Molecular Imaging. Markus Rudin

coordinate around R_c, an “avoided crossing pattern” or “avoided crossing seam” limited by two short branches of hyperbola showing a narrow energy splitting between them. The reason for this is explained, for instance, in [14, pp. 123 ff.]. We want only mention here that the splitting is due to the “Pauling resonance” [17, p. 204], see “the prototype case of

[17, p. 204] in Ref. [4, p. 14 ff.] (and see Appendix) between the ionic and covalent forms of NaCl, usually symbolized

which happen to be in a “perfect” resonance at R_c because at that R the two forms have the same energy. Joining now the two “I” branches with a dashed line along a hyperbola’s asymptote, we have a new curve which is called the ionic “diabatic” curve and which away from the avoided crossing seam “will merge and be essentially the same” [14, p. 153] with two branches of the adiabatic curves, see Fig. 1* [cf. 16, Figs. 14.2 and 14.3, p. 537]. Likewise, the two C branches and a dashed line connecting them across the avoided crossing region represent the “diabatic” covalent curve. The two C branches merge out of the avoided crossing seam and become essentially the same with two branches of the adiabatic curves. If we now imagine starting with (Na⁺ +Cl⁻)_R=∞ and follow the ionic diabatic curve, jumping from the above adiabatic curve to the one below in the avoided crossing region, is then possible to go from the ions directly to the molecule, with its mainly ionic bond, without the need of an intermediate ET. Processes like the first two we considered, in which the system follows a single adiabatic PEC, are called “adiabatic.” The word “adiabatic” is almost a transliteration from the ancient Greek “αδιάβατoς” meaning “nonpassable,” “noncrossable.” Those in which the system follows first a branch of an adiabatic curve, jumps then on another adiabatic curve across an avoided crossing and subsequently follows the second curve, are called “nonadiabatic.” The nonadiabatic processes follow a diabatic curve. “Diabatic” comes from “διαβαíνω,” “I pass.” In other words, the ET process happens when the system follows an adiabatic curve, and it doesn’t if it follows a diabatic curve. If the system follows a branch of one of the two adiabatic curve and then, in the interaction region of the avoided crossing, jumps to the other adiabatic curve, we have a nonadiabatic process, no ET has happened, it is as if the process had happened along a single diabatic curve.

      The question as to when we have nonadiabatic processes and when adiabatic ones is answered by the celebrated Landau–Zener–Stueckelberg–Majorana formula for nonadiabatic transition probabilities:

The formula is more usually designated simply as the Landau–Zener formula. For a discussion see Refs. [14, 16, 18, 19]. A short summary of the Landau–Zener theory is given in Ref. [18]. In Eq. (1.16), S_c ≡ S(R_c) is the splitting between the adiabatic curves at the avoided crossing, equal to twice the resonance interaction between the resonant structures NaCl and Na⁺Cl⁻ when Na and Cl (and likewise Na⁺ and Cl⁻) are at a distance R_c (see Appendix). F = F_ion − F_cov where F_ion and F_cov are the slopes of the ionic and covalent diabatic curves at R_c and υc is the velocity with which the system passes across the avoided crossing region. Looking at Eq. (1.16) we see that “in those region of the nuclear configuration space where adiabatic potential surfaces are close together or intersect, where electronic wave-functions are changing very rapidly for varying nuclear coordinates, where nuclei are moving with high velocity, non-adiabatic transitions become probable” [14, p. 22], and so if the system passes through the avoided crossing with high velocity, it jumps through the splitting with a nonadiabatic transition probability close to 1 and no ET happens because the electronic structure of Na + Cl doesn’t have time to change to that of Na⁺ + Cl⁻ and the system follows the diabatic path of the covalent PEC, that is, for collisions velocities high enough the diabatic terms can be interpreted as potential surfaces which govern the motion of the nuclei [14, p. 153]. On the other hand, when Na and Cl approach each other with a vanishingly small velocity, the probability P of a nonadiabatic jump from the lower to the upper curve in the avoided crossing region is almost zero and so the probability 1 − P of staying on the lower curve is almost 1. This probability is the probability of the adiabatic ET reaction Na + Cl → Na⁺ + Cl⁻.

The term “diabatic” has been introduced in the physical literature only since 1963 by Lichten [20]. Kauzmann [16] calls the adiabatic and diabatic curves “slow” and “fast” curves. A recent description of diabatic curves is given in Ref. [21].

The above adiabatic curves do not cross because of the Wigner–Witmer noncrossing rule [13, p. 563]. Similar curves are also reported on p. 77 of Ref. [4]. There Pauling used valence bond calculations and the curves cross because of the approximate calculations. This is why the avoided crossing is also designated as pseudocrossing, that is, false or spurious crossing. Pauling’s valence bond energies are diabatic. The curves are also reported on p. 372 of Ref. [22]. Note that noncrossing rule is only true for diatomics. Conical intersections are the rule for polyatomics.

One very important point needs here to be emphasized. We observe that the crossing point distance R_c ∼ 10.15Å is much larger than the sum of covalent radii of the atoms in NaCl (∼2.53Å) or of the ionic radii in Na⁺Cl⁻ (∼2.45Å) [15]. This means that the charge transfer inducing interaction, measured by the small energy splitting between the curves around R_c, and proportional to the overlap of the orbitals of the atomic wave functions, is a weak electronic interaction (see Appendix) if compared to the strong chemical interaction typical of the covalent bonds, where there is a much greater orbitals’ overlap. A. C. Wahl et al. [23] in a calculation reported in Refs. [15, 24] showed that when the lithium and fluorine atoms approach, to form the lithium fluoride molecule, an ET occurs at R_c ∼ 7.35Å and the lithium cation and fluorine anions form. In this case, that distance is about four to five times larger the sum of the atomic or ionic radii!

      I want to end up this section citing the fascinating study by A. Zewail of the reaction, analogous to Eq. (1.11)

      In Ref. [25, p. 264], one finds that curves similar to the ones discussed above have been used to follow processes (1.17) studied by femtoseconds