Bifurcated geometries

The HjO HP-HF complex was reexamined more recently in conjunction with a comparison with H O H O -HF. The focus of this work was an exploration of the potential energy surface to identify all the local minima. In addition to the cyclic structure of Fig. 5.27, a bifurcated geometry, in which the water serves as double proton acceptor, was also located as a minimum of the PES of H O -HF—HF. In the complex containing a pair of water molecules and one HF, the only minimum located was of cyclic type. Both of these structures are illustrated in Fig. 5.28. 

Whereas the equilibrium structure itself is pretty much agreed on for the water dimer, there are a number of other geometries that one might suppose should be comparable in energy. For example, the notion of a cyclic structure has been advanced over the years as have various types of bifurcated geometries. Smith et al. recently completed the most comprehensive survey of the ab initio potential energy surface of the water dimer to date. They examined the question of a number of possible minima on the surface and potential transition states that connect them. 

Finally, a switching of the two hydrogens in the proton donor molecule passes through a Ca, bifurcated geometry illustrated in Figure 11. This pathway... 

Gas-phase spectroscopic measurements [115] had been unable to clearly determine the structure of the H2S dimer, but had offered the suggestion of a linear or bifurcated geometry. Frisch et al. [94] compared these two theoretically and found indications that the bifurcated structure represents a transition state on the surface, about 1 kJ/mol higher in energy than the linear structure. The dimer is only very weakly bound, by less than 6 kJ/mol and by an even smaller amount (3.3 kJ/mol) when zero-point vibrational energies are considered. Accentuating the weakness of the interaction, the S S distance was calculated to be quite long at 451.4 pm. 

A few numerical studies of flow and transport in the carotid artery bifurcation have been reported recently as summarized in Figure 12.4. The carotid artery bifurcation is a major site for the localization of atherosclerosis, predominantly on the outer wall (away from the flow divider) in the flow separation zone, which is a region of low and oscillating wall shear stress [26]. Perktold et al. [27] simulated Oj transport in a realistic pulsatile flow through an anatomically realistic three-dimensional carotid bifurcation geometry using a constant wall concentration boundary condition. Ma et al. [28] simulated... 

As in the sudden expansion and stenosis geometries, the bifurcation geometry can induce flow separation on the outer wall with reattachment downstream. Again there is a region of attenuated transport near the separation point and amplified transport near the reattachment point. Perktold et al. [27] predicted minimum Sherwood numbers close to zero in the flow separation zone. Ma et al. [28] predicted the same general spatial distribution, but the minimum Sherwood number was 25. Differences in the minimum Sherwood number may be due to differing entry lengths upstream of the bifurcation as well as differences in flow pulsatility. 

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

The set of the reaction-diffusion equations (78) can be solved by different methods, including bifurcation analysis [185,189-191], cellular automata simulations [192,193], or numerical integration [194—197], Recently, two-dimensional Turing structures were also successfully studied by Mecke [198,199] within the framework of integral geometry. In his works he demonstrated that using morphological measures of patterns facilitates their classification and makes possible to describe the pattern transitions quantitatively. 

An important open question relates to whether an optimal AR exists with regard to entrainment enhancement. Laboratory jet experiments with pseudo-elliptical geometries [27] suggest that an optimal AR with regard to nozzle-geometry-enhanced entrainment might be at a value AR = 3. However, the experiments are not conclusive since they involved AR up to 3.5 and nonuniform momentum-thickness distributions, which are known to also affect the entrainment process [5]. Moreover, the possible effects on jet entrainment of other more complicated interactions such as vortex-ring bifurcation still need to be established. 

The largest number of hydrogen bonds in crystal structures of alkyl hydroperoxides refer to intermolecular bonds between the hydroperoxide proton and functionalities of the type 0=X, where X denotes a sulfur (e.g. 27), carbon (e.g. 30) or a phosphorous atom (e.g. 32, Figure 14, Table 7)93,108,115 geometry of [l,2-bis(diphenylphosphinoyl)ethane] bis(2,2-dihydroperoxypropane) (32) in the solid state is a rare example of a bifurcated hydrogen bond between an OOH donor and an 0=X proton acceptor. 

The pancake theory today is perceived by mathematicians as a chapter contributed by Ya.B. to the general mathematical theory of singularities, bifurcations and catastrophes which may be applied not only to the theory of large-scale structure formation of the Universe, but also to optics, the general theory of wave propagation, variational calculus, the theory of partial differential equations, differential geometry, topology, and other areas of mathematics. 

Three general dimer types were considered for both nitroanilines A, the optimal dimer with two distinct H-bonds, each between one amino-hydrogen and one nitro-oxygen B, a relaxed geometry, with at least one bifurcated H-bond that is the local minimum closest to the crystal structure, and, C, the structure obtained by fixing the H-bonds at their experimental (crystal structure) distances and optimizing the rest of the dimer within the same constraints as A and B. Structure C is closest to the experimental structure. 

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