Critical interatomic distance

As is to be expected, inherent disorder has an effect on electronic and optical properties of amorphous semiconductors providing for distinct differences between them and the crystalline semiconductors. The inherent disorder provides for localized as well as nonlocalized states within the same band such that a critical energy, can be defined by distinguishing the two types of states (4). At E = E, the mean free path of the electron is on the order of the interatomic distance and the wave function fluctuates randomly such that the quantum number, k, is no longer vaHd. For E < E the wave functions are localized and for E > E they are nonlocalized. For E > E the motion of the electron is diffusive and the extended state mobiHty is approximately 10 cm /sV. For U <, conduction takes place by hopping from one localized site to the next. Hence, at U =, )J. goes through a... 

This step is important in that it prevents the large-amplitude oscillations (typically present at low k) from dominating over the smaller ones (typically at high k). This is critical since the determination of interatomic distances depends on the frequency and not the amplitude of the oscillations. Other approaches having the same effect have also been employed.37,47... 

Fig. 10. Molecular and electronic structure of cation 74. (a) Perspective view on the cation 74. (b) Coordination sphere of the Pb atom, selected interatomic distances (pm) and angles (deg) Pb-(center CC, C7), 281.7 Pb-(center C6 CT), 280.7 Pbl-C2, 227(2) Pbl-C2 221.1(15) Pbl-C8, 231.0(19) C6-C7, 133(2) CC-CT 132(2) (center C6, C7)-Pb-(center C6, C7 163.6. (c) Contour plots of the Laplacian distribution [V p(r)] in the plane containing the atoms Pb, C6 and C7. Solid and dotted lines designate regions of local charge concentration and depletion, respectively. The bond paths are indicated by the solid back lines, bond critical points are marked with a black square. (Reprinted with permission from Ref. 53. Copyright 2003, Wiely-VCH.)...

The overlay of the key amino acid residues and the CDNB molecule docked in the H-site of each GST-pi variant protein are depicted in Plate 1 (see color insert). The interatomic distances between critical H-site residues and the docked CDNB are presented in Table 15.2. 

Two methods appear to be very powerful for the study of critical phenomena field theory as a description of many-body systems, and cell methods grouping together sets of neighboring sites and describing them by an effective Hamiltonian. Both methods are based on the old idea that the relevant scale of critical phenomena is much larger than the interatomic distance and this leads to the notion of scale invariance and to the statistical applications of the renormalization group technique.93... 

In this inequality all quantities depend on interatomic distance. For this reason the character of the bond may (as it is well known, see [12] for the recent analysis of this situation in similar terms) change in the prescription of its formation or cleavage. The critical distance given by the relation ... 

The formulation of spatially separated a and 7r interactions between a pair of atoms is grossly misleading. Critical point compressibility studies show [71] that N2 has essentially the same spherical shape as Xe. A total wave-mechanical model of a diatomic molecule, in which both nuclei and electrons are treated non-classically, is thought to be consistent with this observation. Clamped-nucleus calculations, to derive interatomic distance, should therefore be interpreted as a one-dimensional section through a spherical whole. Like electrons, wave-mechanical nuclei are not point particles. A wave equation defines a diatomic molecule as a spherical distribution of nuclear and electronic density, with a common quantum potential, and pivoted on a central hub, which contains a pith of valence electrons. This valence density is limited simultaneously by the exclusion principle and the golden ratio. 

It was pointed out that there probably is a critical distance Rc(n,l) such that for interatomic distances It < Rc n,l), outer electrons with... 

Differences in CC bond lengths of the aromatic moiety, which are less than 0.025 A are indeed small if compared to distances of 1.48 A and 1.33 A characterising C(sp2)-C(sp2) single and C(sp2)=C(sp2) double bonds, respectively, as usually pointed out by X-ray researchers. However, they are quite significant for many properties which critically depend on interatomic distances like e.g. molecular quadrupole moments, electric field gradients, spin-spin coupling constants of directly bonded atoms etc. A... 

Consider a J-electron system, such as a transition metal compound. The valence d atomic orbitals do not range far from the nucleus, so COs comprised of Bloch sums of d orbitals and, say, O 2p orbitals, tend to be narrow. As the interatomic distance increases, the bandwidth of the CO decreases because of poorer overlap between the d and p Bloch SUMS. In general, when the interatomic distance is greater than a critical value, the bandwidth is so small that the electron transfer energy becomes prohibitively large. Thus, the condition for metallic behavior is not met insulating behavior is observed. 

The value of the rotational angle will have a direct influence on the extent of steric interactions between substituent atoms (Figure 1). Specifically, as increases from 0 (trans) to 180 (cis), the interatomic distances 1...3 and 2...4 will increase while 1...4 and 2...3 will decrease. Of these, the 1...4 interaction in the c conformation is particularly critical since here contacts are encountered with the potential for severe steric conflicts. In fact, for (-CC1-CC1) the c conformation is essentially precluded since the 1...4 interactions are exclusively Cl...Cl. In (-CH-CC1) and (-CH CH-CCl CCl), the Cl...Cl interactions are replaced by... 

G8. L. E. Sutton, ed. Interatomic Distances, Chem. Soc. Spec. Publ. 11 (1958), supplement, ibid. 18 (1965). A number of organometallic compounds are listed in this critical compilation of bond lengths and angles. See also B3.5. 

The question of how accurately a given structural parameter need be known is difficult to answer in a general way. Most would agree that a measurement of the interatomic distance in the HC1 molecule to the nearest 0.00001 A is mainly a tour de force that has limited bearing on the correlation of this distance with other properties. On the other hand, it is easy to find examples in the literature where very shaky conclusions have been drawn from structural data that were not as reliable as implied by the original investigator. In particular, when one tries to interpret differences in a certain structural parameter—variations from one molecule to another-the question of accuracy becomes critical. The one point that cannot be disputed is that a realistic assessment of the accuracy of a structural parameter is just as important as the value of that parameter itself. 

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