Multidimensional molecular systems

Due to the large-level density of the lower-lying adiabatic electronic state, the chances of a back transfer of the adiabatic population are quite small for a multidimensional molecular system. To a good approximation, one may therefore assume that subsequent to an electronic transition a random walker will stay on the lower adiabatic potential-energy surface [175]. This observation suggests a physically appealing computational scheme to calculate the time evolution of the system for longer times. First, the initial decay of the adiabatic population is calculated within the QCL approach up to a time to, when the... 

Considering the practical application of the mapping approach, it is most important to note that the quantum correction can also be determined in cases where no reference calculations exist. That is, if we a priori know the long-time limit of an observable, we can use this information to determine the quantum correction. For example, a multidimensional molecular system is for large times expected to completely decay in its adiabatic ground state, that is. 

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

In this review, we have discussed the Feshbach-Lowdin PT as a tool for studying multidimensional quantum dynamics of (molecular) systems. The central element in this approach is the emergence of overlapping resonances through the application of the PT on the Hilbert space of the system under study, and the possibility that such resonances ultimately interfere. The TOR, which is the result of this approach, provides a fruitful method to understand and conceptually link diverse physical phenomena and processes. We have tried to demonstrate this by discussing various examples, as FIT and ORIT, the suppression of spontaneous decay in atoms and molecules, and the CC of IC in pyrazine and / -carotene, as well as of IVR in the OCS molecule. 

Catalyst Characterization. The three SAPO molecular sieves employed in this study represents the three pore sizes of molecular sieves, ranging tom 0.4 nm to 0.8 nm. While SAPO-5 and SAPO-11 have unidimensional pores, SAPO-34 has a multidimensional pore system with supercages. The chemical composition and total ammonia uptake of the tifiree SAPO molecular sieves are listed in Table I. 

All intermolecular interactions can be adequately described, at least in principle, by multidimensional scalar and vector fields representing the energetics of a molecular system as functions of both intermolecular distances and orientations as well as intramolecular structure data. The visualization of these fields, however, has to be based on a three-dimensional picture or a two-dimensional projection because human pattern recognition ability is strongly related to the two- and three-dimensional world. Consequently, the multidimensional field has to be reduced to a two- or three-dimensional representation. In molecular science this can be done in many different ways. 

Katritzky et al. [88] and Jug [89] and others [14,22,57] have shown recently that aromaticity is a multidimensional phenomenon. For this purpose they applied statistical methods (the principal component or factor analyses [51,90,91]) which need application of many indices of aromaticity estimated (experimentally or calculated theoretically) for many molecular systems. Thus their conclusion is of great general importance but no information may be extracted for any individual molecule or its fragment. The method presented above, the separation of HOMA value into EN and GEO terms, allows us to describe numerically which of these two factors is... 

One of the most significant advances made in applied quantum chemistry in the past 20 years is the development of computationally workable schemes based on the analytical energy derivatives able to determine stationary points, transition states, high-order saddle points, and conical intersections on multidimensional PES. The determination of equilibrium geometries, transition states, and reaction paths on ground-state potentials has become almost a routine at many levels of calculation (SCF, MP2, DFT, MC-SCF, CCSD, Cl) for molecular systems of chemical interest. 

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