(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, Ex∗ = Ex, 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, ϕx∗ + cϕx, 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(Hα − Hβ) and R(Hβ − Hγ). 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+