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. Pu refers to X∗, not to x∗. Every microscopic state has its own contribution towards Pu 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,
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
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