Arbitrary character

The determination of atomic charges has been a controversial subject because there has been disagreement on how the atoms in a molecule should be defined or, in other words, how the electronic charge should be apportioned between the atoms. Proposed orbital methods for determining atomic charges include the natural bond orbital (NBO) method (Reed et al., 1985, 1988) and the Mulliken method (1985). However, these orbital methods have some unsatisfactory features. There is a certain arbitrary character associated with them because, as we saw in Chapter 3, the choice of orbitals to describe a given molecule is not unique and... 

Recognition has been made of some rather strongly worded criticism, from various sides, of the treatment of resonance of molecules among alternative valence-bond structures, as presented in earlier editions of this book, on the basis of its idealistic and arbitrary character, by the introduction of a section (Sec. 6-6) in which it is pointed out that the theory of resonance involves only the same amounts of idealization arid arbitrariness as the classical valence-bond theory. 

The NBO method can be nsed for ab initio calcnlations at the HF and any correlated levels as well as for DFT methods. The results do not strongly depend on the size of the basis set, which is one reason that the NBO method has replaced the ontdated Mnlliken popnlation analysis, for example, for the calculation of atomic partial charges. But there is a price that has to be paid for the advantages. Like any orbital-based method for partitioning the electronic charge into atomic and bonding domains, the choice of the selection procedure has some arbitrary character that needs to be known in order to jndge the qnality of the results. 

The argument so far has all been about fitness components, whose partial correlation with fitness is positive by definition. For any other trait, such as size, bristle number, or the g factor of psychometrics, fitness will not automatically increase if an individual has more of it. In a cruel world, the probability of such positive relationships with arbitrary characters is higher, as maladapted individuals... 

Reid [26]. But the low-load one-hour wear data, also listed in Table 11-8, are not in the same relative rank order as the EP data. Because of the empirical and arbitrary character of the four-ball test, it is probable that the contact and rubbing parameters have not been identified precisely enough for a quantitative formulation of the mechanism of additive action. 

Weinstein et al., 1997 Eisen et al., 1998). Inspection of the dendrogram and color plot will reveal samples with closely related patterns of gene expression as well as clusters of genes with similar expression pattern. Software for this type of display is freely available. This approach has been productively applied to yeast and human expression data. It is important to note that there is an arbitrary character to the manner in which a dendrogram is drawn. Clusters can be rotated about the point of bifurcation affecting the apparent proximity of the edges of a cluster with adjacent clusters. The important information is contained in the cluster contents and their similarity. 

The whole theory hangs upon the mathematical conceptions which this communal system suggests. They are not without certain arbitrary characters, and it is therefore of importance that the underlying ideas should find further confirmation in another major phenomenon, namely that of ferromagnetism. 

Note that this result holds for every a and a" (since A vanishes) as well as for every a and a" (because of the completely arbitrary character of B). 

This means that there is a cross-over temperature defined by (1.7) at which tunneling switches off , because the quasiclassical trajectories that give the extremum to the integrand in (2.1) cease to exist. This change in the character of the semiclassical motion is universal for barriers of arbitrary shape. 

The polynomial (1.5) which I called cycle index is, if H is the symmetric group, equal to the principal character of H in representation theory. Professor Schur informed me that the cycle index of an arbitrary permutation group being really a subgroup of a symmetric group is of importance for the representation of this symmetric group. We will, however, not expand on the relationship between representation theory and our subject. 

There is an extremely wide range of potentially useful chemical treatments available, and for any boiler system, proper selection, utilization, and control are vital considerations that may largely determine the ultimate success of the overall program. These chemicals usually are organized by type of compound, function, mode of action, or similar classification, but, because many chemicals are multifunctional in character, may be used in either a primary or supplementary (adjunct or conjunctional treatment) role, and additionally may be branded (especially many modem polymers) or otherwise disguised, such classifications may be quite arbitrary. 

The proposed specification of the kernel for m- and J-diffusion models is mathematically closed, physically clear and of quite general character. In particular, it takes into consideration that any collisions may be of arbitrary strength. The conventional m-diffusion model considers only strong collisions (0(a) = 1 /(27c)), while J-diffusion considers either strong (y = 0) or weak (y = 1) collisions. Of course, the particular type of kernel used in (1.6) restricts the problem somewhat, but it does allow us to consider kernels with arbitrary y < 1. 

Relations (2.46) and (2.47) are equivalent formulations of the fact that, in a dense medium, increase in frequency of collisions retards molecular reorientation. As this fact was established by Hubbard within Langevin phenomenology [30] it is compatible with any sort of molecule-neighbourhood interaction (binary or collective) that results in diffusion of angular momentum. In the gas phase it is related to weak collisions only. On the other hand, the perturbation theory derivation of the Hubbard relation shows that it is valid for dense media but only for collisions of arbitrary strength. Hence the Hubbard relation has a more general and universal character than that originally accredited to it. 

That the distinction between sol and gel is by no means an arbitrary one is shown by the character of the distribution as depicted in Figs. 69 and 70. The distribution curves in the former figure always are asymptotically zero at large x, even at the gel point. Extremely large (i.e., almost infinite ) molecules, which might be regarded as intermediate between sol and gel, never account for more than an extremely small fraction of the total. The structural distinction between sol and... 

Only the set of integers 1,1,2 satisfies this relation, the order being arbitrary. In tins group there are two different irreducible representations of order one and one of order two. Thus, tire characters appearing in the column headed... 

An example of the application of Eq. (47) is provided by the group < 3v whose symmetry operations are defined by Eqs. (18). If the same arbitrary function,

symmetry operation can be worked out, as shown in the last column of Table 13. With the use of the projection operator defined by Eq. (47) and the character table (Table 6), it is found (problem 16) that the coordinate z is totally symmetric (representation Ai). However, it is the sum xy + zx that is preserved in the doubly degenerate representation, E. It should not be surprising that the functions xy and zx are projected as the sum, because it was the sum of the diagonal elements (the trace) of the irreducible representation that was employed in each case in the... 

Here we see clearly the large concentration of density around the oxygen nucleus, and the very small concentration around each hydrogen nucleus. The outer contour is an arbitrary choice because the density of a hypothetical isolated molecule extends to infinity. However, it has been found that the O.OOlau contour corresponds rather well to the size of the molecule in the gas phase, as measured by its van der Waal s radius, and the corresponding isodensity surface in three dimensions usually encloses more than 98% of the total electron population of the molecule (Bader, 1990). Thus this outer contour shows the shape of the molecule in the chosen plane. In a condensed phase the effective size of a molecule is a little smaller. Contour maps of some period 2 and 3 chlorides are shown in Figure 8. We see that the electron densities of the atoms in the LiCl molecule are only very little distorted from the spherical shape of free ions consistent with the large ionic character of this molecule. In... 

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