Molecular Imaging. Markus Rudin

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