Triple instability

C.4. 53. Below we present (following [185]) a list of asymptotic normal forms which describe the trajectory behavior of a triply-degenerate equilibrium state near a stability boundary in systems with discrete symmetry. We say there is a triple instability when a dynamical system has an equilibrium state such that the associated linearized problem has a triplet of zero eigenvalues. In such a case, the analysis is reduced to a three-dimensional system on the center manifold. Assuming that (x, y, z) are the coordinates in the three-dimensional center manifold and a bifurcating equilibrium state resides at the origin, we suppose also that our system is equivariant with respect to the transformation (x,y,z) <- (-X, -y, z). 

This model does not say anything about the mechanism of triple-helix formation, because even in the case of an AON mechanism, nucleation may take place at many positions of the chains and may lead to products the chains of which are staggered. The AON model is based on the assumption that these products are too instable to exist in measurable concentration. As already mentioned, Weidner and Engel142 succeeded in proving by relaxation measurements of al CB2 that the kinetics of in vitro triple-helix formation is governed by more than one relaxation time. This rules out an AON mechanism, but the fitting to the experimentally found equilibrium transition curves nevertheless showed good accommodation and AH° computed from these curves could be confirmed by calorimetric measurement. 

S. H. Chung, Stabilization, propagation and instability of tribrachial triple flames, Proc. Combust. Inst. 31 877-892, 2007. 

Their main danger is bonded to the instability of the triple bond and of the metal derivatives when the halogen derivative has a real acetylene functional group. 

Instability, functional polymerisation, additions to triple bond... 

Addition of a boron-boron bond across a carbon-carbon triple bond is known for some 40 years since the finding that diboron tetrahalides add to alkenes and alkynes in the absence of catalysts.36 Although the reaction seemed to be potentially attractive, the instability of diboron tetrahalides was the critical drawback for the practical use in synthesis. In 1993, much more stable pinacol ester derivative of diboron was found to add to alkynes in the presence of platinum catalysts such as Pt(PPh3)4, Pt(CH2=CH2)(PPh3)2, and Pt(CO)2(PPh3)2 (Figure 1, Scheme 2).37,38 Other... 

Very often the carbon framework of the future allene is already present in the substrate and often it is propargylic in nature. For example, base-catalyzed isomeriza-tions of acetylenic hydrocarbons - with the triple bond in a non-terminal (40) or terminal position - were often used to prepare allenic hydrocarbons in the early days of allene chemistry [8]. The disadvantage of this approach consists in the thermodynamic instability of the allenes produced if not prohibited for structural reasons, the isomerizations do not stop at the allene but proceed to the more stable conjugated diene stage. In practice, complex mixtures are often formed [9] (see also Chapter 1). 

As for basis-set convergence, triple- calculations at the MP2 and CCSD levels are provided for comparison to die double- results. For this particular property, the results for p-benzyne are not terribly sensitive to improvements in the Ilexibility of the basis set. In the pyridynium ion case, die CCSD results are also not very sensitive, but a large effect is seen at the MP2 level. This has more to do with the instability of the perturbation expansion than any intrinsic difference between the isoelectronic aryiies. 

The presence of large singles amplitudes can also be problematic for the CCSD(T) method, because the perturbation theory estimate for the triples can become unstable. One possibility to eliminate that instability involves changing the orbitals used to express the reference... 

Taken together, these conclusions created serious problems in physical interpretation. How could neutron capture by a single isotope initiate three such different reaction processes How could the capture of just one neutron create such great instability that multiple beta decays were needed to alleviate it Nuclear isomerism was known, but how to explain the triple isomerism of 239U Worst of all, how could one account for the inherited isomerism - for several generations - in the... 

That is, a perturbation expansion will diverge for any pair of fermions with exactly opposite momenta k and — k (and the same spin, but, being lazy, we are not summing over spin states). This can be shown by Feynman diagrams (but not here ). The instability means that, within the triple sum of the second term of Eq. (8.11.6), the only terms that are significant are those with k = —k thus the summation over k can be neglected, and we are left with the BCS reduced Hamiltonian ... 

C3H3N3 and also the tetrazines C2H2N4 are very stable compounds, pentazine CHN5 and hexazine N6, a polymer of nitrogen, do not exist. This is not due to the instability of these molecules as such but to the very special stability of the triple bond in nitrogen and in HCN, compared with the corresponding single and double bonds, while this holds to a much smaller extent for the triple carbon-carbon bond (Table 18). 

Extension of this approach leads to the conclusion that fluorine attached to a carbon-carbon triple bond would be considerably destabilising, since electron-pair repulsions with fluorine would then be at a maximum (Figure 7.10). This could partly account for the instability of fluoroalkynes, described later (Section lllA, below). Of course, it must not... 

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