Atoms defined

Cross G, Schirmeisen A, Stalder A, Grutter P, Tschudy M and Durig U 1998 Adhesion interaction between atomically defined tip and sample Phys. Rev. Lett. 80 4685... 

The incidence matrix is an n A m matrix where the nodes (atoms) define the columns (n) and the edges (bonds) correspond to the rows [m). An entry obtains the value of 1 if the corresponding edge ends in this particular node (Figure 2-16),... 

The number of protons in an atom defines what element it is. For example carbon atoms have six protons, hydrogen atoms have one, and oxygen atoms have eight. The number of protons in an atom is referred to as the atomic number of that element. The number of protons in an atom also determines the chemical behavior of the element. 

The aldotetroses are the four stereoisomers of 2 3 4 trihydroxybutanal Fischer pro jections are constructed by orienting the molecule m an eclipsed conformation with the aldehyde group at the top The four carbon atoms define the mam chain of the Fischer projection and are arranged vertically Horizontal bonds are directed outward vertical bonds back... 

Ecample You can monitor improper torsion angles to determine which side of a substrate molecule faces the active site of a protein. Select three atoms on the substrate molecule and a fourth in the active site. These atoms define an improper torsion angle. Save this selection as a named selection. Then observe a plot of this improper torsion angle (in the Molecular Dynamics Results dialog... 

Compounds that have the empirical formulas MCr02 and DCr204 where M is a monovalent and D a divalent cation, are known as chromites. These are actually mixed oxides and probably are better written as M20-Cr203 and D0-Cr203, respectively. The oxides of D are largely spinels, ie, the oxygen atoms define a close-packed cubic array having the octahedral holes occupied by the Cr(III) cation and the tetrahedral holes occupied by D (54). Chromite ore is an important member of this class of oxides. 

Atoms defined in this way can be treated as quantum-mechanically distinct systems, and their properties may be computed by integrating over these atomic basins. The resulting properties are well-defined and are based on physical observables. This approach also contrasts with traditional methods for population analysis in that it is independent of calculation method and basis set. 

Consider, for example, the protein shown in Figure 15.7. The bottom left-hand amino acid is valine, which is linked to proline. Suppose for the sake of argument that we wanted to treat this valine quantum-mechanically and the rest of the protein chain according to the methods of molecular mechanics. We would have to draw a QM/MM boundary somewhere between valine and the rest of the protein. The link atoms define the boundary between the QM and the MM regions. A great deal of care has to go into this choice of boundary. The boundary should not give two species whose chemical properties are quite different from those implied by the structural formulae on either side of this boundary. 

Cyclopropane, for example, must be a rigid, planar molecule because three points (the carbon atoms) define a plane. No bond rotation can take place around a cyclopropane carbon-carbon bond without breaking open the ring (Figure 4.1). 

A molecule that contains just one chiral carbon atom (defined as a carbon atom connected to four different groups also called an asymmetric or stereogenic carbon atom) is always chiral, and hence optically active. As seen in Figure 4.1, such a molecule cannot have a plane of symmetry, whatever the identity of W, X, Y, and Z, as long as they are all different. However, the presence of a chiral carbon is neither a necessary nor a sufficient condition for optical activity, since optical activity may be present in molecules with no chiral atom and since some molecules with two or more chiral carbon atoms are superimposable on their mirror images, and hence inactive. Examples of such compounds will be discussed subsequently. 

An interesting mechanistic issue was raised by Firestone on the aqueous Diels-Alder reaction between 2-methylfuran and maleic acid in water, which is found to be 99.9% stereospecific.80 By adding heavy atom (defined as any below the first complete row of the periodic table) salts to the aqueous media, it was found that addition of heavy but not light atom salts reduced the degree of stereospecificity significantly in the retrodiene reaction. The results suggest that a large portion of the Diels-Alder reaction occurs via diradical intermediates (Scheme 12.2). 

An important advantage of the finite atoms defined by AIM is that they do not overlap, which is not generally true for orbital-defined atoms. Each atom has a sharp and well-defined boundary inside the molecule, given by its interatomic surfaces. The atoms fit exactly into each other, leaving no gaps. In other words, the shape and the volume of the atoms are additive. This is true also for other physical properties of an atom, such as the electron population and the charge, as seen in Table 6.2 and as indeed has been shown to be true for all other properties. (Bader 1990, Popelier 1999). 

The atoms defined in the quantum theory of atoms in molecules (QTAIM) satisfy these requirements [1], The atoms of theory are regions of real space bounded by a particular surface defined by the topology of the electron density and they have all the properties essential to their role as building blocks ... 

The equatorial positions of the octahedron are occupied by the CN moieties of the [Ag(CN)2] groups. As in the pz derivative, each [Ag(CN)2] group connects two iron atoms defining the edges of a Fe4[Ag(CN)2]4 rhombus. However, the edge-shared rhombuses define 2D corrugated nets in contrast to the pz derivative, due to the three-coordination of the Ag atoms (see Fig. 19). A schematic view of one 3D network is depicted in Fig. 20. 

One now finds that the orbital radius us restricted to certain values, termed orbits, having values equal to h2/me2Z, 4fj2/me2Z, 9ft2/me2Z, etc. For the smallest allowed orbit of a hydrogen atom (defined by Z = 1 and n — 1), one finds that ... 

A notation for the various sets of atoms defined in the last paragraph will now be introduced. Let NJ denote the set of... 

A cluster may be considered, in a sense, the antithesis of a complex even if there are many similarities between the two groups of compounds due to common symmetry properties. In both cases a set of atoms defines the vertices of a polyhedron in the complex, however, these atoms may be considered as bound to another, central, atom and not to each other. In the cluster, on the other hand, there is not necessarily a central atom and the atoms at the vertices can be described as directly connected to each other. 

A large number of solids with stoichiometry AB form the CsCl structure. In this structure, atoms of A define a simple cubic structure and atoms of B reside in the center of each cube of A atoms. Define the cell vectors... 

Figure 6.13 Schematic illustration of a lattice model for diffusion of Ag atoms on Pd doped Cu(100). The diagrams on the right show the fourfold surface sites in terms of the four surface atoms defining the site. The section of the surface shown on the left includes two well separated Pd atoms in the surface. The Pd atoms are located at the centers of the two grey squares in the diagram on the left.

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

A particular atom can be both a stereogenic center and one of the atoms defining a stereogenic axis. In such cases the stereogenic center must first be specified. 

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