Successive steps

A straightforward derivation (not reproduced here) shows that the effect of the diree successive steps embodied in equation (b3.3.7), with the above choice of operators, is precisely the velocity Verlet algorithm. This approach is particularly usefiil for generating multiple time-step methods. 

Each of these operators is unitary U —t) = U t). Updating a time step with the propagator Uf( At)U At)Uf At) yields the velocity-Verlet algorithm. Concatenating the force operator for successive steps yields the leapfrog algorithm ... 

Fig. 7.13, this shifts the vacancy—represented by the square-in the coordination sphere of the titanium to a different site. Syndiotactic regulation occurs if the next addition takes place via this newly created vacancy. In this case the monomer and the growing chain occupy alternating coordination sites in successive steps. For the more common isotactic growth the polymer chain must migrate back to its original position. 

Refining and Isomerization. Whatever chlorination process is used, the cmde product is separated by distillation. In successive steps, residual butadiene is stripped for recycle, impurities boiling between butadiene (—5° C) and 3,4-dichloto-l-butene [760-23-6] (123°C) are separated and discarded, the 3,4 isomer is produced, and 1,4 isomers (140—150°C) are separated from higher boiling by-products. Distillation is typically carried out continuously at reduced pressure in corrosion-resistant columns. Ferrous materials are avoided because of catalytic effects of dissolved metal as well as unacceptable corrosion rates. Nickel is satisfactory as long as the process streams are kept extremely dry. 

The reactions described here are not synchronous and accordingly can be divided into two successive steps, the last step belonging to one of the three classes of Section 4.04.3.1.1. However, several syntheses of pyrazoles result from the reaction of two compounds, one component contributing four atoms to the ring and the other the fifth atom. There are three possibilities for 4 + 1 reactions shown in Scheme 46, but only those represented by (ii) are described in the literature. 

SO that the batch may be dumped into a tank for processing by successive steps, which normally include centrifugation, filtration, and/or drying. In some cases, a small heal of slurry is left in the batch ciys-tallizer to act as seed for the next batch. 

Successive steps in this distribution can then be obtained to calculate the values of Q at a given value of x in a sequence of time intervals. 

Application of these tests at successive steps will give a good indication of whether or not the purification is satisfactory and will also show when adequate purification has been achieved. Finally elemental analyses, e.g. of carbon, hydrogen, nitrogen, sulfur, metals etc. are very sensitive to impurities (other than with isomers), and are good criteria of purity. 

The synthesis of 1-ethoxy-1-butjme has been reported previously, but the preparations have required multistep sequences. Two of the procedures use 1,2-dibromo-l-ethoxy butane which is dehydrohalogenated in two successive steps, first by an amine base and then by either powdered potassium hydroxide or sodium amide no yields are given. The... 

Given a certain release of a given fuel, the procedure of vapor cloud explosion blast modeling according to HSE can be subdivided into a number of successive steps ... 

In the preparations, the nitrobenzene is con erted by successive steps into azo y-, azo- and hydrazo-beiizene but, by suitably modifying the conditions, the intermediate steps may be omitted. Thus, nitrobenzene may be converted with alcoholic caustic soda and zinc dust directly into hydrazobenzeiie. 

The oxidation of the side-chain of an aiomatic hydiocaibon, when more than one is present, takes place in successive steps. Thus, mesitylene is converted into the following compounds on oxidation ... 

The pyruvate dehydrogenase complex (PDC) is a noncovalent assembly of three different enzymes operating in concert to catalyze successive steps in the conversion of pyruvate to acetyl-CoA. The active sites of ail three enzymes are not far removed from one another, and the product of the first enzyme is passed directly to the second enzyme and so on, without diffusion of substrates and products through the solution. The overall reaction (see A Deeper Look Reaction Mechanism of the Pyruvate Dehydrogenase Complex ) involves a total of five coenzymes thiamine pyrophosphate, coenzyme A, lipoic acid, NAD+, and FAD. 

More often than what has been mentioned above regarding the cyclization of paraffins over the platinum catalyst, the formed olefin species reacts with the acid catalyst forming a carbocation. Carbocation formation may occur by abstraction of a hydride ion from any position along the hydrocarbon chain. However, if the carbocation intermediate has the right configuration, cyclization occurs. For example, cyclization of 1-heptene over the alumina catalyst can occur by the following successive steps ... 

Putting it another way, the adds in successive steps become progressively weaker. This is reasonable it should be more difficult to remove a positively charged proton, H+, from a negatively charged species like H2P04- than from a neutral molecule like H3PO4. 

In Figure 5-12 is a set of core level spectra shown, which have been recorded between successive steps during the growth of an aluminum metallic overlayer on top of PPV [59]. The thickness of the aluminum layer for the lowest and highest coverage correspond to 2 and 20 A, respectively. 

The space environment, seen as beginning in the center of the earth, extends to infinity. In the past few decades outer space has been penetrated. These initial successful steps depended on a number of factors, one of which was the use of plastics. As in terrestrial uses, plastics have their place in space. 

The stoichiometry of decomposition of [Ni(NH3)4](NCS)2 was dependent on the method of salt preparation [1126]. Ammonia was lost in three successive steps (—NH3, —NH3, —2 NH3) from the solution-prepared salt, but the first intermediate could not be isolated from the similar reaction of material prepared by heterogenous reaction. The difference in behaviour was ascribed to differences in perfection of the crystallites resulting from the alternative preparative methods. 

In order to isolate adipic acid, nylon-6,6 fibers were depolymerized under reflux with a 50% NaOH solution in the presence of catalytic amounts of benzyltrimethylammonium bromide. The oligomers formed in successive steps were depolymerized under similar conditions. The yields in steps 1, 2, and 3 were... 

The sensitivity of the relevant rate constants to the groups at the different sites is demonstrated in Table 7.1. The cleavage of amides in the active site of serine protease can be described formally by the two successive steps ... 

The vibrations of the free molecule can be correlated with the vibrations of the crystal by group theoretical methods. Starting with the point group of the molecule Did)> the irreducible representations (the symmetry classes) have to be correlated with those of the site symmetry (C2) in the crystal and, as a second step, the representations of the site have to be correlated with those of the crystal factor group (D2h) [89, 90]. Since the C2 point group is not a direct subgroup of of the molecule and of D211 of the crystal, the correlation has to be carried out in successive steps, for example ... 

Fig. 4.4 Atomistic representation of successive steps in the ECALE synthesis of CdTe on an Au substrate. Observe the deposition and stripping of Te for assembling the correct atomic planes of the zinc blende structure. (Adapted from [27])...

The starting step of the present work is a specific analysis of the solution of the Schrodinger equation for atoms (section 1). The successive steps for the application of this analysis to molecules are presented in the section 2 (description of the optimised orbitals near of the nuclei), 3 (description of the orbitals outside the molecule), and 4 (numerical test in the case of H ). The study of other molecules will be presented elsewhere. 

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