In theoretical chemistry, 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 ${\displaystyle \Delta G^{\circ }}$ 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 ${\displaystyle \Delta G^{\circ }}$ 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.^[6]

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.

Outer vs inner ET

In a redox reaction an electron donor D must diffuse to the acceptor A, forming a precursor complex, which is labile but allows electron transfer to give successor complex. The pair then dissociates. For a one electron transfer the reaction is

${\displaystyle {\ce {{D}+A<=><=>->{D+}+{A^{-}}}}}$

(D and A may already carry charges). Here k₁₂, k₂₁ and k₃₀ are diffusion constants, k₂₃ and k₃₂ are 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). The ligand shells around A and D are retained. This process is called outer sphere electron transfer. Outer sphere ET is the main focus of traditional Marcus Theory. The other kind or redox reactions is inner sphere where A and D are covalently linked by a bridging ligand. Rates for such ET reactions depend on ligand exchange rates.

The problemedit

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 the aquo complexes Fe(H₂O)₆²⁺ and Fe(H₂O)6³⁺. Redox occurs with Gibbs free reaction energy ${\displaystyle \Delta G^{\circ }=0}$.

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,

${\displaystyle k_{\text{act}}=A\cdot e^{-{\frac {\Delta G^{\ddagger }}{RT}}}}$

${\displaystyle \Delta G^{\ddagger }}$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 modeledit

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 S_N2 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).^[7] 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 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 theoryedit

The macroscopic system: two conducting spheresedit

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 P_e and the slow atom and orientation polarisation P_u 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 W_I 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.^[8] Then the spheres in this state of charge give rise to a defined electric field in the solvent which creates the total solvent polarization P_u + P_e. By the same token this polarization of the solvent interacts with the charges.

In a second step the energy W_II of the reversible (back) transfer of the charge to the first sphere, again via the vacuum, is calculated. However, the atom and orientation polarization P_u is kept fixed, only the electron polarization P_e may adjust to the field of the new charge distribution and the fixed P_u. After this second step the system is in the desired state with an electron polarization corresponding to the starting point of the redox reaction and an atom and orientation polarization corresponding to the "transition state". The energy W_I + W_II 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 non-equilibrium 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

${\displaystyle G=\left({\frac {1}{2r_{1}}}+{\frac {1}{2r_{2}}}-{\frac {1}{R}}\right)\cdot \left({\frac {1}{\epsilon _{\text{opt}}}}-{\frac {1}{\epsilon _{\text{s}}}}\right)\cdot (\Delta e)^{2}}$

Where r₁ and r₂ are the radii of the spheres and R is their separation, ε_s and ε_opt 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 λ_o, 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.^[9] In terms of exchange direction the system is symmetric.

The microscopic system: the donor-acceptor pairedit

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 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)^‡ = λ_o/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^[10] e.g.

${\displaystyle {\ce {{Fe^{II}(CN)6^{4-}}+{Ir^{IV}Cl6^{2-}}<=>{Fe^{III}(CN)6^{3-}}+{Ir^{III}Cl6^{3-}}}}}$

and they have positive (endergonic) or negative (exergonic) Gibbs free energies of reaction ${\displaystyle \Delta G^{\circ }}$.

As Marcus calculations refer exclusively to the electrostatic properties in the solvent (outer sphere) ${\displaystyle \Delta G^{\circ }}$ and ${\displaystyle \lambda _{0}}$ are independent of one another and therefore can just be added up. This means that the Marcus parabolas in systems with different ${\displaystyle \Delta G^{\circ }}$ are shifted just up or down in the ${\displaystyle G}$ vs. ${\displaystyle \Delta e}$ diagram (Fig. 2). Variation of ${\displaystyle \Delta G^{\circ }}$ can be affected in experiments by offering different acceptors to the same donor.

Simple calculation of the intersection point between the parabolas i ${\displaystyle (y=x^{2})}$ and ${\displaystyle f_{i}}$ ${\displaystyle (y=(x-d)^{2}+c)}$ give the Gibbs free energy of activation

${\displaystyle \Delta G^{\ddagger }={\frac {(\lambda _{0}+\Delta G^{\circ })^{2}}{4\lambda _{0}}}}$,

where ${\displaystyle \lambda _{0}}$ = ${\displaystyle d^{2}}$ and ${\displaystyle \Delta G^{\circ }}$= c. 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. Zdroj:https://en.wikipedia.org?pojem=Marcus_theory
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