Spatial formulas

Optical isomerism is possible in the cyclopropane series and if the spatial formulae of tae thrse dibasic acids,1,1 and two 1,2 acids,are examined it... 

As (2, 3i )-(+)-tartaric acid can be genetically connected by chemical interactions with the D-series of organic compounds originating from D-glyceraldehyde, also (+)-tartaric acid can be considered a more refiable structure than D-glyceraldehyde. It turned out that the spatial formula adopted arbitrarily by Fischer proved to be the correct absolute configuration. On the other hand, Waser ascribed the opposite configuration for (+)-tartaric acid, but that was not confirmed. 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

Structural Formula and Prefixes. In the structural formula the sequence and spatial arrangement of the atoms in a molecule are indicated. 

Spatial congruence of C-H graphs is applied essentially only in chemical formulas which represent a compound of carbon atoms and atoms of valence 1 (or radicals of valence 1). In this case condition (IV), besides (I), (II), (III), adds another restriction not only the relationships are important but also the spatial arrangement of the bonds. The spatial interpretation of the congruence of C-H graphs coincides with the interpretation of the chemical formula as stereoformula. I use stereoisomers in this sense. For example, the number of different stereoisomers is equal to the number of spati-... 

Two pictures of two spatial (three-dimensional) models can represent the same structural formula without representing the same stereoformula they describe the same structural formula if they exhibit the same relationships (if they are topologically congruent, i.e., they satisfy conditions (I), (II), (III)). In order to describe the same stereoformula they must display the same relationships and the same spatial orientation [they satisfy (I), (II), (III), and in addition (IV) (with A ), that is, be spatially congruent]. If two formulas viewed as stereoformulas are equal then they are certainly equal when they are treated as structural formulas. Consequently there are at least as many stereoisomers as there are structural isomers. This fact is reflected by (2.8). It is true particularly for paraffins and monosubstituted paraffins. 

Not all chemistry textbooks provide a clear description of what stereo- and structural formulas imply. Maybe the concepts of spatial and topological congruence of graphs could contribute to the clarification of the chemical terminology. 

The coefficient of xV in the three expressions indicates that there exist four different derivatives of cyclopropane of the form (disubstituted cyclopropane with two identical substituents). Two of the four derivatives are mirror images of each other, that is, they form a pair of optical antipodes. If the spatial arrangement is disregarded, only two distinct cyclopropanes with formula are... 

The physical and chemical properties of complex ions and of the coordination compounds they form depend on the spatial orientation of ligands around the central metal atom. Here we consider the geometries associated with the coordination numbers 2,4, and 6. With that background, we then examine the phenomenon of geometric isomerism, in which two or more complex ions have the same chemical formula but different properties because of their different geometries. 

Two or more species with different physical and chemical properties but the same formula are said to be isomers of one another. Complex ions can show many different kinds of isomerism, only one of which we will consider. Geometric isomers are ones that differ only in the spatial orientation of ligands around the central metal atom. Geometric isomerism is found in square planar and octahedral complexes. It cannot occur in tetrahedral complexes where all four positions are equivalent... 

Although they are built from the same numbers and kinds of atoms, structural isomers have different chemical formulas, because the formulas show how the atoms are grouped in or outside the coordination sphere. Stereoisomers, on the other hand, have the same formulas, because their atoms have the same partners in the coordination spheres only the spatial arrangement of the ligands differs. There are two types of stereoisomerism, geometrical and optical. 

In octahedral symmetry, the F term splits into Aig + T2g + Tig crystal-field terms. Suppose we take the case for an octahedral nickel(ii) complex. The ground term is 2g. The total degeneracy of this term is 3 from the spin-multiplicity. Since an A term is orbitally (spatially) non-degenerate, we can assign a fictitious Leff value for this of 0 because 2Leff+l = 1. We might employ Van Vleck s formula now in the form... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

Monosaccharides can differ in their formulas, their ring sizes, and the spatial orientations of their hydroxyl groups. To analyze the differences between two monosaccharides, begin with structural drawings of the molecules, oriented so the ether linkages are in comparable positions. Then examine the stmctures to locate differences in constituents and bond orientations. 

Proceeding in a manner paralleling the derivation of the excluded volume for a pair of molecules, we consider the polymer molecule to consist of a swarm of segments distributed on the average about the molecular center of gravity in accordance with the Gaussian formula (see Eq. XII-51). This spatial distribution in the unperturbed molecule, as it would exist on the average in the total absence of inter-... 

Two or more substances that have the same molecular formula but different structures and properties are called isomers. Two main types of isomers exist. Structural isomers are ones in which the atoms are bonded in different orders. In stereoisomers, all the bonds in the molecule are the same, but the spatial arrangements are different. 

In order to specify the structure of a chemical compound, we have to describe the spatial distribution of the atoms in an adequate manner. This can be done with the aid of chemical nomenclature, which is well developed, at least for small molecules. However, for solid-state structures, there exists no systematic nomenclature which allows us to specify structural facts. One manages with the specification of structure types in the following manner magnesium fluoride crystallizes in the rutile type , which expresses for MgF2 a distribution of Mg and F atoms corresponding to that of Ti and O atoms in rutile. Every structure type is designated by an arbitrarily chosen representative. How structural information can be expressed in formulas is treated in Section 2.1. 

In Table IV some physical data and spectral characteristics of 6,7-secoberbines are listed. Only methyl corydalate (55) is optically active. Formula 55 presents the spatial structure of this compound, deduced by Nonaka et al. (65) and confirmed by Cushman et al. by both correlation with (+)-mesotetrahydrocorysamine (72) (<5S) and total synthesis (69). It is difficult to find common characteristic features in both the mass and H-NMR spectra of these alkaloids because they differ significantly from each other in their structures. On one hand, corydalic acid methyl ester (55) incorporates a saturated nitrogen heterocycle, while the three aromatic bases (56-58) differ in the character of the side chain nitrogen. For example, in mass fragmentation, ions of the following structures may be ascribed to the most intensive bands in the spectrum of 55 ... 

The computation of the curvatures from the bulk field differential geometry has proven to be rather imprecise. The errors produced by the use of the approximate formulas (100)-(104) are especially big if the spatial derivatives of the field sharp peaks at the phase interface. This is a common situation in the late-stage kinetics of the phase separating/ordering process, when the order parameter is saturated and the domains are separated by thin walls. Here, to calculate the curvatures, we propose a much more accurate method. It is based on the observation that the local curvatures are quantities that can be inferred solely from the shape of the interface, without appealing to the properties of the bulk field [Pg.212]

Instead of nodal lines in closed systems we are interested in the statistics of NPs for open chaotic billiards since they form vortex centers and thereby shape the entire flow pattern (K.-F. Berggren et.al., 1999). Thus we will focus on nodal points and their spatial distributions and try to characterize chaos in terms of such distributions. The question we wish to ask is simply if one can find a distinct difference between the distributions for nominally regular and irregular billiards. The answer to this question is clearly positive as it is seen from fig. 3. As shown qualitatively NPs and saddles are both spaced less regularly in chaotic billiard in comparison to the integrable billiard. The mean density of NPs for a complex RGF (9) equals to k2/A-k (M.V. Berry et.al., 1986). This formula is satisfied with good accuracy in both chaotic and integrable billiards. 

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