the very
negative
G0 domain. Scientists searched the inverted region for proof of a slower electron transfer rate for 30 years until
it was unequivocally verified experimentally in 1984.
R.A. Marcus received the Nobel Prize in Chemistry in 1992 for this theory. Marcus theory is used to describe a number of
important processes in chemistry and biology, including photosynthesis, corrosion, certain types of chemiluminescence,
charge separation in some types of solar cells and more. Besides the inner and outer sphere applications, Marcus theory
has been extended to address heterogeneous electron transfer.
Contents
The one-electron redox reaction
Outer sphere electron transfer
The problem
The Marcus model
Marcus theory
The macroscopic system: two conducting spheres
The microscopic system: the donor-acceptor pair
Inner sphere electron transfer
The probability of the electron jump
Experimental results
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See also
References
Marcus's key papers
The one-electron redox reaction
Chemical reactions may lead to a substitution of a group in a molecule or a ligand in a complex, to the elimination of a
group of the molecule or a ligand, or to a rearrangement of a molecule or complex. An electron-transfer reaction may,
however, also cause simply an exchange of charges between the reactants, and these redox reactions without making or
breaking a bond seem to be quite simple in inorganic chemistry for ions and complexes. These reactions often become
manifest by a change of colour, e.g. for ions or complexes of transition metal ions, but organic molecules, too, may change
their colour by accepting or giving away an electron (like the herbicide Paraquat (N,N-dimethyl-4,4'-bipyridinium
dichloride) which becomes blue when accepting an electron, thence the alternative name of methyl viologen). For this type
of electron-transfer reactions R.A. Marcus has developed his theory. Here the trace of argument and the results are
presented. For the mathematical development and details the original papers[6][7] should be consulted.
In a redox reaction one partner acts as an electron donor D the other as an acceptor A. For a reaction to take place D and A
must diffuse together. They form the precursor complex, usually a kinetic, unstable, solvated encounter complex, which by
electron transfer is transformed to the successor complex, and finally this separates by diffusion. For a one electron
transfer the reaction is
(D and A may already carry charges). Here k12, k21 and k30 are diffusion constants, k23 and k32 rate constants of activated
reactions. The total reaction may be diffusion controlled (the electron transfer step is faster than diffusion, every
encounter leads to reaction) or activation controlled (the "equilibrium of association" is reached, the electron transfer step
is slow, the separation of the successor complex is fast).
Outer sphere electron transfer
Redox reactions are preferably run in polar solvents. Donor and acceptor then have a solvent shell and the precursor and
successor complexes are solvated also. The closest molecules of the solvent shell, or the ligands in complexes, are tightly
bound and constitute the "inner sphere". Reactions in which these participate are called inner sphere redox reactions. The
free solvent molecules constitute the "outer sphere". Outer sphere redox reactions do not change the inner sphere, no
bonds are made nor broken.
It was R.A. Marcus who realized the role of the solvent when he worked on the nature and magnitude of the Gibbs free
energy of activation for redox reactions, more precisely: one-electron transfer reactions of the outer sphere type. He
published two fundamental papers.[6][7] The ideas of these two papers are often referred to Marcus Theory, although
Marcus’ later work goes much beyond them.[1]
The problem
In outer sphere redox reactions no bonds are formed or broken; only an electron transfer (ET) takes place. A quite simple
example is the Fe2+/Fe3+ redox reaction, the self exchange reaction which is known to be always occurring in an aqueous
solution containing both FeSO4 and Fe2(SO4)3 (of course, with equal and measurable rates in both directions and with
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Gibbs free reaction energy ΔG0 = 0).
From the reaction rate's temperature dependence an activation energy is determined, and this activation energy is
interpreted as the energy of the transition state in a reaction diagram. The latter is drawn, according to Arrhenius and
Eyring, as an energy diagram with the reaction coordinate as the abscissa. The reaction coordinate describes the minimum
energy path from the reactants to the products, and the points of this coordinate are combinations of distances and angles
between and in the reactants in the course of the formation and/or cleavage of bonds. The maximum of the energy
diagram, the transition state, is characterized by a specific configuration of the atoms. Moreover, in Eyring's TST[4][5] a
quite specific change of the nuclear coordinates is responsible for crossing the maximum point, a vibration in this
direction is consequently treated as a translation.
For outer sphere redox reactions there cannot be such a reaction path, but nevertheless one does observe an activation
energy. The rate equation for activation-controlled reactions has the same exponential form as the Eyring equation,
is the Gibbs free energy of the formation of the transition state, the exponential term represents the probability of its
formation, A contains the probability of crossing from precursor to successor complex.
The Marcus model
The consequence of an electron transfer is the rearrangement of charges, and this greatly influences the solvent
environment. For the dipolar solvent molecules rearrange in the direction of the field of the charges (this is called
orientation polarisation), and also the atoms and electrons in the solvent molecules are slightly displaced (atomic and
electron polarization, respectively). It is this solvent polarization which determines the free energy of activation and thus
the reaction rate.
Substitution, elimination and isomerization reactions differ from the outer sphere redox reaction not only in the structural
changes outlined above, but also in the fact that the movements of the nuclei and the shift of charges (charge transfer, CT)
on the reactions path take place in a continuous and concerted way: nuclear configurations and charge distribution are
always "in equilibrium". This is illustrated by the SN2 substitution of the saponification of an alkyl halide where the rear
side attack of the OH− ion pushes out a halide ion and where a transition state with a five-coordinated carbon atom must
be visualized. The system of the reactants becomes coupled so tightly during the reaction that they form the activated
complex as an integral entity. The solvent here has a minor effect.
By contrast, in outer sphere redox reactions the displacement of nuclei in the reactants are small, here the solvent has the
dominant role. Donor-acceptor coupling is weak, both keep their identity during the reaction. Therefore, the electron,
being an elementary particle, can only "jump" as a whole (electron transfer, ET). If the electron jumps, the transfer is
much faster than the movement of the large solvent molecules, with the consequence that the nuclear positions of the
reaction partners and the solvent molecules are the same before and after the electron jump (Franck–Condon
principle).[8] The jump of the electron is governed by quantum mechanical rules, it is only possible if also the energy of the
ET system does not change "during" the jump.
The arrangement of solvent molecules depends on the charge distribution on the reactants. If the solvent configuration
must be the same before and after the jump and the energy may not change, then the solvent cannot be in the solvation
state of the precursor nor in that of the successor complex as they are different, it has to be somewhere in between. For the
self-exchange reaction for symmetry reasons an arrangement of the solvent molecules exactly in the middle of those of
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precursor and successor complex would meet the conditions. This means that the solvent arrangement with half of the
electron on both donor and acceptor would be the correct environment for jumping. Also, in this state the energy of
precursor and successor in their solvent environment would be the same.
However, the electron as an elementary particle cannot be divided, it resides either on the donor or the acceptor and
arranges the solvent molecules accordingly in an equilibrium. The "transition state", on the other hand, requires a solvent
configuration which would result from the transfer of half an electron, which is impossible. This means that real charge
distribution and required solvent polarization are not in an "equilibrium". Yet it is possible that the solvent takes a
configuration corresponding to the "transition state", even if the electron sits on the donor or acceptor. This, however,
requires energy. This energy may be provided by the thermal energy of the solvent and thermal fluctuations can produce
the correct polarization state. Once this has been reached the electron can jump. The creation of the correct solvent
arrangement and the electron jump are decoupled and do not happen in a synchronous process. Thus the energy of the
transition state is mostly polarization energy of the solvent.
Marcus theory
The macroscopic system: two conducting spheres
On the basis of his reasoning R.A. Marcus developed a classical theory with the aim of calculating the polarization energy
of the said non-equilibrium state. From thermodynamics it is well known that the energy of such a state can be determined
if a reversible path to that state is found. Marcus was successful in finding such a path via two reversible charging steps for
the preparation of the "transition state" from the precursor complex.
Four elements are essential for the model on which the theory is based:
1. Marcus employs a classical, purely electrostatic model. The charge (many elementary charges) may be transferred in
any portion from one body to another.
2. Marcus separates the fast electron polarisation Pe and the slow atom and orientation polarisation Pu of the solvent on
grounds of their time constants differing several orders of magnitude.
3. Marcus separates the inner sphere (reactant + tightly bound solvent molecules, in complexes + ligands) and the outer
sphere (free solvent )
4. In this model Marcus confines himself to calculating the outer sphere energy of the non-equilibrium polarization of the
"transition state". The outer sphere energy is often much larger than the inner sphere contribution because of the far
reaching electrostatic forces (compare the Debye-Hückel theory of electrochemistry).
Marcus’ tool is the theory of dielectric polarization in solvents. He solved the problem in a general way for a transfer of
charge between two bodies of arbitrary shape with arbitrary surface and volume charge. For the self-exchange reaction,
the redox pair (e.g. Fe(H2O)63+ / Fe(H2O)62+) is substituted by two macroscopic conducting spheres at a defined distance
carrying specified charges. Between these spheres a certain amount of charge is reversibly exchanged.
In the first step the energy WI of the transfer of a specific amount of charge is calculated, e.g. for the system in a state
when both spheres carry half of the amount of charge which is to be transferred. This state of the system can be reached by
transferring the respective charge from the donor sphere to the vacuum and then back to the acceptor sphere.[9] Then the
spheres in this state of charge give rise to a defined electric field in the solvent which creates the total solvent polarization
Pu + Pe. By the same token this polarization of the solvent interacts with the charges.
In a second step the energy WII of the reversible (back) transfer of the charge to the first sphere, again via the vacuum, is
calculated. However, the atom and orientation polarization Pu is kept fixed, only the electron polarization Pe may adjust
to the field of the new charge distribution and the fixed Pu. After this second step the system is in the desired state with an
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electron polarization corresponding to the starting point of the redox reaction and an atom and orientation polarization
corresponding to the "transition state". The energy WI + WII of this state is, thermodynamically speaking, a Gibbs free
energy G.
Of course, in this classical model the transfer of any arbitrary
amount of charge Δe is possible. So the energy of the nonequilibrium state, and consequently of the polarization energy of
the solvent, can be probed as a function of Δe. Thus Marcus has
lumped together, in a very elegant way, the coordinates of all
solvent molecules into a single coordinate of solvent polarization
Δp which is determined by the amount of transferred charge Δe.
So he reached a simplification of the energy representation to
only two dimensions: G = f(Δe). The result for two conducting
spheres in a solvent is the formula of Marcus
Fig. 1. The parabolas of outer-sphere
reorganisation energy of the system two spheres
in a solvent. Parabola i: the charge on the first,
transfer to the second, parabola f: the charge on
the second, transfer to the first. The abscissa is
the transferred amount of charge Δe or the
induced polarization P, the ordinate the Gibbs free
energy. ΔG(0)‡ = λo/4 is the reorganization energy
at Δe = 0.5, it corresponds to the activation
energy of the self-exchange reaction.
Where r1 and r2 are the radii of the spheres and R is their separation, εs and εopt are the static and high frequency (optical)
dielectric constants of the solvent, Δe the amount of charge transferred. The graph of G vs. Δe is a parabola (Fig. 1). In
Marcus theory the energy belonging to the transfer of a unit charge (Δe = 1) is called the (outer sphere) reorganization
energy λo, i.e. the energy of a state where the polarization would correspond to the transfer of a unit amount of charge, but
the real charge distribution is that before the transfer.[10] In terms of exchange direction the system is symmetric.
The microscopic system: the donor-acceptor pair
Shrinking the two-sphere model to the molecular level creates the problem that in the self-exchange reaction the charge
can no longer be transferred in arbitrary amounts, but only as a single electron. However, the polarization still is
determined by the total ensemble of the solvent molecules and therefore can still be treated classically, i.e. the polarization
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energy is not subject to quantum limitations. Therefore, the energy of solvent reorganization can be calculated as being
due to a hypothetical transfer and back transfer of a partial elementary charge according to the Marcus formula. Thus the
reorganization energy for chemical redox reactions, which is a Gibbs free energy, is also a parabolic function of Δe of this
hypothetical transfer, For the self exchange reaction, where for symmetry reasons Δe = 0.5, the Gibbs free energy of
activation is ΔG(0)‡ = λo/4 (see Fig. 1 and Fig. 2 intersection of the parabolas I and f, f(0), respectively).
Up to now all was physics, now some chemistry enters. The self exchange reaction is a very specific redox reaction, most of
the redox reactions are between different partners[11] e.g.
and they have positive (endergonic) or negative (exergonic) Gibbs free energies of reaction ΔG0.
As Marcus calculations refer exclusively to the electrostatic properties in the solvent (outer sphere) ΔG0 and λo are
independent of one another and therefore can just be added up. This means that the Marcus parabolas in systems with
different ΔG0 are shifted just up or down in the G vs. Δe diagram (Fig. 2). Variation of ΔG0 can be affected in experiments
by offering different acceptors to the same donor.
Simple calculations of the points of intersection of the parabolas i
,
and
to
give the
Gibbs free energy of activation
The intersection of those parabolas represents an activation energy and not the energy of a transition state of fixed
configuration of all nuclei in the system as is the case in the substitution and other reactions mentioned. The transition
state of the latter reactions has to meet structural and energetic conditions, redox reactions have only to comply to the
energy requirement. Whereas the geometry of the transition state in the other reactions is the same for all pairs of
reactants, for redox pairs many polarization environments may meet the energetic conditions.
Marcus’ formula shows a quadratic dependence of the Gibbs free energy of activation on the Gibbs free energy of reaction.
It is general knowledge from the host of chemical experience that reactions usually are the faster the more negative is ΔG0.
In many cases even a linear free energy relation is found. According to the Marcus formula the rates increase also when
the reactions are more exergonic, however only as long as ΔG0 is positive or slightly negative. It is surprising that for redox
reactions according to the Marcus formula the activation energy should increase for very exergonic reaction, i.e. in the
cases when ΔG0 is negative and its absolute value is greater than that of λo. This realm of Gibbs free energy of reaction is
called "Marcus inverted region". In Fig. 2 it becomes obvious that the intersection of the parabolas i and f moves upwards
in the left part of the graph when ΔG0 continues to become more negative, and this means increasing activation energy.
Thus the total graph of ln k vs. ΔG0 should have a maximum.
The maximum of the ET rate is expected at ΔG ‡ = 0. Here Δe = 0 and q = 0 (Fig. 2) which means that the electron may
jump in the precursor complex at its equilibrium polarization. No thermal activation is necessary: the reaction is
barrierless. In the inverted region the polarization corresponds to the difficult to imagine notion of a charge distribution
where the donor has received and the acceptor given off charge. Of course, in real world this does not happen, it is not a
real charge distribution which creates this critical polarization, but the thermal fluctuation in the solvent. This polarization
necessary for transfer in the inverted region can be created – with some probability – as well as any other one.[12] The
electron is just waiting for it for jumping.
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Inner sphere electron transfer
In the outer sphere model the donor or acceptor and the tightly
bound solvation shells or the complex’ ligands were considered to
form rigid structures which do not change in the course of
electron transfer. However, the distances in the inner sphere are
dependent on the charge of donor and acceptor, e.g. the central
ion-ligand distances are different in complexes carrying different
charges and again the Franck–Condon principle must be obeyed:
for the electron to jump to occur, the nuclei have to have a
configuration which is an identical one of as well the precursor as
the successor complexes, of course highly distorted. In this case
the energy requirement is fulfilled automatically.
In this inner sphere case the Arrhenius concept holds, the
transition state of definite geometric structure is reached along a
geometrical reaction coordinate determined by nuclear motions.
No further nuclear motion is necessary to form the successor
complex, just the electron jumps, which makes a difference to the
TST theory. The reaction coordinate for inner sphere energy is
governed by vibrations and they differ in the oxidized and
reduces species.[13]
For the self-exchange system Fe2+/Fe3+ only the symmetrical
breathing vibration of the six water molecules around the iron
ions is considered.[13] Assuming harmonic conditions this
vibration has frequencies
fA are
and
, the force constants fD and
and the energies are
where q0 is the equilibrium normal coordinate and
Fig. 2 Marcus-Parabolas for different redox
reactions: f1 such with positive ΔG0, f(0) for the
self-exchange reaction with ΔG0 = 0 (broken line),
f2 for moderately negative ΔG0 (selected so, that
ΔG‡ = 0) and f3 for strongly negative ΔG0. The
free energy of activation ΔG‡ decreases from f1
(b1) via f(0) (a) to f2 (zero) and increases again for
f3 („Marcus inverted region").
the displacement along the normal coordinate, the
factor 3 stems from 6 (H2O)·½. Like for the outer-sphere reorganization energy potential energy curve is quadratic, here,
however, as a consequence of vibrations.
The equilibrium normal coordinates differ in Fe(H2O)62+ and Fe(H2O)63+. By thermal excitation of the breathing
vibration a geometry can be reached which is common to both donor and acceptor, i.e. the potential energy curves of the
breathing vibrations of D and A intersect here. This is the situation where the electron may jump. The energy of this
transition state is the inner sphere reorganization energy λin.
For the self-exchange reaction the metal-water distance in the transition state can be calculated[13]
This gives the inner sphere reorganisation energy
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It is fortunate that the expressions for the energies for outer and inner reorganization have the same quadratic form. Inner
sphere and outer sphere reorganization energies are independent, so they can be added to give
and inserted
in the Arrhenius equation
‡
Here, A can be seen to represent the probability of electron jump, exp[-ΔGin /kT] that of reaching the transition state of the
‡
inner sphere and exp[-ΔGo /kT] that of outer sphere adjustment.
For unsymmetrical (cross) reactions like
the expression for
can also be derived, but it is more complicated.[13] These reactions have a free reaction enthalpy
ΔG0 which is independent of the reorganization energy and determined by the different redox potentials of the iron and
cobalt couple. Consequently, the quadratic Marcus equation holds also for the inner sphere reorganization energy,
including the prediction of an inverted region. One may visualizing this by (a) in the normal region both the initial state
and the final state have to have stretched bonds, (b) In the Δ G‡ = 0 case the equilibrium configuration of the initial state is
the stretched configuration of the final state, and (c) in the inverted region the initial state has compressed bonds whereas
the final state has largely stretched bonds. Similar considerations hold for metal complexes where the ligands are larger
than solvent molecules and also for ligand bridged polynuclear complexes.
The probability of the electron jump
The strength of the electronic coupling of the donor and acceptor decides whether the electron transfer reaction is
adiabatic or non-adiabatic. In the non-adiabatic case the coupling is weak, i.e. HAB in Fig. 3 is small compared to the
reorganization energy and donor and acceptor retain their identity. The system has a certain probability to jump from the
initial to the final potential energy curves. In the adiabatic case the coupling is considerable, the gap of 2 HAB is larger and
the system stays on the lower potential energy curve.[14]
Marcus theory as laid out above, represents the non-adiabatic case.[15] Consequently, the semi-classical Landau-Zener
theory can be applied, which gives the probability of interconversion of donor and acceptor for a single passage of the
system through the region of the intersection of the potential energy curves
where Hif is the interaction energy at the intersection, v the velocity of the system through the intersection region, si and sf
the slopes there.
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Fig. 3 Energy diagram for Electron Transfer including inner and outer sphere reorganization and electronic coupling: The
vertical axis is the free energy, and the horizontal axis is the "reaction coordinate" – a simplified axis representing the
motion of all the atomic nuclei (inclusive solvent reorganization)
Working this out, one arrives at the basic equation of Marcus theory
where
is the rate constant for electron transfer,
is the electronic coupling between the initial and final states,
is the reorganization energy (both inner and outer-sphere), and
transfer reaction (
is the Boltzmann constant and
is the total Gibbs free energy change for the electron
is the absolute temperature).
Thus Marcus's theory builds on the traditional Arrhenius equation for the rates of chemical reactions in two ways: 1. It
provides a formula for the activation energy, based on a parameter called the reorganization energy, as well as the Gibbs
free energy. The reorganization energy is defined as the energy required to "reorganize" the system structure from initial
to final coordinates, without making the charge transfer. 2. It provides a formula for the pre-exponential factor in the
Arrhenius equation, based on the electronic coupling between the initial and final state of the electron transfer reaction
(i.e., the overlap of the electronic wave functions of the two states).
Experimental results
Marcus published his theory in 1956. For many years there was an intensive search for the inverted region which would be
a proof of the theory. But all experiments with series of reactions of more and more negative ΔG0 revealed only an increase
of the reaction rate up to the diffusion limit, i.e. to a value indicating that every encounter lead to electron transfer, and
that limit held also for very negative ΔG0 values (Rehm-Weller behaviour).[16] It took about 30 years until the inverted
region was unequivocally substantiated by Miller, Calcaterra and Closs for an intramolecular electron transfer in a
molecule where donor and acceptor are kept at a constant distance by means of a stiff spacer (Fig.4).[17]
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A posteriori one may presume that in the systems where the reaction partners
may diffuse freely the optimum distance for the electron jump may be sought,
i.e. the distance for which ΔG‡ = 0 and ΔG0 = - λo. For λo is dependent on R, λo
increases for larger R and the opening of the parabola smaller. It is formally
always possible to close the parabola in Fig. 2 to such an extent, that the fparabola intersects the i-parabola in the apex. Then always ΔG ‡ = 0 and the
rate k reaches the maximum diffusional value for all very negative ΔG0. There
are, however, other concepts for the phenomenon,[1] e.g. the participation of
excited states or that the decrease of the rate constants would be so far in the
inverted region that it escapes measurement.
R. A. Marcus and his coworkers have further developed the theory outlined
here in several aspects. They have included inter alia statistical aspects and
quantum effects,[19] they have applied the theory to chemiluminescence[20]
and electrode reactions.[21] R. A. Marcus received the Nobel Prize in Chemistry
in 1992, and his Nobel Lecture gives an extensive view of his work.[1]
See also
Fig.4. Marcus behaviour in a
molecule, which is composed of a
biphenyl entity, whose anion
(produced by means of pulse
radiolysis) acts as a donor, a steroid
entity, which is a rigid spacer and
different aromatic hydrocarbons
(1−3) and quinones (4−8), which are
the acceptors.[18]
Hammond's postulate
Solvated electron
Free-energy relationship
References
1. "Electron Transfer Reactions in Chemistry: Theory and Experiment" (http://nobelprize.org/nobel_prizes/chemistry/laur
eates/1992/marcus-lecture.pdf) (PDF). Nobelstiftung. Retrieved 02.04.2007. Check date values in: |access-date=
(help)
2. Contrary to Marcus' approach the inner sphere electron transfer theory of Noel S. Hush refers to a continuous change
of the electron density during transfer along a geometrical coordinate (adiabatic case), and takes also into account
the solvent influence as did Marcus. Hush's formulation is known as Marcus-Hush theory.
3. Hush, N.S. Trans. Faraday Soc. 1961, 57,557
4. P. W. Atkins: Physical Chemistry, 6. Ed., Oxford University Press, Oxford 1998 p.830
5. R.S. Berry, S. A. Rice, J. Ross: Physical Chemistry, Wiley, New York 1980, S. 1147 ff,
6. Marcus, R.A. "On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer I" J.Chem.Phys.1956, 24,
966. doi:10.1063/1.1742723 (https://doi.org/10.1063%2F1.1742723) or Free Text (http://www.cce.caltech.edu/faculty/
marcus/publications/16.pdf)
7. Marcus.R.A. "Electrostatic Free Energy and Other Properties of States Having Nonequilibrium Polarization I.
J.Chem.Phys.1956, 24, 979. doi:10.1063/1.1742724 (https://doi.org/10.1063%2F1.1742724) or Free Text (http://www.
cce.caltech.edu/faculty/marcus/publications/17.pdf)
8. W.F. Libby, "Theory of Electron Exchange Reactions in Aquous Solution" J.Phys.Chem. 1952, 56, 863
9. Marcus takes the vacuum state of the reactants as the zero energy point. Therefore many of his equations contain
also the solvation energy of the isolated species Wiso and the electrostatic energy of formation of the precursor and
successor complexes.
10. Note: The quadratic dependence of outer sphere reorganization energy is not a consequence of vibrations in
reactants or solvent!
11. they are often called Marcus cross reactions.
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12. The reverse reaction may support understanding: for this reaction the polarization due to the hypothetical transfer of a
unit electron charge is not sufficient to reach a polarization where the polarization energies of A/D and A−/D+ are
equal. This can only happen on the hypothetical transfer of more that one electron charge.
13. N. Sutin, 'Theory of Electron Transfer Reactions: Insights and Hindsights', Progr. Inorg. Chem. 1083, 30, 441-448
14. In normal chemical reactions, like substitutions, which proceed via a transition state the upper potential energy curve
is so far up that it is neglected
15. The theory of adiabatic electron transfer with participation of nuclear movement (which may be considered as a
transfer of charge, not an electron jump), has been worked out by Hush.
16. Rehm, D., Weller, A. "Kinetik und Mechanismus der Elektronenübertragung bei der Fluoreszenzlöschung in
Acetonitril" Ber. Bunsenges.Physik.Chem. 1969, 73, 834-839 characterized this behaviour by the empirical formula
17. Miller J.R., Calcaterra L.T., Closs G.L.: "Intramolecular long-distance electron transfer in radical anions. The effects of
free energy and solvent on the reaction rates", J.Am.Chem.Soc. 1984, 106, 3047, doi:10.1021/ja00322a058 (https://d
oi.org/10.1021%2Fja00322a058)
18. We recommend reference to the original paper, JACS does not license Wikipedia for graphs.
19. Siders, P., Marcus, R. A. "Quantum Effects in Electron-Transfer Reactions" J.Am.Chem.Soc. 1981,103,741; Siders,
P., Marcus, R. A. "Quantum Effects for Electron-Transfer Reactions in the 'Inverted Region'" J.Am.Chem.Soc.
1981,103,748
20. Marcus. R.A. "On the Theory of Chemiluminescent Electron-Transfer Reactions" J.Chem.Phys. 1965,43,2654
21. Marcus, R. A. "On the Theory of Electron-Transfer Reaction VI. Unified Treatment of Homogeneous and Electrode
Reactions" J.Chem.Phys. 1965, 43,679
Marcus's key papers
Marcus, R.A (1956). "On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer. I" (https://authors.li
brary.caltech.edu/45509/1/MARjcp56b.pdf) (PDF). J. Chem. Phys. 24 (5): 966–978. Bibcode:1956JChPh..24..966M
(https://ui.adsabs.harvard.edu/abs/1956JChPh..24..966M). doi:10.1063/1.1742723 (https://doi.org/10.1063%2F1.174
2723).
Marcus, R.A (1956). "Electrostatic Free Energy and Other Properties of States Having Nonequilibrium Polarization. I"
(https://authors.library.caltech.edu/45494/1/MARjcp56a.pdf) (PDF). J. Chem. Phys. 24 (5): 979–989.
Bibcode:1956JChPh..24..979M (https://ui.adsabs.harvard.edu/abs/1956JChPh..24..979M). doi:10.1063/1.1742724 (ht
tps://doi.org/10.1063%2F1.1742724).
Marcus, R.A (1957). "On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer. II. Applications to
Data on the Rates of Isotopic Exchange Reactions" (https://authors.library.caltech.edu/45505/1/MARjcp57b.pdf)
(PDF). J. Chem. Phys. 26 (4): 867–871. Bibcode:1957JChPh..26..867M (https://ui.adsabs.harvard.edu/abs/1957JChP
h..26..867M). doi:10.1063/1.1743423 (https://doi.org/10.1063%2F1.1743423).
Marcus, R.A (1957). "On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer. III. Applications to
Data on the Rates of Organic Redox Reactions". J. Chem. Phys. 26 (4): 872–877. Bibcode:1957JChPh..26..872M (htt
ps://ui.adsabs.harvard.edu/abs/1957JChPh..26..872M). doi:10.1063/1.1743424 (https://doi.org/10.1063%2F1.174342
4).
Marcus, R.A (1960). "Exchange reactions and electron transfer reactions including isotopic exchange. Theory of
oxidation-reduction reactions involving electron transfer. Part 4.—A statistical-mechanical basis for treating
contributions from solvent, ligands, and inert salt" (http://authors.library.caltech.edu/59464/1/df9602900021.pdf)
(PDF). Disc. Faraday Soc. 29: 21–31. doi:10.1039/df9602900021 (https://doi.org/10.1039%2Fdf9602900021).
Marcus, R.A (1963). "On The Theory Of Oxidation--Reduction Reactions Involving Electron Transfer. V. Comparison
And Properties Of Electrochemical And Chemical Rate Constants". J. Phys. Chem. 67 (4): 853–857.
doi:10.1021/j100798a033 (https://doi.org/10.1021%2Fj100798a033).
Marcus, R.A (1964). "Chemical and Electrochemical Electron-Transfer Theory". Annu. Rev. Phys. Chem. 15 (1): 155–
196. Bibcode:1964ARPC...15..155M (https://ui.adsabs.harvard.edu/abs/1964ARPC...15..155M).
doi:10.1146/annurev.pc.15.100164.001103 (https://doi.org/10.1146%2Fannurev.pc.15.100164.001103).
Marcus, R.A (1965). "On the Theory of Electron-Transfer Reactions. VI. Unified Treatment for Homogeneous and
Electrode Reactions" (https://authors.library.caltech.edu/33657/1/MARjcp65b.pdf) (PDF). J. Chem. Phys. 43 (2): 679–
https://en.wikipedia.org/wiki/Marcus_theory
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Marcus theory - Wikipedia
701. Bibcode:1965JChPh..43..679M (https://ui.adsabs.harvard.edu/abs/1965JChPh..43..679M).
doi:10.1063/1.1696792 (https://doi.org/10.1063%2F1.1696792).
Marcus, R.A.; Sutin N (1985). "Electron transfers in chemistry and biology". Biochim. Biophys. Acta. 811 (3): 265.
doi:10.1016/0304-4173(85)90014-X (https://doi.org/10.1016%2F0304-4173%2885%2990014-X).
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