Classical path

This nomially refers to the use of the straight-line trajectory/ (t) = (b +v t ), 0(t) = arctan(b/vt) within the classical path treatment. See Bates [18,19] for examples and fiirtlier discussion. 

Note the meaning of this expression for each choice of the initial and final position a and a , calculate the classical path that takes you from x to x" m time t. Specifically, calculate tire momentum along the path and the final momentum, p", and find out how p" varies with the initial position. This would give, for a multidimensional problem, a matrix dp"-Jdx"- whose absolute detenninant needs to be inverted. 

In this seiniclassical calculation, we use only one wavepacket (the classical path limit), that is, a Gaussian wavepacket, rather than the general expansion of the total wave function. Equation (39) then takes the following form ... 

The standard semiclassical methods are surface hopping and Ehrenfest dynamics (also known as the classical path (CP) method [197]), and they will be outlined below. More details and comparisons can be found in [30-32]. The multiple spawning method, based on Gaussian wavepacket propagation, is also outlined below. See [1] for further infomiation on both quantum and semiclassical non-adiabatic dynamics methods. 

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. 

Another popular and convenient way to study the quantum dynamics of a vibrational system is wave packet propagation (Sepulveda and Grossmann, 1996). According to the ideas of Ehrenfest the center of these non-stationary functions follows during a certain time classical paths, thus representing a natural way of establishing the quantum-classical correspondence. In our case the dynamics of wave packets can be calculated quite easily by projection of the initial function into the basis set formed by the stationary eigen-... 

The above results can be qualitatively understood. In spherical cavity, the semi classical paths can bounce back from any point of the wall. However a path ending for example at the summit of the conical cavity has no direction to be reflected back. Thus one can speculate that the configuration space available for a field in the conical cavity is less than the spherical cavity. 

As in scattering theory in general, one can treat the role of V in either a time independent or a time dependent point of view. The latter is simpler if the perturbation V is either explicitly time dependent or can be approximated as such, say by replacing the approach motion during the collision by a classical path. Algebraic methods have been particularly useful in that context,2 where an important aspect is the description of a realistic level structure for H0. Figure 8.3 is a very recent application to electron-molecule scattering. 

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 ... 

III. MEAN-FIELD TRAJECTORY METHOD A. Classical-Path Approximation... 

As explained in the Introduction, most mixed quantum-classical (MQC) methods are based on the classical-path approximation, which describes the reaction of the quantum degrees of freedom (DoF) to the dynamics of the classical DoF [9-22]. To discuss the classical-path approximation, let us first consider a diabatic... 

Equation (28) is still exact. To introduce the classical-path approximation, we assume that the nuclear dynamics of the system can be described by classical trajectories that is, the position operator x is approximated by its mean value, namely, the trajectory x t). As a consequence, the quantum-mechanical operators of the nuclear dynamics (e.g., Eh (x)) become classical functions that depend parametrically on x t). In the same way, the nuclear wave functions dk x,t) become complex-valued coefficients dk x t),t). As the electronic dynamics is evaluated along the classical path of the nuclei, the approximation thus accounts for the reaction of the quantum DoE to the dynamics of the classical DoF. 

In order to introduce the classical-path approximation in the adiabatic electronic representation, we expand the total wave function in terms of... 

As long as no approximation is introduced, it is clear that the equations of motion are equivalent in the diabatic and adiabatic representations. This is no longer true, however, once the classical-path approximation is employed the resulting classical-path equations of motion in the adiabatic representation are... 

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... 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

As a starting point, we consider the Schrodinger equation (30) in the adiabatic classical-path approximation. This equation can be recast in a density-matrix... 

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