Classical path methods

The TDSCF approximation is a good starting point for a mixed quantum mechanical/classical treatment. Let us assume that R is the classical and r the quantum mechanical mode. Then, the wavefunction r(r t) describing the vibration of the fragment molecule is a solution of the time-dependent Schrodinger equation 

Equation (4.37) is identical to (4.34) with the exception that the kinetic energy operator Tr is replaced by its classical analogue. 

Expanding r(r t) in terms of vibrational states pn(r) of the free oscillator with energies en, 

The combination of quantum mechanical and classical modes is particularly important for larger systems which prohibit a complete quantal treatment. It is most suitable for direct processes with short interaction times and it is less applicable for long-lived intermediate complexes. The ultimate step in the hierarchy of time-dependent approximations is a complete classical treatment of all degrees of freedom. This is the topic of the next chapter. 

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Solving the Eqs. (C.6-C.8,C.12,C.13) comprise what is known as the Ehrenfest dynamics method. This method has appealed under a number of names and derivations in the literatnre such as the classical path method, eilconal approximation, and hemiquantal dynamics. It has also been put to a number of different applications, often using an analytic PES for the electronic degrees of freedom, but splitting the nuclear degrees of freedom into quantum and classical parts. 

Tully has discussed how the classical-path method, used originally for gas-phase collisions, can be applied to the study of atom-surface collisions. It is assumed that the motion of the atomic nucleus is associated with an effective potential energy surface and can be treated classically, thus leading to a classical trajectory R(t). The total Hamiltonian for the system can then be reduced to one for electronic motion only, associated with an electronic Hamiltonian Jf(R) = Jf t) which, as indicated, depends parametrically on the nuclear position and through that on time. Therefore, the problem becomes one of solving a time-dependent Schrodinger equation ... 

The classical-path approximation introduced above is common to most MQC formulations and describes the reaction of the quantum DoF to the dynamics of the classical DoF. The back-reaction of the quantum DoF onto the dynamics of the classical DoF, on the other hand, may be described in different ways. In the mean-field trajectory (MFT) method (which is sometimes also called Ehrenfest model, self-consistent classical-path method, or semiclassical time-dependent self-consistent-field method) considered in this section, the classical force F = pj acting on the nuclear DoF xj is given as an average over the quantum DoF... 

Classical Path. Another approach to scattering calculations uses a quantum-mechanical description of the internal states, but classical mechanics for the translational motion. This "classical path" method has been popular in line-shape calculations (37,38). It is almost always feasible to carry out such calculations in the perturbation approximation for the internal states (37). Only recently have practical methods been developed to perform non-perturbative calculations in this approach (39). 

Several other path integral-based approaches to compute KIE exist, which however do not use QI approximation. These include, for example, approaches using other quantum TSTs [55-60] or the quantized classical path method [7,61,62]. 

ABSTRACT. Exact quantum mechanical calculations on energy transfer processes are possible or numerical feasible only for a few mainly hydrogen containing systems. However, often the semiclassical (classical path) method will be sufficiently accurate to allow a quantitative determination of cross sections and rate constants for inelastic as well as reactive processes. A brief review of the classical path method is given and several numerical aspects are discussed. 

Improvements in the classical path method may be introduced by using a Gaussian wave packet description for some of the degrees of freedom which are treated classically. This quanturo trajectory" approach has recently been used to establish the validity of the velocity... 

However, for a large class of problems it will be possible to use the simple classical path scheme, and we have in table 1 shown how it is possible to reduce the dimensionality of the g matrix. But often the dimension of g is large and the integration of eqs. (1) will then be the timedetermining part of the classical path method. We shall below give a numerically convenient method for solving the eqs. (1). 

In the classical path method one usually propagates a number of vectors rather than a full matrix. The reason being that the effective potential which couples the quantum and classical subsystems depends upon the initial state or even upon the specific transition under consideration. However, in an approximation one may use a trajectory which is governed only by the elastic part of the interaction potential. Here the propagator becomes independent of the initial quantum state and it is convenient to propagate a full matrix. One should keep in mind, however, that the... 

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