Always Well Behaved

A Failed Thorpe-Dieckmann Cyclisation "Obvious" Reactions are not Always Well Behaved. 

The two main concerns are (1) Cl scales badly (N5 to N8) and exacts too large a penalty in term of computational resources and (2) Cl is not always well-behaved in as much as an apparently systematic improvement of the Cl expansion does not necessarily lead to a systematic increase in the accuracy of the computed results. 

In many applications of quantum mechanics to chemical systems, a knowledge of the ground-state energy is sufficient. The method is based on the variation theorem-, if 0 is any normalized, well-behaved function of the same variables as and satisfies the same boundary conditions as then the quantity = (p H (l)) is always greater than or equal to the ground-state energy Eq... 

The density functional (DF) method has been successful and quite useful in correlating experimental results when model densities are used in the calculations. In fact, the equations characteristic of the DF method can be derived from a variational approach as Kohn and Sham showed some time ago. In this approach, when model densities are introduced, it is not always possible to relate such densities to corresponding wave functions this is the N-representability problem. Fortunately, for any normalized well behaved density there exists a Slater single determinant this type of density is then N-representable. The problem of approximately N-representable density functional density matrices has been recently discussed by Soirat et al. [118], In spite... 

Heteroboranes are those in which one or more non-boron atoms replace a BH vertex, together with groups that may be attached to these heteroatoms. Boranes that contain CH vertices constitute the vast family of carbaboranes. The possibility for carbon to participate in electron-deficient frameworks contradicted the former prejudice of the always electron-precise carbon as the well-behaved brother of naughty boron. So far, most elements have been introduced as heteroatoms into borane frameworks, with the exception of the halogens and the noble gases. 

The value of 2Ecai will always decline in value as more components are calculated, whereas that of lEcai has the possibility of increasing slightly in size although, in most well behaved cases, will also reduce with number of components. If the lEcai does increase against component number it is... 

Arene activation by metal complexation is not always necessary for the cyclisation of samarium ketyls on to aromatic systems. Studies carried out by Reissig demonstrated the utility of these reactions for the synthesis of a wide variety of polycyclic systems, although yields and stereoselectivities depend highly on the substrate structure.68 One well-behaved system is represented by the cyclic y-naphthyl ketones 50-54, which cyclise with excellent diaster-eocontrol, thereby incorporating four- to eight-membered rings into the tetracyclic structures 55-59 (Scheme 5.38).68 Tetracyclic product 56 (n = 1) is of particular interest because of its steroid-like structure. 

In evaluating the effect of pH, it is always well to examine for possible indicator action which could be misinterpreted as an adsorption effect. Many colors in agricultural and industrial products behave as acid-base indicators, in which case a change in pH may alter the hue or change the intensity of the color, e.g., the color of a molasses solution diminished by one-third in going from pH 9 to 4. When it is desired to compare decolorizations conducted at different pH levels, all filtrates should be adjusted to an identical pH when reading the residual color intensities. 

In our experience, a necessary but insufficient condition for a well-behaved critical isotherm is that, at the critical point, the slope of the critical isochore from the equation of state be equal to the slope of the vapor-pressure equation, 6P/6T = dP /dT. This constraint always is applied in the following work via the least-squares program (7). 

Multi-valued and discontinuous functions often present difficulties for differentiation and integration. They are mathematically not well-behaved. There is always great difficulty, for example, in fitting an equation of state in the vicinity of critical points (critical points being both mathematical and physical discontinuities). 

In terms of a comparison between PFRs and CSTRs, a little reasoning plus common sense should tell us that for well-behaved kinetics of the reaction(s) involved (positive power law), the PFR will always be more efficient than n corresponding CSTR on a per-volume basis. For such reactions the rate increases with the concentration level of the reactant, and since the CSTR always operates at the lowest level of reactant concentration (i.e., the reactor exit concentration), the PFR will always win. We can support this conclusion more precisely by comparing the residence times required by each to attain the same conversion level, which for a set feed rate then determines the reactor volume required. As an example, for a first-order reaction at constant density, we have... 

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