Bimolecular superpositions

Chapter 5 deals with derivation of the basic equations of the fluctuation-controlled kinetics, applied mainly to the particular bimolecular A + B 0 reaction. The transition to the simplified treatment of the density fluctuation spectrum is achieved by means of the Kirkwood superposition approximation. Its accuracy is estimated by means of a comparison of analytical results for some test problems of the chemical kinetics with the relevant computer simulations. Their good agreement permits us to establish in the next Chapters the range of the applicability of the traditional Waite-Leibfried approach. 

As was demonstrated by Kikuchi and Brush [88], using the Ising model as an example, an increase of mo in the expansion in the form secures the monotonic approach of the calculated critical parameters to exact results, except for the critical exponents which cannot be reproduced by algebraic expressions. It is important to note here that the superposition approximation permits exact (or asymptotically exact) solutions to be obtained for models revealing the critical point but not the phase transition. This should be kept in mind when interpreting the results of the bimolecular reaction kinetics obtained using approximate methods. 

A general expression for the superposition approximation (2.3.55) has to be specified for a reaction under study. For instance, let us do it for the actual case of the bimolecular reaction employing many-particle densities Pm,m Single-particle densities are nothing but macroscopic concentrations... 

Kuzovkov and Kotomin [89-91] (see also [92, 93]) were the first to use the complete Kirkwood superposition approximation (2.3.62) in the kinetic calculations for bimolecular reaction in condensed media. This approximation allows us to cut off the infinite hierarchy of equations for the correlation functions describing spatial distribution of particles of the two kinds and to restrict ourselves to the treatment of minimal set of the kinetic equations which realistically could be handled (Fig. 2.21). In earlier studies [82, 84, 91, 94-97] a shortened superposition approximation was widely used... 

In this Section we consider again the kinetics of bimolecular A + B -A 0 recombination but instead of the linearized approximation discussed above, the complete Kirkwood superposition approximation, equation (2.3.62) is used which results in emergence of two new joint correlation functions for similar particles, Xu(r,t), v = A,B. The extended set of the correlation functions, nA(f),nB(f),Xfi,(r,t),Xa(r,t) and Y(r,t) is believed to be able now to describe the intermediate order in the particle spatial distribution. 

All the above-said demonstrates well that there are arguments for and against applicability of the superposition approximation in the kinetics of bimolecular reactions. Because of the absence of exactly solvable problems, it is computer simulation only which can give a final answer. Note at once some peculiarities of such computer simulations. The largest deviations from the standard chemical kinetics could be expected at long t (large ). Unlike computer simulations of equilibrium phenomena [4] where the particle density is constant, in the kinetics problems particle density n(t) decays in time which puts natural limits on time of reaction. An increase of the standard deviation at small values of N(t) = (N) when calculating the mean concentration in computer simulations compel us to interrupt simulations at the reaction depth r = Io 3, where... 

Computer simulations of bimolecular reactions for a system of immobile particles (incorporating their production) has a long history see, e.g., [18-22]. For the first time computer simulation as a test of analytical methods in the reaction kinetics was carried out by Zhdanov [23, 24] for d, = 3. Despite the fact that his simulations were performed up to rather small reaction depths, To < 1, it was established that of all empirical equations presented for the tunnelling recombination kinetics (those of linear approximation - (4.1.42) or (4.1.43)) turned out to be mostly correct (note that equations (5.1.14) to (5.1.16) of the complete superposition approximation were not considered.) On the other hand, irrespective of the initial reactant densities and space dimension d for reaction depths T To his theoretical curves deviate from those computer simulated by 10%. Accuracy of the superposition approximation in d = 3 case was first questioned by Kuzovkov [25], it was also... 

In summary, we have shown that the kinetics of the bimolecular reaction A + B —> 0 with immobile reactants follows equation (6.1.1), even on a fractal lattice, if d is replaced by d, equation (6.1.29). Moreover, the analytical approach based on Kirkwood s superposition approximation [11, 12] may also be applied to fractal lattices and provides the correct asymptotic behaviour of the reactant concentration. Furthermore, an approximative method has been proposed, how to evaluate integrals on fractal lattices, using the polar coordinates of the embedding Euclidean space. 

In our opinion, this book demonstrates clearly that the formalism of many-point particle densities based on the Kirkwood superposition approximation for decoupling the three-particle correlation functions is able to treat adequately all possible cases and reaction regimes studied in the book (including immobile/mobile reactants, correlated/random initial particle distributions, concentration decay/accumulation under permanent source, etc.). Results of most of analytical theories are checked by extensive computer simulations. (It should be reminded that many-particle effects under study were observed for the first time namely in computer simulations [22, 23].) Only few experimental evidences exist now for many-particle effects in bimolecular reactions, the two reliable examples are accumulation kinetics of immobile radiation defects at low temperatures in ionic solids (see [24] for experiments and [25] for their theoretical interpretation) and pseudo-first order reversible diffusion-controlled recombination of protons with excited dye molecules [26]. This is one of main reasons why we did not consider in detail some of very refined theories for the kinetics asymptotics as well as peculiarities of reactions on fractal structures ([27-29] and references therein). 

The variety of manifestations in time of coherent development of molecular dynamics also includes such phenomena as mono- and bimolecular chemical reactions. Thus, Seideman et al [342] suggest the idea of governing the yield of a reaction by suddenly creating coherent superposition of two states of the transient complex and applying a second pulse with fixed delay for the dissociation of the complex. The appearance of coherent beats in femtochemistry , in particular, at photodissociation, has been analyzed by Zewail (review [404]). 

Control of bimolecular collisions is achieved by constructing an initial state E, q, am ) composed of a superposition of N energetically degenerate asymptotic states E, q, m 0) ... 

This scenario opens up a wide range of possible experimental studies of control bimolecular collisions. Specifically, we need only prepare A and A in a control superposition of two states [e.g., by resonant laser excitation of A(1))] to produf superposition with r/>A(2)), direct them antiparallel in the laboratory, and vary t coefficients in the superposition to affect the reaction probabilities. Control -originates in quantum interference between two degenerate states associated wiili, r the contributions of 0A(1)) A<(2)> and I[Pg.154]

We can readily extend bimolecular control to superpositions composed of more than two states. Indeed, we can introduce a straightforward method to optimize the reactive cross section as a function of am for any number of states [252], Doing so is an example of optimal control theory, a general approach to altering control parameters to optimize the probability of achieving a desired goal, introduced in Chapter 4. 

Optimized Bimolecular Scattering Total Suppression of ReadthA Event In Section 7.3.1 we considered optimizing reactive scattering by varyj the coefficients a,- of a superposition of states. In this section we show that whefi j number of initial open states in the reactant space exceeds the number of open st8jj ... 

The spectrum of Nph form on aerosil is not resolved. The wide-band fluorescence contribution relative to the molecular emission is large, afterglow is not observed. The wideband excitation spectrum at 400 nm is shifted relatively to that of a molecular form by 10 nm. For zeolites this is mainly CTC and for aerosil a bimolecular associate of Nph. When adsorbing from a vapor phase, the emission spectrum of Nph in a zeolite consists of a continuous structureless band which is a superposition of CTC and dimers adsorbed at the outer surface (Fig. 3a).In the case of co-adsorption of water vapor or hexane the spectrum transforms with the appearance of structured fluorescence and phosphorescence components (Fig.3b). The coadsorbate seems to promote breaking up of dimers and diffusion of molecules in zeolite cages. 

Formally, this process may be thought of as a superposition of bimolecular activation and unimolecular decay. The bimolecular step occurs in the collisional activation of an AB+ ion above one (or more) of its dissociation thresholds A+ fragment ions appear as products of the unimolecular decay of the activated ion. In the process noted above, Rg is often a rare gas atom. 

The functions in equations (95) and (96) are monotone, increasing with increasing z, but T(z)fko increases much more rapidly than T(z)cm- These dependences appear to represent extremes in behavior a number of other proposed approximate functions give plots vs, z that lie between these two cases. It will be noted that these two functions also correspond to extremes in symmetry of models for the bimolecular cluster, i.e, the interpenetration of two soft spheres in the FKO model describes inherently asymmetric configurations of segments, while the CM model makes every configuration a spherically symmetrical superposition of the segments of the two participant chains. 

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