Radiationless potential energy curves

Besides a transition to a continuum level of an excited electronic state, dissociation can occur by another mechanism in electronic absorption spectroscopy. If the potential-energy curve of an excited electronic state A that has a minimum in UA(R) happens to be intersected by the U(R) curve of an unstable excited state B with no minimum in U, then a vibrational level of A whose energy lies near the point of intersection of UA and UB has a substantial probability to make a radiationless transition to state B, which then dissociates. This phenomenon is called predissociation. Predissociation shortens the lifetimes of those vibrational levels of A that are involved, and therefore by the uncertainty principle gives broad vibrational bands with rotational fine structure washed out. 

Predissociation is governed not only by the intersection of the potential energy curves (Franck-Condon principle) but by the selection rules which specify the types of state between which transitions may take place. These are treated fully by Herzberg. Accidental predissociation is said to occur when the dissociation takes place by two radiationless transitions via the intermediacy of a third state. 

Figure 5.7. Franck-Condon factors for radiationless transitions between different potential energy curves of a diatomic molecule a) for a large and b) for a small energy gap, such as those observed, for instance, between S and S, or between S, and S respectively, and c) for the case that the potential energy curves (e.g., S, and T,) cross.

This review shows how the photochemistry of ketones can be rationalized through a single model, the Tunnel Effect Theory (TET), which treats reactions of ketones as radiationless transitions from reactant to product potential energy curves (PEC). Two critical approximations are involved in the development of this theory (i) the representation of reactants and products as diatomic harmonic oscillators of appropriate reduced masses and force constants (ii) the definition of a unidimensional reaction coordinate (RC) as the sum of the reactant and product bond distensions to the transition state. Within these approximations, TET is used to calculate the reactivity parameters of the most important photoreactions of ketones, using only a partially adjustable parameter, whose physical meaning is well understood and which admits only predictable variations. 

Figure 7. Barrier widths (horizontal lines with double arrows) for radiationless transition for cases in which the vibronic coupling is weak (left), strong (center), and very strong (right). (From ref. [32] with permission.) The dashed curves represent the potential energy curves in the absence of vibronic coupling.

The theory of dissociative recombinadon between diatomic positive ions and electrons was formulated many years ago (Bates 1950). The process takes place through a radiationless transition in which the frre dectron enters a bound orbital of a state having a rqiulsive potential energy curve (Figure 1) so that the two atoms move apart and thoeby prevent die inverse radiationless transition from occurring ... 

The vibrational wavefunction depends only on the nuclear coordinates qnuc. The nuclei move in a fixed electronic potential that is often approximated as a harmonic potential (dashed curve in Figure 1.10) to determine vibrational wavefunctions near the potential s bottom. The electronic wavefunction carries the complete information about the motion and distribution of electrons. It still depends on both sets of coordinates qe and qnuc, but the latter are now fixed parameters rather than independent variables the nuclei are considered to be fixed in space. The BO assumption is a mild approximation that is entirely justified in most cases. Not only does it enormously simplify the mathematical task of solving Equation 1.5, it also has a profound impact from a conceptual point of view The notions of stationary electronic states and of a PES are artefacts of the BO approximation. It does break down, however, when two electronic states come close in energy and it must be abandoned for the treatment of radiationless decay processes (see Section 2.1.5). 

Another important factor, considered by Franck and Sponer(i2) and by Herzberg(i6), is the relative position of the curves A and P. Of the vibration-rotation levels of B with J — 0, i.e. in the non-rotating molecule, only for those which lie at about the same height as an intersection point of the two energy curves A and B and simultaneously above PP is the chance of radiationless transitions considerable. For the vibration-rotation levels of B with J 0 we must first add to the energy curves A and B the potential energy of the centrifugal forces h J J+l)... 

The radiationless decay of a quasidiscrete excited state of an atom or molecule into an ion and electron of the same total energy is called autoionization. The quasidiscrete state must, of course, lie above the first ionization potential of the atom or molecule. The occurrence of autoionization may be inferred from the appearance of absorption spectra or ionization cross-section curves which exhibit line or band structure similar to that expected for transitions between discrete states. However, in the case of autoionization the lines or bands are broadened in inverse proportion to the lifetime of the autoionizing state, as required by the uncertainty principle. In the simple case of one quasidiscrete state embedded in one continuum, the line profile has a characteristic asymmetry which has been shown to be due to wave-mechanical interference between the two channels, i.e., between autoionization and direct ionization. In an extreme case the line profile may appear as a window resonance, i.e., as a minimum in the absorption cross section. 

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