Quasiclassical State Method

It has been argued, however, that the high-spin states experience some exchange repulsions which are not present in the ground states and that this could more or less influence the force constants of the o bonds. To rule out this criticism, we have proposed the quasiclassical state method which is described below. 

The quasiclassical trajectory method disregards completely the quantum phenomenon of superposition (13,18,19) consequently, the method fails in treating the reaction features connected with the interference effects such as rainbow or Stueckelberg-type oscillations in the state-to-state differential cross sections (13,17,28). When, however, more averaged characteristics are dealt with (then the interference is quenched), the quasiclassical trajectory method turns out to be a relatively universal and powerful theoretical tool. Total cross-sections (detailed rate constants) of a large variety of microscopic systems can be obtained in a semiquantitative agreement with experiment (6). 

Abstract A generalization of the Landau-Teller model for vibrational relaxation of diatoms in collisions with atoms at very low energies is presented. The extrapolation of the semiclassical Landau-Teller approach to the zero-energy Bethe-Wigner limit is based on the quasiclassical Landau method for calculation of transition probabilities, and the recovery of the Landau exponent from the classical collision time. The quantum suppression-enhancement probabilities are calculated for a general potential well, which supports several bound states, and for a Morse potential with arbitrary number of states. The model is applied to interpretation of quantum scattering calculations for the vibrational relaxation of H2 in collisions with He. 

If the initial diatomic is heteronuclear, the collision complex can separate into two possible product channels. The branching ratio for the two channels reflects the properties of the complex. We have investigated a series of reactions involving the state of oxygen and carbon using the quasiclassical trajectory method to elucidate these mechanistic and dynamical features. 

Oscillations in time of quantal states are usually much faster than those of the quasiclassical variables. Since both degrees of freedom are coupled, it is not efficient to solve their coupled differential equations by straightforward time step methods. Instead it is necessary to introduce propagation procedures suitable for coupled equations with very different time scales short for quantal states and long for quasiclassical motions. This situation is very similar to the one that arises when electronic and nuclear motions are coupled, in which case the nuclear positions and momenta are the quasiclassical variables, and quantal transitions lead to electronic rearrangement. The following treatment parallels the formulation introduced in our previous review on this subject [13]. Our procedure introduces a unitary transformation at every interval of a time sequence, to create a local interaction picture for propagation over time. 

In order to test our methods, we consider a model of two quantum states coupled to a quasiclassical variable, suitable for describing the near resonant electron transfer of an alkali atom such as Na as it approaches a metal surface M at thermal energies. The two asymptotic potential energies correspond to a ground state of neutral components Na + M (state 1) and an excited state for ionic components Na" " + M (state 2), which cross at a short distance. The model consists of two diabatic surfaces and a coupling term given by [21,22]... 

It should be noted that the adiabatic approximation may be violated during the passage of the region x x. if the nuclear velocity dx/dt is high. However, this approximation is certainly valid outside it, i.e., in the initial and final states of the system. Under these conditions can be evaluated using the quasiclassical method of... 

The last method (section V.D) employs a reference state called the quasiclassical (QC) state which represents the o frame and avoids thereby the task of dissecting terms. The QC approach is versatile and can be used at any computational level, including density functional theory. The method can also be... 

The paper is arranged in the following way. In the next section a few aspects of the RIOSA which are relatively new are discussed and a detailed procedure is given for doing the actual calculations. The third section deals with numerical results as obtained by us in the last five years. However, in contrast to what has been published elsewhere, we concentrate only on those results which can be compared either with findings obtained by reliable methods, i.e. exact quantum mechanical (EQM), coupled states (CS) and quasiclassical trajectory (QCT) methods or with experiment. A discussion and a summary are given in the last section. 

The paper by Karplus, Porter and Sharma (KPS) [1] is, from my perspective, the most important early (pre-1970) piece of computational work in gas-phase chemical reaction dynamics. In it, the commonly used quasiclassical trajectory (QCT) method was described for three-dimensional atom-diatom reactive collisions (i.e., A -f BC —> AB + C), and was applied to the H -f H2 reaction to determine cross sections and thermal rate constants. In 35 years of subsequent work on gas-phase reaction dynamics, the QCT method has remained largely the same, and it continues to be a standard tool for studying quantum state-resolved dynamical processes. 

By definition, a mixed quantum-classical method treats the various degrees of freedom (DoF) of a system on a different djmamical footing, e.g. quantum mechanics for the electronic DoF and classical mechanics for the nuclear DoF. As was discussed above, some of the problems with these methods are related to inconsistencies inherent in this mixed quantum-classical ansatz. To avoid these problems, recently a conceptually different way to incorporate quantum mechanical DoF into a semiclassical or quasiclassical theory has been proposed, the so-called mapping approach. " In this formulation, the problem of a classical treatment of discrete DoF such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. In this section we briefly introduce the general concept of the mapping approach and discuss the quasiclassical implementation of this method as well as applications to the three models introduced above. The semiclassical version of the mapping approach is discussed in Sec. 7. 

Pig. 8. Time-dependent (a) diabatic and (b) adiabatic electronic excited-state populations and (c) vibrational mean positions as obtained for Model I. Shown are results of the mean-field trajectory method (dotted lines), the quasiclassical mapping approach (thin full lines), and exact quantum calculations (thick full lines). 

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