Multi-state interactions

The present analysis relies on - and extends - the comprehensive theoretical study of Refs. [23,24] on the multi-state interactions in the manifold of the X — E states of Bz+. Like this recent work, it utilizes an ab initio quantum-dynamical approach. In Refs. [23,24] we have, in addition, identified strong coupling effects between the B — C and B — D electronic states, caused by additional conical intersections between their potential energy surfaces. A whole sequence of stepwise femtosecond internal conversion processes results [24]. Such sequential internal conversion processes are of general importance as is evidenced indirectly by the fluorescence and fragmentation dynamics of organic closed-shell molecules and radical cations [49,50]. It is therefore to be expected that the present approach and results may be of relevance for many other medium-sized molecular systems. 

In these (and other) solid superacid catalyst systems, bi- or multi-dentate interactions are thns possible, forming highly reactive intermediates. This amounts to the solid-state equivalent of protosolvation resulting in superelectrophilic activation. 

It is important to select stoichiometric co-reductants or co-oxidants for the reversible cycle of a catalyst. A metallic co-reductant is ultimately converted to the corresponding metal salt in a higher oxidation state, which may work as a Lewis acid. Taking these interactions into account, the requisite catalytic system can be attained through multi-component interactions. Stereoselectivity should also be controlled, from synthetic points of view. The stereoselective and/or stereospecific transformations depend on the intermediary structure. The potential interaction and structural control permit efficient and selective methods in synthetic radical reactions. This chapter describes the construction of the catalytic system for one-electron reduction reactions represented by the pinacol coupling reaction. 

Another type of ligand, capable of forming multi-hapto interactions with metals, are boron and boron-carbon cage compounds of which in particular the carboranes have been used extensively in organometallic chemistry . The structures in the solid state of a few magnesacarboranes have been determined by X-ray crystallographic studies. 

This section will first deal with the phases in particle-fluid two-phase flow by developing a mathematical model to quantify local hydrodynamic states. This analysis will reveal the insufficiency of the conditions for the conservation of mass and momentum alone in determining the hydrodynamic states of heterogeneous particle-fluid systems, and calls for a methodology different from what is used in analyzing dilute uniform flow. For this purpose the concept of multi-scale interaction between particles and fluid and the principle of energy minimization are proposed. 

In this article we have given an overview over our theoretical studies on tlie multi-mode multi-state vibronic interactions in the benzene cation and several of its fluoro derivatives. These are all associated with multiple conical intersections between the... 

The processes responsible for most of the reactions Identified are oxidation, dehydration, sulfur fixation, solid-state interactions and vaporization. Isolating specific reactions occurring in a multi-component system is difficult the use of model systems was helpful in tracing mineral transformations. Additional work planned in this area includes research on non-equilibrium systems and experiments in reducing atmospheres. 

A high intensity of the incident radiation enhances the probability for simultaneous multi-photon interactions with a single molecule, i.e. two or more photons are annihilated and absorbed by the molecule in a single quantum mechanical process. The frequency of the scattered photon does in such cases not have to be equal that of the absorbed photons, e.g. two quanta with frequency m may be annihilated, creating a third photon with frequency 2a). As indicated in Fig. 2, the system returns to its ground state 0> after the interaction has taken place, and the intermediate virtual state is separated from the first excited state by an energy A . This is an example of a nonlinear optical process known as second-harmonic generation, which can... 

The total number of independent variables appearing in Fq. (4.32) is thus quite large, and in fact too large for practical applications. However, as mentioned earlier, by coupling Eq. (4.32) with the Navier-Stokes equation to find the forces on the particles due to the fluid, the Ap-particle system is completely determined. Although not written out explicitly, the reader should keep in mind that the mesoscale models for the phase-space fluxes and the collision term depend on the complete set of independent variables. For example, the surface terms depend on all of the state variables A[p ( x ", ", j/p" j, V ", j/p" ). The only known way to determine these functions is to perform direct numerical simulations of the microscale fluid-particle system using all possible sets of initial conditions. Obviously, such an approach is intractable. We are thus led to reduce the number of independent variables and to introduce mesoscale models that attempt to capture the average effect of multi-particle interactions. 

Beyond a certain system size, even DFT methods using conventional basis sets become computationally very intensive. In such situations, one has to take recourse to the use of solid-state physics methods like the pseudopotential plane wave or tight-binding methods [28,29]. As the systems become larger, Monte Carlo (MC) simulations and molecular dynamics simulations based on effective pair potentials (including two-body to multi-body interactions) are carried out. 

On the other hand, the symbolism (2) also includes the generalization to a multi-state discrete spectrum interacting with a multichannel continuous one. This case was also admirably tackled by Fano formally [17, 29], albeit only for fhe confinuous spectrum, without the inclusion of fhe closed channels. 

The use of the form (32), in the single- or in the multi-state version, for the solution of the CESE, has been applied in the context of the SSA to a variety of problems. Its extension to problems involving the interaction with external fields is discussed in Section 11. 

In the paper the multi-state approach to the analysis and evaluation of systems reliability risk and availability has been practically applied. The pipeline transportation system has been considered in varying in time operation conditions. The system reliability structure and its components reliability functions were changing in variable operation conditions. The paper proposed an approach to the solution of practically very important problem of linking the systems reliability and their operation processes. To involve the interactions between the systems operation processes and their varying in time reliability structures... 

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