Fundamentals of Reaction TheoryFundamentals of Reaction Theorylast modified by MM, 25/12/05

So far, we have primarily dealt with the equilibrium properties of a chemical systems, developing theories and methods to describe and predict the thermodynamical behaviour of a many-body system starting from the knowledge of the interaction between the particles (either atoms or molecules).Chemistry, however, is mainly focused on transformations, i.e. on processes that change the chemical nature of a system (in this case even as small as a single molecule), and because of this it is of chief interest to develop ways for predicting the rate and mechanism of a specific reaction (i.e. how quickly the reaction takes place and what is the sequence of “dynamical steps” to go from the reactant R to the product P). Thus, it is our goal to provide a mathematical description of a reactive event on the basis of the fundamental laws governing the motion of atoms and molecules.To do so, we start from the simple case of a reaction in gas phase and build on that to provide models for more complicate situations.

The simple ideas of a bimolecular reaction in the gas phaseThe simple ideas of a bimolecular reaction in the gas phase:consider the gas phase bimolecular reaction of the form

The transformation of A and B (R's) into C and D (P's) it is likely to happen when A and B are in close contact; only in this situation they could easily exchange atoms between themselves.

A � B � C � D

If this idea is correct, then, in order to predict the rate of the reaction (i.e. the rate at which A and B disappear while C and D appear), one must provide an estimate for the collision frequency between A and B. To do so, let us now suppose that both A and B are spherical objects with radius r

A and r

B, respectively, and that A molecules

are stationary while B molecules are moving with a velocity distribution described by the Boltzmann distribution at temperature T.

B

A

From the picture, we see that B, while traveling across the gas for a time ∆t, may collide with all the molecule whose centre of mass is inside a cylinder of radius d=r

A+r

B, and of height ∆t.

The number of A molecules inside this volume (hence the number of collisions) is given by their concentration (C

A=N

A/V) times the

volume of the cylinder:σ=cross sectionv

�

t � r A

� r B2 C A

� v

�

t � C A

v �

m B2 � k B T

3

�

2 �

d v v e

� m B v2

2k B T � 4 �

m B2 � k B T

3

�

2 �

d v v 3 e

� m B v2

2k B T �

8 k B T

� m B

1

�

2

Substituting in the previous formula and dividing by the time span to get the number of collision for unit time, one gets

There are, however, two adjustments that we should make to this equation: first, A molecules are not standing still; second, the above formula gives the frequency of collision for a single B molecule, but the system contains many more B molecule.To correct for the first point, it is enough to use the reduced mass of A and B (1/µ

AB=1/µ

A+1/µ

B) that comes from considering the “relative velocity” of the two

species; for the second point, it is enough to multiply the above formula by the concentration of B to obtain the collision frequency density (for unit volume):

This is similar to the familiar expression for the rate of a bimolecular reaction as a function of the concentrations of A and B; in other words, we can write k

AB as

(assume σ=12.5*10-20 m2, T=300 K, µAB

=100*10-3 kg/mol -> 3.15*10-17 m3 s-1

z � � C A8 k B T

� m B

1

�

2

Z AB

� � C A C B8 k B T

��B

1

�2

k ABkin � � 8 k B T

��

B

1

2

Moving away from simple collision theory- activated and oriented collisionsMoving away from simple collision theory- activated and oriented collisions:the equation just derived indicates that, assuming a reactive event every collision, the rate of the reaction should increase with T following a T1/2 law. This is far from being correct, as shown in the following experimental results for the reactions

Indeed, the semi-logarithmic plot in the figure indicates that the rate constants increase following an exponential behavior against 1/k

BT, which is not present in our

“naive” prediction using simple collision theory.

CH 3

� O2

� CH 3O

� OCH 3

� O2

� CH 2 O

� OH

Arrhenius proposed that the difference in behaviour could be explained by assuming that reactive encounters take place only between molecules that posses at least a minimum amount of energy relative kinetic energy E

A; the fraction of

molecules that posses such amount of energy is proportional to exp[-E

A/k

BT] (prove it!). The slow

change proportional to T1/2 is hiddenby the fast variation due to the exponential term over small temperature ranges.

With this assumption, one can measure the rate constant for a reaction at various T and extract the activation energy from a semi-logarithmic plot as the one shown before (mmmhhh, what we really want to do is to compute the activation energy and predict the rate, but let us explore this possibility first).There is still an ingredient missing, which is the molecular cross section. A simple way to estimate this quantity is to use information coming from transport measurements that give an indication of how large is a molecule.Let us assume for the time being that we posses all these pieces of information and let us compute the rate for a given reaction to compare with experimental results. In doing do we would discover that our predicted rates are always too large with respect to the experimental ones.This fact is usually explained using the concept of oriented collision: a collisional a collisional event can be reactive if and only if the two molecules are oriented in a suitable event can be reactive if and only if the two molecules are oriented in a suitable wayway (e.g. an exchanging atom should found itself facing the incoming acceptor molecule). Very often, this spatial requirement is quite demanding (think about a relatively large molecule that has many possible conformations), so that the ratio between predicted and measured constants can be quite large. In other words, it seems quite difficult to make rate constant predictions unless we explore the fundamentals of the dynamics involved in the reactive encounter. This is the topic of the remaining part of the lecture.

The dynamics involved in the reactive encounterThe dynamics involved in the reactive encounter:let us now consider the global PES over which the nuclei of molecule A and molecule B move assuming that (magically?) we know its analytical form as a function of the nuclear positions. It is generally possible to separate the phase space of this composite system indicating a region that describes the R's and a region that describes the P's.

The blue line in the PES V2 shown on the side

represents the surface separating the P from the R for the dissociation reaction

In this case the separatrix surface is locate in such a way that it passes through the TS of the PES, a natural choice whenever a TS is present.Intuitively, one could compute the rate for this transformation by running MD trajectories on the PES with initial condition such that they represent the R (e.g. position of the molecule away from the surface with the H-H distance around 0.8 A), and by counting the number of trajectories per unit time that end up being on the P side. The idea is clear, let us now sort out the details.

H 2�

Cu surfaceH � H

R

P

R P

We begin by assuming that the state of the R is described by an (almost) equilibrium situation defined by the proper statistical ensemble (it is not necessary to specified what is the ensemble right now, because the ideas are quite general, but for the sake of discussion let us say that it is the canonical ensemble).We also assume that we can find a property of the system q that is able to discriminate between R and P (it may be a distance, an angle or a linear combination of geometrical variables) so that we can say:

� if qq* we have P

q*q

Using the reaction coordinate q, we can define the R and P region of space using the Heavyside function Ξ(x):

� Ξ(x)=1 if x>0

� Ξ(x)=0 if x

We have already stated that we assume to be able to integrate the equations of motion for the system under study; in other words, we have access to all the dynamical information needed to specify the state of the system at any time through the trajectory [x(t),p(t)]. Let us now define a new time dependent quantity, the conditional probability to find the system in region P at time t assuming it was in region R at time 0

Here, the angular brackets indicate that we are computing this expectation value over all the possible instances of [x(0),p(0)] provided by the probability distribution function of the statistical ensemble we are simulating. Also, remember that [x(0),p(0)] are sampled inside the region R. Again, the correlation function is computed by starting trajectories from R and accumulating statistics about the time in which the trajectories end up being in P.This correlation function give us information about the total probability of finding the system in P, but we are instead interested in computing its time derivatives which will give the rate of changerate of change of this probability, i.e. the k

RP reaction rate.

C t ��

R x 0 , p 0

�

P x t , p t

�

R x 0 , p 0

�

�

R 0�

P t

�R 0

k RP

� dC tdt

�

�

R 0d

�

P t

dt

�

R 0

Using the identity

(remember, time correlation functions depend only on the time difference for an equilibrium system), we get

The time derivatives in the previous equation can be easily rewritten using the chain rule as

so that one obtains

What is the meaning of this equation? It suggest that we should sample our statistical ensemble and start trajectories whenever we have q=q*; we can then measure the flux of particles (the number of particle times their velocity along the reaction coordinate) that end up being in P at time t having started at q=q* at time 0.

A t A t ' � A 0 A t ' � t � A t � t ' A 0

�

R 0d

�

P t

dt

� � d

�

R 0

dt

�

P t

d

�

q 0dt

� d q 0dt

� �

q 0

�

q 0�� d q 0

dt

q � q

k RP

�

d q 0dt

�

q �� q 0 � P q t�

R 0

It is important to notice that the rate constant so computed is formally time-dependent. In other words, it value can depend on the chosen t, in particular at short time when the dynamics of the transformation (i.e. the behaviour of the trajectories) is important. Instead, our macroscopical kinetic model for a reaction requires that the rate constant does not change with time, whatever is concentration of R and P.This discrepancy can be easily resolved by taking into account that one should compute a reaction rate only after the initial transient behaviour has relaxed (i.e. t must be larger than the microscopic time require to relax or surmount the barrier).

The picture represents the behaviour of the time derivative of the correlation function C(t) for a simple model composed by 7 particles. The oscillations present in the plot indicate changes in slope of C(t) as a function of time, i.e. changes in the flux of particles going from R to P. Eventually, for t large enough, the oscillations disappear and the time derivatives reaches a stationary value (a plateau) that we associate with the reaction rate. Notice that at very short time the rate is larger than its limiting value.

Details of the dynamics on a given PESDetails of the dynamics on a given PES:the calculation of the correlation function C(t) using trajectory simulations provides us with the possibility of predicting the rate of a chemical reaction. However, it does not provide us with a clear understanding of what is actually happening on the surface during the dynamics. To get such insight, it is more useful to analyze trajectories one by one, in order to extract possible patterns of behaviour. An even better approach is provided by selecting one of the possible variables involved in the process (e.g., the relative kinetic energy, the amount of vibrational energy in a mode, the angle of impact, etc) and to study the dependence of the reaction rate on this variable. This allows us to infer more about the dynamical features of the process and, for instance, to control the efficiency of the process by selecting dynamical variables in the right range.

A relevant question that one may wish to answer is:given a total amount of energy to be partitioned given a total amount of energy to be partitioned among the various degrees of freedom, is it among the various degrees of freedom, is it better to have more energy in the relative better to have more energy in the relative translation of the two species or in a specific translation of the two species or in a specific vibrational modevibrational mode?

As an example, we shall consider the case of the collinear reaction between Br and H

2 , whose PES is

presented in the picture aside.

Details of the dynamics on a given PESDetails of the dynamics on a given PES:the calculation of the correlation function C(t) using trajectory simulations provides us with the possibility of predicting the rate of a chemical reaction. However, it does not provide us with a clear understanding of what is actually happening on the surface during the dynamics. To get such insight, it is more useful to analyze trajectories one by one, in order to extract possible patterns of behaviour. An even better approach is provided by selecting one of the possible variables involved in the process (e.g., the relative kinetic energy, the amount of vibrational energy in a mode, the angle of impact, etc) and to study the dependence of the reaction rate on this variable. This allows us to infer more about the dynamical features of the process and, for instance, to control the efficiency of the process by selecting dynamical variables in the right range.

Br + HBr + H22 -> HBr + H -> HBr + H:

we start with considering the above reaction. Let's assume that the hydrogen molecule does not have any vibrational energy (at least, no more than the ZPE) and that all the available energy is put in th relative motion of the two species. A typical trajectory for this situation is provided by the green line in the picture aside.

Because of the shape of the PES, the trajectory does not feel any steering force driving it toward the TS, so that the most likely outcome is to bounce back from the repulsive wall at short Br-H distances.Conversely, the presence of some vibrational energy in the relative motion of the two H's may help to surmount the barrier as shown by the red line. It should be made clear that not ALL the trajectories having vibrational energy will pass; the probability for this to happen depends on the phase of the vibration which should be averaged over.

H + HBr -> Br + HH + HBr -> Br + H22:

it is easy to guess that the reverse reaction may behave in different way. The TS region can be easily reached by an almost linear trajectory on the bottom of the PES valley are shown in the picture below (blue line).

Indeed, it is also seen that it is likely for the hydrogen molecule to leave the reaction region with some of the relative kinetic energy transformed in vibrational energy. In other words, the newly produced molecules are vibrationally hot, a feature that can be detected experimentally.Because of the bottleneck shape of the PES (also indicated by saying that the PES is attractive), it is likely for a vibrationally excited HBr molecule to be “bounced back” with no reaction (purple line).

The barrier-less case of HThe barrier-less case of H22O + DO + D

33OO++ -> D -> D

22O + HO + H

22DODO++:

occasionally, reactions take place on a PES with no barriers and with very deep wells, such as the case of the title reaction. What happen in these cases is very different. First of all, the presence of a deep well suggest the possibility that a long lasting intermediate is formed after the collision between the two reactants.

Because of the (attractive) shape of the PES, the two reactants are always attracted together, so that the amount of vibrational energy stored in specific modes should not be particularly important. What is likely to happen is that a trajectory would be trapped in the PES well for some time if its relative kinetic energy is not large enough to escape the attraction.During the lifetime of the complex, the reaction may take place as shown in the picture aside. Of course, it is also possible for the complex to dissociate back into the reactant (in this case, if the lifetime is long, there is a 50:50 chance of coming back with no reaction).

Transition State Theory – the short time approach to reactive flux calculationTransition State Theory – the short time approach to reactive flux calculation:in the previous section, we have derived the statistical mechanics equation needed to compute the reactive flux (the rate constant) for a chemical reaction starting from the assumption that we could sample the probability distribution for our system and that we can integrate enough trajectories to collect enough samples to have a set of representative transition from R to P. However, this may be a computationally expensive task, especially if the system contains many particles, there is an high barrier to be surmounted, or if the cost of integrating a single trajectory is high.To circumvent these difficulties, an approximation can be introduced in the exact formula derived above, and this is made substituting Ξ

P(q(t)) with

By making this substitution, we are implicit assuming that ALL the trajectories started from the separatrix with a velocity along the reaction coordinate pointing in the direction of the products are going to reach the P region and stay there for a long time. Also, ALL the trajectories initially moving toward R will not contribute to the reaction rate. In other words, we assume that the faith of a trajectory is defined only by its initial velocity along the reaction coordinate.

�

PTST q t � 1 if

d q 0dt

� 0

�

PTST q t � 0 if

d q 0dt

� 0

goes toward P

goes toward R

Introducing the above equations, the approximate formula for the reaction rate reads

which provides us with an alternative approach to rate constant calculation. It suggest that we should sample the correct (equilibrium) probability distribution function for our system and that whenever q=q* we should collect the flux (number of samples times their velocity along the reaction coordinate) of trajectories that would be started with a velocity directed from R to P. This avoids us to integrate the trajectories, and hopefully would provide with rate constant values that are accurate enough at a cost similar to the one of an equilibrium calculation. The result presented above is usually referred to as the Transition State Theory (TST) reaction rateTransition State Theory (TST) reaction rate, the reason being that often the separatrix passes through the TS separating R from P.

Can we say anything about the accuracy of this approximation?Indeed, it is possible to state on general ground that TST rate constants are larger than their dynamical counterparts. This is due to the fact that a trajectory, although started in the right direction (from R to P), may recross the separatrix at later time due to dynamical effects. This is not taken into account by the TST equation and it is strongly system-dependent. In other words, it is difficult to predict beforehand how much larger the TST rate is going to be. However, it is not necessary to have the separatrix passing through the TS, and its position can be varied to variationally minimize the TST rate (VTST).

k RPTST �

d q 0dt

�

q � � q 0

�

PTST q t

�

0

For the cases in which a numerical comparison is possible, the TST theory is usually found to provide accurate values, being generally off by a factor of 2-3 with respect to the dynamical rates. A good example is provided by the dissociation of the acetyl radical.

We see that, over all range of energies explored, the dynamical and TST (statistical) rates agrees reasonably well. It is also seen that the agreement deteriorates slightly at higher energies. This is due to the fact that TST assumes the system to be always at equilibrium (i.e. of being distributed accordingly to the ensemble distribution function). Conversely, the dynamical approach makes a similar assumption only for the initial condition of the trajectories.In this case, trajectories that are fast and/or start very close to the separatrix would dissociate rapidly, bringing the system away from equilibrium (the corrisponding regions of phase space will be less densely populated). Since the re-population of those regions requires some time, the dissociation process is slowed down because the system has no dissociating trajectories available.

Notice that, although quite simplified, the evaluation of the TST expression for the rate constant may still be quite demanding numerically. What we ought to do now is to make some other approximation so that, perhaps, the calculation of the reaction rate could be carried out using only local information on the PES. If that were possible, in principle we could find some strategy based on ab initio calculations to predict the rate of reactions.To do that, we need to rewrite the equation for the TST rate constant. We start by noticing that the Heavyside function can be eliminated if the integration over the velocity along the reaction coordinate was carried out only from 0 to infinity (i.e. on the part of speed distribution that points in the right direction). Making the assumption that we are working with the canonical ensemble, this allows us to write

In the above equation, the first expectation value is the average value of the velocity along the reaction coordinate (this depends on the statistical ensemble), whereas the second term is a ratio of configurational partition function between for the free system (well, it is constrained to stay in R anyway) and the system constrained to have q=q*.Here comes the fundamental assumption: we assume that the potential energy surface of both the free system and the constrained one are well described by a quadratic expansion around two different stationary points, the global minimum and the TS, respectively.

k RPTST � 1

2

d q 0dt

�

q � � q 0

�

0� 1

2

d q 0dt

�

q � � q 0

�

0

If we select the coordinates so that, locally, the quadratic form is diagonal (normal modes analysis)

(notice that both coordinates and frequencies used in the two equations are different), the two coordinate dependent terms can be easily integrated to give

How many frequency do we have? It depends on the system. If we are in gas phase, N is three times the number of atoms minus three translational and three rotational degrees of freedom. Other way, N is three times the number of atoms in the system.

There is another possible approach to derive a TST-Harmonic rate constant, this being quite useful in studying bi-molecular reactions. We start from the original equation, recognizing that the R state is represented by two non interacting molecules A and B, so that

where Q is the vibrational-rotational configuration partition function and V the volume.

V � V TS ��

i � 1, N � 112

2 � � ' i2 q ' i

2

V � V Min ��

j � 1, N12

2 � � j2 q j

2

k RPTST � Harmonic � j

1, N

j

i 1, N � 1

' ie

� V TS� V Min

k B T

�

R 0

� V 2 Q A QB

Similarly, for the TS one can write:

where the expectation value is computed over one less degree of freedom, the latter being constrained to be q=q*. The TST equation then becomes

The remaining expectation value is a simple mono-dimensional integral whose value can be easily compute obtaining

�

q � � q � V QTS

k RPTST � 1

2

d q 0dt

QTSVQ A QB

k RPTST �

k B T2 �

QTSVQ A QB

Monomolecular transformations - Energy-dependent rate constantsMonomolecular transformations - Energy-dependent rate constants:in the previous sections, we have developed the vast majority of the mathematical machinery which is necessary to describe and predict gas phase kinetics. Insofar, this machinery has been applied (and specialized) only to bimolecular reactions, and it is now the time to tackle the topic of mono-molecular reactions.Experimentally, it is seen that the decomposition of a metastable system (A) in a diluent gas (M) follows a first order kinetic equation of the form

The explanation of the mechanism for this kind of reaction was proposed by Lindemann and Hinshelwood, and it is based on the three reactions scheme:

Assuming steady state for the activated species A', the rate law reads

d Adt

� � k mono A

A � M � A ' � M d A 'dt

� k act A M

A ' � M � M � A d A 'dt

�� k deact A ' M

A ' � P d A 'dt

�� k dec A '

d Pdt

� k act k dec A M

k deact M

� k dec

�

k deact M

k dec

k act k dec A

k deact

This result indicate that, to predict the rate of product formation, one needs to be able to predict the rate at which the energized complex decomposes or transform Differently from what is usually done in bi-molecular reactions, here the emphasis is posed on the energy dependency of the reaction rate; this is due to the fact that it is not easy to define the temperature of the energized species, while it is less complicate to define the amount of energy accumulated during the collision with the bath gas M.During the derivation of the TST rate equation from the correlation function formalism, we made no reference to the statistical ensemble. In other words, the result

is equally valid for the canonical and microcanonical ensemble, and it is just a matter of introducing the appropriate distribution function. Specializing the above equation to the case of the microcanonical system, one gets

where we recognize that the denominator is proportional to the number of states available to the reactant at energy E, whereas the numerator gives the TST flux for that energy.

k RPTST �

d q 0dt

�

q � � q 0

�

PTST q t

�

0

k decTST E �

�

Rd R d P

�

E � H R , Pd q 0

dt

�

q � � q

PTST q t

�R

d R d P

�

E � H R , P

The numerator has also another interpretation, which is occasionally more useful when it comes to predicting the energy dependency of the rate constant. Assuming that we can write a new set of coordinates {q

i, i=1,F(=3N)} so that one of these (q

F)

measures the distance from the separatrix, the Hamiltonian function can be re-written including explicitly the reaction coordinate and the equation becomes

where qF=0, R'=(q

1, q

2, q

3,...,q

F-1) and P' is the corresponding vector of linear

momenta. In other words, the numerator gives the number of state accessible to the system when it is sitting on the separatrix, and the rate constant is fundamentally given by the ratio between the number of state at the TS and the total number of state in R.Again, one can say that the probability for unit time of leaving the R phase space is proportional to the probability for the system to occupy states belonging to the TS region, an intrinsically statistical results. This result is usually indicate as the Rice-Ramsperberg-Kassel-Marcus (RRKM) theory.An advantage of the previous equation is that it gives the opportunity introduce the quantum description of the reaction; it is simply necessary to estimate the number of quantum states available to R and TS. Of course, this requires the solution of two Schroedinger equations and it can be a very difficult task.Comments similar to the one made for the bi-molecular case apply to this case as well. In other words, microcanonical TST is fairly accurate if no approximations are made in the evaluation of the number of states for R and TS.

k decTST E �

�

Rd R ' d P '

�

E � H R ' , P '

2 ��

Rd R d P

�

E � H R , P

Electron transfer rate in condensed phase – Marcus' TheoryElectron transfer rate in condensed phase – Marcus' Theory:a very important process in many branches of chemistry is represented by the transfer of one electron between two species (a redox reaction) in condensed phase. Usually the species are ions, around which the solvent molecules are distributed to minimize the free energy of the species (e.g. is the ions where Fe2+ and Fe3+, water molecules would be oriented to present the negative side to the ion).

e

ε (ω) Because of the mass difference between the transferring electron and the nuclei involved in the process, it is sensible to consider that the process takes place following the Franck-Condon condition, i.e. with a frozen nuclear geometry. In other words, no time is left to the solvent molecules to reorganize during the process.If the transfer happened with the solvent molecules distributed with an equilibrium configuration, a large amount of energy would be necessary to send the electron from the donor to the acceptor, and this is due to the “out of equilibrium” solvation of the systems end state (e.g. the newly created Fe3+ would find itself surrounded with a less than optimal distribution of water molecules). Because of the large energy requirement, the transfer will be an extremely rare event.

φR(t)

The polarization of the solvent molecules, however, fluctuates with time due to thermal effects, therefore somewhat reducing the stabilization of the ions in solution and increasing their free energy.Marcus proposed to use a collective reaction coordinate to describe the solvent fluctuations around an ion, and that the change in free energy due to the fluctuations could be described by a quadratic form (a parabola) of the reaction coordinate itself.

Notice that for this idea to be applicable the two species involved in the redox process need not to share any common ligand.Marcus also showed that it is possible to use as reaction coordinate the amount of charge transferred from one species to the other, but the formulation considering the organization of the solvent molecules is more effective.In order to minimize the energy requirement (thus increasing the probability) of the transfer, the fluctuations of the solvent polarization around the ions

G n

� eq q �

12

Aq�

q 2

Charge transfered

Non

-equ

ilibr

ium

fre

e en

ergy

ωs2

δ q

Reactants

Products

Free

ene

rgy

Nuclear coordinateXR XPXC

In order to minimize the energy requirement (thus increasing the probability) of the transfer, the fluctuations of the solvent polarization around the reactant ions should be large enough to intersect with the free energy curve of the products. At the intersection point, the free energy of R and P are identical and the system can “slide” down the free energy surface generating P with no other energetic costs. At the crossing point, the electron is “delocalized” between the two species, and thermal fluctuations of the nuclear positions will decide which species (R or P) is generated.

The energy curves for the R and P are given by:

and one is left with the task of finding their intersection QC.

Ga

∆G0

G R

� G Req � 1

2Aq Q � Q R

2

G P

� G Peq � 1

2Aq Q � QP

2

The results are

The last equation gives the free energy difference between the equilibrium solvation of R and the non-equilibrium solvation at the crossing point. It is convenient to the to introduce a new quantity, the reorganization energy

which gives the free energy difference between the equilibrium solvation of P and a “R-like” solvated P. Using this quantity, the free energy difference is written as

which represent the energy barrier that needs to be surmounted for the electron exchange to take place. Marcus also suggested the reaction rate to be given simply by

where χ is the transmission coefficient that depends on the details of the electronic structure of the species.

QC

� 12

QP

� Q R

�

�

G 0Ad QP � Q R

G a

� 12

Aq12

QP

� Q R

�

�

G 0Ad QP � Q R

2

� � 12

Aq QP � Q R2

G a

�

� � � G 02

4�

k et

e

�� � G 0

2

4

�

The magnitude of the activation free energy Ga depends on the relative position of the

free energy minima for the two species. Depending on ∆G0 and on XP-X

R, three

different situations are possible:

∆ G0

Normal: increase of ∆ G0decreases the rate

No activation barriermaximum kET

Inverted: decrease of ∆ G0 decreases the rate

invertednormal

In reality other factors also play a role inthe electron transfer problem: distance between donor and acceptor, diffusion can be rate limiting step. The invertedregime has only been observed in rigidsystems, such as proteins.