Atom distances

Despite the many advantages of these physical models, they have several essential deficiencies. As the size of the structure increases, the assembly of a model becomes progressively more unmanageable and more complex. Furthermore, d.ata such as atom distances and atom angles can be determined only with difficulty, or not at all. 

Hydrogen bond geometries may be reproduced or predicted fairly weH with reasonable, but sometimes underestimated heavy atom—heavy atom distances radial dependence of the hydrogen bond may be in error. 

Here and below, T , 1, , and e, i, j = 1,. . . , 5, denote atomic position vectors, atom-atom distances, and the corresponding unit vectors, respectively. In order to construct a correctly closed conformation, variables qi,. . . , q4 are considered independent, and the last valence angle q is computed from Eq. (7) as follows. Variables qi,.. ., q4 determine the orientation of the plane of q specified by vector 634 and an in-plane unit vector 6345 orthogonal to it. In the basis of these two vectors, condition (7) results in... 

Figure 3 Model building by Modeller [31], First, spatial restraints in the form of atomic distances and dihedral angles are extracted from the template stmcture(s). The alignment is used to determine equivalent residues between the target and the template. The restraints are combined into an objective function. Finally, the model for the target is optimized until a model that best satisfies the spatial restraints is obtained. This procedure is technically similar to the one used in structure determination by NMR.

Charges and atomic distances in MP2/6-31G transition state stmcture for ketene + ethene cycloaddition... 

Atom numbering as in Figure 11-1.1 thank my colleague J. D. Dunitz (ETH Zurich) for providing the appropriate computer program for the calculation of these atomic distances and the various N O distances mentioned below. 

It is worth noting that each Na atom appears to perturb the electron density of the Pt(lll) surface over large ( 12) atomic distances. This can explain nicely the observed long-range promotional effect of Na on Pt surfaces. It is strongly reminiscent of the IR spectroscopic work of Yates and coworkers who showed that a single adsorbed alkali atom can affect the IR spectra of up to 27 coadsorbed CO molecules.80... 

Spillover-backspillover phenomena can take place over enormous ( mm) atomic distances. 

Both Ir02 and Ru02 are metallic oxides with high density of states at the Fermi level. In this respect they are very similar to metals. On the other hand the fact that backspillover 08 ions originating from YSZ can migrate (backspillover) enormous (mm) atomic distances on their surface, as proven experimentally by Comninellis and coworkers, is not at all obvious. 

The crystal structure data are taken from Wyckoff, "International Critical Tables, except where otherwise noted. Inter-atomic distances referred to Goldschmidt are from Goldschmidt, Skrifter Det. Norske Videnskaps-Akad. Oslo I. Maiem.-Naturvid. Klasse, 1926, No. 2. 

In the past, ionic radii have often been compared with observed interatomic distances without much regard to the nature of the crystal from which they were derived. Recently several investigators19 have concluded that in many crystals the bond between atoms does not consist of the electrostatic attraction of only slightly deformed ions. Goldschmidt in particular has divided crystals into two classes, ionic and atomic crystals, and has shown that ionic radii (using Wasastjema s set) do not account for the observed inter-atomic distances in atomic crystals. In the following pages our crystal radii will be compared with the experimental dis-... 

The agreement is satisfactory, except in the cases where there are deviations from additivity. This fact is a verification of our treatment and of the correctness of our screening constants, for the arbitrary selection of only one ionic radius in a series of salts showing additivity in inter-atomic distances is permitted, and our screening constants fixed four radii independently. 

The experimental values for the lithium halides are high. This is due to two different phenomena. In the case of the chloride, bromide and iodide the anions are in mutual contact, that is, the repulsive forces operative are those between the anions, and the anion radius alone determines the inter-atomic distances. The geometry of the sodium chloride structure requires that, for less than 0.414, the anions come into contact... 

In the case of lithium fluoride the ratio R i /Rj>- is 0.44. In this crystal there is double repulsion the repulsive forces between anion and anion and those between anion and cation are simultaneously operative. The inter-atomic distances are determined neither by the sum of the radii for the anion and cation nor by the radius of the anion alone, but are larger than those calculated by either method. Thus the lithium-fluoride distance is 0.05 A. (2.5%) larger than the sum of the radii, and one-half the fluorine-fluorine distance is 0.06 A. larger than the fluoride radius. 

Inter-Atomic Distances for Cesium Chloride Type Crystals... 

The Alkaline-Earth Oxides, etc.—The observed and calculated inter-atomic distances for the alkaline-earth oxides, sulfides, selenides and tellurides are given in Table VIII. Except for the magnesium com-... 

The agreement is also satisfactory for lithium and sodium sulfide. The oxide was used in calculating the lithium radius, 0.60 A., for in this compound it is safe to assume that the anions are not in mutual contact. It is further highly pleasing to note that even in zirconium and cerium oxide, containing quadrivalent cations, our theoretical radii are substantiated by the experimental inter-atomic distances for this makes it probable that even in these crystals the ions are not greatly deformed. 

The Rutile Structure.—A large number of compounds MX crystallize with the tetragonal structure of rutile, TiCfe. In this structure the position of the ion X is fixed only by the determination of a variable parameter by means of the intensity of reflection of x-rays from various crystal planes. In accordance with the discussion in a following section, we shall assume the parameter to have the value which causes the distances between X and the three ions M surrounding it to be constant. With this requirement the inter-atomic distance R and the edges a and c of the unit of structure are related by the equation R = (a/4 /2) [2 + (c/o)2]. In this way the inter-atomic distances in Table XII are obtained. In the case of magnesium fluoride the agreement is satisfactory. 

Inter-Atomic Distances in Rutile Type Crystals and in Anatase... 

Probably the high calculated inter-atomic distances in the oxides are due to our method of using the crystal radii. The substitution in Equation 13 of z = 4 for the cation and z = 2 for the anion, instead of z = V8 for each, would lead to high calculated values in case the anion is much smaller than the cation, as in the rutile type crystals and anatase. 

Goldschmidt has classed also with the ionic crystals the C-modification of the sesqui-oxides, cubic crystals with 16 M2O3 in the unit of structure. The inter-atomic distances reported by him are 2.16-2.20 A. for scandium oxide and 2.34-2.38 A. for yttrium oxide, in good agreement with the radius sums 2.21 A. for Sc+3-0= and 2.33 A. for Y+3-0". 

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