Active Subsystem

Consider a typical Sn2 reaction as an example (eq 9). The reaction consists of the breaking of a C-F bond followed by the formation of a new C-Cl bond. 

In the above example, this would mean consideration of the full set of the six VB structures (3-8) that one can possibly construct for a system of four electrons in three orbitals. 

The active electrons are thus explicitly correlated, while the inactive electrons are not. One expects that the lack of correlation in the inactive subsystem will result in a constant error throughout the potential surface and therefore just uniformly shift the calculated energies relative to fully correlated surfaces. Note that in this model the inactive electrons are still affected by the progress of the reaction, since their orbitals rearrange and optimize at all points of the reaction coordinate. It is simply their mutual correlation that is considered as constant. 

The above definitions of active/inactive subsystems is of course not restricted to the study of reactions but can be generalized to all static systems 

After the choice of the relevant Lewis structures has been made, the following step involves their quantum mechanical formulation. Each Lewis structure corresponds to a set of atomic orbitals which are singly or doubly occupied, as illustrated in 9-11 for the F2 molecule. 

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Now we pass to the formal derivations of a hybrid method. We assume that the orbitals forming the basis for the entire molecular system may be ascribed either to the chemically active part of the molecular system (reactive or R-states) or to the chemically inactive rest of the system (medium or M-states). In the present context, the orbitals are not necessarily the basis AO, but any set of their orthonormal linear combinations thought to be distributed between the subsystems. The numbers of electrons in the R-system (chemically active subsystem) Nr and in the M-system (chemically inactive subsystem) NM = Ne — Nr, respectively, are good quantum numbers at least in the low energy range. We also assume that the orbital basis in both the systems is formed by the strictly local orbitals proposed in [59]. The strictly local orbitals are orthonormalized linear combinations of the AOs centered on a single atom. In that sense they are the classical hybrid orbitals (HO) ... 

In order to increase availability redundant subsystems may be deployed. The redundant controllers may work in l-out-of-Z principle (or dual cross wired) comprise two subsystems of identical design. They could be kept electrically isolated from one another, and are synchronized over fiber-optic cables. In the event of a fault, there should be a bumpless transfer from the active subsystem to the backup subsystem. Though it is possible to keep two subsystems but in the same rack, or spatially it is better to keep them apart so that in case of fire both systems may not be lost. 

Again, our enquiry centers around the instability of the homogeneous reference state, which in this case is the state of continuous wave (CW) laser operation. The movement of the photons relative to the stationary laser medium and absorber is intrinsic, hence it provides a required differential flow. The saturable absorber, whose absorption coefficient + hFhi) (the symbols are defined below) depends on the photon flux density F, gives rise to an unstable subsystem as follows. The absorber causes a localized fluctuation of the light field to grow, since the transmission of the partly saturated absorber increases (decreases) in response to an increasing (decreasing) photon flux. In the three-variable description of the laser, in terms of photon density and the populations of absorber and laser medium, the unstable (or activator) subsystem is therefore formed by the photon density and the population of the absorber. As shown below, the three-variable description reduces to the classical two-variable case [10]. 

Our search procedures represent a departure from the above type of paradigm. Rather than simply accepting and implementing a decision policy found by DUg, that optimizes an overall measure of performance, the infimal subsystems and corresponding plant personnel play an active role in the construction and validation of solutions. One tries to build a consensus decision policy, Xpp, validated by all subsystems, DU , k = I,..., K, as well as by the whole plant, DUg, and only when that consensus has been reached does one move toward implementation. Within this context, the upper-level decision unit, DUg, assumes a eoordination role. 

In the ONIOM(QM MM) scheme as described in Section 2.2, the protein is divided into two subsystems. The QM region (or model system ) contains the active-site selection and is treated by quantum mechanics (here most commonly the density functional B3LYP [31-34]). The MM region (referred to as the real system ) is treated with an empirical force field (here most commonly Amber 96 [35]). The real system contains the surrounding protein (or selected parts of it) and some solvent molecules. To analyze the effects of the protein on the catalytic reactions, we have in general compared the results from ONIOM QM MM models with active-site QM-only calculations. Such comparisons make it possible to isolate catalytic effects originating from e.g. the metal center itself from effects of the surrounding protein matrix. 

Smaller values of the activation free energy due to (i) the distortion of the shape of the free energy surfaces and (ii) the increase of the resonance splitting, AJF, of the potential free energies for the classical subsystem due to the increased overlapping of the wave functions of the quantum particles. 

The height of the potential barrier separating the initial and final states of the nuclear subsystem decreases and, hence, the Franck-Condon factor increases (Fig. 6). In the classical limit, this results in a decrease of the activation free energy. 

Changes in the degrees of freedom in a reaction can be classified in two ways (1) classical over the barrier for frequencies o) such that hot) < kBT and (2) quantum mechanical through the barrier for two > kBT. In ETR, only the electron may move by (1) all the rest move by (2). Thus, the activated complex is generated by thermal fluctuations of all subsystems (solvent plus reactants) for which two < kBT. Within the activated complex, the electron may penetrate the barrier with a transmission coefficient determined entirely by the overlap of the wavefunctions of the quantum subsystems, while the activation energy is determined entirely by the motion in the classical subsystem. 

The book thus embraces an extended study on a variety of issues within the theory of orientational ordering and phase transitions in two-dimensional systems as well as the theory of anharmonic vibrations in low-dimensional crystals and dynamic subsystems interacting with a phonon thermostat. For the sake of readability, the main theoretical approaches involved are either presented in separate sections of the corresponding chapters or thoroughly scrutinized in appendices. The latter contain the basic formulae of the theory of local and resonance states for a system of bound harmonic oscillators (Appendix 1), the theory of thermally activated reorientations and tunnel relaxation of orientational... 

Fermentation may take place in the three major microbial subsystems of a sewer, i.e., the wastewater, the biofilm and the sediments (Figure 3.2). Sulfate-reducing bacteria are slow growing and are therefore primarily present in the biofilm and in the sediments, where sulfate from the wastewater may penetrate (Nielsen and Hvitved-Jacobsen, 1988 Hvitved-Jacobsen et al., 1998 Bjerre et al., 1998). However, as a result of biofilm detachment, sulfate reduction may, to some minor extent, take place in the wastewater. Methanogenic microbial activity normally requires absence of sulfate — or at least a low... 

The separation of a reactant system (solute) from its environment with the consequent concept of solvent or surrounding medium effect on the electronic properties of a given subsystem of interest as general as the quantum separability theorem can be. With its intrinsic limitations, the approach applies to the description of specific reacting subsystems in their particular active sites as they can be found in condensed phase and in media including the rather specific environments provided by enzymes, catalytic antibodies, zeolites, clusters or the less structured ones found in non-aqueous and mixed solvents [1,3,6,8,11,12,14-30],... 

Component Any active element that performs a useful task we call a component. Components can be individual objects or large subsystems they can even be the departments in an organization In general, components differ from plain objects in being packaged more robustly. 

Subsystems can interact with each other in many ways. This pattern defines a consistent scheme governing those interactions. For every subsystem, you may choose to uniformly have a distinguished head object that controls the connections between its children s ports and those in other subsystems based on a naming scheme. The head object also mediates all control and asynchronous communication between the subsystem and its parent system and coordinates the activities of its child components (see Figure 12.3). This arrangement gives a consistent structure for every subsystem a head object, a defined role relative to its children, and a consistent protocol regardless of actual subsystem function. 

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