Total propagator

Further extensions of chemical mapping within a fixed bed lie in quantifying mass transfer between the intra-pellet and inter-pellet pore space. One approach to measuring mass transfer processes is to use displacement propagator measurements (see Section II.C). The data acquired for the esterification reaction described previously are shown in Fig. 31. In this propagator measurement, the total propagator measured for the system has been separated into two... 

From the expression it can be concluded that the spatial resolution is not merely equal to the ratio of the time resolution of the time digitizer to the total propagation delay of the line. 

In Section IV it was noted that for a given spin system with Hamiltonian and a given multiple-pulse sequence with. f(t), it is always possible to calculate the total propagator U r ), a corresponding effective Hamiltonian and the evolution of the density operator ait) (see Fig. 5). However, because the relationship between t) + [ t) and U(t )... 

Here r(i ) = 0.1(i — lO) 0(i — 10) 2) 2 with 2) being the electronic excited state, and 6 is the Heaviside step function. Time step dt was set to 1, and total propagation time was 1200. The derivative operator and position operator was treated in sine discrete variable representation [92] with 600 grid points equally distributed from R = —3 to 14. We have checked the convergence of the results with respect to grid-density and time step. 

The two important convergence parameters introduced by the power series expansion of the ABC Green s function are the total propagation time T, and the time step At. The total time T represents the time required for reaction and absorption by e(q). The time step At is the duration for which the STP is a faithful representation of the propagator. These are both a function of the dynamics and the choice of absorbing potential. We measure At in units of a fundamental small time given by... 

Figure 3.4 Convergence of N(E) at JE = 0.52 eV with respect to the total propagation time T. The solid curve uses At = 1 aw = 0.024 fs and the dashed curve uses At = 20.2 au = 0.49 fs. Both curves give smooth convergence, suggesting independent optimization of the paxameters At and T. The optimal T = 5000 au = 121 fs.

Figure 4.12 Convergence of initial state reaction probabilities for various energies and initial states of the D+H2(v,i) system with respect to the total propagation time convergence parameter S. The systems studied are the same as in Fig. 4.10. The thick solid line is what would occur if S perfectly controlled the error in the reaction probabilities. The solid line with squares is the average of the error, and the dashed line with circles is the maximum error from the systems studied all as a function of S. We see the remarkable result that, even in the worst case of maximum error, S rehably controls the error in the reaction probability.

Convergence of N(B) with respect to total propagation time. Effect on total propagation time of strong absorbing potential... 

Convergence of probabilities with total propagation time. . . . ... 

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