8 Triplet-Energy (and Electron) Transfer Rates

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orders are more sensitive to bond-order changes than large bond orders. For the same reason, the product nav|loxϪlred| must be divided by the sum of the equilibrium bond lengths.

The success of this procedure of reducing the effective displacement to a common reference state can be assessed from the average and standard deviation of the reduced bondlength changes: 0.106 Ϯ 0.016. A similar procedure was followed by the ISM to obtain the

reduced bond-length changes of bond-breaking–bond-making reactions. It was shown in

Chapter 7 that the fundamental equation obtained for isothermic processes was, eq. (7.9),

deff =



aЈln ( 2)

n



‡



(l D + l A ) =



0.108

(l D + l A )

n‡



(15.135)



where n‡ is now nav, because no bonds are broken in the electronic transitions involved in

Tables 15.1 and 15.2.

The scaling factor aЈ ϭ 0.156 of hydrogen-atom and proton transfers seems to be transferable to triplet-energy and electron transfers. This is not totally unexpected because the

reference system chosen for bond-breaking–bond-making reaction, H ϩ H2, does not have

significant steric effects and the resonance at the transition state is minimal. These are also

properties of triplet-energy and electron transfers. The effective displacements calculated

with eq. (15.135) are presented in Table 15.2 and compared with the spectroscopic data in

Figure 15.23. For the systems selected for this test, the performance of eq. (15.135) is particularly encouraging. However, it must be emphasised that these systems were selected



Figure 15.23 Correlation between the effective displacements calculated by the ISM and the experimental data on the diatomic molecules of Table 15.2.



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15. Transitions between Electronic States



for their adherence to Pauling’s relation in discrete increments of 0.5 in the bond order. For

3 −

example, the electronic transition from the Σg state to the 1⌬g state of molecular oxygen,

shown in Table 15.1, does not involve the same type of change in bond order and its effective displacement cannot be calculated from eq. (15.135).

The generalisation of this method to polyatomic systems is straightforward. For example, benzene formally has three double bonds and three single bonds, that is n ϭ 1.5. Its

cation has one bonding electron less and the total bonding is reduced to n ϭ 1.42, thus

nav ϭ 1.44. The CC bond length of benzene is 1.397 Å and normal mode analysis gives

f ϭ 3.87ϫ103 kJ molϪ1 ÅϪ2. With these values eq. (15.135) gives deff ϭ 0.207 Å and (benzene) ϭ 82.7 kJ molϪ1. Another example is TCNE, with its four triple bonds, one double

bond and four single bonds, which lead to n ϭ 2.0. Its anion has n ϭ 1.94 and, consequently,

nav ϭ 1.97. With the average bond length of 1.301 Å for the neutral and 1.291 Å for the anion,

and the corresponding average force constants of 6.42ϫ103 and 6.19ϫ103 kJ molϪ1 ÅϪ2,

respectively, the reorganisation energy for this redox couple is (TCNE) ϭ 63.7 kJ molϪ1.

The charge transfer in the HMB/TCNE charge-transfer complex should be intermediate

between these two values. As mentioned before, resonance Raman spectroscopy suggests

that the total reorganisation energy is between 45 and 56 kJ molϪ1 for this system, which

may be increased upon consideration of the role of the CH bonds. Again, this is an encouraging agreement between eq. (15.135) and the experimental data.

The heuristic validation of eq. (15.135) for the calculation of effective displacements for

triplet-energy and electron transfers under the specified conditions has interesting consequences. The isomorphism of eqs. (15.135) and (7.9), suggests that the energy dependence

of the effective displacement should have the form of eq. (7.8), or, more precisely,



deff



(

(



⎧

‡

0

aЈ ⎪ 1 + exp 2 n E

⎪

= ‡ ln ⎨

2n

⎪1 − ⎡1 + exp 2 n‡ E 0

⎪ ⎢

⎩ ⎣



)

)⎤⎥⎦



⎫

⎪

⎪

l + lA )

−1 ⎬ ( D

⎪

⎪

⎭



(15.136)



where is a parameter that accounts for the coupling of the reaction energy with vibrational

modes other than the high-frequency mode selected for the reaction coordinate. An empirical choice of , guided by the best fit to the energy dependence of a series of reactions,

gives the correct reorganisation energy and, together with the high-frequency mode and the

electronic coupling, leads to good estimates of the reaction rates. A detailed interpretation

of eq. (15.136) is postponed to Chapter 16, where it is applied to electron-transfer reactions.

For the moment, it is just emphasised that deff increases with ⌬E0, that is, the reorganisation energy tends to increase with the driving force of the reaction.

The most convincing argument for the relation between molecular structure, spectroscopy and kinetics developed in this chapter, is to calculate triplet-energy transfer rates

using the Franck–Condon factors and electronic couplings given by the molecular models.

An interesting system to test these relations is the triplet-energy transfer from a biacetyl

molecule trapped inside a hemicarcerand cage to aromatic hydrocarbons in free solution.

This system has two very convenient features: first, biacetyl is one of the few organic molecules phosphorescing at room temperature and second, it is possible to study experimentally an extended range of reaction energies [53,54]. The fluorescence of biacetyl overlaps



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433



Figure 15.24 Biacetyl phosphorescence and anthracene triplet excitation spectra. The simulated

spectra (dotted lines) used ED ϭ 19,300 cmϪ1, v D ϭ 1470 cmϪ1, D ϭ 480 cmϪ1 and SD ϭ0.35 for

biacetyl, and EA ϭ 14,720 cmϪ1, v A ϭ 1390 cmϪ1, A ϭ 280 cmϪ1 and SA ϭ 1.25 for anthracene.



Figure 15.25 Rate constants for the triplet energy transfer from biacetyl, through a hemicarcerand

cage, to aromatic hydrocarbons in solution. The full line represents the Golden Rule calculations and

the other lines the ISM calculations differing in the parameters shown in the plot.



with its phosphorescence in steady-state emission spectra, but registering the spectrum

with a 10 sec delay gives enough time for the biacetyl fluorescence to disappear, as illustrated in Figure 15.24 [27]. This figure also shows the triplet-excitation spectrum of

anthracene, already previously shown in Figure 15.11. The simulation of this spectrum

requires at least two modes, with frequencies 1390 and 380 cmϪ1, but only the single,

high-frequency approximation for each reactant will be pursued here.

Golden Rule calculations using eq. (15.101) with the reorganisation energy given by

eq. (15.100) and the parameters indicated in Figure 15.24 give the rates presented in



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15. Transitions between Electronic States



e

Figure 15.25 when the electronic coupling is fitted (Vks ϭ 0.1 cmϪ1) to reproduce the

fastest experimentally observed triplet-energy transfer rate from biacetyl, through a hemicarcerand cage, to an aromatic hydrocarbon in solution. The Golden Rule reproduces the

general energy dependence of the rates, although this formulation tends to overestimate the

rates in the range of reaction energies where the energy-gap law is followed. The same figure also presents calculations with the ISM using as an adjustable parameter. The molecular parameters of the aromatic hydrocarbon are those of benzene discussed above, and for

biacetyl the following parameters were employed: f ϭ 5.91ϫ103 kJ molϪ1 ÅϪ2, l ϭ 1.30 Å,

n ϭ 1.58 [27]. With ϭ 90 kJ molϪ1 the ISM reproduces the Golden Rule calculations, but

a lower value of yields a better agreement with the experimental data. The implications

of these results for electron transfers are discussed in the next chapter.



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SJ Formosinho, LG Arnaut, Adv. Photochem. 16 (1991) 67.

EG Azenha, AC Serra, M Pineiro, MM Pereira, J Seixas de Melo, LG Arnaut, SJ Formosinho,

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SJ Formosinho, J. Chem. Soc., Farad. Trans. 2:72 (1976) 1313–1331.

JB Birks: Photophysics of Aromatic Molecules, Wiley, London, 1970.

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JJ Hopfield, Proc. Natl. Acad. Sci. USA 71 (1974) 3640–3644.

C Serpa, LG Arnaut, SJ Formosinho, KR Naqvi, Photochem. Photobiol. Sci. 2 (2003) 616.

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RA Marcus, Farad. Discuss. Chem. Soc. 29 (1960) 21.

JP Claude, KM Omberg, DS Williams, TJ Meyer, J. Phys. Chem. A 106 (2002) 7795–7806.

R Schmidt, J. Photochem. Photobiol. A Chem. 80 (1994) 1–5.



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MJ Lang, XJ Jordanides, X Song, GR Fleming, J. Chem. Phys. 110 (1999) 5884–5892.

SA Kovalenko, R Schanz, VM Farztdinov, H Hennig, NP Ernsting, Chem. Phys. Lett. 323

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AB Myers, Chem. Rev. 96 (1996) 911–926.

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BM Britt, JL McHale, DM Friedrich, J. Phys. Chem. 99 (1995) 6347–6355.

M Bixon, J Jortner, J Cortes, H Heitele, ME Michel-Bayerle, J. Phys. Chem. 98 (1994)

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– 16 –

Electron Transfer Reactions



16.1



RATE LAWS FOR OUTER-SPHERE ELECTRON EXCHANGES



Electron transfers are another prototype of chemical reactions as paradigmatic as H-atom

or proton-transfer reactions. Furthermore, these are chemical processes ubiquitous in

chemistry and biology. They are important from a historical point of view in the development of chemical kinetics, and are scientifically relevant in many physical, chemical, biological and technological processes.

The field of electron-transfer reactions in solution has only developed since World War

II and initially was centred on inorganic processes. Two kinds of mechanisms are invoked

for ET reactions in transition-metal complexes. One, the inner-sphere mechanism, where

the two reactants share one or more ligands in their first coordination shell in the activated

complex and the electron transfer is considered to occur via the bridging ligand. This

requires bond breaking followed by bond forming. The other is the outer-sphere mechanism, where the first coordination shells of the two reactants remain intact with respect to

the number and kind of ligands present. The latter mechanism is the focus of this chapter,

because bond-breaking–bond-forming reactions have already been extensively discussed

in this book.

Let us consider the isothermal process for the transfer of one electron between hydrated

iron (II) and (III) species

d ⎡ * Fe 3+ ⎤

⎣

⎦

dt



= k2 ⎡ Fe 3+ ⎤ ⎡ * Fe 2+ ⎤ − k −2 ⎡ Fe 2+ ⎤ ⎡ * Fe 3+ ⎤

⎣

⎦⎣

⎦

⎣

⎦⎣

⎦



(16.I)



This is a self-exchange reaction, where the products are identical to the reactants. For it to

be followed experimentally, the iron species normally are marked radioactively (represented by *). If the reaction is followed, for example, in terms of *Fe3ϩ , the rate law can

(aq)

be written as:

d ⎡ * Fe 3+ ⎤

⎣

⎦

dt



= k2 ⎡ Fe 3+ ⎤ ⎡ * Fe 2+ ⎤ − k −2 ⎡ Fe 2+ ⎤ ⎡ * Fe 3+ ⎤

⎣

⎦⎣

⎦

⎣

⎦⎣

⎦



(16.1)



by simplifying the representation of the ionic species in terms of the chemical symbol of

the metallic element and of the electric charge. Since second-order rate constants in the

437



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438



16. Electron Transfer Reactions



forward and reverse directions are equal, k ϭ k2 ϭ kϪ2. If we represent the initial concentration of reactants by a and the concentration of products by x, we can write

dx

2

= k ( a − x ) − kx 2

dt



(16.2)



dx

= ka 2 − 2 kax

dt



(16.3)



dx

= ka ( a − 2 x )

dt



(16.4)



dx

= ka dt

a − 2x



(16.5)



which can be rearranged to



or



As a consequence, we have



This equation can be integrated



∫



x



0



t

dx

= ∫ ka dt

0

a − 2x



(16.6)



The result requires knowledge of the value of the standard integral



∫



x



0



dx

1

= ln ( a + bx )

a − bx b



(16.7)



and thus

1

− ln ( a − 2 x ) = kat + constant

2



(16.8)



The value of the constant of integration is obtained by solving eq. (16.8) for t ϭ 0,

where x ϭ 0,

ln



a − 2x

= −2 kat

a



(16.9)



or

a − 2x

= exp ( −2 kat )

a



(16.10)



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Figure 16.1 Change in the concentration of the products with time in a reaction following mechanism 16.I. If the concentrations of the two reactants are identical, the concentration of the products

tends to half of the initial concentration of the reactants.

Table 16.1

Rate constants of electron self-exchange reactions in transition-metal

complexes, measured in water at room temperaturea

Metal complexes



k (molϪ1 dm3 secϪ1)



Fe(OH2)63ϩ/2ϩ

Fe(CN)63Ϫ/4Ϫ

Cr(OH2)63ϩ/2ϩ

Co(en)33ϩ/2ϩ

Fe(phen)63ϩ/2ϩ

Co(bpy)32ϩ/ϩ



1.1

19

Յ2 ϫ10Ϫ5

7.7 ϫ10Ϫ5

3.3 ϫ108

2 ϫ109



Data from ref. [1]. Ligands: en ϭ ethylenediamine; phen ϭ 1,10-phenanthroline; bpy ϭ

2,2Ј-bipyridine.

a



Figure 16.1 displays the profile of the concentration of the products as a function of time

in a self-exchange reaction, according to eq. (16.9).

Table 16.l lists typical values of rate constants for self-exchange reactions of transitionmetal complexes with different kinds of ligands in aqueous solutions; the rates cover 14

orders of magnitude. The collection of the vast majority of such data became experimentally feasible only through the availability of several metal radioactive isotopes due to the

Manhattan project. The surprising result for the chemists at the end of the 1940s and beginning of the 1950s of the twentieth century was the slowness of many such processes.

In contrast, in the beginning of the 1980s, Gray [2] and collaborators showed that

electrons can be transferred in proteins over distances of 15–20 Å on a biologically

relevant timescale of milliseconds to microseconds. This appears to be consistent with



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16. Electron Transfer Reactions



a significant effect producing slow rates for electron transfers due to bulk solvent.

Anticipating an alternative view, one should be aware that the electronic nature of the

reactive bonds in hydrated transition-metal ions and in donor and acceptor complexes in

proteins are entirely different. As it will be shown in due course, such a direct comparison is therefore not meaningful from a theoretical point of view.



16.2

16.2.1



THEORIES OF ELECTRON-TRANSFER REACTIONS



The classical theory of Marcus



ET reactions between molecular species in cases where no bond-breaking–bond-forming

processes occur are usually considered to be very different in nature from atom-transfer reactions. For the latter case, we can use a Born–Oppenheimer adiabatic potential-energy surface

and there is a strong interaction with the attacking atom, which effectively determines the

barrier for the reaction. In contrast, for the former processes the interaction between the electron donor and its acceptor is very weak, the nuclear configuration of the products resembles

that of the reactants and, in retrospect with a certain naivety, chemists expected that ET

would in general be fast processes, and would mainly be diffusion controlled.

The current view is that the electron-transfer event itself is a fast activationless process;

the barrier for the reaction stems from the necessity to adjust the orientation of the solvent

dipoles around ions and the lengths of some bond in the inner-coordination shells prior to

the transfer step. According to this view, which was due largely to Rudolph Marcus [3],

for the solvent, and to Noel Hush [4], for the metal-ligand bond lengths, there are no

proper transition states in electron-transfer reactions, because the solvent molecules are

not in equilibrium distribution with the charges of the oxidised and reduced species.

The theoretical formalism proposed to estimate the rates of ET reactions is known as the

theory of Marcus (TM) [3,5,6]. However it is relevant to make a distinction between two

components of the theory. One component is concerned with the estimation of the intrinsic

barrier, G(0)‡, for homonuclear reactions in terms of molecular parameters of the reactants, which we will call TM-1. The other component of the theory addresses the effect of

the reaction energy, G0, on the reaction rates, presented in Chapter 7 in terms of the quadratic expression of Marcus (eq. (7.6)); this will be called TM-2. It is currently employed to

estimate the rates of cross-reactions, when the reaction energies of the heteronuclear reactions are known together with the rates of the corresponding homonuclear reactions.

In general terms, the theory of Marcus involves a model for ET reactions based on the

approximation that the inner-coordination sphere energy is independent of the outer-sphere

reorganisation. In its classical formulation, TM provides the rate for a self-exchange reaction such as reaction (16.I)



(



*

kMT = κ el Z exp ⎡ − wr + ΔGv + ΔGs*

⎣



)



RT ⎤

⎦



(16.11)



which involves collision frequency factor, Z (taken as Z ϭ 1011 MϪ1 secϪ1); el is an electronic transmission factor ( el ϭ 1 for adiabatic reactions and el Ͻ 1 for non-adiabatic



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Theories of Electron-Transfer Reactions



441



reactions). The reaction energy barrier is made up of three contributions: (i) a small electrostatic term, wr, the electrostatic work required to bring the two reactants together; (ii) an

external reorganisation of the solvent dipoles, Gs*; and (iii) an internal reorganisation of

the metal-ligand bonds in the inner-coordination shell, Gv*. Within the framework of

TM-1, the explicit formulae of these terms, taking energies in kJ molϪ1 and radii in Å, are:

wr =



ΔGs* =



e2 zox zred

⎡

8 N e I⎤

r ⎢1 + r

⎥

1033 RT ⎥

⎢

⎣

⎦

2 2

A



=



1389 zox zred

⎡

r ⎢1 + 50.3r

⎣



I ⎤

⎥

T⎦



⎛ 1 1 ⎞ ⎛ 1 1⎞

1

1 ⎞ ⎛ 1 1⎞

e2 ⎛ 1

− ⎟ = 347.3 ⎜ − ⎟ ⎜ 2 − ⎟

+

−

4 ⎜ 2rox 2rred r ⎟ ⎜ nD

⎝

⎠⎝ 2

⎝ rav r ⎠ ⎝ nD

⎠

⎠

*

ΔGv =



f f

1

2

cn ox red (lox − lred )

2 fox + fred

e2 =



2

e0 N A

4 0



(16.12)



(16.13)

(16.14)

(16.15)



In the above expressions e0 is the elementary charge, NA the Avogadro constant, 0 the

permittivity of vacuum, nD and are the refractive index and the static dielectric constants

of the solvent, respectively, I the ionic strength of the solution and r the distance between

the centres of the reactants in the collisional complex, assumed to be equal to

2ravϭ(roxϩrred)/2; fox and fred are the stretching vibrations for the two reactants in the oxidised and reduced forms, and lox and lred are the equilibrium lengths of the metal-ligand

bonds in the same species. The parameter r corresponds to the diameter of the spherical symmetric transition-metal complexes which can be estimated from the relation r ϭ (rx ry rz)1/3.

Finally, cn is the coordination number, i.e. the number of metal-ligand bonds involved in

the reaction coordinate.

By end of the 1950s, the experimental data on ET in transition-metal complexes was

insufficient for a reliable comparison with the theoretical ideas of Marcus and Hush.

However, by the beginning of the 1960s, Norman Sutin [7] had obtained a considerable

amount of kinetic data on such systems and was able to test the theoretical ideas available.

The idea of accounting for an internal reorganisation is due to N. Hush, but eq. (16.14) is

not, in fact, the expression originally proposed. Instead, Sutin employed another expression available from spectroscopy to account for the minimal energy rearrangement for

bond length changes and this expression was empirically found to be more convenient to

express Gv*.

The rates were calculated according to TM-1 assuming the following approximations:

(i) near adiabaticity ( el 1); (ii) dielectric continuum model for the solvent; (iii) intermediate distortions between oxidised and reduced species for the internal modes; and

(iv) separation between internal modes and medium modes (either high-frequency vibrational modes or low frequency). This last approximation is implicit in eqs. (16.12)–(16.15).

The separation between the inner-sphere, Gv*, and the outer-sphere, Gs*, reorganisation