Z coordinates

In the active picture adopted here the (X,Y,Z) axis system remains fixed in space and a translational synnnetry operation changes the (X,Y,Z) coordinates of all nuclei and electrons in the molecule by constant amounts, (A X, A Y, A Z) say,... 

Now let us use a more specific coordinate system shown in Figure 2 for H2 molecule. The z axis is taken to be along the intemuclei vector R. Adapting this coordinate-system, the consequence is that only the z coordinate of the nucleus are to be expanded in powers of k. That is, we define... 

Basically, two different methods arc commonly used for representing a chemical struchiive in 3D space. Both methods utilize different coordinate systems to describe the spatial arrangement of the atoms of a molecule under con.sidcration. The most common way is to choose a Cartesian coordinate system, i.e., to code the X-, y-, and z-coordinates of each atom, usually as floating point numbers, For each atom the Cartesian coordinates can be listed in a single row. giving consecutively the X-, )> , and z-valnc.s. Figure 2-90 illustrates this method for methane. 

To avoid having the wave function zero everywhere (an unacceptable solution ), the spin orbitals must be fundamentally difl erent from one another. For example, they cannot be related by a constant factor. You can write each spin orbital as a product of a space function W hich depen ds on ly on the x, y, and z. coordin ates of th e electron—and a spin fun ction. The space function is usually called themolecnlarorbitah While an in finite number of space functions are possible, only two spin funclions are possible alpha and beta. 

We start with the governing equations of the Stokes flow of incompressible Newtonian fluids. Using an axisymraetric (r, z) coordinate system the components of the equation of motion are hence obtained by substituting the shear-dependent viscosity in Equations (4.11) with a constant viscosity p, as... 

This equation is important in interpreting the results of calculations. In ab initio and semi-empirical calculations, atomic orbitals are functions of the x, y, and z coordinates of the electron that closely resemble the valence orbitals of the isolated atoms. 

After stretching, the z coordinate is increased by a, while x and y are decreased... 

Coordinate Systems The commonly used coordinate systems are three in number. Others may be used in specific problems (see Ref. 212). The rectangular (cartesian) system (Fig. 3-25) consists of mutually orthogonal axes x, y, z. A triple of numbers x, y, z) is used to represent each point. The cylindricm coordinate system (/ 0, z Fig. 3-26) is frequently used to locate a point in space. These are essentially the polar coordinates (/ 0) coupled with the z coordinate. As... 

Consider the behavior of various tensors under the transformation of coordinates in Figure A-1 where a rotation about the z-axis is made. That is, the x, y, z coordinates are transformed to the x , y , z coordinates where the z-direction coincides with the z -direction. The direction cosines for this transformation are... 

FIG. 5 (a) Fraction of adsorbed monomers (i.e., those with z-coordinate less than 6) vs ejk T for four different chain lengths, (b) The same for the second Legendre polynomial P2 cosd). (c) Scaling plot of and P2 cos9) vs distance from the adsorption threshold, using = —1.9 [13]. 

Here is a graph showing the positions of the carbon and chlorine atoms in the XY-plane, ignoring their common Z-coordinate ... 

The superscript t denotes a transposition of the r-vector, i.e. converting it from a column to a row vector. The rr notation for the quadmpole moment therefore indicates a 3 x 3 matrix containing the products of the x-, y- and z-coordinates, e.g. the Qxy component is calculated as the expectation value of xy. 

Cartesian and cylindrical polar atomic coordinates of the structural repeating unit of 31 polysaccharide helices are provided in Tables A1 to A31. Errors, if any, in the original publications have been corrected. The coordinates of hydrogen atoms are given in a majority of structures. If missing, they are not available in the references cited in Table I. Each table caption contains the structure number and polymer name assigned in Table I. Refer to Table II for its chemical repeating unit. Cartesian (x, y, z) and cylindrical (r, , z) coordinates are related by x r cost ), y = r sin<(> and z is the same in both systems. 

The interaction between particle and surface and the interaction among atoms in the particle are modeled by the Leimard-Jones potential [26]. The parameters of the Leimard-Jones potential are set as follows pp = 0.86 eV, o-pp =2.27 A, eps = 0.43 eV, o-ps=3.0 A. The Tersoff potential [27], a classical model capable of describing a wide range of silicon structure, is employed for the interaction between silicon atoms of the surface. The particle prepared by annealing simulation from 5,000 K to 50 K, is composed of 864 atoms with cohesive energy of 5.77 eV/atom and diameter of 24 A. The silicon surface consists of 45,760 silicon atoms. The crystal orientations of [ 100], [010], [001 ] are set asx,y,z coordinate axes, respectively. So there are 40 atom layers in the z direction with a thickness of 54.3 A. Before collision, the whole system undergoes a relaxation of 5,000 fsat300 K. 

Display Values) gives the X-, T-, and Z-coordinates and the X - and y -projection coordinates for each point. 

A sphere has no preferred axis, so it has no directionality in space. When there is a preferred axis, as for a football, Figure 7-18 shows that the axis can point in many different directions relative to an X J Z coordinate system. Thus, objects with preferred axes have directionality as well as shape. 

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