For k2 we have the following:
The above number of times per second is given by the frequency of motion of the valence electron in the ground state of the ferrous ion, which is of the order of the frequency of excitation of this electron to the next higher principal quantum number (for a brief discussion of the correlation between classical frequency of revolution in electron orbits and quantum frequency of energy emission see, e.g., Ref. [30]). From data on the energy levels of the ferrous ion [31] M. estimated the frequency to be 2 × 1015 sec−1, so that:
which is of the same order of magnitude as k−1 considering the approximations which have been made.
2.8.Validity of the Assumed Small-Overlap Transition State
In the Marcus’ theory of ET reactions, states X∗ and X are supposed to be isoenergetic with weak electronic interaction of the reacting particles, that is, with a small overlap of their electronic orbitals.
A fundamental limitation to the statement of equal energy for X∗ and X comes from the energy–time uncertainty relation (a very nice discussion of it can be found in Ref. [32]. See also, e.g., Refs. [33, 34]). The energy–time uncertainty relation relates the lifetime τ of a state to the uncertainty δε of its energy:
which means that the energy of state X∗ is broadened by an amount δε and the same is true for state X. The energies of the two states can then be equal only within an energy interval of 2δε prescribed by the above uncertainty principle. The energy broadening 2δε is then like a minimal splitting of PESs in the neighborhood of the nuclear configurations space region where the ET process may happen. Such a splitting is related, as we have seen, to the orbitals’ overlap in the activated complex and “The greater the overlap the shorter will be the lifetimes of X∗ and X.” But 2δε is related to τ in Eq. (2.38) and τ depends, on its turn, on the rate constants k2, k−2, k3, and k−1 making up the factor 1/[1 + (1 + k−2/k3)k−1/k2] in Eq. (2.7). Their knowledge allows then—through knowledge of the lifetime of the activated complex—an estimate of the splitting 2δε and so of the extent of the overlap in the complex.
From Eq. (2.38), we see then that overlaps and lifetimes are inversely proportional. Let us consider the lifetime of X∗ which can disappear through the reactions:
Its lifetime is then equal to 1/(k−1 + k2) and is essentially the same as that of state X. For
which was derived supposing exactly equal energies for X∗ and X, we must replace ΔF0 with ΔF0 ± 0.15kcal mole−1, a negligible correction with negligible effects on the calculated ΔF∗.
Even if, as a result of large values of k2 and k−2, the lifetimes were as small as 10−14 sec, 2δε would be only 1.5kcal mole−1, with a relatively small effect on ΔF∗.
A large-overlap activated complex would be characterized by a switching of electronic structures from X∗ to X of the order of the electronic frequency in molecules. These frequencies are of about 1015 sec−1. The lifetime τ would be about a femtosecond, 10−15 sec and 2δε would have the value of 15kcal mole−1, that is, 0.65eV, a large value. In Eq. (2.24), ΔF0 should be replaced by ΔF0 ± 15kcal mole−1, and the restraint involved in the equation would not be very strong anymore.
As a result of the earlier discussion, we have now a quantitative measure to distinguish a small-overlap activated complex from a large overlap one. If the lifetimes of the intermediate states X∗ and X are greater than 10−14 sec, we have a small-overlap activated complex. “On the basis of the calculations for k3 and k2 given previously we infer that a small-overlap activated complex complex may well prevail for many ET reactions.” (M.)
NOTE: This first formulation of the theory was later replaced by a later derivation. In this first paper, M16, there is a quasiequilibrium between X∗ and X with rate constants in the opposite directions of the reaction. In the later derivation in M53 of 1965, Marcus has a more modern formalism which applies the Transition State Theory in the spirit of Wigner in his paper in the Transactions of the Faraday Society of 1938 and of Eyring, Walter, and Kimball in their book “Quantum Chemistry” of 1944. There is there a ballistic treatment and a nonadiabatic formalism. The k−1 or the equivalent for the electron going through is the rate constant for the fastest velocity at which an ET could occur, with the ET happening in 100 femtoseconds according to the more recent experiments.
2.9.The Interionic Distance R
The interionic distance R of the reactants in the intermediate states X∗ and X affects the overall reaction rate because it appears in (i) the tunneling probability κe, Eq. (2.37), and (ii) in ΔF∗, Eq. (2.27).
(i)κe determines the rate constant k2 in (2.37b). We saw that when k2 ≥ 1013 sec−1, the overall reaction rate is independent of k2. But κe depends exponentially on R (rab in Eq. (2.37)) and, for large R, k2 will become small and will then affect the overall reaction rate. Because of the exponential decrease of the electron jump probability with R, the reaction rate k2 will be maximal for minimum R.
(ii)R