State-selected reactions

In a third step the S-matrix is related to state-selected reaction cross sections a., in principle observable in beam scattering experiments [28, 29, 30, 31, 32, 33, 34 and 35], by the fiindamental equation of scattering theory... 

The methodology presented so far allows the calculations of state-to-state. S -matrix elements. However, often one is not interested in this high-level of detail but prefers instead to find more average infomiation, such as the initial-state selected reaction probability, i.e. the probability of rearrangement given an initial state Uq. In general, this probability is... 

One of the most in ortant properties of zeolites is their ability to cany out shqie selective reactions [5]. These can be cl sified as, firstfy, product shape selective reactions in which the only products formed are those which can diffiise out of e pores of die zeolite, second, reactant shape selective reactions which occur when some of the molecules in a reactant mbcture are too large to diffiise through the catalyst pores, and, thirdfy, restricted transition-state selective reactions in which the only reactions which occur are those in which qiace exists in the pores or cavities to allow the formation of the activated transition state con lex. In some cases where the zeolite is three dimensional the gze of the channel intersections will also be a determining ictor. This unique catalytic property is related to the pore size of the zeolite and has led to the synthesis of zeohtes with a very w e range of pore gzes. 

We can often understand the dynamics in greater detail by studying 7-specific state-selected reaction probabilities P n (E), which are related to 7-specific state-to-state reaction probabilities via... 

We also find it useful to define the corresponding densities of state-selected reaction probability p ( ) ... 

Assignment of the remaining fitted features was made [14] largely on the basis of densities of state-selected reaction probability, p° and pln, presented in Fig. 5. Each of the panels (b) through (e) and (g) through (j) displays densities of state-selected reaction probability for a given value of v or v and for each value of j or j open up to 1.9 eV. 

Many but not all of the quantized transition states observed in the densities of state-selected reaction probability are observed as peaks in the total density of reactive states. Some highly bend excited states (e.g., [0 12°], and [0 14°]) are observed as peaks only in the state-selected dynamics. If the closely spaced features in the stretch-excited manifolds for p(i are indicative of supernumerary transition states more closely spaced in energy than the variational transition states (which adiabatic transition state theory also suggests), then only some of the supernumerary transition states, in particular S[20°], 5[22°], 5[24°], 5[30°], 5[32°], 5[34°], and 5[36°], are observed in the total density, i.e., only some are of the first kind. The other supernumerary transition states identified in the state-selected dynamics are of the second kind. 

We discussed the implications of the O + H2 reaction s multiple bottleneck regions in terms of variational and supernumerary transition states. We related the observed features to the scattering results for asymmetrical Eckart potentials. We emphasized that global control is maintained to very high energy (1.9 eV) and very high levels of v2. We demonstrated the influence of quantized transition states at the level of state-selected reaction probability for this reaction. 

The discussion so far has concentrated on the calculation of bimolecular rate constants for gas-phase reactions under thermal conditions. Many extensions, e.g., unimolecular reactions [10], microcanonical ensembles [8j,81,10,29], and state-selected reactions [19c,30] are described elsewhere but are not reviewed here. 

Ervin, K. M. Armentrout, P. B. Spin-orbit State-selected Reactions of Krfi Pj and... 

However, being interested in less detailed information, one might ask for the probability of reaction if the reactants are prepared in a given initial state Up. These initial state-selected reaction probabilities are obtained by summing all state-to-state transition probabilities corresponding to the initial state... 

In contrast to S-matrix elements, these observables do not depend on the entire potential energy surface. Pn is infiuenced only by the potential energy surface on the reactant side of the reaction barrier and the region in the vicinity of the barrier. The potential energy surface further on the product side has no impact on pn It only decides upon the specific quantum states in which the products are formed. Thus, the calculation of initial state-selected reaction probabilities only requires the treatment of a reduced portion of the potential energy surface. It causes less numerical effort than a full S-matrix calculation. 

We discuss thoroughly in Chapter 4 the feasibility of time independent approaches for computing the ABC Green s function. We eventually come full circle, in the end developing a technique called the Newton algorithm similar in spirit to the PSG, but vastly improved in efficiency. We test the Newton algorithm on the calculation of initial state selected reaction probabilities for the three dimensional D-I-H2 reaction, and find both rapid convergence and strict accuracy control. 

Figure 4.9 The initial state selected reaction probability for D+H2(u,i) — DH+H at the total energy E = 0.9 eV, from all energetically accessible initial states. The solid hnes are the present calculation, and the dashed Hnes are from the 5-matrix Kohn variational principle calculations of Groenenboom and Colbert, in which the state-to-state reaction probabiHties are summed for comparison. The larger probabilities are from the u = 0 vibrational state, and the smaller (multiplied by 20) are from the = 1 vibrational state. Excellent agreement is obtained for all initial states, even those with small probabilities which are most challenging for the ABC method.

We have derived the ABC formulation of quantum reactive scattering theory, and applied it to the calculation of the initial state selected reaction probability. By exploiting the highly localized nature of forces in reactive scattering, the ABC formulation facilitates the direct calculation of detailed or averaged reaction probabilities while sampling only a finite region of space. 

The Newton algorithm wa.s applied to calculating initial state selected reaction probabilities for three dimensional D-fH2(u,i) — DH-I-H with zero total angular momentum. We found that the probabilities with initial j = 1 were the largest, and attributed this effect to a small amount of orbital angular momentum helping to focus the system into more reactive geometries. 

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