Coordinates, atomic distance

X-ray and neutron diffraction methods and EXAFS spectroscopy are very useful in getting structural information of solvated ions. These methods, combined with molecular dynamics and Monte Carlo simulations, have been used extensively to study the structures of hydrated ions in water. Detailed results can be found in the review by Ohtaki and Radnai [17]. The structural study of solvated ions in lion-aqueous solvents has not been as extensive, partly because the low solubility of electrolytes in 11011-aqueous solvents limits the use of X-ray and neutron diffraction methods that need electrolyte of -1 M. However, this situation has been improved by EXAFS (applicable at -0.1 M), at least for ions of the elements with large atomic numbers, and the amount of data on ion-coordinating atom distances and solvation numbers for ions in non-aqueous solvents are growing [15 a, 18]. For example, according to the X-ray diffraction method, the lithium ion in for-mamide (FA) has, on average, 5.4 FA molecules as nearest neighbors with an... 

Their conclusion is that a rescaling of the CN is necessary with increasing bond distances as indicated in Fig. 15. This situation may be encountered in cases where metal-coordinating atom distances can take values very different from those in the model structure. It is to be noticed that the results for CN are not improved either by variations of the disorder parameter in the curve fitting or by measurements at low temperature. 

Spanned by tbc atoms 4, 2, and 1, and 2, 1, and 3 (tlic ry-planc), Except of the first three atoms, each atom is described by a set of three internal coordinates a distance from a previously defined atom, the bond angle formed by the atom with two previous atoms, and the torsion angle of the atom with three previous atoms. A total of 3/V - 6 internal coordinates, where N is the number of atoms in the molecule, is required to represent a chemical structure properly in 3D space. The number (,3N - 6) of internal coordinates also corresponds to the number of degrees of freedom of the molecule. 

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

The interatomic distances involving the single 4-coordinate atoms at 2(b) were only 160 pm this is unusually short for B-B but reasonable for B-C or B-N distances. 

The niobium atom has a slightly distorted octahedral coordination. Interatomic distances between the niobium atom and the two oxygen atoms in trans positions, O-Nb-O are 1.81 and 2.14 A. The niobium atom is shifted from the base plane of the octahedron by 0.23 A, and this shift, in adjacent chains, is in opposite directions. Pakhomov and Kaidalova [204] concluded that the shorter Nb-O bond (1.81 A) is an intermediate between a single and double bond. 

The first step in the DG calculations is the generation of the holonomic distance matrix for aU pairwise atom distances of a molecule [121]. Holonomic constraints are expressed in terms of equations which restrict the atom coordinates of a molecule. For example, hydrogen atoms bound to neighboring carbon atoms have a maximum distance of 3.1 A. As a result, parts of the coordinates become interdependent and the degrees of freedom of the molecular system are confined. The acquisition of these distance restraints is based on the topology of a model structure with an arbitrary, but energetically optimized conformation. 

Fig. 8.1 Representation of the potential wells for the and T2 states of an iron(II) SCO system, the nuclear coordinate being the metal-donor atom distance...

When specifying atomic coordinates, interatomic distances etc., the corresponding standard deviations should also be given, which serve to express the precision of their experimental determination. The commonly used notation, such as d = 235.1(4) pm states a standard deviation of 4 units for the last digit, i.e. the standard deviation in this case amounts to 0.4 pm. Standard deviation is a term in statistics. When a standard deviation a is linked to some value, the probability of the true value being within the limits 0 of the stated value is 68.3 %. The probability of being within 2cj is 95.4 %, and within 3ct is 99.7 %. The standard deviation gives no reliable information about the trueness of a value, because it only takes into account statistical errors, and not systematic errors. 

The PES is only a function of the shape of the molecule, as described by the internal coordinates (there are ZN — 6 of them for a non-collinear molecule of N atoms). There are many possible choices of internal coordinates, including atom-atom distances (bond lengths), bond angles, dihedral or out-of-plane angles, or some combination of these. However, the PES can be expressed solely in terms of the atom-atom distances (a result that follows from the group theory of functions like the PES which are invariant to rotation).48,49 For N atoms there are NCi = N(N — l)/2 such distances, which are easily calculated from the Cartesian coordinates. Rather than use these atom-atom distances, Rn, we actually use the reciprocal distances, Zn ... 

Problems arise in two situations firstly, if the molecule is linear, no complete set of 3N — 6 internal coordinates is possible. For this case, a method for constructing PES in terms of Cartesian coordinates could be used.56 Secondly, if the molecule is planar, atom-atom distances (or their reciprocals) cannot provide a complete set of internal coordinates, since they cannot describe out-of-plane motion. However, we have found the coordinates Zn so useful that we retain these coordinates and avoid planar geometries (except for three atoms, when only linear geometries are taboo). That is,... 

Like many other chemical concepts the concept of strain is only semi-quantitative and lacks precise definition. Molecules are considered strained if they contain internal coordinates (interatomic distances (bond lengths, distances between non-bonded atoms), bond angles, torsion angles) which deviate from values regarded as normal and strain-free . For instance, the normal bond angle at the tetra-coordinated carbon atom is close to the tetrahedral value of 109.47°. In the course of force field calculations these normal values are defined more satisfactorily, though in a somewhat different way, as force field parameters. 

In the REC model, the ligand is modelled through an effective point charge situated in the axis described by the lanthanide-coordinated atom axis, at a distance R, which is smaller than the real metal-ligand distance (Figure 2.6). To account for the effect of covalent electron sharing, a radial displacement vector (Dr) is defined, in which the polar coordinate R is varied. At the same time, the charge value (q) is scanned in order to achieve the minimum deviation between calculated and experimental data, whereas 9 and cp remain constant. 

The Fe(III) complexes of R-substituted salicylaldehyde thiosemicarbazone (R-thsa2- Fig. 6) are among the most studied spin crossover materials of this family. The crystal structures of several of them have been determined at various temperatures. The iron-donor atom distances are compiled in Table 2. The Fe(III) ion is in a distorted FeS2N202 octahedron formed by two thiosemicarbazone ligands, which are geometrically arranged in such a way that the S and O atoms are located in cis positions, whereas the N atoms occupy trans positions, i.e. each tridentate molecule coordinates in an equatorial plane [101]. 

Once the protein interaction pattern is translated from Cartesian coordinates into distances from the reactive center of the enzyme and the structure of the ligand has been described with similar fingerprints, both sets of descriptors can be compared [25]. The hydrophobic complementarity, the complementarity of charges and H-bonds for the protein and the substrates are all computed using Carbo similarity indices [26]. The prediction of the site of metabolism (either in CYP or in UGT) is based on the hypothesis that the distance between the reactive center on the protein (iron atom in the heme group or the phosphorous atom in UDP) and the interaction points in the protein cavity (GRID-MIF) should correlate to the distance between the reactive center of the molecule (i.e. positions of hydrogen atoms and heteroatoms) and the position of the different atom types in the molecule [27]. 

Notice however that in the analysis of a given structure, in order to evaluate coordination and atomic distances, the atomic positions in several adjacent cells may be relevant, and therefore values of x, y, z higher than 1 or lower than 0 should be... 

Fig. 2. Relationship between the bond lengths of neighboring bonds in sulfur rings indicating strong bond-bond interaction. The distance, d, of a bond between two-coordinated atoms is. a function of the arithmetic mean of the lengths of the two neighboring bonds, V2(di dj). The values were taken from the compounds indicated in the Figure. The curve ends on the right side at d2 = 189 pm, the bond length of the diatomic molecule Sj...

At this stage, sulfoximines had been shown to function as chiral ligands for various palladium and copper catalysts which led to enantioselectivities of >95% ee in various reactions. Furthermore, most of those catalyzed reactions were C-C bond formations. Obvious questions were, therefore, whether sulfoximines could also be applied in combination with other metals and whether reductions and oxidations could be catalyzed as well. A structural comparison of the sulfoximines leading to high ee values such as 55, 60, 81, and 85 revealed that all of them had a two-carbon distance between the two coordinating atoms (which were all nitrogen in these cases). 

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