Split operator technique

This procedure is then repeated after each time step. Comparison with Eq. (2) shows that the result is the velocity Verlet integrator and we have thus derived it from a split-operator technique which is not the way that it was originally derived. A simple interchange of the Ly and L2 operators yields an entirely equivalent integrator. 

In order to better understand the experimental results, we performed quantum mechanical calculations using the fast Fourier transform (FFT) split-operator technique, which was previously employed by Meier and Engel [38]... 

The time-dependence of the wavepacket evolving on any potential surface can be numerically determined by using the split operator technique of Feit and Fleck [10-15]. A good introductory overview of the method is given in Ref. [12]. We will discuss a potential in two coordinates because this example is relevant to the experimental spectra. The time-dependent Schrodinger equation in two coordinates Qx and Qy is... 

The time evolution of the wave packet given by Eq. (27) can be evaluated by an eigenfunction expansion or by a split-operator technique. With the latter technique, the wave packet X t + St) after a small increment of time St can be expanded approximately as... 

Chemical dynamics determination (time-dependent Schrodinger equation) Split operator technique... 

It is obvious that the shorter At gives the more accurate results, but with higher computational costs. Also, the split operator technique have been extended to the higher order product forms [410, 497], and the higher order schemes generally lead to the more accurate results for a given time interval At, again the more computational labor in return. 

As At is small enough, one may split H(t) into the time-dependent and time-independent parts, or to a pure diagonal matrix and a pure off-diagonal matrix. The split operator technique is then used to propagate the wave function. One may alternatively expand U(t) with Chebyshev polynomials for the propagation.Ishii et have found numerically that the former... 

Note that Equation 3.10 is different from the conventional split-operator techniques [49,50], where is usually chosen to be the kinetic energy operator and V is the remaining Hamiltonian, depending on the spatial coordinates only. The use of the energy representation in Equation 3.10 allows explicit elimination of the undesirable fast-oscillating high-energy components and speeds up considerably the... 

The coupled equations have been integrated by the split-operator technique [44]. In matrix form, the system reads... 

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

In the present work, we monitor the laser-driven dynamics designed by the present formulation by numerically solving the time-dependent Schrodinger (5.2). It is solved by the split operator method [52] with the fast Fourier transform technique [53]. In order to prevent artificial reflections of the wavepacket at the edges, a negative imaginary absorption potential is placed at the ends of the grid [54]. The envelope of the pulses employed is taken as... 

On the other hand, approximations to Eq. (8) and time-integration techniques, suitable especially for time-independent Hamiltonians, under the requirement of only a few degrees of freedom and short-time evolution, have been developed and applied extensively in connection with grid-type techniques (see Section 2), by focusing on appropriate algebraic expansions of fhe exponenfial form. For example, such a approach is effected by the split-operator method [4] and references there in. 

Interdigital mixers and micromixer with the split-recombine technique Single microchannel operating in annular flow regime Microstructured falling film reactor Mesh microreactor... 

Reactor configurations involved in continuous emulsion polymerization include stirred tank reactors, tubular reactors, pulsed packed reactors, Couett-Taylor vortex flow reactors, and a variety of combinations of these reactors. Some important operational techniques developed for continuous emulsion polymerization are the prereactor concept, start-up strategy, split feed method, and so on. The fundamental principles behind the continuous emulsion polymerizations carried out in the basic stirred tank reactor and tubular reactor, which serve as the building blocks for the reaction systems of commercial importance, are the major focus of this chapter. 

Prominent representatives of the first class are predictor-corrector schemes, the Runge-Kutta method, and the Bulir-sch-Stoer method. Among the more specific integrators we mention, apart from the simple Taylor-series expansion of the exponential in equation (57), the Cayley (or Crank-Nicholson) scheme, finite differencing techniques, especially those of second or fourth order (SOD and FOD, respectively) the split-operator, method and, in particular, the Chebychev and the shoit-time iterative Lanczos (SIL) integrators. Some of the latter integration schemes are norm-conserving (namely Cayley, split-operator, and SIL) and thus accumulate only... 

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