Classical-path approximation

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... 

The standard translation of the full quantum formula of the absorbance, Eq. (31), to a mixed quantum classical description (see, e.g., [16-18]) is similar to what is the essence of the (electronic ground-state) classical path approximation introduced in the foregoing section. One assumes that all involved nuclear coordinates behave classically and their time-dependence is obtained by carrying out MD simulations in the systems electronic ground state. This approach when applied to the absorbance is known as the dynamical classical limit (DCL, see, for example, Res. [17]). 

In contrast to the computations of the preceding section we directly calculate the expectation value of the CC dipole operator (finally linearized with respect to the external field) applying the classical path approximation for nuclear dynamics. Such a direct calculation of the dipole operator expectation value becomes of particular interest when focusing on ultrafast nonlinear optical properties (transient absorption, photon echo signal, etc.). 

According to the size of the system the MD runs have to be carried out in a way not to notice the actual CC excited state (electronic ground state classical path approximation). However, it seems rather reasonable that any back reaction of the actual excited electronic state should be of minor importance since even in the largest studies complexes only singly excited states (single exciton states) are incorporated. 

Fig.3.11a-c. Phase perturbation of an oscillator by collisions (a) classical path approximation of colliding particles (b) frequency change of the oscillator A(t) during the collision (c) resulting phase shift... 

Before introducing the classical path approximation in reactive systems it is necessary to switch to a coordinate system which does not discriminate between the reaction channels. This condition is fulfilled in the so-called hyperspherical coordinates in which the atom-atom distances for the three-body system are expressed in terms of the hyperradius p and the two hyperangles 6, by [193 ... 

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