Non-classical behaviour

Transition state theory has been useful in providing a rationale for the so-called kinetic isotope effect. The kinetic isotope effect is used by enzy-mologists to probe various aspects of mechanism. Importantly, measured kinetic isotope effects have also been used to monitor if non-classical behaviour is a feature of enzyme-catalysed hydrogen transfer reactions. The kinetic isotope effect arises because of the differential reactivity of, for example, a C-H (protium), a C-D (deuterium) and a C-T (tritium) bond. 

Relationships between reaction rate and temperature can thus be used to detect non-classical behaviour in enzymes. Non-classical values of the preexponential factor ratio (H D i 1) and difference in apparent activation energy (>5.4kJmoRi) have been the criteria used to demonstrate hydrogen tunnelling in the enzymes mentioned above. A major prediction from this static barrier (transition state theory-like) plot is that tunnelling becomes more prominent as the apparent activation energy decreases. This holds for the enzymes listed above, but the correlation breaks down for enzymes... 

Bohm s failure to give an adequate explanation to support the pilot-wave proposal does not diminish the importance of the quantum-potential concept. In all forms of quantum theory it is the appearance of Planck s constant that signals non-classical behaviour, hence the common, but physically meaningless, proposition that the classical/quantum limit appears as h —> 0. The actual limiting condition is Vq —> 0, which turns the quantum-mechanical... 

The distinction between classical and non-classical descriptions of a system should not be confused with the classical and non-classical behaviours of a single system that become identical in the classical limit of the Bohmian interpretation. The unacceptable procedures outlined above amount to extrapolation of the classical description into the non-classical domain, beyond the limit, to where it has no validity. 

Quantum phenomena arise from the presence of the interfacial potential. To appreciate its effect, consider a sequence of classical particles in order of decreasing size. They become increasingly susceptible to the influence of the interfacial potential in the same order. At the interface, the potential gradient is very sharp and any assumption of constant potential, on the scale of the smallest particle, becomes totally untenable. This is the precise condition that differentiates between classical and non-classical behaviour. 

Systems with vanishing Vq [i.e. h/m 0) must clearly behave classically. A photon of zero mass is a pure quantum an electron less so and a marble is classical. There is no discontinuity from classical to non-classical behaviour. 

Physico-chemical behaviour in aqueous solution was studied by viscosity measurements. Expected associative behaviour has been evidenced in pure water, while in salt media, the associative behaviour strongly depends on PCL chains length. For shorter PCL chains, intramolecular hydrophobic interactions are predominant, even in the semi-dilute regime. This non-classical behaviour for an associative polyelectrolyte opens the way to the conception of amphiphilic matrices with hydrophobic clusters for controlled release applications. 

Some investigators have reported the isotope effect of migration by replacing protons with deuterons . Scherban et al. observed non-classical behaviour in Yb-doped SrCeOs the ratio of conductivity o-h/<7d is about 2.5 which is significantly greater than the classical value of /l, although the reason for this is not yet clear. 

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

Apparent non-RRKM behaviour occurs when the molecule is excited non-randomly and there is an initial non-RRKM decomposition before IVR fomis a microcanonical ensemble (see section A3.12.2). Reaction patliways, which have non-competitive RRKM rates, may be promoted in this way. Classical trajectory simulations were used in early studies of apparent non-RRKM dynamics [113.114]. 

This is a recurrent theme of quantum theory. Many quantum systems can be formulated exactly in terms of a wave equation and the behaviour of the system will be described exactly by the wavefunction, the solution to the wave equation. What is not always appreciated is that this is a mathematical description only, which does not ensure understanding of the event in terms of a comprehensible physical model. The problem lies therein that the description is only possible in terms of a wave formalism. Understanding of the physical behaviour however, requires reduction to a particle model. The wave description is no more than a statistically averaged picture of the behaviour of many particles, none of which follows the actual statistically predicted course. The wave description is non-classical, and the particle model is classical. Mechanistic understanding is possible only in terms of the classical approach, and a mathematically precise description only in terms of the wave formalism. The challenge of quantum theory is to reconcile the two points of view. 

The quantum theory of the previous chapter may well appear to be of limited relevance to chemistry. As a matter of fact, nothing that pertains to either chemical reactivity or interaction has emerged. Only background material has been developed and the quantum behaviour of real chemical systems remains to be explored. If quantum theory is to elucidate chemical effects it should go beyond an analysis of atomic hydrogen. It should deal with all types of atom, molecules and ions, explain their interaction with each other and predict the course of chemical reactions as a function of environmental factors. It is not the same as providing the classical models of chemistry with a quantum-mechanical gloss a theme not without some common-sense appeal, but destined to obscure the non-classical features of molecular systems. 

The importance of the quantum potential lies therein that it defines the classical limit with Vq —> 0, or more realistically where the quantity h/m —> 0, which implies h/p = A —> 0. It means that quantum effects diminish in importance for systems with increasing mass. Massless photons and electrons (with small mass) behave non-classically, and atoms less so. Small molecules are at the borderline, and macro molecules approach classical behaviour. When the system is in an eigenstate (or stationary state) of energy E, the kinetic energy E — V = k) is by definition equal to zero. 

The classical idea of molecular structure gained its entry into quantum theory on the basis of the Born Oppenheimer approximation, albeit not as a non-classical concept. The B-0 assumption makes a clear distinction between the mechanical behaviour of atomic nuclei and electrons, which obeys quantum laws only for the latter. Any attempt to retrieve chemical structure quantum-mechanically must therefore be based on the analysis of electron charge density. This procedure is supported by crystallographic theory and the assumption that X-rays are scattered on electrons. Extended to the scattering of neutrons it can finally be shown that the atomic distribution in crystalline solids is identical with molecular structures defined by X-ray diffraction. 

The weird properties that came to be associated with quantum systems, because of the probability doctrine, obscured the simple mathematical relationship that exists between classical and quantum mechanics. The lenghthy discussion of this aspect may be of less interest to chemical readers, but it may dispel the myth that a revolution in scientific thinking occured in 1925. Actually there is no break between classical and non-classical systems apart from the relative importance of Planck s action constant in macroscopic and microscopic systems respectively. Along with this argument goes the realization that even in classical mechanics, as in optics, there is a wave-like aspect associated with all forms of motion, which becomes more apparent, at the expense of particle behaviour, in the microscopic domain. 

Nevertheless only the classical Coulomb law appears as the interaction law the special feature of the atomic bond thus does not depend on a special non-classical interaction but on the particular wave-mechanical behaviour of the electrons, so that in the symmetrical solution a piling up of negative charge can be produced at a place where the potential energy for the electrons is low. In the individual configuration H>H the repulsion of the two nuclei predominates, as can be seen from Fig. 13 K, over the attraction of the one proton for the cloud of negative charge around the other proton, at all distances. 

Williams et al. [55] and Pfeiffer and Jewett [56] independently studied the structure of the ethenium cation which they assume to model the behaviour of carbenium ions reliably. They found equilibrium to exist between the classical and non-classical forms, the former being more stable by about 40 kJ mol-1 than the species on the right-hand side of the equation. 

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