Unavoided crossing

Now, to have degenerate solutions (i.e., an unavoided crossing), the discriminant must vanish, and it is necessary to satisfy two independent conditions ... 

Different topological situations are possible for unavoided crossings between surfaces. One can have intersections between states of different spin multiplicity [an (n - l)-dimensional intersection space in this case, since the interstate coupling vector vanishes by symmetry], or between two singlet surfaces or two triplets [and one has an n - 2)-dimensional conical intersection hyperline in this case]. We have encountered situations in which both types of... 

In the many-dimensional case, the situation is more complicated (Figure 6.5). The arrival of a nuclear wave packet into a region of an unavoided or weakly avoided conical intersection (Section 4.1.2) still means that the jump to the lower surface will occur with high probability upon first passage. However, the probability will not be quite 100%. This can be easily understood qualitatively, since the entire wave packet cannot squeeze into the lip of the cone when viewed in the two-dimensional branching space, and some of it is forced to experience a path along a weakly avoided rather than an unavoided crossing, even in the case of a true conical intersection. 

Fernandez-Alberti S, Roitberg AE, Nelson T, Tretiak S (2012) Identification of unavoided crossings in nonadiabatic photoex-cited dynamics involving multiple electronic states in polyatomic conjugated molecules. J Chem Phys 137 014512... 

Nelson T, Femandez-Alberti S, Roitberg AE, Tretiak S (2013) Artifacts due to trivial unavoided crossings in the modeling of photoinduced energy transfer dynamics in extended conjugated molecules. Chem Phys Lett 590 208... 

Examples of approaches followed by a small selection of the major RM developers are provided below and summarized in Table 3.2. It must be emphasized that these assignments are based on the author s interpretation of approaches described in the literature that placing into one of the approaches defined in this paper is not always feasible due to cross-over between different modes, fusion of ideas from one and another, elimination of steps, selective choices of parameters, modification of parameters, and streamlining of the overall procedure. Variations on a theme are unavoidable. In assigning certification approaches, the numbers refer to the approaches defined above multiple numbers indicate a blending of two or more modes with a asterisk indicating the author s assignment of the dominant mode of certification. 

Generally, during hydroboration polymerization of dicyano compounds, the formation of the borazine structures that have a six-membered boron-nitrogen ring (scheme 19a) and dihydroborated end groups (scheme 19b) as a structural defect is unavoidable. The borazine cross-linked structures often cause the gelation, and dehydroboration causes a decrease in molecular weight. 

This general situation has led to a burst of research activities into the role Fe oxides play in environmental managment practices.The results have, in turn, widened our basic understanding of the nature and properties of Fe oxides. Some overlap between this chapter and almost every other chapter in the book is, therefore, unavoidable, but will be kept to a minimum by cross references. 

The remaining diethyl ether is fractionally distilled off from the dimethyl-cadmium at atmospheric pressue (bp 34-35°C). Distillation is continued until pure dimethylcadmium has started to cross the still head (bp 105-106°C)—at which point the argon flow through the paraffin-oil bubbler is increased to a rapid purge while the oil bath is lowered quickly to prevent distillation of further product. The system is allowed to stand for several minutes to allow any dimethylcadmium in the Vigreux column to drain back down into the distillation flask and valve D is then closed. A small amount of dimethylcadmium will decompose during this distillation process, as witnessed by the cadmium metal residue formed in the distillation flask, but this is unavoidable and does not significantly affect the subsequent yield of product. 

This is the most desirable situation for a fuel cell, because the ohmic resistance is minimized. However, due to the disposition of the current collectors, this situation is not achievable, and a deviation along the plane of the cell is unavoidable. For this reason, the resistance generated if current flows in a straight line is usually named cross-plane resistance while the term in-plane resistance refers to the contribution due to the directional deviation. Even for the planar configuration, where expression (A3.1) is usually employed, small in-plane resistance is present (Figure A3.2). 

According to the detection principle of PAS, the interference between two gases sometimes is unavoidable. A monitor, which is able to detect several gases in a single sample, can reduce, and sometimes overcome, the problem by cross-... 

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