υe is estimated as υe ≈ 1au ≈ 108 cm/s” [13].
A slow electronically adiabatic diatomic collision is a collision such that “the electron cloud is able to adjust its state instantaneously to the nuclear framework” and, on the other hand, “the nuclear motion will not be influenced by the momentary spatial arrangement of the single electrons but only by the mean force field (averaged over many periods of the motion) of the whole electron cloud” [14, p. 7]. To discuss the energetics of such a collision, it is necessary to represent the total electronic energy of the Na + Cl system as a function of the internuclear distance. By total electronic energy, we mean the sum of the electronic energy of the electrons plus the repulsive potential energy of the nuclei [15, p. 3]. The curve representing this function is called adiabatic potential energy curve (PEC), or adiabatic molecular interaction potential, because the total electronic energy plays the role of potential energy governing the motion of atoms, that is, of the nuclei and of the attached electron clouds, when discussing the dynamics of the process, see, for example, Refs. [14, 15]. Such a curve is schematically shown by the lowest solid line in Fig. 1∗, [cf. 16, Fig. 14.3, p. 537], we see that it is almost a straight line parallel to the axis of the internuclear R distances until, at Rc ∼ 10.15Å, it suddenly turns down changing into the segment of hyperbola which goes down in energy until it changes into the segment of a parabola around the minimum. Beyond that, the curve goes up steeply to higher energies. Let’s now explain this behavior. Starting with atoms infinitely far apart and with zero kinetic energy, that is, in their asymptotic state, the potential energy of the system decreases very slightly, remaining almost constant, because between neutral atoms there is only a small Van der Waals attraction. At internuclear distances in the neighborhood of Rc it so happens that an electron transfer may occur and Na + Cl changes then into Na+ + Cl−. The two ions will attract each other with an electronic potential energy equal to:
Fig. 1∗. (Adapted from Ref. [22]).
This function represents a hyperbola. We mark the segment of the curve corresponding to R > Rc with a C (from “covalent”) and the segment for R < Rc with an I (from “ionic”). When the atoms are close enough, a Pauli repulsion (exchange repulsion) sets in between the atomic cores and the combination of Pauli repulsion, of nuclear repulsion and of nuclear-electronic attraction gives the segment of parabola around the minimum. At minimum energy, the molecule Na+Cl− is formed, whose bonding distance we denote with Rb. Going to shorter R, the Pauli repulsion—to which one should also add the internuclear repulsion—overwhelms the attraction and the curve shows a steeply rising repulsive branch. At the minimum potential energy, the system reaches its maximum stability and the parabola at the bottom of the curve describes the low energy vibrational motion of the Na+Cl− molecule like that of a harmonic oscillator. As a whole we can then symbolically describe in the following way the process of formation of the ionic bond:
Notice that the systems above are considered each with a total energy equal to the potential energy at the various values of R. Na and Cl or Na+ and Cl− are fixed at the various distances R. The symbols are not describing a scattering process in which kinetic energy would be present.
The description of the process using only the lower PEC in the figure is unsatisfactory because we cannot explain why the nature of the two branches of the lower solid curve changes right around R ∼ Rc, that is, why Rc has that particular value and why just around there an ET becomes possible. Moreover, a question comes naturally to mind: how would one describe the formation of NaCl starting not with atoms but rather with the ions Na+ and Cl−? To describe this process and to answer the above two questions, we need to consider also another PEC, the upper solid line in Fig. 1*.
At the right panel side of the upper curve, we see the system of the ions in their asymptotic state. Their energy differs from that of the atoms by the difference of the ionization potential of Na, IP (Na), and the electron affinity of Cl, EA (Cl). These energies correspond to the processes:
The energy associated with the process (Na + Cl)R=∞ → (Na+ + Cl−)R=∞ is IP − EA and the ET is radiative at large distances between the atoms because a photon is needed to ionize the Na atom. The new PEC correlated to (Na+ + Cl−)R=∞ begins with a hyperbola of equation:
using as zero of energy the energy of the asymptote of the lower solid curve. The upper curve decreases to reach Rc, in whose neighborhood an ET may happen:
and the curve flattens because the strong Coulombic attraction has disappeared due to the formation of neutral atoms. Going to shorter distances, the Pauli and nuclear repulsions set in, and the curve goes up steeply. Even in this case a letter “I” marks the branch of the curve corresponding to the ions and a “C” the one corresponding to the atoms.
Now we can understand why the ET reaction (1.11) occurs right in the neighborhood of Rc. It so happens because at that interatomic distance in the narrow region of the avoided crossing shown in the figure the systems Na+ + Cl− and Na + Cl may have the same energy because there the large amount of energy required to transfer an electron from Na to Cl forming Na+ and Cl− is completely counterbalanced by the mutual Coulomb energy of the ions [4, p. 73] and it is therefore possible to go from the atoms to the ions and vice versa without the help of photons making up for the energy difference between reagents and products. We have here an example of a resonant ET. In reaction (1.7), we have an example of a thermal resonant ET because the resonant condition is reached thanks to an appropriate thermal fluctuation of the solvent.
The second question—about the possibility of forming the molecule starting from the ions—has thus far received a negative answer: if we move only on the higher curve, we cannot directly go from NaCl in its electronic ground state. The two curves we have been considering until now are “adiabatic” curves, the meaning of the term “adiabatic” being explained shortly below. They can be obtained by accurate ab initio quantum mechanical calculations, in which the total electronic energy at each point of the curve is computed at the fixed nuclear distance given by the abscissa of the point and each curve has either a “C” character before the avoided crossing and an “I” character after the covalent–ionic crossing or vice versa. Looking at the two