The Geometric Criterion

The main computational criteria used for detomining aromaticity are (a) the geometric criterion, (b) the energetic criterion, (c) the magnetic critaion. These criteria are discussed below. 

This widely used criterion is manifested in (a) bond length equalization due to cyclic delocalization and (b) planarity of the cyclic systems . The planarity of cyclic conjugated systems is regarded as a manifestation of aromaticity, while the non-planarity of cyclic conjugated systems is often taken as an indication of antiaromaticity. 

TT-electron delocalization stabilizes planar strnctures, but other effects such as cr-skeleton strain may cause deviation from planarity, diminishing the degree of aromaticity . Deviation from planarity is fonnd in many silicon compounds snch as hexasUabenzene and silole anions which, according to other criteria, are believed to be aromatic (see below). 

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If a molecule is strained, atoms may not be ver y close to the minimum of their individual potential energy wells when the best compromise geometry is reached. In such a case, the geometric criterion does not provide an exit from the loop. Programs are usually written so that they can automatically switch from a geometric minimization criterion to an energy minimization procedure. 

The geometric criterion is also rather simple. If we have the bond equalization in a cycHc structure, with the bond length being between a single and a... 

An additional encouragement of using the geometric criterion stems from the well-known statement by R. Hoffmann There is no more basic enterprise in chemistry than the determination of the geometrical structure of a molecule. Such a determination, when it is well done, ends all speculation as to the structure and provides us with the starting point for the understanding of every physical, chemical and biological property of the molecule . 

Within the intersecting-state model, the reaction energy barrier is determined by the shape of the potential energy curves of AB and BC and the geometric criterion for the configuration of the transition state given by eqs(17) and (29). 

The criterion is sufficient but not necessary as the flow rate of the leachate is crucial to the erosion process. Even if the geometrical criterion is not fulfilled, erosion occurs only if the forces of flowing water promote materials transport. Only a few studies are available from which a filter criterion could be derived applicable to the actual boundary condition, i.e. horizontal flow in the boundary of the two layers. For instance, research results of J. Brauns on the erosion behaviour of layered soil with horizontal flow may be considered (Brauns 1985). The filter criterion derived from this leaves more clearance than the pure geometrical approach, but even... 

It is also evident that the limiting amomit of a filler depends on the size and shape of its particles. To increase the amoruit of filler, the polydisperse filler can be used. For such fillers, the geometric criterion of filling can be calcrdated, which is defined as the difference between the volume fraction of filler at its limiting packing density, (p the unitary volrune, and its volume fraction in real... 

Restraints due to artifacts may, by chance, be completely consistent with the correct structure of the molecule. However, the majority of incorrect restraints will be inconsistent with the correct structural data (i.e., the correct restraints and information from the force field). Inconsistencies in the data produce distortions in the structure and violations in some restraints. Structural consistency is often taken as the final criterion to identify problematic restraints. It is, for example, the central idea in the bound-smoothing part of distance geometry algorithms, and it is intimately related to the way distance data are usually specified The error bounds are set wide enough that all data are geometrically consistent. 

One criterion of aromaticity is the ring current, which is indicated by a chemical shift difference between protons, in the plane of the conjugated system and those above or below the plane. The chemical shifts of two isomeric hydrocarbons are given below. In qualitative terms, which appears to be more aromatic (Because the chemical shift depends on the geometric relationship to the ring current, a quantitative calculation would be necessary to confirm the correctness of this qualitative impression.) Does Hiickel MO theory predict a difference in the aromaticity of these two compounds ... 

Referring to Table 6.3, it can be seen that with geometric similarities in self controls there is no mixing variable. In practice, we would select the important criterion that needs to be controlled and then size the vessel accordingly. 

The geometrical interpretation of MLR is given in Fig. 29.10. The n rows (objects) of X form a pattern P" of points (represented by x,) which is projected upon an (unknown) axis b. This causes the axis b in S to be imaged by X in the dual space S" at the point y. The vector of observed measurements y has dimension n and, hence, is also represented as a point in 5". Is it possible then to define an axis b in S " such that the predicted y coincides with the observed y Usually this will not be feasible. One may propose finding the best possible b such that y comes as close to y as possible. A criterion for closeness is to ask for the distance between y and y, which is equal to the normlly - yll, to be as small as possible. 

Work in groups of three. Each person should select one of the three composite candidate materials listed in Table 8.11. Using the data in Table 8.11 and the geometric data in Table 8.10, calculate the critical frequency of rotation, Wr, for each type of vehicle listed in Table 8.10 for your composite. Using this criterion only, how many of the vehicles will your composite be suitable for Compare your answers with the other group members for their composites. 

Boudart (223) suggested that all reactions might not be equally sensitive to the geometric arrangements in various metal surfaces or to the differences in the electronic structure of sites in different geometric environments (coordination). Boudart divided the reactions into two groups (I) structure insensitive and (II) structure sensitive. The operational criterion of structure sensitivity is the specific activity (the rate per unit surface area) or, the turnover numbers (TONs) (the rate per site) TONs should differ by more than a factor of 5-10 when the dispersion D is varied sufficiently. Bond (224) formulated similar ideas and also suggested several reasons why the variations of TONs with D can monotonically decrease (antipathic), mono-tonically increase (sympathetic), or show a maximum. 

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