Molecular Imaging. Markus Rudin

and so a resonance energy appears that separates the two diabatic curves. If ϕ₁ and ϕ₂ denote the electronic wave functions of the “pure” covalent and ionic valence bond structures, then the diabatic energy V₁ of structure ϕ₁ at the internuclear distance R is given by V₁(R) = ∫ϕ₁ H_elϕ₁dx, where H_el is the electronic Hamiltonian of the system depending parametrically on the internuclear distance R, the diabatic energy V₂ of structure ϕ₂ is given by V₂(R) = ∫ϕ₂ H_elϕ₂dx. The interaction energy of the two structures is given by the resonance (or exchange) integral V₁₂(R) = ∫ϕ1 H_elϕ₂dx = ∫ϕ₂ H_elϕ₁dx, a perturbation which, in the Landau–Zener theory is assumed to be constant and equal to V₁₂ in the narrow avoided crossing (or pseudocrossing) region [14, p. 151]. The energies W_± of the adiabatic curves are then given by:

where 2V₁₂ is the separation of the levels due to as shown in Fig. 5.7 on p. 150 of Ref. [14] or on Fig. 8.9(b) on p. 173 of Ref. [52] and in the interaction region of the avoided crossing the wave functions representing the two states system become linear combinations of the resonant forms, that is,

      Using perturbation theory involving time, it is possible to show that the probabilities P₁ and P₂ of finding the system in state 1 and 2, respectively, at time t are:

      that is, “We see that these probabilities vary harmonically between the values 0 and 1. The period of a cycle (from P₁ = 1 to 0 and back to 1 again) is seen to be h/2V₁₂ and the frequency 2V₁₂/h, this being . . . just 1/h times the separation of the levels due to the perturbation” [53, p. 323], see also Ref. [16, p. 534 ff.]. The situation is then the following: if the system were to remain static in the pseudocrossing region, there would be no final electron transfer: the electron would simply jump back and forth between the two resonant structures 1 and 2. But the system moves across the pseudocrossing region with some velocity, the electronic and nuclear motions are coupled in the avoided crossing region, both PECs govern the dynamics there through their splitting and through the difference of their slopes and following the Landau–Zener theory there may be two possible outcomes. If the system will move very rapidly across the interaction region, the electronic cloud will not have time to change from structure 1 to structure 2, no ET will happen, the system will simply go along a diabatic curve. But if the system will pass through the pseudocrossing at a velocity such that the electron will jump from 1 to 2 but it will not have time to go back, then we shall have an electron transfer. As a matter of fact in the so-called Massey parameter where l(R_c), the linear dimension of the pseudocrossing region, is “a characteristic length over which the corresponding electronic wave-functions substantially change and u some average (classical) velocity of the nuclei at point R_c” [14, p. 21, 22], we see that we have the ratio of the passage time of the nuclei through the interaction region to the transfer time h/V₁₂.

      “We may state the general rule that exchange (resonance) integrals will tend to be large only if the orbitals concerned overlap effectively” [16, p. 298]. Which means that the smaller the distance R between the colliding partners the greater the orbitals overlap, the greater the exchange (resonance) integrals, that is, the greater V₁₂ and the greater the splitting 2V₁₂ between the adiabatic curves at the pseudocrossing. An interesting example is presented by the PECs for the system HF and H⁺F⁻ reported on Fig. 3.4, p. 74 in Ref. [4]. There we see that the crossing of the ionic and covalent curves happen at somewhat less than 1Å and the splitting is then very large. On the other hand, the crossing of the curves for NaCl and Na⁺Cl⁻ happens at about 10.15Å and the splitting is very small, as reported on p. 536 with Fig. 14.3 on p. 537 of Ref. [16]. The first case is an example of Pauling’s resonance between Lewis structures, while the second is an example of the weak interactions considered in Marcus’ theory of electron transfer. Another example is that, previously cited in the text, of the LiF system. In the case of Pauling’s resonance, the frequency with which the electron jumps between the two structures is “the frequency of resonance among structures . . . is very large, of the order of magnitude of electronic frequencies in general . . .” [4, p. 186] so high that there is no chemical equilibrium between the two electronic tautomers and the resonance between the Lewis structures is sometimes indicated with a double-pointed arrow, a symbol suggested by Fritz Arndt and Bern Eistert to indicate resonance [4, p. 187]. We then have but in the case of electron transfer reaction supposed to happen at fixed internuclear distance there is a real chemical equilibrium between the electronic tautomers which exchange the electron at a lower frequency than that in the case of the usual Pauling resonance.

      In the first paper on electron transfer M. considered the states X^∗ and X to be two forms of the activated complex, a static formulation then like the one considered above, and abandoned since 1964 in favor of the dynamic Landau–Zener treatment of ET probability.

In the first paper M. considered two methods of calculating the splitting 2V₁₂, in one of them making use, as above, of time-dependent perturbation theory and in the other, already used by previous authors [54, 55] considering the electron tunneling of the transferred electron, as explained below.

Once the Na + Cl colliding system has reached the pseudocrossing region, in order to transfer from Na to Cl the electron must pass through an electronic potential energy barrier due to the attraction exerted on the leaving electron by the Na⁺ ion left behind. The penetration of the barrier is classically impossible but “It is possible in quantum mechanics to sneak quickly across a region which is illegal energetically” [56, 8.12]. So that the electron tunnels through the potential barrier separating them: In Fig. 7*(b), I have used a wiggly arrow to symbolically represent the electron “worming its way”—in the words of Feynman [56, 10.2]—through the barrier.⁽⁸⁾ In Fig. 7*(a) and 7*(c), the potential wells are represented, as in Ref. [24], Fig. 6.4, p. 153 for the symmetric electronic potential energy barrier to be penetrated for electron transfer reaction (1.3) when H and H⁺ are at a large enough distance, in Ref. [57], Fig. 3.6, p. 58 for electron tunneling between two metal nuclei, and in Ref. [36], Fig. 3.2, p. 91. In Fig. 7*(a), the electron symbolically represented by an arrow