Semiclassical analysis theory

The semiclassical Ehrenfest theory coupled with this representation was applied in an electron flux analysis in chemical reactions where large charge transfer occurs caused by significant nonadiabatic transition. The chemical systems treated in the summaries below are Na - - Cl and formic acid dimer (FAD). The time shift flux operator stated in the previous subsection was utilized in an analysis of the microscopic electron d3mamics in this chemically representative case. 

We have carrried out an analysis of the multilevel structure of the tunneling centers that goes beyond a semiclassical picture of the formation of those centers at the glass transition, which was primarily employed in this chapter. These effects exhibit themselves in a deviation of the heat capacity and conductivity from the nearly linear and quadratic laws, respectively, that are predicted by the semiclassical theory. 

The purpose of this chapter is to review some properties of isomerizing (ABC BCA) and dissociating (ABC AB + C) prototype triatomic molecules, which are revealed by the analysis of their dynamics on precise ab initio potential energy surfaces (PESs). The systems investigated will be considered from all possible viewpoints—quanmm, classical, and semiclassical mechanics—and several techniques will be applied to extract information from the PES, such as Canonical Perturbation Theory, adiabatic separation of motions, and Periodic Orbit Theory. 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

V. Aquilanti and G. Capecchi, Harmonic analysis and discrete polynomials from semiclassical angular momentum theory to the hyperquantization algorithm. Theor. Chem. Accounts, 104 183-188, 2000. 

D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Spectral analysis method of obtaining molecular spectra from classical trajectories, J. Chem. Phys. 67 404 (1977) M. L. Koszykowski, D. W. Noid, and R. A. Marcus, Semiclassical theory of intensities of vibrational fundamentals, overtones, and combination bands, J. Phys. Chem. 86 2113 (1982) D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Quasiperiodic and stochastic behavior in molecules, Ann. Rev. Phys. Chem. 32 267 (1981). 

The Cl + HC1 quantized transition states have also been studied by Cohen et al. (159), using semiclassical transition state theory based on second-order perturbation theory for cubic force constants and first-order perturbation theory for quartic ones. Their treatment yielded 0), = 339 cm-1 and to2 = 508 cm"1. The former is considerably lower than the values extracted from finite-resolution quantal densities of reactive states and from vibrationally adiabatic analysis, 2010 and 1920 cm 1 respectively (11), but the bend frequency to2 is in good agreement with the previous (11) values, 497 and 691 cm-1 from quantum scattering and vibrationally adiabatic analyses respectively. The discrepancy in the stretching frequency is a consequence of Cohen et al. using second-order perturbation theory in the vicinity of the saddle point rather than the variational transition state. As discussed elsewhere (88), second-order perturbation theory is inadequate to capture large deviations in position of the variational transition state from the saddle point. 

There are other even less conventional approaches to deriving kinetic energy functionals. There are recent approaches using hydrodynamic tensors,semiclassical expansions,information theory, ° ° theory of moments, ° " analysis of quantum fluctuations," and higher-order electron distribution functions. ° "" " ... 

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