Three-state system trajectory calculation

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. 

The simplest way of taking aceount of vibrational effeets is to assume vibrational adiabatieity during the motion up to the eritieal dividing surface [27]. As mentioned aheady in the Introduetion, mueh of the earlier work on vibrational adiabatieity was concerned with its relationship to transition-state theory, espeeially as applied to the prediction of thermal rate constants [24-26], It is pointed out in [27] that the validity of the vibrationaUy adiabatie assumption is supported by the results of both quasielassieal and quantum seattering ealeulations. The effeetive thresholds indicated by the latter for the D -I- H2(v =1) and O + H2(v =1) reactions [37,38] are similar to those found from vibrationaUy adiabatic transition-state theoiy, which is a strong evidence for the correctness of the hypothesis of vibrational adiabatieity. Similar corroboration is provided by the combined transition-state and quasielassieal trajectory calculations [39-44]. For virtually all the A + BC systems studied [39-44], both collinearly and in three... 

This model has several advantages. First, since all degrees of freedom are treated equally, it is possible to observe resonance phenomena that cannot be seen if certain degrees of freedom are treated with different approximations. Second, since the method basically involves the addition of two degrees of freedom, that is, four equations of motion, to the standard classical trajectory calculation, it is possible to handle cases where quantum mechanical treatments are presently impossible (e,g., three-dimensional two-state collision systems). 

Eaker and Muzyka [39] have performed a trajectory-surface hopping calculation on the D+ — H2 system, but restricted to the two lowest potential energy surfaces. They observed the electron transfer between the two molecules prior to dissociation, but their CID cross section for D+(v = 3) + H2 at 4 eV was only 0.9 A2, about three times smaller than our result. In addition, their calculated ratio for H+ to D+ products was 8 1, whereas our result was 1 1. A possible explanation of the discrepancy [15] is a mechanism in which the second excited state of the [H2H2]+ system is excited during the collision. This state, which was not considered by Eaker, is repulsive for the H2+ moiety, and would give slow (fast) H+ ions if electron transfer did (did not) occur during the first part of the collision. This suggestion remains to be tested by theory. 

Finally, we consider Model III, which describes an ultrafast photoin-duced isomerization process. Figure 13 shows quantum mechanical results as well as results of the ZPE-corrected mapping approach for three different observables the adiabatic and diabatic population of the excited state, and Pcis> the probability that the system remains in the initially prepared cis-conformation. A ZPE correction of 7 = 0.5 has been used in the mapping calculation, based on the criterion to reproduce the quantum mechanical long-time limit of the adiabatic population. It is seen that the ZPE-corrected mapping approach represents an improvement compared to the mean-field trajectory method (cf. Fig. 3), in particular for the adiabatic population. The influence of the ZPE correction on the dynamics of the observables is, however, not as large as in the two other models. 

This develops the general algorithm of calculation of minimum reflux mode for the columns with two feed inputs at distillation of nonideal zeotropic and azeotropic mixtures with any number of components. The same way as for the columns with one feed, the coordinates of stationary points of three-section trajectory bundles are defined at the beginning at different values of the parameter (L/V)r. Besides that, for the intermediate section proper values of the system of distillation differential equations are determined for both stationary points from the values of phase equihbrium coefficients. From these proper values, one finds which of the stationary points is the saddle one Sm, and states the direction of proper vectors for the saddle point. The directions of the proper vectors obtain linear equations describing linearized boundary elements of the working trajectory bundle of the intermediate section. We note that, for sharp separation in the top and bottom sections, there is no necessity to determine the proper vectors of stationary points in order to obtain linear equations describing boundary elements of their trajectory bundles, because to obtain these linear equations it is sufficient to have... 

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