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), . . . , qN(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 UR and UP of reactants and products cross at a point {R, q2(R), . . . , qN(R)} in N-dimensional space where UR(R, q2(R), . . .) = UP(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 = Rc at which they electronically interact, the PESs cross at the point {Rc, q2(Rc), . . . , qN (Rc)} 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