Classical dynamics

In this section we interpret the Hamiltonian (10.2.4) as a classical Hamiltonian function of the conjugate variables xi,pi) and x2,P2)- Transforming to action and angle variables with the help of the canonical transformation (6.1.18), the Hamiltonian (10.2.4) becomes  

In order to be useful, (10.3.3) has to be regularized. Defining a new time r according to 

For all practical purposes the set (10.3.6) is regularized. But there remains a problem when both r/i and 7/2 simultaneously tend to zero modulo 7T. This means that xi and X2 simultaneously. We call this situation the triple collision case . 

Closed orbits are of special interest for the semiclassical quantization of the helium atom. For a closed orbit F we have 

Currently available numerical results indicate that the one-dimensional heUum atom is completely chaotic. The best-known semiclassical quantization procedure for completely chaotic systems is Gutzwiller s trace formula (see Section 4.1.3), which is based on classical periodic orbits. Therefore we search for simple periodic orbits of the one-dimensional he-hum atom. Since a two-electron orbit is periodic if the orbits ni t), 0i t)) and (ri2(t), 2( )) of the first and second electron have a common period, the periodic orbits of the one-dimensional model can be labelled with two integers, m and n, which count the 27r-multiplicity of the angle variables 0i and 02 after completion of the orbit. Therefore, if for some periodic orbit 

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Another question is the nature of the changes in the classical dynamics that occur with the breakdown of the polyad number. In all likelihood there are farther bifiircations. Apart from the identification of the individual polyad-breaking resonances, the bifiircation analysis itself presents new challenges. This is partly becanse with the breakdown... 

In coimection with the energy transfer modes, an important question, to which we now turn, is the significance of classical chaos in the long-time energy flow process, in particnlar the relative importance of chaotic classical dynamics, versus classically forbidden processes involving dynamical tuimelling . 

The question of non-classical manifestations is particularly important in view of the chaos that we have seen is present in the classical dynamics of a multimode system, such as a polyatomic molecule, with more than one resonance coupling. Chaotic classical dynamics is expected to introduce its own peculiarities into quantum spectra [29, 77]. In Fl20, we noted that chaotic regions of phase space are readily seen in the classical dynamics corresponding to the spectroscopic Flamiltonian. Flow important are the effects of chaos in the observed spectrum, and in the wavefiinctions of tire molecule In FI2O, there were some states whose wavefiinctions appeared very disordered, in the region of the... 

It should be emphasized that the existence of energy transfer modes hypotliesized earlier with the polyad breakdown is completely consistent with the energy transfer being due to non-classical, dynamical tiumelling processes. This is evident from the observation above that the disorder in the FI2O spechnm is attributable to non-classical effects which nonetheless are accompaniments of cte.s/c CT/bifiircations. 

Finally, new mathematical developments in the study of nonlinear classical dynamics came to be appreciated by molecular scientists, with applications such as the bifiircation approaches stressed in this section. 

The above is a comprehensive, readable introduction to modern nonlinear classical dynamics, with quantum applications. 

A3.12.6 CLASSICAL DYNAMICS OF INTRAMOLECULAR MOTION AND UNIMOLECULAR DECOMPOSITION... 

Sibert E L III, Reinhardt W P and Hynes J T 1982 Classical dynamics of energy transfer between bonds in ABA triatomics JCP77 3583-94... 

The calculation of the time evolution operator in multidimensional systems is a fomiidable task and some results will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics as demonstrated, among others, by Heller [86, 87 and 88], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-fimction distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

V. I. Arnold, Mathematical methods of classical dynamics. Chap. 7, Springer, New York, 1978,... 

The principal idea behind the CSP approach is to use input from Classical Molecular Dynamics simulations, carried out for the process of interest as a first preliminary step, in order to simplify a quantum mechanical calculation, implemented in a subsequent, second step. This takes advantage of the fact that classical dynamics offers a reasonable description of many properties of molecular systems, in particular of average quantities. More specifically, the method uses classical MD simulations in order to determine effective... 

U. Schmitt and J. Brinkmann. Discrete time-reversible propagation scheme for mixed quantum classical dynamics. Chem. Phys., 208 45-56, 1996. 

Example Brady investigated classical dynamics of a-d-glucose in water.In this simulation, 207 water molecules surrounded one a-d-glucose. The system was in a cubic box with periodic boundary conditions. During the simulation, several hydroxyl group transitions occurred. These transitions are normally unlikely with an in vacuo simulation. 

Reality suggests that a quantum dynamics rather than classical dynamics computation on the surface would be desirable, but much of chemistry is expected to be explainable with classical mechanics only, having derived a potential energy surface with quantum mechanics. This is because we are now only interested in the motion of atoms rather than electrons. Since atoms are much heavier than electrons it is possible to treat their motion classically. Quantum scattering approaches for small systems are available now, but most chemical phenomena is still treated by a classical approach. A chemical reaction or interaction is a classical trajectory on a potential surface. Such treatments leave out phenomena such as tunneling but are still the state of the art in much of computational chemistry. 

Second, the classical dynamics of this model is governed by the generalized Langevin equation of motion in the adiabatic barrier [Zwanzig 1973 Hanggi et al. 1990 Schmid 1983],... 

The only feasible procedure at the moment is molecular dynamics computer simulation, which can be used since most systems are currently essentially controlled by classical dynamics even though the intermolecular potentials are often quantum mechanical in origin. There are indeed many intermolecular potentials available which are remarkably reliable for most liquids, and even for liquid mixtures, of scientific and technical importance. However potentials for the design of membranes and of the interaction of fluid molecules with membranes on the atomic scale are less well developed. 

Classical dynamics is studied as a special case by analyzing the Ehrenfest theorem, coherent states (16) and systems with quasi classical dynamics like the rigid rotor for molecules (17) and the oscillator (18) for various particle systems and for EM field in a laser. 

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