Time-dependent theoretical treatment

Deumens, E., Diz, A., Longo, R. and Ohm, I. Time-dependent theoretical treatments ofthe dynamics of electrons and nuclei in molecular systems, Rev.Mod.Phys., 66 (1994), 917-983... 

The time-dependent theoretical treatment of the electronic absorption spectrum is very similar to that of the emission spectrum because the same two... 

The time-dependent theoretical treatment of the electronic emission spectrum is very similar to that of the absorption spectrum because the two potential surfaces involved are the same. The principal difference is that the initial wavepacket starts on the upper (excited state) electronic surface and propagates on the ground electronic state surface. The overlap of the initial wavepacket with the time-dependent wavepacket is given by Eq. (4). The emission spectrum is given by (49)... 

R. B. Bernstein (ed.), Atom-Molecule Collision Theory, Plenum, New York, 1979 J. Broeck-hove and L. Lathouwers (eds.), Time Dependent Quantum Molecular Dynamics NATO ASI Series B, Vol. 299, Plenum Press, New York, 1992 R. Kosloff, Propagation methods for quantum molecular dynamics, Ann. Rev. Phys. Chem. 45 145 (1994) E. Deumens, A. Diz, R. Longo, and Y. Ohrn, Time-dependent theoretical treatments of the dynamics of electrons and nuclei in molecular systems, Rev. Mod. Phys. 66, 917 (1994). 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

When dehydration occurs as a consecutive reaction, its effect on polarographic curves can be observed only, if the electrode process is reversible. In such cases, the consecutive reaction affects neither the wave-height nor the wave-shape, but causes a shift in the half-wave potentials. Such systems, apart from the oxidation of -aminophenol mentioned above, probably play a role in the oxidation of enediols, e.g. of ascorbic acid. It is assumed that the oxidation of ascorbic acid gives in a reversible step an unstable electroactive product, which is then transformed to electroinactive dehydroascorbic acid in a fast chemical reaction. Theoretical treatment predicted a dependence of the half-wave potential on drop-time, and this was confirmed, but the rate constant of the deactivation reaction cannot be determined from the shift of the half-wave potential, because the value of the true standard potential (at t — 0) is not accessible to measurement. 

Traditionally, experimental values of Zeff have been derived from measurements of the lifetime spectra of positrons that are diffusing, and eventually annihilating, in a gas. The lifetime of each positron is measured separately, and these individual pieces of data are accumulated to form the lifetime spectrum. (The positron-trap technique, to be described in subsection 6.2.2, uses a different approach.) An alternative but equivalent procedure, which is adopted in electron diffusion studies and also in the theoretical treatment of positron diffusion, is to consider the injection of a swarm of positrons into the gas at a given time and then to investigate the time dependence of the speed distribution, as the positrons thermalize and annihilate, by solving the appropriate diffusion equation. The experimentally measured Zeg, termed Z ), is the average over the speed distribution of the positrons, y(v,t), where y(v,t) dv is the number density of positrons with speeds in the interval v to v + dv at time t after the swarm is injected into the gas. The time-dependent speed-averaged Zef[ is therefore... 

Unlike the case of catalytic mechanism discussed in the previous section, the theoretical study of CE and EC mechanisms (see reaction scheme 4.IVb, c) in double potential pulse techniques is much more complex than that corresponding to a single potential pulse since the surface concentrations of the species involved in these reaction schemes corresponding to the application of the first potential pulse are time dependent (see also Sects. 3.4.2 and 3.4.3). Due to this, only simplified situations of these mechanism are considered in this section under planar diffusion conditions. The treatment of both mechanisms at other geometries can be found in [76-79]. 

In principle, the physics involving unstable states ought to engage descriptions that are time dependent. Yet, in the formulation and practical solution of related problems, both time-dependent and time-independent treatments are pertinent and necessary. Furthermore, in certain theoretical approaches, the phenomenologies as well as the computational methodology are based on constructions that are non-Hermitian. We add that the Hamiltonians may... 

Similar dependence of catalyst deactivation on coke or catalyst residence time is suggested by Corella et al. (5-7). The authors give details on possible mechanisms of catalyst deactivation by coke, and also suggest, based on their data, that the deactivation order n may not be a constant. For our analysis, however, we will assume that n is constant and a function of catalyst type. Further theoretical treatment of catalyst decay is given by Wojdechowski (8.9). 

So far, we have considered various theoretical treatments of time-dependent wave packets controlled by laser pulses to produce a desired product in a chemical reaction. Another type of problem, based on adiabatic behavior of wave functions, is the transfer of population from one state to another. 

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