Bound potential curve

In indirect photodissociation processes, the initial absorption occurs into a bound discrete excited state which subsequently interacts with the continuum of a final dissociating state. The process of predissociation, in which the bound potential curve is crossed by a repulsive state of a different symmetry, is illustrated in Figure 16. The cross section consists in this case of a series of discrete peaks, broadened by the predissociation process. 

Figure B3.4.12. A schematic ID vibrational pre-dissociation potential curve (wide flill line) with a superimposed plot of the two bound fimctions and the resonance fimction. Note that the resonance wavefiinction is associated with a complex wavevector and is slowly increasing at very large values of R. In practice this increase is avoided by iismg absorbing potentials, complex scaling, or stabilization.

Promotion of an electron in Hc2 from the (7 15 to a bonding orbital produces some bound states of the molecule of which several have been characterized in emission spectroscopy. For example, the configuration ((J l5 ) ((7 l5 ) ((7 25 ) gives rise to the 2i and bound states. Figure 7.24(a) shows the form of the potential curve for the state. The A-X transition is allowed and gives rise to an intense continuum in emission between 60 nm and 100 nm. This is used as a far-ultraviolet continuum source (see Section 3.4.5) as are the corresponding continua from other noble gas diatomic molecules. 

The examples of ArF (193 nm), KrF (248 nm), XeF (351 nm), KrCl (222 nm), XeCl (308 nm) and XeBr (282 nm) indicate the range of wavelengths from excimer lasers. Because the ground states of these molecules are not totally repulsive but very weakly bound, there is a very shallow minimum in the potential curve, as illustrated in Figure 9.15. In the case of XeF the potential energy minimum is relatively deep, about 1150 cm and supports a few vibrational levels. As a result the laser may be tuned over several transitions. 

A computational method which is suitable for studies of this nature should fulfill certain basic requirements (a) it should be sufficiently economical to allow computation of full potential-energy curves for comparatively large number of states, (b) the calculated potential curves for bound states should give rise to vibrational and rotational constants which are in reasonable agreement with experiment when a comparison is possible, (c) the calculated total energies of all the states should be of comparable accuracy, and (d) the ordering of the states should be correct. 

The analysis of spectroscopic data for bound states of diatomic molecules gives accurate potential curves if one follows the semi-classical Rydberg-Klein-Rees method. For a review of this see Ref. 126). It is sufficient to note that this gives the two values of r as a function of potential energy by considering the dependence of the total spectroscopic energy on the vibrational and rotational quantum numbers n and J. A somewhat simpler procedure, and the only one plicable to polyatomic molecule, is to use the Dunham expansion of the potential 127). 

Fig. 1.2. Intermolecular potential curves and radiative transitions of the complex of molecules 1 and 2. The energy spacing AE = Ef — E-, is the difference of the rotovibrational energies of initial and final states of the complex, , = l7l +EV2j2 and Ef = E VlJl + E V2j2, respectively. The bound and free designate bound and free state energies of the complex a prime indicates final states. Also shown are representative radiative transitions hv from bound state to bound state, and from free state to free state, involving rotovibrational transitions in one or both molecules.

Theoretical distribution curves are synthesized from known potential curves (of the ground state, bound metastable, and repulsive excited states) of the molecular ion and from Franck-Condon transition probabilities. The necessity of including contributions from excited metastable states in the ion beam is indicated when a fit is obtained between the calculated... 

Our analysis of the valence states of 02 is based on available spectroscopic data and on several extensive sets of Cl calculations.24 A more recent study of the valence states is that of Beebe et al.55 A composite potential curve for the bound states of 02 is shown in Fig. 24. Separate curves showing all states for each symmetry, are given in Figs. 25 to 41. The molecular correlations for the valence states of 02 are given in Table... 

A composite potential curve for the bound stales of NO is shown in Fig. 42. Potential curves, including repulsive slates, are shown in Figs. 43 to 52 for each symmetry type. The low-lying molecular slates of NO and their dissociation limits are given in Table VII. The most important electronic... 

It is evident that the square wave charge-potential curves corresponding to surface-bound molecules behave in a similar way to the normalized current-potential ones observed for a soluble solution reversible redox process in SWV when an ultramicroelectrode is used (i.e., when steady-state conditions are attained), providing the analogous role played by 2sw (surface-bound species) and (soluble solution species), and also 2f (Eq- (7.93)) and the steady-state diffusion-limited current (7 css), see Sect. 2.7. This analogy can be made because the normalized converted charge in a surface reversible electrode process is proportional to the difference between the initial surface concentration (I ) and that... 

To illustrate this phenomenon, we return to the molecular hydrogen ion H2+. The ground vibrational state of the system is bound in the potential depicted in Figure 1.13. Suppose now that we expose the system to a monochromatic electromagnetic radiation with a frequency radiation field now couples between the ground electronic state and the excited electronic state of the system. The excited electronic state of the hydrogen molecular ion is a dissociative potential curve, which is well approximated by [48] ... 

Fig. 6.4. Schematic illustration of the multi-dimensional reflection principle in the adiabatic limit. The left-hand side shows the vibrationally adiabatic potential curves en(R). The independent part of the bound-state wavefunction in the ground electronic state is denoted by

Figure 23 Predissociation of the v = 2 vibrational level of the bound electronic state by a vibrational continuum wave function of the dissociative electronic state after radiative excitation (arrows) from the electronic ground state Po- The circle around the potential curve crossing point indicates an area of large overlap between the vibrational wave functions.

Figure 24 Model cases for the potential energy curve crossing between a bound and dissociative state, (a) The potential energy curves cross approximately at right angles. This is often the case when the dissociative state intersects the bound state on its outer limb, i.e., at bond distances longer than its equilibrium internuclear separation, (b) Bound and dissociative state exhibit similar slopes and cross on the inner limb of the bound state, (c) The dissociative state crosses the bound potential both on the inner and outer limbs.

Fig. 1. Potential curve for (4He)2 (---------). There exists no bound state for this complex.

The second type of predissociation observed for diatomic molecules is known as electronic predissociation the principles are illustrated in figure 6.28. A vibrational level v of a bound state E lies below the dissociation asymptote of that state, but above the dissociation asymptote of a second state E2. This second state, E2, is a repulsive state which crosses the bound state E as shown. The two states are mixed, and the level v can predissociate via the unbound state. It is not, in fact, necessary for the potential curves of the two states to actually cross. It is, however, necessary that they be mixed and there are a number of different interaction terms which can be responsible for the mixing. We do not go into the details here because electronic predissociation, though an important phenomenon in electronic spectroscopy, seldom plays a role in rotational spectroscopy. Since it involves excited electronic states it could certainly be involved in some double resonance cases. 

Recombination may also proceed via an electronically excited state if during the course of a bimolecular collision the system may transfer from the nonquantized part of the potential curve associated with one electronic state to a second state from which emission is allowed. This process is called preassociation or inverse predissociation, and the selection rules that control the probability of crossing in both directions are well known [109]. In such encounters total angular momentum must be conserved. For diatomic molecules, the system can pass only into the rotational level of the excited bound state which corresponds to the initial orbital angular momentum in the collision. 

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