Predissociation selection rules

Not only do the experimental vibrational predissociation lifetimes require interpretation, so do the increasingly sophisticated theoretical calculations whose results often fall out of a web of coupled differential equations or the convoluted algebra of quantum mechanics. In order to offer a qualitative overview of dynamical processes in van der Waals molecules, we shall introduce a selection rule which can provide insight into possible relaxation channels of vibrationally excited molecules. This selection rule concerns the change in a quantum number, Anj., which is to remain small for efficient vibrational predissociation processes. It bears a close analogy to the selection rules of optical spectroscopy which require small changes in quantum numbers Au, AJ, AS, etc. for efficient transitions between molecular states. Let us review the origin of the vibrational predissociation selection rule which has been developed in more detail elsewhere. ... 

The remark made previously about the applicability of the selection rules for predissociation reactions now becomes clearer, since these selection rules merely describe properties of the matrix element v2. That is, although no assumption about the decay process has been directly introduced, the lifetime against decay will take a form similar to that obtained from first-order, time-dependent perturbation theory, and therefore be proportional to p2v2. 

The probability of the radiationless transition leading to predissociation is governed by the Franck Condon principle and by a group of selection rules first given by Kronig (1930). One reason why predissociation is so widespread is that, especially for non-linear molecules, the selection rules are extremely permissive. A striking example of a forbidden predissociation from the linear 2H (2I I) Renner state of the HCO radical has been described by Ramsay (1959). 

Ewing, G.E. (1987). Selection rules for vibrational energy transfer Vibrational predissociation of van der Waals molecules, J. Phys. Chem. 91, 4662-4671. 

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. 

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. 

The selection rules and v, J-dependence of predissociation effects depend on the identity of the operator responsible for the predissociation. From knowledge of the selection rules, qualitative information can immediately be obtained from the variation of the total interaction with v or J. For example, if lines from low-7 levels are missing in emission, the predissociation is certainly not due to a gyroscopic (Afi 0) interaction, which would be zero for J = 0, but must arise from a homogeneous (Afl = 0) interaction. 

More precisely the two energy surfaces A and B will intersect each other along a curve. The chance of predissociation for a vibration-rotation level of B will be great—apart from the fulfilment of selection rules perhaps introduced by the symmetry properties of the molecule—if its energy is about the same as that of a point on the intersection curve. For then the vibratory motion of the molecule represented by a sort of Lissajous figure on the surface V pa) come somewhere near the line of intersection with the surface V (p, pa) of A, making it easy for the molecule to jump from the former to the latter surface. 

Fig. 7. Perturbation-Facilitated OODR fluorescence scheme used to obtain b IIy predissociation data. Perturbation and optical selection rules are indicated. Dashed lines between b II and a Zu" " levels depict Interactions which are forbidden by the selection rules cited adjauenLly to them.

We are now prepared to set down the effective quantum numbers for use of the selection rule expression of eq. 5. Application of the analytical expression for vibrational predissociation rates of A-B C for a wide variety of van der Waals molecules bound by Morse intermolecular potential functions like those shown in Fig. 2 reveals the effective translational quantum number change... 

Figure 3. The total quantum number change, An, and lifetimes, r, for vibrational predissociation. The line is the selection rule expression of eq. 5. Experimental measurements are indicated by the open circles described in the text.

The problem lies in the assumption required to derive the selection rule that the u=l and u=0 surfaces are the same shape and are merely displaced vertically as we have illustrated in Fig. 2. For HF HF on the contrary, the intermolecular potential is highly anisotropic and rotational excitation of the fragments results in an effective potential which is shallow and may actually cross other surfaces. This has been demonstrated in calculations of Halberstadt et al.. The surfaces taken from their work are shown in Fig. 4. The curve crossing yields relaxation times orders of magnitude more efficient than those calculated by our selection rule. It is a challenge to the theorists to model the predissociation process, consistent with experiment, that allows both HF molecules to rotate on fragmentation. Clearly anisotropic effects will play an important role in understanding vibrational predissociation in other systems as well-for example, in the electronically excited state of OH Ar by Lester et al.. ... 

It has been shown that Fermi resonances between chemical bond vibrational levels and van der Waals modes can dramatically reduce vibrational predissociation lifetimes. The selection rule becomes altered because the definitions of the quantum numbers become blurred by the Fermi resonances. This is illustrated in the recent study of Tiller, Peet and Clary and shown in Fig. 5. Here 5 0,0,0> mixes with nearby 17 0,0,10> whose vibrational predissociation lifetime is calculated to be two orders of magnitude shorter than the prepared state. 

Here NaCl(lOO) is the face of a single crystal of salt and the dots represent the physisorbed bond dominated by electrostatic contributions as in the case of van der Waals molecules. The vibrational energy in CO of 2100 cm exceeds the physisorbed bond strength of 1500 cm so predissociation is possible and relaxed CO flies away with kinetic energy AE 600 cm. For the same reason vibrational predissociation is inefficient for N2 -N2 so it is for the relaxation of eq. 12. Indeed attempts to photodesorb CO from NaCl(lOO) have failed and the quantum yield for this relaxation channel is < 10". Theory to account for relaxation channels from excited molecules physisorbed to surfaces is developing but the paucity of experimental data prevents a calibration of the calculations. The selection rule of eq. 5 can be easily extended to qualitatively account for photodesorption and becomes... 

In gas-phase dynamics, the discussion is focused on the TD quantum wave packet treatment for tetraatomic systems. This is further divided into two different but closed related areas molecular photofragmentation or half-collision dynamics and bimolecular reactive collision dynamics. Specific methods and examples for treating the dynamics of direct photodissociation of tetraatomic molecules and of vibrational predissociation of weakly bound dimers are given based on different dynamical characters of these two processes. TD methods such as the direct projection method for direct photodissociation, TD golden rule method and the flux method for predissociation are presented. For bimolecular reactive scattering, the use of nondirect product basis and the computation of the initial state-selected total reaction probabilities by flux calculation are discussed. The descriptions of these methods are supported by concrete numerical examples and results of their applications. 

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