Classical motion

The method of molecular dynamics (MD), described earlier in this book, is a powerful approach for simulating the dynamics and predicting the rates of chemical reactions. In the MD approach most commonly used, the potential of interaction is specified between atoms participating in the reaction, and the time evolution of their positions is obtained by solving Hamilton s equations for the classical motions of the nuclei. Because MD simulations of etching reactions must include a significant number of atoms from the substrate as well as the gaseous etchant species, the calculations become computationally intensive, and the time scale of the simulation is limited to the... 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

Hence, as the second class of techniques, we discuss adaptive methods for accurate short-term integration (Sec. 4). For this class, it is the major requirement that the discretization allows for the stepsize to adapt to the classical motion and the coupling between the classical and the quantum mechanical subsystem. This means, that we are interested in discretization schemes which avoid stepsize restrictions due to the fast oscillations in the quantum part. We can meet this requirement by applying techniques recently developed for evaluating matrix exponentials iteratively [12]. This approach yields an adaptive Verlet-based exponential integrator for QCMD. 

QCMD describes a coupling of the fast motions of a quantum particle to the slow motions of a classical particle. In order to classify the types of coupled motion we eventually have to deal with, we first analyze the case of an extremely heavy classical particle, i.e., the limit M —> oo or, better, m/M 0. In this adiabatic limit , the classical motion is so slow in comparison with the quantal motion that it cannot induce an excitation of the quantum system. That means, that the populations 6k t) = of the... 

For larger values of m/M, we have to expect nonadiabatic redistribution of the populations induced by the classical motion. 

The splitting of the quantum propagator negatively effects the efficiency of the scheme especially if m/M is small, i.e., if the quantum oscillation are much faster than the classical motion and the number n of substeps is becoming inefficiently large. 

The classical motion of a particle interacting with its environment can be phenomenologically described by the Langevin equation... 

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

The relationship (equation (5.81)) between M and L depends only on fundamental constants, the electronic mass and charge, and does not depend on any of the variables used in the derivation. Although this equation was obtained by applying classical theory to a circular orbit, it is more generally valid. It applies to elliptical orbits as well as to classical motion with attractive forces other than dependence. For any orbit in any central force field, the angular... 

In the high-field limit (F > 1 atomic unit meaning that it is greater than the binding potential) the smoothed Coulomb potential in Eq. (2) can be treated as a perturbation on the regular, classical motion of a free electron in an oscillating field. So, let us first consider the Hamiltonian for the one-dimensional motion of a free electron in the... 

The classical motion is considerably simplified if we consider variables in the oscillating frame... 

The classical motion corresponding to the quantum dynamics generated by the Dirac-Hamiltonian (2) can most conveniently be obtained by considering the limit h — 0 in the Heisenberg picture Consider an operator B that is a Weyl quantisation of some symbol (see (Dimassi and Sjostrand, 1999))... 

This experimental work on the dissociation of excited Nal clearly demonstrated behavior one could describe with the vocabulary and concepts of classical motions.The incoherent ensemble of molecules just before photoexcitation with a femtosecond laser pump pulse was transformed through the excitation into a coherent superposition of states, a wave packet that evolved as though it represented a single vibrationally activated molecule. 

As we see in Fig. 38, transitions leading from classical order to quantum order are also possible. For example, for 9 = 6.7 the quasiperiodic classical motion is reduced to periodic motion after the quantum correction. 

Figure 7. (a) Concept of time-dependent alignment as a method for structural determination. Top Initial alignment at t = 0, dephasing, and recurrence of alignment at later times. Bottom Classical motion of a rigid prolate symmetric top. (b) Structures of stilbene and tryptamine-water complex from rotational coherence spectroscopy transients are shown, [see ref. 13]. 

G. Casati In the first part of your talk you discussed the longtime propagation of an initially localized quantum packet. Did you check whether in such situations the corresponding classical motion is in dynamically stable regions or instead is chaotic ... 

---