Propagation schemes

Leforestier C et a/1991 A comparison of different propagation schemes for the time dependent Schrddinger equation J. Comput. Phys. 94 59... 

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

In a standard back-propagation scheme, updating the weights is done iteratively. The weights for each connection are initially randomized when the neural network undergoes training. Then the error between the target output and the network predicted output are back-propa-... 

Diene monomers with suitably disposed double bonds may undergo intramolecular ring-closure in competition with propagation (Scheme 4.12). The term cyclopolymcrization was coined to cover such systems. Many systems which give cyclopolymerization to the exclusion of normal propagation and crosslinking are now known. The subject is reviewed in a series of works by Butler.98 102... 

Molecular dynamics has also been used to replace the MC moves for conformational advancement [43]. In the molecular dynamics version of parallel tempering, often referred to as replica exchange molecular dynamics, momenta are used in the propagation scheme such that a constant temperature is maintained between the swaps. After the swap in conformational space (with the same acceptance criterion as in the MC implementation), a readjustment in momentum space is also needed. This is done by renewing the momenta for replica i by the transformation... 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

In practice one does not proceed as we did in the above derivation. Instead of calculating first all stationary wavefunctions and then constructing the wavepacket according to (4.3), one solves the time-dependent Schrodinger equation (4.1) with the initial condition (4.4) directly. Numerical propagation schemes will be discussed in the next section. Since 4 /(0) is real the autocorrelation function fulfills the symmetry relation... 

The propagation of the wavepacket is thereby reduced to the solution of coupled first-order differential equations for the parameters representing the Gaussian wavepacket, with the true potential being expanded about the instantaneous center of the wavepacket [i2(<),f(<)]. This propagation scheme is very appealing and efficient provided the basic assumptions are fulfilled. The essential prerequisite is that the locally quadratic approximation of the PES is valid over the spread of the wavepacket. This rules out bifurcation of the wavepacket, resonance effects, or strong an-harmonicities. 

It is interesting, however, to compare the above mechanisms in terms of the different polymerisation behaviour observed for systems with catalysts containing multinuclear species and mononuclear species. In the case of polymerisations with catalysts possessing mononuclear species, two catalyst molecules must approach one another to form the active site for the propagation [scheme (7)]. Such a requirement is not needed for propagation involving the active sites... 

See Ref. [148] for a physically appealing derivation.) Equation (12) yields an efficient, accurate and formally exact scheme to perform quantum dynamics. It involves the action of a handed, sparse and Toeplitz matrix on a vector. The propagation scheme has been shown to accurately represent [147] all quantum dynamical features including zero-point effects, tunneling as well as over-barrier reflections and in this sense differs from standard semi-classical treatments. The approach also substantially differs from other formalisms such as centroid dynamics [97,98,168-171], where the Feynman path centroid is propagated in a classical-like... 

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