Numerical time propagation

For time propagation of vibrational wavefunctions in three dimensions, we employ the Jacobi coordinates (r, R, 9), respectively the bond distance for one NO moiety, the distance between the center of mass of this NO moiety and the other O, and the associated angle. Numerical time propagation of the vibrational wavefunction within this coordinate system was discussed in Sec. 3.2.2. 9 = tt places the second O on the same side as N with respect to the center of mass of NO. Although this choice does not reflect the symmetry of the NO2 molecule, it is convenient as the molecule can dissociate into NO-l-0 in the energy range of interest. 

The propagator nature of the Chebyshev operator is not merely a formality it has several important numerical implications.136 Because of the similarities between the exponential and cosine propagators, any formulation based on time propagation can be readily transplanted to one that is based on the Chebyshev propagation. In addition, the Chebyshev propagation can be implemented easily and exactly with no interpolation errors using Eq. [56], whereas in contrast the time propagator has to be approximated. 

Obtaining the exact rate (which is independent of qds), necessitates a real time propagation. A numerically exact solution is feasible for systems with a few degrees of freedom,already discussed above, there is still a way to go before one can rigorously implement the time evolution in a liquid. 

It was indicated numerous times that the SHG intensity is dependent on both the magnitude of x<2) tensor elements as well as the phase relationships between fundamental and harmonic fields in the crystal. Under certain circumstances, it is possible to achieve phase matched propagation of the fundamental and harmonic beams. Under these conditions, power is continually transferred from the fundamental to harmonic beam over a path length, which is only limited by the ability... 

We remark that the simulation scheme for master equation dynamics has a number of attractive features when compared to quantum-classical Liouville dynamics. The solution of the master equation consists of two numerically simple parts. The first is the computation of the memory function which involves adiabatic evolution along mean surfaces. Once the transition rates are known as a function of the subsystem coordinates, the sequential short-time propagation algorithm may be used to evolve the observable or density. Since the dynamics is restricted to single adiabatic surfaces, no phase factors... 

Once chlorine atoms are produced (initiation), the propagation steps provide a closed cycle that can be repeated numerous times (e.g., 10 ) prior to the recombination of the chlorine atoms (termination). 

Abstract. The Chebyshev operator is a diserete eosine-type propagator that bears many formal similarities with the time propagator. It has some unique and desirable numerical properties that distinguish it as an optimal propagator for a wide variety of quantum mechanical studies of molecular systems. In this contribution, we discuss some recent applications of the Chebyshev propagator to scattering problems, including the calculation of resonances, cumulative reaction probabilities, S-matrix elements, cross-sections, and reaction rates. 

Exponential divergence in systems that are chaotic prevents accurate long-time trajectory calculations of their dynamics. That is, numerical errors18 propagate exponentially during the dynamics so that accuracy beyond 100 characteristic periods of motion is extraordinarily difficult to achieve thus, accurate long-time dynamics is essentially uncomputable for chaotic classical systems. This serves as additional... 

In many physical applications the Hamiltonian is explicitly time dependent. The common solution for propagation in these explicitly time-dependent problems is to use very small grid spacing in time, such that within each time step the Hamiltonian H(r) is almost stationary. Under these semistationary conditions a short-time propagation method in the time-energy phase space is employed. The drawback of this solution is that it is based on extrapolation therefore the errors accumulate. Moreover, time ordering errors add with the usual numerical dispersion errors (108). 

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