Coordination of atoms

For each combination of atoms i.j, k, and I, c is defined by Eq. (29), where X , y,. and Zj are the coordinates of atom j in Cartesian space defined in such a way that atom i is at position (0, 0, 0), atomj lies on the positive side of the x-axis, and atom k lies on the xy-plaiic and has a positive y-coordinate. On the right-hand side of Eq. (29), the numerator represents the volume of a rectangular prism with edges % , y ., and Zi, while the denominator is proportional to the surface of the same solid. If X . y ., or 2 has a very small absolute value, the set of four atoms is deviating only slightly from an achiral situation. This is reflected in c, which would then take a small absolute value the value of c is conformation-dependent because it is a function of the 3D atomic coordinates. 

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... 

The third integral vanishes because the derivative of the dipole operator itself p = Zi e rj + Za Za e Ra with respect to the coordinates of atomic centers, yields an operator that contains only a sum of scalar quantities (the elementary charge e and the... 

Algorithms that displace the Cartesian coordinates of atoms have also been... 

Coordinates of atoms can be set by normal translation or rotation of HyperChem molecules. To set initial velocities, however, it is necessary to edit the HIN file explicitly. The unit of velocity in the HIN file is Angstroms/picosecond. Areact.hin file and a script react.scr are included with HyperChem to illustrate one simple reacting trajectory. In order to have these initial velocities used in a trajectory the Restart check box of the Molecular Dynamics Options dialog box must be checked. If it is not, the initial velocities in the HIN file will be ignored and a re-equilibration to the temperature T of the Molecular Dynamics Options dialog box will occur. This destroys any imposed initial conditions on the molecular dynamics trajectory. 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

A complete and detailed description of molecular structure includes statements concerning the metric coordinates ) of atomic nuclei supplemented by electron density distribution data. Although a large amount of data is involved, such representation is not particularly suitable for the chemically relevant structural features of molecules. 

Note the subscript C to indicate classical (or high temperature). In Equation 4.81 the p s are momenta and the q s the associated coordinates (not to be confused with q s previously used to symbolize molecular partition functions). In Cartesian coordinates dpjdqj = dpxidpyidpzidxidy1dzi with xi, yi, zi, the coordinates of atom... 

In principal one can calculate the electronic energy as a function of the Cartesian coordinates of the three atomic nuclei of the ground state of this system using the methods of quantum mechanics (see Chapter 2). (In subsequent discussion, the terms coordinates of nuclei and coordinates of atoms will be used interchangeably.) By analogy with the discussion in Chapter 2, this function, within the Born-Oppenheimer approximation, is not only the potential energy surface on which the reactant and product molecules rotate and vibrate, but is also the potential... 

In the example above, we placed atoms in our slab model in order to create a five-layer slab. The positions of the atoms were the ideal, bulk positions for the fee material. In a bulk fee metal, the distance between any two adjacent layers must be identical. But there is no reason that layers of the material near a surface must retain the same spacings. On the contrary, since the coordination of atoms in the surface is reduced compared with those in the bulk, it is natural to expect that the spacings between layers near the surface might be somewhat different from those in the bulk. This phenomenon is called surface relaxation, and a reasonable goal of our initial calculations with a surface is to characterize this relaxation. 

The coordinates of atoms in chain-polymer crystals cannot be deter-... 

Tj, Zj, Fj cylindrical coordinates of atom j Jn Bessel function of order n. 

Extended X-ray absorption fine structure (EXAFS) spectrum Part of an X-ray absorption spectrum that is used to identify the coordination of atoms, estimate bond lengths, and determine the adsorption complexes on the surfaces of adsorbents. EXAFS spectra may provide useful information on the speciation (valence state), surface complexes, and the coordination of arsenic on adsorbents (e.g. (Randall, Sherman and Ragnarsdottir, 2001 Ladeira, et al. (2001) Teixeira and Ciminelli (2005) Kober, et al. (2005)) (compare with X-ray absorption spectroscopy (XAS), X-ray absorption near edge structure (XANES) spectra, and X-ray absorption fine structure spectroscopy (XAFS)). 

Here Z is the core charge on A (e. g. the nuclear charge less the number of inner core electrons), Pu is the total charge on atom A, is the cartesian coordinate of atom A, and Z x is the Slater orbital exponent for 2 s and 2 p orbitals of atom A. 

In this type of equations the forces depend on not only the coordinates of atoms, but also their velocities (as effective friction force) therefore, numerical integration of equation of motion should be performed self-... 

Many ternary, quaternary, and multicomponent metal sulfides also exhibit tetrahedral coordination of atoms and possess structures similar to that of cubic... 

Figure 2.9. Location of the second peptide unit. Dipeptides are constructed by knowing the coordinates of atoms in the first peptide unit (using X-ray diffraction) and then translating along the line between the first and second Ca by a dis-...

From a structural viewpoint, a polypeptide is composed of planar peptide units as shown in Figure 2.8. The usefulness of considering the peptide unit as opposed to the amino acid is that the peptide unit is almost planar as opposed to the amino acid, which has atoms that are in more than one plane. To illustrate this point, the coordinates of atoms in the peptide unit are given in Table 2.2 and nonbonded atoms cannot be closer than the sum of the minimum atomic distances (Table 2.3). Note that all the atoms from the first alpha carbon (Ca) to the second alpha carbon do not have a z-coordinate. These coordinates come from X-ray diffraction studies on proteins and represent the average coordinates found among many pro-... 

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