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CHAPTER 1

Introduction

Charge transfer is present in the phenomenon of contact electrification at the very beginning of the history of electricity, with Thales of Miletus, as well as at the dawn of electricity as science, with William Gilbert. Triboelectricity, the contact electrification by tribocharging, was first observed, as far as we know, by Thales of Miletus when he rubbed amber with wool, the Greek word for friction being τρι′βoς and that for amber η′λεκτρoν (or η′λεκτρoς). Tribocharging is due, we today know [1], to ions or electrons transferring from one to the other of two surfaces in contact. In a sense, we can consider Thales’ tribocharging also as the first experiment in chemical kinetics because, as Marcus remarked, that rubbing is overcoming the activation energy barrier for transfer of an ion or of an electron.

We have an example of charge transfer that people are familiar with in the form of proton transfer, in acid–base reactions such as

If we now focus on electron transfer, then probably the first one we meet in Physics is the transfer of the electron in a hydrogen atom from an atomic orbital in the ground state to another [2] in an excited state

a process we might call photon induced intra-atomic ET.

The simplest molecule of Chemistry is, as we all know, And the simplest ET reaction [3] is

More in general we have gas phase interatomic ET reactions like

The simplest ET reaction between neutral atoms giving a cation and an anion [4, p. 26] is

In the last two reactions, the reactants and the products are different. Reaction (1.3) in which reactants and products are the same is the simplest example of an electron exchange reaction or self-exchange reaction.

A much more complicated kind of ET occurs when an ion dissolved in a polar solvent absorbs light. In this case, as in Eq. (1.2), the electron jumps to an excited state almost instantaneously, but a considerable longer time is required for the slower moving solvent dipoles to readjust to the new electronic configuration, electron density and associated electric field of the ion. Thus, for some time after the absorption act, the overall electrical polarization of the solvent medium in the neighborhood of the ion will not be in electrostatic equilibrium with the ion’s electric field, that is, the solvent polarization will not be the one dictated by the ion’s new charge distribution. This concept served as a basis of a theory of the absorption spectrum of various halide ions in solution [5].

Even more complicated ET’s occur in the usual oxidation–reduction reactions. For instance, in:

five electrons are altogether exchanged between and Fe⁺², in successive elementary reaction steps, with an extensive rearrangement of chemical bonds among the atoms and of solvent molecules around the ions. In this case, the ET is nonradiative, that is, it doesn’t happen because of absorption or emission of light, but it is thermal: it happens because of suitable thermal fluctuations in nuclear configurations of reactants and solvation molecules. This last statement may sound obscure to the uninitiated but it will be made clear in the following.

The simplest oxidation–reduction reactions in solution are those in which no bonds are broken or formed when the electron is transferred between reagents. Consider one such reaction:

The symbol “aq” means that the ions in water solution are solvated, the number of water dipoles and/or their orientations around the ions being clearly dependent on the ionic charges. In this electron exchange or “self-exchange” reaction, two isotopes of iron are used—one of them, Fe^∗, radioactive—to follow the ET between the ions because without the use of isotopes the reagents and products systems look the same. It was “in this small corner of inorganic chemistry” (M.), that of isotopic exchange reactions, where from the story of ET in polar solvents began. In reactions such as these, in which reactants and products are the same, the standard free energy difference between final and initial states is zero and the thermodynamic control on the reaction is missing. They are very interesting because it is in such reactions that those “intrinsic factors” which control their chemical kinetics, that is, the structure of the transition state (TS) and the nature of the reaction coordinate, come to the fore.

As for reactions in which reactants and products are the same, we may also remember the simplest of all bimolecular ones, the hydrogen atom exchange reaction:

the simplest reaction of Chemistry where a bond breaks while another bond forms [6, 7]. Here reactants and products are the same although a reaction really happens because a covalent bond breaks while another one forms. The Greek subscripts correspond here to the isotopic labels in (1.7).

If we now turn to Electrochemistry looking for examples of elementary reactions in which an electron is transferred from the electrode to the ion without formation or rupture of chemical bonds, we could consider for instance the reduction of permanganate to manganate [8]:

or of, say, protonated nitrogen oxide to NOH [9]:

where M is the metal electrode.

Today, among the topics included in the field of ET, are: inorganic, organic and biological ET, charge transfer spectra, ET at interfaces other than metal electrode–liquid, like semiconductor–liquid, modified electrode–liquid, polymer–liquid and liquid–liquid, ET at colloids and micelles and many others, see Fig. 1, p. 51 in [10]. Among them a recent one is ET in quantum dots [11].

Electron transfer processes are also implicitly present when the cohesive energy is defined for ionic or metallic crystals in solid state physics [12].

1.1.Description of Electron Transfer Reactions with Potential Energy Curves

Adiabatic and Nonadiabatic Processes

Before delving into the treatment of bimolecular ET reactions in polar solvents, we first consider—as a warm up and to introduce some fundamental concepts—the much simpler case of a diatomic ET in vacuo. Let us consider the reaction:

Without in general being aware of it, we pass by this reaction when studying the formation of the ionic bond in alkali halides. Let us then consider the process of formation of the ionic bond in the molecule of NaCl starting with an atom of Na far apart from one of Cl and letting the two atoms slowly (adiabatically) approach each other to form a chemical bond. “Slow collisions of atoms and molecules (neutral or charged) are defined as collisions in which the velocity v of relative motions of colliding particle is substantially lower than the velocity of valence electrons υ_e,

υ_e is estimated as υ_e ≈ 1au ≈ 10⁸ cm/s” [13].

A slow electronically adiabatic diatomic collision is a collision such that “the electron cloud is able to adjust its state instantaneously to the nuclear framework” and, on the other hand, “the nuclear motion will not be influenced by the momentary spatial arrangement of the single electrons but only by the mean force field (averaged over many periods of the motion) of the whole electron cloud” [14, p. 7]. To discuss the energetics of such a collision, it is necessary to represent the total electronic energy of the Na + Cl system as a function of the internuclear distance. By total electronic energy, we mean the sum of the electronic energy of the electrons plus the repulsive potential energy of the nuclei [15, p. 3]. The curve representing this function is called adiabatic potential energy curve (PEC), or adiabatic molecular interaction potential, because the total electronic energy plays the role of potential energy governing the motion of atoms, that is, of the nuclei and of the attached electron clouds, when discussing the dynamics of the process, see, for example, Refs. [14, 15]. Such a curve is schematically shown by the lowest solid line in Fig. 1^∗, [cf. 16, Fig. 14.3, p. 537], we see that it is almost a straight line parallel to the axis of the internuclear R distances until, at R_c ∼ 10.15Å, it suddenly turns down changing into the segment of hyperbola which goes down in energy until it changes into the segment of a parabola around the minimum. Beyond that, the curve goes up steeply to higher energies. Let’s now explain this behavior. Starting with atoms infinitely far apart and with zero kinetic energy, that is, in their asymptotic state, the potential energy of the system decreases very slightly, remaining almost constant, because between neutral atoms there is only a small Van der Waals attraction. At internuclear distances in the neighborhood of R_c it so happens that an electron transfer may occur and Na + Cl changes then into Na⁺ + Cl⁻. The two ions will attract each other with an electronic potential energy equal to:

Fig. 1^∗. (Adapted from Ref. [22]).

This function represents a hyperbola. We mark the segment of the curve corresponding to R > R_c with a C (from “covalent”) and the segment for R < R_c with an I (from “ionic”). When the atoms are close enough, a Pauli repulsion (exchange repulsion) sets in between the atomic cores and the combination of Pauli repulsion, of nuclear repulsion and of nuclear-electronic attraction gives the segment of parabola around the minimum. At minimum energy, the molecule Na⁺Cl⁻ is formed, whose bonding distance we denote with R_b. Going to shorter R, the Pauli repulsion—to which one should also add the internuclear repulsion—overwhelms the attraction and the curve shows a steeply rising repulsive branch. At the minimum potential energy, the system reaches its maximum stability and the parabola at the bottom of the curve describes the low energy vibrational motion of the Na⁺Cl⁻ molecule like that of a harmonic oscillator. As a whole we can then symbolically describe in the following way the process of formation of the ionic bond:

Notice that the systems above are considered each with a total energy equal to the potential energy at the various values of R. Na and Cl or Na⁺ and Cl⁻ are fixed at the various distances R. The symbols are not describing a scattering process in which kinetic energy would be present.

The description of the process using only the lower PEC in the figure is unsatisfactory because we cannot explain why the nature of the two branches of the lower solid curve changes right around R ∼ R_c, that is, why R_c has that particular value and why just around there an ET becomes possible. Moreover, a question comes naturally to mind: how would one describe the formation of NaCl starting not with atoms but rather with the ions Na⁺ and Cl⁻? To describe this process and to answer the above two questions, we need to consider also another PEC, the upper solid line in Fig. 1*.

At the right panel side of the upper curve, we see the system of the ions in their asymptotic state. Their energy differs from that of the atoms by the difference of the ionization potential of Na, I_P (Na), and the electron affinity of Cl, E_A (Cl). These energies correspond to the processes:

The energy associated with the process (Na + Cl)_R=∞ → (Na⁺ + Cl⁻)_R=∞ is I_P − E_A and the ET is radiative at large distances between the atoms because a photon is needed to ionize the Na atom. The new PEC correlated to (Na⁺ + Cl⁻)_R=∞ begins with a hyperbola of equation:

using as zero of energy the energy of the asymptote of the lower solid curve. The upper curve decreases to reach R_c, in whose neighborhood an ET may happen:

and the curve flattens because the strong Coulombic attraction has disappeared due to the formation of neutral atoms. Going to shorter distances, the Pauli and nuclear repulsions set in, and the curve goes up steeply. Even in this case a letter “I” marks the branch of the curve corresponding to the ions and a “C” the one corresponding to the atoms.

Now we can understand why the ET reaction (1.11) occurs right in the neighborhood of R_c. It so happens because at that interatomic distance in the narrow region of the avoided crossing shown in the figure the systems Na⁺ + Cl⁻ and Na + Cl may have the same energy because there the large amount of energy required to transfer an electron from Na to Cl forming Na⁺ and Cl⁻ is completely counterbalanced by the mutual Coulomb energy of the ions [4, p. 73] and it is therefore possible to go from the atoms to the ions and vice versa without the help of photons making up for the energy difference between reagents and products. We have here an example of a resonant ET. In reaction (1.7), we have an example of a thermal resonant ET because the resonant condition is reached thanks to an appropriate thermal fluctuation of the solvent.

The second question—about the possibility of forming the molecule starting from the ions—has thus far received a negative answer: if we move only on the higher curve, we cannot directly go from NaCl in its electronic ground state. The two curves we have been considering until now are “adiabatic” curves, the meaning of the term “adiabatic” being explained shortly below. They can be obtained by accurate ab initio quantum mechanical calculations, in which the total electronic energy at each point of the curve is computed at the fixed nuclear distance given by the abscissa of the point and each curve has either a “C” character before the avoided crossing and an “I” character after the covalent–ionic crossing or vice versa. Looking at the two curves, we see that in the neighborhood of R_c they come very close to each other but they don’t cross forming, in a very small range of the R coordinate around R_c, an “avoided crossing pattern” or “avoided crossing seam” limited by two short branches of hyperbola showing a narrow energy splitting between them. The reason for this is explained, for instance, in [14, pp. 123 ff.]. We want only mention here that the splitting is due to the “Pauling resonance” [17, p. 204], see “the prototype case of [17, p. 204] in Ref. [4, p. 14 ff.] (and see Appendix) between the ionic and covalent forms of NaCl, usually symbolized which happen to be in a “perfect” resonance at R_c because at that R the two forms have the same energy. Joining now the two “I” branches with a dashed line along a hyperbola’s asymptote, we have a new curve which is called the ionic “diabatic” curve and which away from the avoided crossing seam “will merge and be essentially the same” [14, p. 153] with two branches of the adiabatic curves, see Fig. 1* [cf. 16, Figs. 14.2 and 14.3, p. 537]. Likewise, the two C branches and a dashed line connecting them across the avoided crossing region represent the “diabatic” covalent curve. The two C branches merge out of the avoided crossing seam and become essentially the same with two branches of the adiabatic curves. If we now imagine starting with (Na⁺ +Cl⁻)_R=∞ and follow the ionic diabatic curve, jumping from the above adiabatic curve to the one below in the avoided crossing region, is then possible to go from the ions directly to the molecule, with its mainly ionic bond, without the need of an intermediate ET. Processes like the first two we considered, in which the system follows a single adiabatic PEC, are called “adiabatic.” The word “adiabatic” is almost a transliteration from the ancient Greek “αδιάβατoς” meaning “nonpassable,” “noncrossable.” Those in which the system follows first a branch of an adiabatic curve, jumps then on another adiabatic curve across an avoided crossing and subsequently follows the second curve, are called “nonadiabatic.” The nonadiabatic processes follow a diabatic curve. “Diabatic” comes from “διαβαíνω,” “I pass.” In other words, the ET process happens when the system follows an adiabatic curve, and it doesn’t if it follows a diabatic curve. If the system follows a branch of one of the two adiabatic curve and then, in the interaction region of the avoided crossing, jumps to the other adiabatic curve, we have a nonadiabatic process, no ET has happened, it is as if the process had happened along a single diabatic curve.

The question as to when we have nonadiabatic processes and when adiabatic ones is answered by the celebrated Landau–Zener–Stueckelberg–Majorana formula for nonadiabatic transition probabilities:

The formula is more usually designated simply as the Landau–Zener formula. For a discussion see Refs. [14, 16, 18, 19]. A short summary of the Landau–Zener theory is given in Ref. [18]. In Eq. (1.16), S_c ≡ S(R_c) is the splitting between the adiabatic curves at the avoided crossing, equal to twice the resonance interaction between the resonant structures NaCl and Na⁺Cl⁻ when Na and Cl (and likewise Na⁺ and Cl⁻) are at a distance R_c (see Appendix). F = F_ion − F_cov where F_ion and F_cov are the slopes of the ionic and covalent diabatic curves at R_c and υc is the velocity with which the system passes across the avoided crossing region. Looking at Eq. (1.16) we see that “in those region of the nuclear configuration space where adiabatic potential surfaces are close together or intersect, where electronic wave-functions are changing very rapidly for varying nuclear coordinates, where nuclei are moving with high velocity, non-adiabatic transitions become probable” [14, p. 22], and so if the system passes through the avoided crossing with high velocity, it jumps through the splitting with a nonadiabatic transition probability close to 1 and no ET happens because the electronic structure of Na + Cl doesn’t have time to change to that of Na⁺ + Cl⁻ and the system follows the diabatic path of the covalent PEC, that is, for collisions velocities high enough the diabatic terms can be interpreted as potential surfaces which govern the motion of the nuclei [14, p. 153]. On the other hand, when Na and Cl approach each other with a vanishingly small velocity, the probability P of a nonadiabatic jump from the lower to the upper curve in the avoided crossing region is almost zero and so the probability 1 − P of staying on the lower curve is almost 1. This probability is the probability of the adiabatic ET reaction Na + Cl → Na⁺ + Cl⁻.

The term “diabatic” has been introduced in the physical literature only since 1963 by Lichten [20]. Kauzmann [16] calls the adiabatic and diabatic curves “slow” and “fast” curves. A recent description of diabatic curves is given in Ref. [21].

The above adiabatic curves do not cross because of the Wigner–Witmer noncrossing rule [13, p. 563]. Similar curves are also reported on p. 77 of Ref. [4]. There Pauling used valence bond calculations and the curves cross because of the approximate calculations. This is why the avoided crossing is also designated as pseudocrossing, that is, false or spurious crossing. Pauling’s valence bond energies are diabatic. The curves are also reported on p. 372 of Ref. [22]. Note that noncrossing rule is only true for diatomics. Conical intersections are the rule for polyatomics.

One very important point needs here to be emphasized. We observe that the crossing point distance R_c ∼ 10.15Å is much larger than the sum of covalent radii of the atoms in NaCl (∼2.53Å) or of the ionic radii in Na⁺Cl⁻ (∼2.45Å) [15]. This means that the charge transfer inducing interaction, measured by the small energy splitting between the curves around R_c, and proportional to the overlap of the orbitals of the atomic wave functions, is a weak electronic interaction (see Appendix) if compared to the strong chemical interaction typical of the covalent bonds, where there is a much greater orbitals’ overlap. A. C. Wahl et al. [23] in a calculation reported in Refs. [15, 24] showed that when the lithium and fluorine atoms approach, to form the lithium fluoride molecule, an ET occurs at R_c ∼ 7.35Å and the lithium cation and fluorine anions form. In this case, that distance is about four to five times larger the sum of the atomic or ionic radii!

I want to end up this section citing the fascinating study by A. Zewail of the reaction, analogous to Eq. (1.11)

In Ref. [25, p. 264], one finds that curves similar to the ones discussed above have been used to follow processes (1.17) studied by femtoseconds laser spectroscopy.

1.2.Molecular Polarization, Reactants Model

In the beginning. . . there were experimental kinetics studies. “It was found that isotopic exchange between ions differing only in their valency are generally slow if single cations are involved and fast if the ions are relatively large, such as complex ions” [26].

W. Libby [27] surmised that this behavior was dependent on the orientation of the solvent dipoles around the ions. Immediately after an ET event, the charges on the ions change and the solvent dipoles around them are not anymore in electrostatic equilibrium with the charges (vide infra). This means that the new state, not being in stable equilibrium, is of higher energy than the original and this fact could explain the slow reaction velocity due to the high activation energy barrier for the small ions’ reactions, being more highly solvated than the big ones. The insight of Libby, that the barrier depended on the nonequilibrium polarization of the solvation molecules, was correct. Moreover, he was right in thinking that the quantitative explanation should have been found in applying the Franck–Condon (FC) principle. But he was wrong in the way he applied it [28].

At this point, a brief reminder on molecular polarization is in order.

An electric field—that of an ion in particular—induces in a molecule electronic, atomic and orientational polarization. The electronic polarization is due to the shift of electrons relative to the nuclei, atomic polarization means that atoms are displaced relative to one another [29], with consequent variation of interatomic distances, bond lengths, and angles. The orientational polarization is due to the orienting effect on the molecular dipole by the directing electric field. Marcus uses the symbol P_e for the electronic polarization that he designated as of “E type,” while the symbol P_u is collectively used for atomic and orientation polarization, which is of “U type.”

The relaxation times are of the order of 10⁻¹⁵ sec for electronic polarization, 10⁻¹³ sec for the atomic and 10⁻¹¹ sec or slower for orientational polarization [26].

The potential energy of orientation between an ion and a dipole [30] is given by:

where θ is the angle between the direction of the electric field E of the ion and that of the molecular dipole p. The ion–dipole system is in electrical equilibrium when θ = 0, that is, when the dipole is lined up with the field. In this case, the direction of the dipole is the one dictated by the electric field and U is at its minimum value.

When a molecule is in a medium, the orienting effect of the electric field is counteracted by the thermal agitation of the molecules, by their mutual interactions and orientation correlation of neighboring molecules [29] so that at temperature T the average equilibrium θ of the solvation molecules in the ion’s field may be different from zero. Moreover—and this fact is of paramount importance for the ET theory—the instantaneous polarization continuously fluctuates around its equilibrium value because of thermal agitation, a situation reminiscent of the vibrations of a harmonic oscillator around its equilibrium geometry.

In the first model used by Marcus for the reacting ions, they are supposed to be spheres of radii a₁ and a₂. The spheres are rigid, formed by the bare ions and possibly by a spherical region of saturated dielectric made up by solvent molecules fully oriented in the ions’ fields. Outside these saturated spheres are the molecules whose P_u polarization is determined, as described above, by the counteracting ordering and disordering effects of electric fields and thermal agitation.

Initially, under the influence of Born’s description of the charging of ions in solution, M. considered the spheres as conducting. This restrictive hypothesis was dropped later on.

1.3.Solvation Molecules’ Contribution to the Barrier in ET Reactions

In order to understand—on a qualitative level—the barrier to ET reactions due to the orientation of the solvent molecules around the ions, let us first consider the barrier to reaction for the most simple threecenter atom transfer reaction involving a linear activated complex, that is the H exchange reaction:

The energy barrier to reaction is due to the fact that an activation energy is necessary to go from the reagents to the products because a chemical bond between two hydrogen atoms is to be broken while another one forms. If we imagine, for simplicity, that the reaction happens on a line, we may represent schematically the process as:

Fig. 2^∗. (Adapted from Ref. [31]).

where R stands for the reagents, P for the products and the double dagger symbol “‡” for the transition state at the nuclear configuration intermediate between that for R and P.

The energy barrier to reaction is represented in the simplest possible way as in Fig. 2* [31–33]. The potential energy at the maximum along the reaction coordinate corresponds to the nuclear configuration of the TS. Note though that, contrary to the gradual process in Eq. (1.19), the jump of the electron in the case of ET is an abrupt process.

Imagine now having a system of two ions of charges +3 and +2, for example, of iron, in a polar solvent, and that an ET reaction happens between them. The distribution of orientations of solvation dipoles around ions with different charge is different because the directing electric field is greater for more highly charged ions and so the average angle θ is smaller for the dipoles around the ion with charge +3 than for those around the one of charge +2. After the charge exchange reaction, the equilibrium orientational distribution of the dipoles around the ions is also switched. This is described in the ET literature by means of figures where the authors either pictorially show the different orientation of the solvent molecules around the ions [34] or more simply represent the dipoles by swarms of small ellipses [35] or arrows [36–38]. The number of arrows varies from many to three used by Krishtalik and two by Kuznetsov. It is possible to give the reader the sense of the shift in dipole orientation using a single arrow, as suggested by Eq. (1.18), to represent the average dipoles distribution.

Fig. 3^∗.

In the left side of Fig. 3* panel A, small circles represent two ions with charges +3 and +2 and, above them, are arrows representing, for simplicity, only the orientational part of P_u (in order to represent the atomic part of it one should use arrows of different length), all at an infinite separation distance of the two ions, for simplicity. The angles between field and average dipole directions are only meant to be illustratively effective.

The left side of the figure represents the reagents system R. After ET, the system relaxes to the new equilibrium system P at the right side of the figure. As in the case of the H atom transfer reaction, the nuclear configuration in the TS will be intermediate between those of R and P. In the words of Marcus the TS: “. . . can be reached by any suitable fluctuation of atomic coordinates to produce some atomic configuration which is usually a compromise between the stabler ones of the redox orbitals. . . . Fluctuations of this nature involve simultaneous changes in orientation, position and atomic polarization of the solvent molecules, in internuclear distances in the coordination shell, in relative motion of the reactants and in configuration of the ionic atmosphere.” [39, p. 22] Such a configuration is represented in the middle of the figure by arrows with orientations intermediate between those in R and P, and is indicated by e.e., for “equivalent equilibrium” [39, p. 25] P_u polarization. Such polarization would be the one in equilibrium with fictitious charges +2.5 on each ion. But the system so designated cannot be the real TS for the following reasons. First of all the fractional +2.5 charges are obviously only hypothetical. Moreover, the transition state is a nonequilibrium state whereas in the e.e. system there is equilibrium between charges and polarization, like in the R and P systems. Finally, and most importantly, the P_e polarization is always in equilibrium with the charges and so we have here, in a misleading depiction, a fictitious electronic polarization in equilibrium with fictitious charges. This hypothetical intermediate system is nonetheless of great importance in ET theory because it has the correct P_u polarization equivalent to that in the transition state and, moreover, it suggests a way to find it: P_u is that polarization which is in equilibrium with a hypothetical intermediate charge and can so be found through a suitable charging process of the ions [40].

In Fig. 3* panel B, the real ET process is schematically described. Between the R and P systems there is, just before the TS, the nonequilibrium state X^∗ and, just after the TS, there is the nonequilibrium state X, the one in which X^∗ transforms immediately after the ET (vide infra). At the TS, the wave function of the system is a linear combination of those of X^∗ and of X. In the following, the R, P, X^∗, and X symbols will represent, as they do in Marcus’ first ET paper, thermodynamic states each one made up, as always in thermodynamics, of very many complexions or microstates and the X^∗ in the figure is really meant to be representative of one of the x^∗ microstates making up the X^∗. Let us now consider two microstates x^∗ and x, one having the electronic structure and the charges of the reagents, belonging to the state X^∗, the other having the electronic structure and the charges of the products and belonging to the successor state X. P_u refers to the polarization of the thermodynamic states. Each of the microstates x^∗ contributes with its own P_u(x^∗) to the P_u polarization.⁽¹⁾ In x^∗, the polarization is P_u(x^∗) and the ions’ charges are those of the reagents. In x, P_u(x) is equal to that of x^∗ but the charges equal to those of the products. Both x^∗ and x are unstable nonequilibrium systems because P_u in them is not in equilibrium with the ionic charges. The TS has equal total energies, including P_e for either electron localization.

We are now in the position of correctly describing the thermal ET process in solution. The process begins with a suitable thermal fluctuation of the nuclear coordinates bringing a microscopic system belonging to R to a system x^∗ belonging to X^∗.⁽²⁾ Such fluctuations are of orientations of solvent molecules and of their bond lengths and angles. At the hypersurface representing the TS, after an electron transfer involving the coupled motion of nuclei and electrons, the successor state x forms, belonging to X, which has the same nuclear configuration of x^∗ but the electronic configuration of the products. The electron transfer probability in this dynamical process is given by the Landau–Zener formula.⁽³⁾ The system x belonging to X, finally relaxes to a microscopic system belonging to P. Of course, each of the above three steps must run in the direction of the products for a successful ET finally to happen.

Marcus has taught us how to apply the FC principle to thermal ET processes. The states FC connected are here states x^∗ and x of equal nuclear configurations and of equal energy contrary to the usual situation in which the FC principle is applied in spectroscopy to vertical transitions between states of equal nuclear configurations but of different energy, like the ones shown, for the NaI system, in Fig. 6.44, p. 356 of Ref. [41]. Marcus applied the FC principle to an energetically horizontal transition between states of equal energy.

We want to emphasize at this point that although the nonequilibrium P_u contribution of the solvation molecules to the reaction barrier is important and was the first to be studied in the development of the Marcus theory, an important contribution is also that of the vibrational motions of the reactants and of the configurations of the ionic atmospheres. The relative importance of the different contributions varies for different reactions. We shall take up later these further contributions to the reaction barrier. In Fig. 3* are represented ions with charges +3 and +2 participating in an isotopic exchange reaction. The ions are supposed to be surrounded by polarized solvent molecules and ionic atmospheres. They are represented by circles with arbitrary different radii intended to simply schematically summarize their different atomic configurations in order to represent the processes illustrated in the figures.

1.4.Electronic Configuration of the Activated Complex

On pp. 967 and 968 of Ref. [26], M. gives a crystal clear description of the electronic configuration of the activated complex. Two remarks are in order. First, M. uses there the older Eyring’s terminology “activated complex” instead of the modern “transition state” which he adopted successively.⁽⁴⁾ Secondly, he imagines that the activated complex is made up of two electronic forms in equilibrium with each other, that is, where X^∗ is the activated complex with the electronic structure of the reactants and X is the one with the electronic structure of the products. Such formulation represents a good approximation but it is “static” and has been superseded by a “dynamical” one in which “X^∗ and X are two electronic participants in the TS. Just before the TS there is the X^∗ form, just after the TS there is the X form, in between there is a combination of the two, you may call in resonance combination. In other words, if a wave function ψ₁ refers to X^∗, a wave function ψ₂ refers to X, at the TS we have ψ₁ +ψ₂” (M, personal communication). The initial state X^∗ goes to the successor state X by a Landau–Zener dynamical process, as discussed in the Appendix.

In the following, I shall briefly mention the main characteristic properties of the activated complex for ET reactions:

(1)In the usual chemical reactions, there are transfers and/or rearrangement of atoms between the reactants. This happens because of strong interactions between atoms as a consequence of considerable spatial overlaps of the atomic and molecular orbitals of the reactants. In the case of ET reactions, there is a slight electronic interaction⁽⁵⁾ which is sufficient to electronically couple the reacting molecules and permit the ET to occur. We have seen above the examples of ET between atoms of Na and Cl or of Li and F atoms happening at interatomic distances such that the interaction between atomic partners is very weak.

(2)In the ET process, the electronic configuration changes, in a successful collision, from the one characteristic of the reactants to that of the products. The process is one of abrupt transfer of an electron, not that of a gradual transfer of electron density from one reactant to the other as, for an example, in the case of the reaction H⁺ + H₂O → H₃O⁺ [42]. The collision process is a dynamical process in which the motion of nuclei (better, of atomic cores) and of electrons (valence or outer electrons) is largely decoupled before reaching the crossing region and the motion is governed by a single PES and the single associated wave function but it is coupled in the crossing region where it is governed by both PESs and by a combination of both associated wave functions, vide infra.

(3)Because of the slight electronic interaction, the energy splitting between the adiabatic potential energy surfaces is small and M. normally approximates the adiabatic surfaces with the diabatic ones, like in this first treatment. But in the diabatic case of zero splitting, there would be no overlap of the electronic orbitals, so the M. approximation is “the better the less the overlap.” He needs the splitting small, or the energy of the TS would be wrong.

(4)Let x^∗ be one of the microstates belonging to the state X^∗ and x a microstate of the state X. The wave function of the electrons in the microstates not only describe the reacting particles but they take account also of the solvent molecules.⁽⁶⁾ The energies of the two states are the same, E_x∗ = E_x, for every x^∗ system in X^∗, there is an x system in X which has the same energy as x^∗, the X^∗ and X, which are made up of x^∗’s and x’s, have the same energy⁽⁷⁾ and the wave function of the system at the TS is a linear combination of the wave functions of the two states, that is, ϕ_x∗ + cϕ_x, like in the case of a wave function describing a quantum mechanical resonant structure built up of Lewis resonant structures. Notice that the frequency of hopping from x^∗ to x and back and forth is slow because of the weak electronic interaction but the time for the single jump is very short (see Appendix).

(5)The average configuration is the same in the two states X^∗ and X and the energy must be the same for the two states. But the charges in X^∗ and X are different, so that the state of the solvent must be one of nonequilibrium. M. so rephrases the foregoing discussion in terms of the FC principle: “When one electron configuration is formed from the other by an electronic transition, the electronic motion is so rapid that the solvent molecules do not have time to move during the electronic jump.”

1.5.Reaction Scheme

An important consequence of the small orbitals overlap in the activated complex is that the ET process may be slow, so determining the rate of the overall process to which the ET step belongs.

After having started from the reactants, A + B, say, once the nonequilibrium state x^∗ is reached, isoenergetic with the x state of the products, there will be a certain probability of the electronic transition x^∗ → x. This transition is discussed in the Appendix. There is also the possibility that the state x^∗ will reform the reactants “by disorganization of some of the oriented solvent molecules.” The state x can either reform x^∗ by an electronic transition [x → x^∗] or, alternatively, the products in this state can merely move apart, say. The detailed reaction scheme for bimolecular ET reactions will be dealt with in the next chapter.

1.6.Potential Energy Hypersurfaces and Schematic Diagrams for ET Reactions—A Summary in Marcus’ Words

“To treat rates of reactions in general , regardless of whether they involve transfers of electrons, atoms, or protons, bond scission, or molecular isomerization, it is useful to plot potential energy curves. The potential energy U is a function of the positions of all the atoms in the system. Thereby, U depends, for example, on all the bond lengths and angles, on orientations of reacting molecules, and on distances and orientations of molecules in the surrounding environment. Because there are so many position coordinates involved, only a profile of U versus some general coordinate can be plotted, which has as components all of the coordinates above. Such a plot is useful for pictorial purposes, although the actual calculations themselves involve all instead of one general coordinate.

The position of each atom in the entire system is subject to thermal fluctuations and the reactive system thereby wanders over the curve R (really surface R) in Fig. 4*. No reaction occurs until the system reaches the coordinates at the intersection of the R and P surfaces. At that intersection, the system can go from the reactants’ surface R to the products’ surface P when there is a coupling between the orbitals of the two reactants. The extent of coupling of two electronic orbitals (one on the reactant, occupied by the electron to be transferred, and an orbital on the other reactant, waiting to be occupied by the transferred electron) is reflected in the splitting 2 of the intersecting R and P curves, as in Fig. 4*.

Fig. 4^∗.

The probability of reaching the intersection can be calculated by statistical mechanics or by some related formalism. The probability of the system’s crossing from the R to the P curve can be calculated by quantum mechanics with a velocity-weighted Landau–Zener transition probability κ. The weaker the electronic coupling of electronic orbitals of the reactants with each other, the smaller is this κ. The probability of transition κ at the intersection increased with increasing ϵ” (from M140, pp. 15–16).

It has a maximum value of unity at strong enough coupling.

Note that the figure refers to a thermoneutral reaction, like that of isotopic exchange reactions, and that the ET is supposed to happen at temperatures high enough that nuclear tunneling is neglected. Such a point will be dealt with later in the book.

In the case of the gas phase ET reaction between Na⁺ and Cl⁻, the dynamics of the process was described using two-dimensional PECs of the kind U = U(R) where the abscissa is the internuclear distance and the ordinate is the total electronic energy. The two-dimensional space of a page is appropriate to represent such curves. In the case of the hydrogen transfer reaction along a line, the potential energy is represented by a two-dimensional potential energy surface (PES) in the three-dimensional space of U and of the internuclear distances R(H_α − H_β) and R(H_β − H_γ). The famous PES for such reaction is described, for instance, in Refs. [14, 31–33, 43].

If one considers a nonlinear three-atomic system like, for instance, the [HO₂]⁺ system, and the gas phase ET reaction between H⁺ and O₂:

three coordinates are necessary to describe the relative positions of the proton and the oxygen atoms in the plane of the nuclei [44]. In these cases, we have then potential energy hypersurfaces (PESs) in four dimensions, one for U and three for the geometrical coordinates, which obviously cannot be visualized as a whole in three-dimensional space. Holding two coordinates fixed, it is possible to compute PECs which are cuts or cross sections, as they are called, of the PES.

When we pass to a system of two “central ions” [26], each one surrounded by solvent molecules and, more in general, even by ionic atmospheres, the N nuclear coordinates necessary to describe the spatial configurations number in the order of thousands [10] if one considers only the reactants and the solvent molecules and the ions closer to the reactants, while the total number of coordinates necessary to describe the whole macroscopic system is of the order of 10²³. The generalized coordinates [45] to describe the configurations space of the system are the distance R between the central ions, the vibrational coordinates, the angles describing molecular orientations and intermolecular distances, all of them parametrically dependent on R, that is, the potential energy hypersurface of the reactants R is U = U(q(R)) where q(R) = {q₁ = R, q₂(R), . . . , q_N(R)}. Because of the high number of nuclear coordinates, the potential energy hypersurface U could only in principle be represented in the same way as for the simple systems considered above.

A schematic potential energy diagram used by Marcus to describe the ET process is reported in Fig. 4*. It is very often found in Marcus’ papers, for example, in Refs. [10, 28, 32, 46]. This figure is really almost a logo of the Marcus theory of electron transfer. Such potential energy “profiles” are not quantitative nor qualitative PECs obtained from the N-dimensional UReactants andUProducts hypersurfaces cutting them as described above. They are just “profiles of the actual potential energy surfaces plotted along the reaction coordinate q,” schematic potential energy diagrams [31, 47] which graphically summarize the information described below.

The R and P surfaces are sketched as two potential wells in parabola-like forms to represent “some sort of vibrational-like motion” [32, 33] where the abscissa is that of “a collective mode of the donor, acceptor and solvent” [48]. They are similar in shape to the unidimensional symmetric bistable PECs used to describe isomerization processes, such as the ammonia inversion [49, 50], or the inversion of its isoelectronic ion H₃O⁺ represented in Fig. 3 of Ref. [51]. In Fig. 5*(a), the “profiles” of PESs U_R and U_P of reactants and products cross at a point {R, q₂(R), . . . , q_N(R)} in N-dimensional space where U_R(R, q₂(R), . . .) = U_P(R, q₂(R), . . .). The systems R and P have the same q configuration and the same energy but the distance R between the central ions is not small enough to allow for an electronic interaction between them, so that there is no splitting of the PESs and no ET.

When the central ions are at a distance R = R_c at which they electronically interact, the PESs cross at the point {R_c, q₂(R_c), . . . , q_N (R_c)} so that:

the ET act is possible and the curves in Fig. 5*(a) change into those of Fig. 5*(b). The energy splitting appears and we shall have a possible adiabatic ET reaction. In this figure, the potential minima of the R and P profiles are equal, so that such curves are suitable to describe, for instance, isotopic exchange reactions in which reactants and products are the same.

Fig. 5^∗.

In Fig. 5*(c), a more general profile considers the possibility that reactants and products are different and that the products are more stable than the reactants. In Fig. 5*(d), the reactants are instead more stable than the products.

In a following development of the theory Marcus introduced a generalized reaction coordinate (the energy difference of the two energy surfaces at each point) and the problem of the multidimensional PESs was reduced, by a statistical mechanical averaging, to a discussion in terms of free energy curves.

Fig. 6^∗.

Using the energy diagrams with the above schematic orientational polarization diagrams, it is possible to show very clearly what was wrong in Libby’s description of thermal ET process.

In Fig. 6*, an arrow shows the vertical excitation of the R system to an excited products state, P* say, wherefrom the system relaxes to the equilibrium state corresponding to the minimum of the P diagram. The vertical excitation corresponds to an optical ET in which the R and P^∗ states have the same nuclear configuration—as expected from the usual way of applying the FC principle to spectroscopic transitions—but while in R the solvent polarization is in equilibrium with the electric field, in P^∗ it is not. The P^∗ system then thermally relaxes to P. The polarization diagrams corresponding to R, P^∗, and P are shown in the Figure. The P^∗ system has two characters in common with the correct activated complex for thermal ET. The P^∗ state is in fact of higher energy than the reactants state and is in a nonequilibrium polarization state. But instead of having a nuclear configuration intermediate between that of reactants and products, as one expects in a TS, it has the same configuration as that of reactants. And in the correct theory not only the excited system relaxes to the equilibrium state with a thermal fluctuation but an a priori suitable thermal fluctuation is necessary for the reactants system to reach the TS nuclear configuration region.

Appendix

In the crossing point of the diabatic curves, the system is degenerate because the two different electronic configurations have the same energy and so a resonance energy appears that separates the two diabatic curves. If ϕ₁ and ϕ₂ denote the electronic wave functions of the “pure” covalent and ionic valence bond structures, then the diabatic energy V₁ of structure ϕ₁ at the internuclear distance R is given by V₁(R) = ∫ϕ₁ H_elϕ₁dx, where H_el is the electronic Hamiltonian of the system depending parametrically on the internuclear distance R, the diabatic energy V₂ of structure ϕ₂ is given by V₂(R) = ∫ϕ₂ H_elϕ₂dx. The interaction energy of the two structures is given by the resonance (or exchange) integral V₁₂(R) = ∫ϕ1 H_elϕ₂dx = ∫ϕ₂ H_elϕ₁dx, a perturbation which, in the Landau–Zener theory is assumed to be constant and equal to V₁₂ in the narrow avoided crossing (or pseudocrossing) region [14, p. 151]. The energies W_± of the adiabatic curves are then given by:

where 2V₁₂ is the separation of the levels due to as shown in Fig. 5.7 on p. 150 of Ref. [14] or on Fig. 8.9(b) on p. 173 of Ref. [52] and in the interaction region of the avoided crossing the wave functions representing the two states system become linear combinations of the resonant forms, that is,

Using perturbation theory involving time, it is possible to show that the probabilities P₁ and P₂ of finding the system in state 1 and 2, respectively, at time t are:

that is, “We see that these probabilities vary harmonically between the values 0 and 1. The period of a cycle (from P₁ = 1 to 0 and back to 1 again) is seen to be h/2V₁₂ and the frequency 2V₁₂/h, this being . . . just 1/h times the separation of the levels due to the perturbation” [53, p. 323], see also Ref. [16, p. 534 ff.]. The situation is then the following: if the system were to remain static in the pseudocrossing region, there would be no final electron transfer: the electron would simply jump back and forth between the two resonant structures 1 and 2. But the system moves across the pseudocrossing region with some velocity, the electronic and nuclear motions are coupled in the avoided crossing region, both PECs govern the dynamics there through their splitting and through the difference of their slopes and following the Landau–Zener theory there may be two possible outcomes. If the system will move very rapidly across the interaction region, the electronic cloud will not have time to change from structure 1 to structure 2, no ET will happen, the system will simply go along a diabatic curve. But if the system will pass through the pseudocrossing at a velocity such that the electron will jump from 1 to 2 but it will not have time to go back, then we shall have an electron transfer. As a matter of fact in the so-called Massey parameter where l(R_c), the linear dimension of the pseudocrossing region, is “a characteristic length over which the corresponding electronic wave-functions substantially change and u some average (classical) velocity of the nuclei at point R_c” [14, p. 21, 22], we see that we have the ratio of the passage time of the nuclei through the interaction region to the transfer time h/V₁₂.

“We may state the general rule that exchange (resonance) integrals will tend to be large only if the orbitals concerned overlap effectively” [16, p. 298]. Which means that the smaller the distance R between the colliding partners the greater the orbitals overlap, the greater the exchange (resonance) integrals, that is, the greater V₁₂ and the greater the splitting 2V₁₂ between the adiabatic curves at the pseudocrossing. An interesting example is presented by the PECs for the system HF and H⁺F⁻ reported on Fig. 3.4, p. 74 in Ref. [4]. There we see that the crossing of the ionic and covalent curves happen at somewhat less than 1Å and the splitting is then very large. On the other hand, the crossing of the curves for NaCl and Na⁺Cl⁻ happens at about 10.15Å and the splitting is very small, as reported on p. 536 with Fig. 14.3 on p. 537 of Ref. [16]. The first case is an example of Pauling’s resonance between Lewis structures, while the second is an example of the weak interactions considered in Marcus’ theory of electron transfer. Another example is that, previously cited in the text, of the LiF system. In the case of Pauling’s resonance, the frequency with which the electron jumps between the two structures is “the frequency of resonance among structures . . . is very large, of the order of magnitude of electronic frequencies in general . . .” [4, p. 186] so high that there is no chemical equilibrium between the two electronic tautomers and the resonance between the Lewis structures is sometimes indicated with a double-pointed arrow, a symbol suggested by Fritz Arndt and Bern Eistert to indicate resonance [4, p. 187]. We then have but in the case of electron transfer reaction supposed to happen at fixed internuclear distance there is a real chemical equilibrium between the electronic tautomers which exchange the electron at a lower frequency than that in the case of the usual Pauling resonance.

In the first paper on electron transfer M. considered the states X^∗ and X to be two forms of the activated complex, a static formulation then like the one considered above, and abandoned since 1964 in favor of the dynamic Landau–Zener treatment of ET probability.

In the first paper M. considered two methods of calculating the splitting 2V₁₂, in one of them making use, as above, of time-dependent perturbation theory and in the other, already used by previous authors [54, 55] considering the electron tunneling of the transferred electron, as explained below.

Once the Na + Cl colliding system has reached the pseudocrossing region, in order to transfer from Na to Cl the electron must pass through an electronic potential energy barrier due to the attraction exerted on the leaving electron by the Na⁺ ion left behind. The penetration of the barrier is classically impossible but “It is possible in quantum mechanics to sneak quickly across a region which is illegal energetically” [56, 8.12]. So that the electron tunnels through the potential barrier separating them: In Fig. 7*(b), I have used a wiggly arrow to symbolically represent the electron “worming its way”—in the words of Feynman [56, 10.2]—through the barrier.⁽⁸⁾ In Fig. 7*(a) and 7*(c), the potential wells are represented, as in Ref. [24], Fig. 6.4, p. 153 for the symmetric electronic potential energy barrier to be penetrated for electron transfer reaction (1.3) when H and H⁺ are at a large enough distance, in Ref. [57], Fig. 3.6, p. 58 for electron tunneling between two metal nuclei, and in Ref. [36], Fig. 3.2, p. 91. In Fig. 7*(a), the electron symbolically represented by an arrow is on the first iron ion, in Fig. 7*(c) on the second.

Fig. 7^∗.

The resonance splitting is one half of the energy splitting produced by electron “tunneling among equivalent Lewis structures” [17, p. 204] (small R’s) and more in general among resonant structures even at larger R’s, like in the case of Na − Cl and Na⁺Cl⁻ at the curves crossing. The tunneling across the potential barrier through which the electron leaving Na − Cl is to pass to reach the state Na⁺Cl⁻ removes the energy degeneracy at the crossing because the separated resonant states combine linearly as in Eq. (A.2) to describe the system present in two potential wells with probabilities changing with time as in Eq. (A.3) and the two states described by wave functions ψ₊ and ψ₋ , with which the time evolution appears, have their energies separated by the above energy splitting. A modern, detailed and very clear description of the tunneling-generated energy splitting can be found, for the analogous two states problem with tunneling of ammonia inversion, in Ref. [50, p. 455 ff.], but there the abscissa coordinate is measuring the distance of the plane containing the three hydrogen atoms from the N atom in NH₃, while here the abscissa is the coordinate of the transferring electron.

A short account of Marcus theory of electron transfer is given in Ref. [58] and a discussion of it in the light of the history of science is given in Ref. [59].

NOTES

1.M: “The different microscopic states contribute differently towards Pu, they are microscopic, Pu is macroscopic. The dielectric polarization involves a number of microscopic states the order of 10²³, a huge number of states. P_u refers to X^∗, not to x^∗. Every microscopic state has its own contribution towards P_u but each contribution is different, it depends on the microscopic state, just like any microscopic property.”

2.M: “You get the fluctuation [of the reactants] at the same electronic configuration that you start with, exactly. You have a whole new distribution and then, provided that the electronic configuration and nuclear configuration is isoenergetic with the other one for the products, then you (may) get a transfer. If it isn’t, you won’t get a transfer.”

3.M: “The whole process involves going to the crossing, going pass it. In LZ you have two electronic states and one nuclear coordinate. It’s the same nuclear coordinate for both, perpendicular to the TS hypersurface. When you treat the system with LZ, you don’t just treat the system at the crossing of the diabatic states, you treat the whole dynamics before and afterwards too. That of Landau and Zener is a semi-classical approximation to handle a full quantum mechanical nuclear and electronic problem.”

When the electron goes from one x* state to an x state on the intersection surface, one has two electronic wells and the electron tunnels going from one to the other.

M: “That just corresponds to what is the probability for going from one of the surfaces to the other in the intersection region, a Landau-Zener type situation. Those wells are not explicitly seen on the potential energy surfaces because they are electronic wells, while the potential energy surfaces are plots of energy versus nuclear configurations, not a plot of potential energy versus electronic configuration. Note that when you are on the intersection surface the bottoms of the two electronic surfaces are of the same height and so you can transfer the electron without changing energy. The molecular energies are equal for every state on that intersection surface, and of course you have a thermal distribution of points on that intersection surface.”

NOTE: This is very well shown in Fig. 3.2, p. 91 of Ref. [36], where the nuclear coordinate is shown in the left panel and the electronic coordinate in the right one.

4.M: “Following Eyring I used the term ‘activated complex’ for the longest time until I finally bent over to the Polanyi’ term ‘transition state,’ which is what everybody uses nowadays. Probably neither terminology is great because the ‘state’ doesn’t exist, it is a particular arrangement in the 10²³-dimensional space, it is a hyper-surface, so that’s not fictitious but typically you don’t observe it.”

5.M. considered, in his first formulation of the theory, a double form for the activated complex that was later abandoned:

M: “Pauling was interested in strongly interacting systems and on how the energy of the strongly interacting system compared with that in one resonant form or another. Here I am interested in weakly interacting systems. I don’t have a strongly interacting system, so formally in the activated complex electrons can hop from one form of the activated complex to the other, and that would be a legitimate description, but in Pauling’s case you don’t think of one resonant structure hopping to another, back and forth, it is not a real equilibrium.”

M: “An activated complex in the ET theory consists really of two electronic structures. The activated complex in this case is an unusual activated complex because usually an activated complex with a single electronic structure was used when one was thinking of the strong interactions systems, but this ET system is a weakly interacting system. So, this is a key difference and it is probably the reason why this type of treatment was missed in earlier work.”

NOTE: As a matter of fact, as discussed in the Appendix, the two forms of the activated complex are in Marcus’ case symbolically connected by two arrows, while in Pauling’s case one often uses the Arndt–Eistert double-pointed arrow:

6.M: “The reactants are really imbedded in the solvent in the fluctuated state. You never speak of the reacting particles alone, you don’t separate them from the other [surrounding particles]. You are talking about what you may call dressed particles, reacting particles with solvent around, influencing them, although it is true that if you are talking of electronic interaction, that’s only weakly influenced by the interaction of the solvent molecules surrounding the reactants except when the solvent molecules come between the reactants, that of course strongly influences the electronic interaction.”

7.NOTE: M. refers below to his M30 paper of 1960 to make clear the relations of the microscopic states x^∗’s to the thermodynamic state X^∗. There the potential energy surface of the reactants crosses that of the products at the surface representing the transition state TS.

M: “In the 1960 paper you have the intersection of two surfaces [order of] 10²³-dimensional. On the intersection, which is one dimension less than 10²³, i.e. 10²³–1, on the intersection of those surfaces then, you can see that if you regard one member of the pair of the surfaces to be that of one electronic configuration and the other member of the pair to be of the other electronic configuration, they share the same nuclear configurations, they share the individual distribution of microscopic energies, they share the same average energy, and they share the same distribution of microscopic states, so they share the same entropy. It is that intersection which is the TS. One of the members of the intersection, one of the surfaces, consists of all the microscopic configurations x^∗ on the intersection, thermally distributed. That is the thermodynamic system X^∗. X^∗ is a thermodynamic ensemble and so it consists of an enormous number of microstates x^∗.”

NOTE: At an “infinite” number of places on the intersection surface, there may be a microscopic reacting system x^∗ with the same nuclear configuration and the same energy of a microscopic products system x where the electron transfer reaction may happen.

M: “The activated complex or the transition state is of 10²³–1 configurations, if the dimension of the space is 10²³, it’s a hyper-surface one dimension less than the entire phase space. That’s a transition state.”

8.M: “The electron tunneling . . . that’s a different way of describing the overlap of electronic wave functions if they don’t overlap well, if you have a weak overlap of wave functions. An alternative description is that to go from one to the other you tunnel and there is a quantitative relation of the two, you can take the overlap of the wave functions, you can use semi-classical theory and get a tunneling probability out of it, so there is a whole theory associated with that.”

NOTE: M. comments below on Pauling’s description of the oscillation back and forth between two configurations in relation with the configuration of the system at the TS and that immediately after the electron transfer:

M: “That description is OK for a static system, but your system isn’t static, so you don’t have that kind of oscillation. If you calculate that way the probability of going from one to the other, you get that right ballpark, but conceptually that theory is not right because the nuclei are moving, so you don’t have that oscillation back and forth, you would have it if the nuclei were static and if you were exactly at the crossing point of those crossing energy curves, but that’s not what you have, so you get a result that is on the right ballpark, but it’s more conceptual than actually describing the process as it occurs when you take into account the combined electron-nuclear motion. But if you had a static picture . . . that gives you sort of the frequency of tunneling. One would use that way if one tried to be moderately modern. You don’t do that way because you don’t have static nuclei, nevertheless you get some result out of it, which is right ballpark, but you have to take into account that [in the case of Pauling’s treatment] you are dealing with static nuclei which don’t exist. This is a question almost of beauty of theory, I mean not that the theory which is unbeautiful gives you answers wrong by any magnitude, it’s just that’s not really the most complete picture. Pauling gives you a picture of what it would be statically and in a ballpark, but the system doesn’t oscillate back and forth, so physically that part isn’t right [for ET] but the oscillation is related to the probability of tunneling. You won’t use that equilibrium description, that wouldn’t be a good quantum mechanical description, you wouldn’t say you have a static system and that there is an equilibrium with the other electronic structure . . . there is no equilibrium there, you would treat the transition. Once you reach the TS you have one electronic configuration and then there is a certain probability that you form the other at the same nuclear configuration, but there is not a back and forth going. Look at how LZ treats it. You go through identical nuclear configurations and you solve the Schrödinger equation that is appropriate for that whole process. In other words, you don’t sit still at the intersection region, it’s quite un-dynamical, it’s quite un-quantum-mechanical. In Pauling’s case the interaction is large and you don’t start with one configuration and go over to the other in his description.”

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