Dimensionality dilemma

This paper reviews recent (and current) work in my research group which is aimed at developing practical methods for describing reaction dynamics in polyatomic systems in as ab initio a framework as possible. To overcome the dimensionality dilemma of polyatomic systems—i.e., the fact that the potential energy surface depends on 3N-6 internal coordinates for an N atom system—we have developed dynamical models based on the intrinsic reaction path", i.e., the steepest descent path which connects reactants and products through the transition state (i.e., saddle point) on the potential energy surface. ... 

In summary, the reaction path does not have direct physical meaning. It is an artificial chemical instrument, a fiction of chemical thinking, but it is extremely valuable in overcoming the dimensionality dilemma. The structure of the reaction valley can be characterized by the RP itself, and by the frequencies of the transverse directions (i.e. by the steepness of the "walls of the valley"). Theories using these kinds of information represent RP concepts. This often works well because the trajectories, or wave-packets, representing the chemical system, are concentrated within the valleys. 

But we have a dilemma. When comparing three-dimensional structures, what should be compared to what In the case of direct comparisons of citrate ions, there W s a central tetrahedral carbon atom in the molecule and all citrate ions could be laid on top of each other, as shown in Figure 16.1, with the central three carbon atoms superimposed. When, however, citrate and isocitrate are compared, it is less clear how the comparison should be made. In the example of aconitase substrates, it was made by reasoning that the enzyme probably holds on to the... 

Figure 1. Catch-and-escape dilemma of SA type algorithms for a discrete two dimensional function X2). To escape local extrema, a certain minimal step-width must be observed by the configuration generator. The finite stepwidth at the same time makes it difficult to approach the exact extreme, (a) GSA configuration generator searches at a predefined radius with respect to the rule...

To resolve this dilemma, some argue for divine creation some invoke the many n-dimensional Universes. Obviously, these latter explanations are far from scientific approaches. Another view is connected with recognition of the inherent limitations of the time estimates above, and propose that the need for an extraordinaryly long incubation period is obtained by invoking a plausibly small pool of amino acids, about 10 or so, short catalytic peptides, 10-20 residues, with some tolerance for sequence variability and non-random searches of only portions of the total sequence space due to the existence of chaotic attractors. Under these assumptions, a much slower reaction rate (for instance, lO -fold slower) in only a fraction of the ocean is more than sufficient to account for rapid, less then 1 million years evolution of complex, possibly living, organic systems. 

We have illustrated the problem of multiplicative noise by referring to motion in one dimension, however the dilemma is present, for practical purposes only for motion in several dimensions since a one-dimensional equation with multiplicative noise can always be transformed into an equation with additive noise. 

Adsorption energies on metals calculated in a cluster approach often show considerable oscillations with size and shape of the cluster models because such (finite) clusters describe the surface electronic structure insufficiently [257-260]. These models may yield rather different results for the Pauli repulsion between adsorbate and substrate, depending on whether pertinent cluster orbitals localized at the adsorption site are occupied or empty. The discrete density of states is an inherent feature of clusters that may prevent a correct description of the polarizability of a metal surface and thus hinders cluster size convergence of adsorption energies [257]. Even embedding of metal clusters does not offer an easy way out of this dilemma [260,261]. Anyway, the form of conventional moderately large cluster models may be particularly crucial. Such models are inherently two-dimensional with substrate atoms from two or three crystal layers usually taken into accormt thus, a large fraction of atoms at the cluster boundaries lacks proper coordination. 

The physics community have distanced themselves from the debate by accepting quantum theory as a mathematically useful tool, without agonizing over the physical interpretation. For the chemist who deals with three-dimensionally structured objects, like molecules, this approach creates a dilemma. Modern chemistry is best understood in terms of experimentally measured electron-density distributions, awkward, if not impossible, to visualize in terms of zero-dimensional objects. The alternative wave model, not only makes intuitive sense, but also eliminates poorly defined concepts such as probability densities and quantum jumps. 

There are alternative studies for ET based on the real-time dynamic approaches. These approaches are very important for dynamic investigation such as electronic population relaxation and its transient non-equilibrium properties. However, the numerical convergence for the rate calculation is still a dilemma because of the multi-dimensional problem in ET. 

A fundamental dilemma in kinetic data analysis can be demonstrated by the well-known static two-dimensional heterogeneous reactor model for the industrially important catalytic fixed-bed reactor (Fig. I) /2/. 

At the same time, mass spectrometry offered exquisite details on the mass and structure of small (<5000 Da) molecules but was unable to efficiently ionize larger ones. The dilemma of the mid-1980s is illustrated in Fig. 1. The results of a two-dimensional gel electrophoresis separation of kidney proteins showed a wealth of information in the >5000 Da range. Results from mass spectrometry, however, left off all molecular species in this region. The lack of efficient ion sources for these molecules started a decade-long race to produce gas-phase ions from ever larger molecules. This quest culminated in the discovery of electrospray ionization (ESI) and matrix-assisted laser desorption ionization (MALDI) by the end of the decade. Almost overnight, molecules with masses in excess of 100 kDa could be studied by... 

Physics has the dilemma of irrefutable evidence for a four-dimensional world, but a genetic inability among physicists to visualize more than three dimensions. It is therefore not surprising to find that those instances, in which reality is badly distorted in three-dimensional projection, inevitably lead to convoluted theories, bordering on the supernatural. Quantum mechanics is a prime example of such a theory. It was inspired by experimental results that defied explanation based on classical theory. It was first recognized in the study of microphysical systems, which in time came to be seen as deviating from the classical and therefore subject to a new theory, without relevance in macrophysics. 

Both cases have much in common in the sense that the imstable set of both bifurcating equilibrium states is one-dimensional. If the unstable set of the critical equilibrimn state is of a higher-dimension, then the subsequent picture may be completely different. Figure 14.3.1 depicts such a situation. When the imstable cycle shrinks into the equilibrium state we have a dilemma the representative point may jump either to the stable node 0 or to the stable node 02- Therefore this dangerous boundary must be classified as dynamically indefinite. 

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