Quantum-thermal correlations

This section is devoted to biological order, organization, and evolution. We have already seen in Appendix F that constructive integration of quantum and thermal correlations under appropriate conditions lead to a so-called CDS, i.e., an optimal spatiotemporal structure formed by the precise relations between time, size, and temperature scales. CDS suggests microscopic selforganization including Godel-like self-referential traits. 

As shown previously analogous equations can be derived in a statistical framework both for localized fermions in a specific pairing mode and/or for bosons subject to a quantum transport environment [7]. The second interconnection regarding the relevance of the basis f is related to the fact that a transformation of form (20) connects canonical Jordan blocks to convenient complex symmetric forms. This will not be explicitly discussed and analysed here except pointing out the possible relationship between temperature scales and Jordan block formation by thermal correlations (see e.g. [7-9,14], for more details). 

In (13.20) and (13.21) the subscripts c and q correspond respectively to the classical and the quantum time correlation functions and denotes a semiclassical approximation. We refer to the form (13.21) as semiclassical because it carries aspects of the quantum thermal distribution even though the quantum time correlation function was replaced by its classical counterpart. On the face of it this approximation seems to make sense because one could anticipate that (1) the time correlation functions involved decay to zero on a short timescale (of order 1 ps that characterizes solvent configuration variations), and (2) classical mechanics may provide a reasonable short-time approximation to quantum time correlation functions. Furthermore note that the rates in (f3.2f) satisfy detailed balance. 

Rigorous results for the thermal rate constant can be obtained by a correct evaluation of (quantum) flux correlation functions. Thus, the flux-position correlation function (2.13), ( =... 

Here. .. )t is the quantum thermal average, (... )t = Tr[e ... ]/Tr[e" ], and Tr denotes a trace over the initial manifold z. We have thus identified the golden rule rate as an integral over time of a quantum time correlation function associated with the interaction representation of the coupling operator. 

Wahnstrom G and Metiu H 1988 Numerical study of the correlation function expressions for the thermal rate coefficients in quantum systems J. Phys. Chem. JPhCh 92 3240-52... 

We can calculate the thermal rate constants at low temperatures with the cross-sections for the HD and OH rotationally excited states, using Eqs. (34) and (35), and with the assumption that simultaneous OH and HD rotational excitation does not have a strong correlated effect on the dynamics as found in the previous quantum and classical trajectory calculations for the OH + H2 reaction on the WDSE PES.69,78 In Fig. 13, we compare the theoretical thermal rate coefficient with the experimental values from 248 to 418 K of Ravishankara et al.7A On average, the theoretical result... 

While for thermal reactions one usually does not correlate the energy input with the amount of product formed, electrochemists and photochemists are certainly more energy-minded . The first ones use the current yield to define the amount of product formed per electrons consumed. The latter ones use the so called quantum yield which is defined as the ratio of number of molecules undergoing a particular process from an excited state over moles of photons absorbed by the system, or in other words, the ratio of the rate constant for the process defined over the sum of all rate constants for all possible processes from this excited state (1.4). Thus, if for every photon absorbed, a molecule undergoes only one chemical process, the quantum yield for this process is unity if other processes compete it will be less than unity. 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

Among the still unanswered questions of the scheme in Figure 14 are the number of steps between I w and I 2, a definitive characterization of the chemical nature of all reaction steps, and the elucidation of the relatively slow processes which thermally regenerate Pr from the intermediates and thus cause the difference of 0.35 between the quantum yields of

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It is however possible to discuss several special cases analytically. The zero temperature correlation length can still be observed as long as this is smaller than the thermal de Broglie wave length At which can be rewritten for K not too close to Ku as t < f/y Kt(, KtK" 1 with tu Lpl, where we defined tk via = jj-, analogously to the definition of At, and used (30). We call this domain the quantum disordered region. 

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