Classical propagator

Integrators Generated from Factorizing the Classical Propagator... 

For an example of curve fitting involving classical propagation of errors in a potentiometric titration setting, see Ref. 142. 

Ab initio simulations of water using classical propagation generally lead to an overstructured liquid compared with experiment. 

The structure of C given above, consists of two distinct components (i) classical propagation on mean surfaces accompanied by quantum mechanical phase oscillations with frequency Lvaa> = (Ea — Ea>)/h, and (ii) nonadiabatic transitions accompanied by changes in the momentum of the environment in order to conserve energy. The classical Liouville operator... 

From left to right, the short-time propagator describes classical propagation on the Sj i surface through a time interval 6/2, a transition Sj 1 —> Sj determined by the elements of A4 and classical propagation on the Sj surface for a time interval 6/2. 

It has been demonstrated in an earlier paper that one can derive a mixed quantum-classical propagation scheme based on the TDSCF approximation [47]. The TDSCF scheme has been extensively used in many studies [48,27,49] and shall be resumed briefly in order to introduce the notations. 

A description, which is somewhat intermediate between a full quantum model and a classical propagation scheme is the Gaussian wave packet approach for the classical-like mode. This picture was first introduced by Heller [44] and adapted by Billing [18,45,46] to the mixed classical quantum system. Therein one mode is described as a time dependent Gaussian wave packet [18,15]... 

According to Trotter s theorem [62], the mixed quantum-classical propagator can be written as... 

In any time segment tj — tj i), using the decomposition of the quantum-classical Liouville operator into diagonal and off-diagonal J parts in (43), the quantum-classical propagator may be written in Dyson form as... 

Now that we know how to evolve the dynamics within a small time segment, we can decide to construct a Monte-Carlo-style stochastic algorithm to account for the quantum transitions that arise from the action of J. At the end of each time segment, the system either may remain in the same pair of adiabatic states or make a transition to a new pair of states. More specifically, for an initial pair of quantum states, (ckockq), the phase point R, P) is evolved for a time At to a new value RAt, PaO (here we use a simplified notation for the time-evolved phase points in the interval At) using the classical propagator and the phase factor is computed. With probability 1/2,... 

The choice of the value of the coherent state width 7 is arbitrary, since this parameter does not affect the mathematical properties of the Gaussian basis set which determine the form of the semi-classical propagator. In fact, the value of this quantity is usually chosen in practical implementations so as to facilitate the numerical convergence. In the following, we shall set 7 = 1/2 since with this choice (25) simplifies considerably and becomes... 

A major difference between the two methods of initiation is that the solvent in y-ray studies is almost inevitably the monomer itself, and these generally have lower dielectric constants than the chlorocarbon solvents most often used in the chemically initiated systems. As a result, it is not possible to compare the values of kp +) obtained from each technique without accounting for this difference in solvation. Classically, propagation involves charge dispersion in forming the transition-state complex and hence a reduction in the polarity of the system. Thus media of lower solvation power should favourably influence the process. (See reference 114 for more detailed discussion.) Experimentally the values of kp(+) from radiation-induced polymerizations are consistently higher than those obtained using stable salts as initiators, and this simplistic picture therefore seems to be confirmed. Dunn has recently carried out a detailed compilation of the available data on / p(+) and readers will find this an excellent distillation of the current position. 

A fundamental theoretical issue for computational photochemistry is the treatment of the hop (nonadiabatic) event. One needs to add the time propagation of the solutions of the time-dependent Schrodinger equation for electronic motion to the classical propagation of the nuclei, thus obtaining the populations of each adiabatic state. The time-dependent wave function for electronic motion is just a time-dependent configuration interaction vector ... 

The time correlation functions that we shall discuss are objects of the form complex conjugate i A =B we refer to a time autocorrelation function. It is often useful to express A t) in terms of the classical propagator,... 

Bf the crack with such a craze at its tip suffers an increasing load without extending, or if it propagates carrying the craze before it, every craze fibril is extended. Almost always, the fibril at the craze mouth (i.e. the crack tip ) extends most rapidly. It is generally accepted that fibrils extend by a process whereby material is cold-drawn from the bulk, so that the cohesive surface corresponds to the shoulders of a classic propagating tensile neck. If it takes place quickly, such a process will concentrate adiabatic heating at the cohesive surface. 

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