Reduced dimensionality methods

Palma, J., Behave, J. and Clary, D.C. (2002) Rate constants for the CH4 + H- - CH3 + H2 reaction calculated with a generalized reduced-dimensionality method. J. Phys. Chem. A 106. 8256-8260. 

The results presented here are in snpport of earlier stndies which have correlated structure in the partial wave reaction probabilities and structure in the differential cross sections. Especially noteworthy among these earlier stndies are the ones of Redmon and Wyatt(21) who first suggested this correlation based on their j -conserving quantum calculations of the F- H2 -> HF-i-H reaction. He applied the CEQ version of the reduced dimensionality method to a study of that reaction( ) as well as the F- HD -> BF+D, DF+H reac-tions(9) and found similar correlations. More recently the BCRLN was used in an extensive study of differential cross sections in the... 

The previous studies of concerted hydrogen atom and proton transfer in hydrogen-bonded systems have been limited to studies of reaction pathways for simple model systems [4-8] with simple, reduced dimensionality methods for including quantum tunneling [13,14,34,35]. The applicability of modem computational methods to such systems is exemplified by more recent studies of the energetics of intramolecular hydrogen atom transfer in molecules such as malonaldehyde [36-39] and models of... 

A major challenge in quantum dynamics is to develop quantitatively accurate methods for practical computational study of chemical reactions involving polyatomic molecules. Currently, rigorous quantum dynamics calculations are limited to those systems involving no more than four atoms. In order to perform a quantitatively accurate quantum dynamics study for the vast majority of chemical reactions that are of chemical or biological interest, it is necessary to develop practical computational methods to treat the reaction dynamics of polyatomic molecules. To this end, some reduced dimensionality methods have been proposed to treat polyatomic systems (tetra-atomic systems in particular) by reducing the dynamical degrees of freedom from six to three. 

This reduced dimensionality method was applied to the H2 + CN reaction treating the reaction path as linear, and calculated rate constants compared quite favorably with experiment. It was also applied to the reaction ... 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

There are a number of alternatives and variations to the reduced mechanism method. The intrinsic low dimensional manifold (ILDM) approach [253] and similar methods [399] seek to decouple the fastest time scales in the chemistry. There is a wide range of time scales for chemical reactions in most high-temperature processes, from 10-9 second to seconds. Fast reactions, or reactions with small time scales, quickly bring composition points down to attracting manifolds in the composition space. Then composition points move along on manifolds. In the ILDM approach it is assumed that any movement of the... 

In this section we examine the primary transient phenomena that are of interest to SOFC analysis, and provide the fundamental model equations for each one. Examples for the use of these models are given in later sections. While the focus is on reduced-order models (lumped and one-dimensional), depending on the needs of the fuel cell designer, this may, or may not be justifiable. Each fuel cell model developer needs to ensure that the solution approach taken will provide the information needed for the problem at hand. For the goal of calculating overall cell performance, however, it is often that one-dimensional methods such as outlined below will be viable. 

Abstract. Over the last deeade, advances in quantum dynamics, notably the development of the initial state selected time-dependent wave packet method, coupled with advances in constructing ab initio potential energy surfaces, have made it possible for some four-atom reactions to be addressed from first principles, in their full six internal degrees of freedom. Attempts have been made to extend the time-dependent wave packet method to reactions with more internal degrees of freedom. Here, we review the full-dimensional theory for the A + BCD four-atom reaction and use it to guide the reduced-dimensionality treatment of the X + YCZ3 reaction. Comparison of rigorous calculations with recent experiments are presented for (a) the benchmark H + H2O abstraction reaction, and (b) the H + CH4 H2 + CH3 reaction. 

The time-dependent wave packet method has been outlined to solve the A + BCD and the X -I- YCZ3 reactions. The tetraatomic system could be solved in full-dimensionality, but to solve the hexaatomie system we followed the method of Palma and Clary,40 where the CD bond is replaced by CZ3 with C3V symmetry throughout the reaction, with reduced dimensionality. Caleulations were performed for the H -l- H2O reaction in full-dimensionality, and the H -l- CFL reaction in seven dimensions. 

Since the shape of one of the electrodes is varied during the machining, as is not known in advance, these problems belong to the class of problems with moving (free) boundaries, and their solution involves great difficulties [4-9], Therefore, approximate quasi-steady-state and local, one-dimensional methods, which enable one to reduce the ECM problems to those of known boundaries, are widely used [1-9]. 

Direct, easy methods are those ones which locate the resonances and only the resonances. Of course the scattering methods give the most detailed information possible about the scattering at a resonance and indeed the two methods can be used in a complementary fashion. Both types of calculations scale exponentially with the number of coupled degrees of freedom and therefore they can quickly become computationally intractable. Thus, for both the scattering and direct methods we have been interested in reduced-dimensionality strategies to reduce the number of coupled degrees of freedom. 

Considerable work has already been carried out using ab initio calculations to predict the photodissociation dynamics of gas-phase metal carbonyls (45). This is a fertile area for computational work, given the extensive experimental results available, which include the use of ultrafast methods to characterize the short time behavior in photoexcited states. There is considerable evidence that surface crossings, especially of a spin-forbidden nature, play a considerable part in the dynamics. Much of the theoretical work so far has focused on reduced-dimensionality models of the PESs, which have been used in quantum mechanical smdies of the nonadiabatic nuclear dynamics, in which spin-forbidden transitions are frequently observed (45). Here, too, the potential benefits to be derived from a proper understanding of the spin-state chemistry are considerable, due to the importance of light-induced processes in organometallic and bioinorganic systems. 

For larger hydrogen-bonded systems, rigorous calculations are far more difficult to carry out, both from the point of view of obtaining full-dimensional potentials and the subsequent quantum vibrational calculations. Reduced dimensionality approaches are therefore often necessary and several chapters in this volume illustrate this approach. With increasing computational power, coupled with some new approaches, it is possible to treat modest sized H-bonded systems in full dimensionality. We have already briefly reviewed the approach we have developed for potentials for the vibrations we have primarily used the code Multimode (MM). The methods used in MM have been reviewed recently [24 and references therein, 25], and so we only give a very brief overview of the method here. 

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