Inversion path

It must be emphasized that the inverse path (induced by heating of natural samples) is more complex, because of kinetic reasons related to the length of heating (see, for instance, Blasi et al., 1983). 

A further development of the Dunitz approach was undertaken by Gilli and his coworkers4 who considered the inversion path not only for enamines, but also for other... 

We have so far not discussed explicitly the path over which the atomic nuclei move during the inversion of a real molecule. Our reference configuration was defined in Section 3.2 with the assumption that the bond lengths do not change during the inversion motion. If we were to choose a different path, the fit to experimental data would lead in general to a different double-minimum potential function Vq p). Obviously physical intuition which is involved in considerations of the inversion path should be supported in further developments by quantum chemical calculations. 

In azobenzene the cis-trans isomerization in the (n.n ) state apparently proceeds along a twisting path whereas in the (n, r ) state it proceeds along the inversion path. This has been suggested by the fact that for azobenzenes such... 

Kinetics studies allowed us to evaluate k0 and k for the Fe(phen)32+/cyanide reaction in relation to the stereochemistry of the Fe(phen)2(CN)2 product (1 ). Although it is evident that the optically active product comes from the cyanide dependent k path, the percent of optically inverted Fe(phen)2(CN)2 product is not directly related to the percent which reacts by the cyanide-dependent path. (To illustrate, at 0.4 M cyanide almost 90% of the reaction proceeds through the cyanide dependent path but the optical activity is only about one-third of the optical activity observed for the 2.0 M cyanide reaction product.) On the other hand, a linear relationship does exist for a plot of (% by kx path)/(% optically active product) vs. 1/[CN-], where the % by the ki path is determined from kx[CN ]/(k0+k1[CN ]). Several possible mechanisms have been considered for the reaction, but the simplest mechanism considered which agreed with the results allowed a separation of retention and inversion paths. The strange linear relationship can be fit to the reaction scheme ... 

Figure 2. Proposed inversion paths (a) and (b) together with retention path (c) for cyanide acting as a nucleophile on an iron(Il)-tris(diimine) complex with rigid ligands such as Fe(phen)32 Intermediates a, b, and c are assumed to add water and cyanide in the vacant octahedral position ( ) without further stereochange, although reinversion of the aquocyano complex appears to compete with anation. Na of a cannot bond because of steric factors (20).

Aped et al. (214) used the MM3 force field to calculate the minimum energy conformations of propellane 120 and the corresponding transition states on the inversion path (Scheme 36). A comparison of the inversion barriers calculated for the overall process (86 kj/mol) and found experimentally by NMR measurements (49 kJ/mol) led the authors to the conclusion that it is difficult to obtain accurate results, since MM3 is currently not well parametrized for the gauche effect also the solvent and entropy effects could not be properly assessed (214). 

Among asymmetric bond-forming reactions, the metal-catalyzed asymmetric ally lie alkylation (AAA) is versatile and has found numerous applications [49], While palladium involves a net retention via a double inversion mechanism with soft nucleophiles and a net inversion path with hard nucleophiles, many other metals also catalyze allylic alkylation (e.g., Rh, Ru, Ir, Mo, W, and Cu), which may involve different stereochemical courses [49]. 

With the aid of the Pearson indexing the inverse path of the crystallographic classification is crossed, i.e., by reducing the classification of the chemical compounds to the set of 230 space groups to the number of 14 Bravais lattice, yet leaving open (like a variable) the total number of atoms present in the elementary cell, a number that varies from case to case. 

The inverse path p l of path p is defined as the path parametrized in the sense opposite to p ... 

The inverse path involves the same image, the same curve in F(A) as the original path p, but the direction of the inverse path is reversed. This implies that the roles of origin and extremity are interchanged. 

Specifically, the inverse path p u) of path p(u) has the very same point set image as the path p(u), however, these paths are considered different, and the inverse path p (u) is defined by the opposite parametrization ... 

By this choice of origin at Kq, the product path necessarily exists for each and every pair of such paths. Specifically, a path multiplied by its inverse path does always exist. We note that a product of an algebraic entity with its inverse usually provides some connection to a formal unit element. In our case, however, there are many such, non-unique products, so, as it is, this choice of multiplication does not lead yet to a unique unit element, consequently, this non-unique result does not lead yet to a group-theoretical structure. 

The unit element [Kq] for these homotopy classes is defined as the homotopy equivalence class that contains the constant path p(u) = Kq. Since all these loop paths, when multiplied by their inverse paths, generate a loop path that is homo-topically contractible to Kq, therefore, these path-products are all homotopically equivalent to the constant path p(u) = Ko, so they must all belong to the same, and unique, homotopy equivalence class. 

During the cathodic period (Figure G.2.3), protons are pushed towards the surface of the metal, while hydroxyl ions will take the inverse path. Since the natural oxide film is... 

Lee et al. described a promising way to prepare multiple emulsion that is derived from the emulsion inversion path (Lee et al., 2002). An oil-in-water microemulsion based on 2-butoxyethanol/n-decane/water system was mixed with oleic acid, and they observed a phase inversion sequence that includes microemulsion-in-oil to oil-in-microemulsion. 

Calculations on transition states of the thermal Z— E isomerizations of push-puU azobenzenes also became available. Ab initio methods, including solvent effects based on continuum models, were applied to 4-(amino)-4 -nitroazobenzene and 4-(dimethylamino)-4 -nitroazobenzene. These calculations showed that the transition-state structure of the rotation path was stabilized by the solvent. However, the transition state of the inversion path of the puU group is lower in energy than the transition state for the rotation path. Moreover, for 4-(dimethylamino)-4 -nitroazobenzene, the authors concluded that the reaction proceeds by a mechanism of a mixture of the two limiting cases inversion and rotation. The expected specific solvent interaction with a dipolar transition state, however, cannot be treated with continuum models. 

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