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n=l and can be easily extended to accommodate unequally spaced elements by including the distance of each element with respect to the first. If several transmitters are detected by the sensors, the received signals can be expressed in the same way for each of the transmitters, thus each of the signals have an associated received signal vector, i.e., if there are K transmitters, K received signal vectors can be determined. This Erequently occurs in practice, such

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Complex capabilities such as massively parallel execution, dynamically adaptive discretizations, and multiphysics coupling have been successfully integrated within the Sierra framework and are being shared by engineering simulation codes at Sandia National Laboratories. Successful development and integration of these capabilities is predicated upon well-de ned mathematical abstractions for the mesh, elds, models, algorithms, and parallel environment of these applications. This approach to managing complexity through a conceptual model based upon mathematical abstractions is broadly applicable and will be fundamental to the emerging discipline of frameworks for scienti c and engineering simulation codes.

A module-level variable is available to all of the procedures in the module. You create a module-level variable by placing the declaration (with the Dim statement) at the top of a module, in a Declarations section. Another way to create a module-level variable is to use the Private statement. Examples are shown here:

A photochemical transformation can be accomplished only if the photoexcited molecule has a chance to do something before it uoresces Intersystem crossing time scales may vary considerably, 10 10 11 s Return to the ground state by means of phosphorescence is comparatively slow, 10 2 10 4 s The rates of chemical processes depend on the magnitudes of barriers hindering the change Intermolecular processes depend additionally on collision frequency between reactant (the photoexcited molecule) and substrate as well as possible orientational (stereoelectronic) criteria E ective barriers hindering reaction tend to be lower in photochemical processes, partly because bonding in the excited molecule is weaker and partly because the excited molecule may be vibrationally excited (hot) The collision frequency for intermolecular reactions may be maximized if the substrate can function as the solvent or if the reaction is intramolecular.

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DAUBEN SALEM TURRO ANALYSIS The following very useful approach for the analysis of photochemical reactions is due to Dauben, Salem, and Turro [13] A bond, a b, made from fragment orbitals fa and fb of molecular fragments, a and b, is broken in the process under consideration The relationship to the orbital interaction diagram is transparent (see Figure 153) The orbitals fa and fb are the left- and right-hand sides of the interaction diagram, while the assembly a b is the middle The ``middle'' constitutes the reactant The noninteracting rightand left-hand sides constitute the products Assume that the orbital fb is lower in energy than fa .

D = 1.5

The two electrons originally in the bonding orbital may be distributed in four distinct ways between the two fragment orbitals, fa and fb , yielding four states, Z1 F F F fb 2 fa 0 , Z2 F F F fb 0 fa 2 , 1 D F F F fb 1 fa 1 , and 3 D F F F fb 1 fa 1 The rst two states are zwitterionic, since the bond dissociates heterolytically Also, since both electrons occupy the same orbital, both zwitterionic states are singlet states and will be totally symmetric with respect to any symmetry operation which may be preserved in the dissociation By our assumption that E fb ` E fa , it is expected that E Z1 ` E Z2 The remaining two states arise from homolytic dissociation of the bond and therefore are diradical in character Both singlet and triplet states arise.

If any residual interaction persists (ie, if the bond broken was a p bond or the products are held together in a solvent cage), then the triplet diradical state is lower than the singlet diradical state Otherwise the two have the same energy Since the electrons end up in di erent orbitals, the spatial symmetry of the diradical states is determined by the symmetry properties of fa and fb The diradical is symmetric (S) if fa and fb are both.