Semiclassical Treatments

It is not possible for the indirect damping mechanism, as considered semiclassically by Robertson and Yarwood [84] and later quantum mechanically by Boulil et al. [90], to be the unique damping mechanism occurring in a hydrogen bond, because the quantum mechanism leads, at the opposite of the less rigorous semiclassical treatment, to a drastic collapse of the lineshapes. 

Fig. 2 Schematic representation of the so-called semiclassical treatment of kinetic isotope effects for hydrogen transfer. All vibrational motions of the reactant state are quantized and all vibrational motions of the transition state except for the reaction coordinate are quantized the reaction coordinate is taken as classical. In the simplest version, only the zero-point levels are considered as occupied and the isotope effect and temperature dependence shown at the bottom are expected. Because the quantization of all stable degrees of freedom is taken into account (thus the zero-point energies and the isotope effects) but the reaction-coordinate degree of freedom for the transition state is considered as classical (thus omitting tunneling), the model is ealled semielassieal.

Kinetic complexity definition, 43 Klinman s approach, 46 Kinetic isotope effects, 28 for 2,4,6-collidine, 31 a-secondary, 35 and coupled motion, 35, 40 in enzyme-catalyzed reactions, 35 as indicators of quantum tunneling, 70 in multistep enzymatic reactions, 44-45 normal temperature dependence, 37 Northrop notation, 45 Northrop s method of calculation, 55 rule of geometric mean, 36 secondary effects and transition state, 37 semiclassical treatment for hydrogen transfer,... 

Because of the complexity of the DLP theory, several semiclassical treatments have been developed to simplify and extend its findings. One such extension is applicable to blocks of nonpolar materials at small separations and is of interest to us because of the possibility it offers for relating surface tension to intermolecular forces. 

We derived (3.54) earlier from our semiclassical treatment, but (3.55) is new information. Quantum electrodynamics (Merzbacher, Chapter 22) confirms the correctness of (3.55). 

Self-adjoint operator, 108 Semiclassical treatment of radiation absorption, 116 Semiempirical methods, 70-76 Separation of variables, 19-20 Shielding constant, 332, 334 calculation of, 337 Shielding tensor, 332 Sigma orbitals, 68... 

The semiclassical treatment just given has the defect of not predicting spontaneous emission. According to (3.13), if there is no outside perturbation, that is, if // (0 = 0, then dcm/dt = 0 for all m if the atom is in the nth stationary state at / = 0, it will persist in that state forever. However, experimentally we find that unperturbed atoms in excited states spontaneously radiate energy and drop to lower states. Quantum field theory does predict spontaneous emission. Since quantum field theory is beyond us, we shall use an argument given by Einstein in 1917 to find the spontaneous-emission probability. 

W. H. Miller Tunneling is surely involved in the present case because the isomerization involves a large amount of hydrogen atom motion. Also, with no tunneling the resonances would be infinitely narrow (and thus unobservable). Our calculations, though, are a fully quantum mechanical treatment and thus do not explicitly identify what is tunneling and what is not. This would only be meaningful in an approximate (e.g., semiclassical) treatment. 

Aside on Barriers In our semiclassical treatment of the properties of charged-particle-induced reaction cross sections, we have equated the reaction barrier B to the Coulomb barrier. This is, in reality, a simplification that is applicable to many but not all charged-particle-induced reactions. 

This condition is identical to the classical conditions (II. 1) and (II.2) for the distances / , at which a transition can occur. For the case where the difference potential is monotonic, so that there exists only one point of stationary phase, R we obtain in this semiclassical treatment immediately the classical result [cf. with equations (11.20) and (11.21)] for a(e), except for an oscillatory factor ... 

Billing, G.D. (1984). The semiclassical treatment of molecular roto-vibrational energy transfer, Com. Phys. Rep. 1, 237-296. 

Two different theoretical approaches have been used to relate the ET rate constant (k ) to the thermodynamic driving force (AG ) and parameters related to molecular structure. The first approach is the semiclassical treatment derived from the early work by Marcus on ET theory [2-4,45],... 

Creutz C, Brunschwig BS, Sutin N. Interfacial charge transfer absorption Semiclassical treatment. J Phys Chem B 2005 109 10251-60. 

Korsch and Mohlenkamp (1983) have used a semiclassical treatment of complex energy states to calculate results corresponding to those in our Table 8.4. The results in their Table 1 are in satisfactory agreement with the first-order approximation of our phase-integral results. 

In most of the more recent classical approaches [18], no allusion to Ehrenfest s (adiabatic) principle is employed, but rather the differential equations of motion from classical mechanics are solved, either exactly or approximately, subject to a set of initial conditions (masses, force constants, interaction potential, phase, and initial energies). The amount of energy, AE, transferred to the oscillator is obtained for these conditions. This quantity may then be averaged over all phases of the oscillating molecule. In approximate classical and semiclassical treatments, the interaction potential is expanded in a Taylor s series and only the first two terms are retained. 

An approximate three dimensional semiclassical treatment of the harmonic oscillator interacting with an impinging atom according to a Morse potential has been presented by Calvert and Amme [23]. 

II SEMICLASSICAL TREATMENT OF NON ADIABATIC COUPLING IN THE SCATTERING PROBLEMS... 

Semiclassical treatment of the relative motion of collision partners. 

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