Coordination value

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

Table 18.1 Crystal data of structures mentioned in Fig. 18.6. The ideal coordinates would apply to an undistorted packing of spheres. Coordinate values fixed by symmetry are stated as 0 or fractional numbers, otherwise as decimal numbers...

In many applications it is customary to define local coordinate systems indirectly by establishing their connection with the Cartesian coordinates in some underlying Euclidean space E if there is one. By labeling the points within each actual space (of local coordinate system) with the coordinate values in the underlying Euclidean space E there is a common reference for all local coordinate systems, and the compatibility conditions can be formulated within the Euclidean space E" of familiar and intuitively simple properties. 

In Fig. 25b, the simulated marker particles were released from the bottom surface, which generates path lines that show more detail of the flow inside the WS, at lower radial coordinate values. The path lines reinforce the trends seen in Fig. 25a, and it is also possible to see some evidence of flow through the center voids of the particles. Most evident is the mix of spiraling and axial flow between the center front and center right particles. 

Let the minimum of the potential U(angular coordinate value

probability density fl co, (p) for a particle to be located at a point with the angular coordinate (p and to have the angular velocity co under thermodynamic equilibrium with a thermostat is given by the Gibbs distribution ... 

Crystallographic studies on S-dimethyl sulfoxide complexes (Table III) show that the geometry of the Me2SO moiety is virtually unaffected by coordination. Values of O—C of —107° compare favorably with the values of 106.7° and 106.8° reported for free dimethyl sulfoxide. Similarly, reported values of C—C between 99.4(6)° and 104.5(7)° compare with the value of 97.4° reported for the free molecule. A slight increase in C— —C would be expected on repulsion... 

In analysing flow problems it is usual to select one coordinate axis (the x-coordinate axis in the above examples) to be parallel to the flow with the coordinate value increasing in the direction of flow so that the fluid s velocity is positive. If the velocity component vx varies in they-direction, it is normal to define the shear rate in terms of the change of vx in the positive y-direction, so... 

Consider a molecule containing N atoms. We can refer to the position of each atom by specifying three coordinates (e.g., X, Y and Z Cartesian coordinates) Thus the total number of coordinate values is 3 N and we say that the molecule has 3 N degrees of freedom since each coordinate value may be specified quite independently of the others. Once all 3 N coordinates have been fixed, the bond distances and bond angles of the molecules are also fixed and no further orbitrary specification can be made. So a molecule which is of finite dimension will thus be made of rotational, vibrational and translational degrees of freedom. 

Figure 2. Initial ( (/a) and final ( J/b) state potential-energy contours for the complete (two-mode) active space the abscissa refers to the inner-sphere mode and the ordinate governs the low-frequency active solvent mode. The difference in frequencies leads to a curved reaction path. Equilibrium coordinate values for the reactant ( j/A) and product ( J/b) states are labeled qA and qB, respectively. For the case of qin, qB° - qA° = Aqin°, as given by Eq. 16.

R , 0,f,x,S, SlSlSl,Sl, (to any desired accuracy). These coordinate values... 

Figure 6 A multi-row detector array is used in a cone-beam system. In this case, the detector rows fall along circular arcs centered on the X-ray source s x, y coordinate values. Detectors of equal size therefore subtend equal angles with respect to the source. This characteristic and other system and data-analysis considerations often make curved detector rows more attractive than straight ones like those of Figures 3 and 4.

This simulated dataset is generated as follows. First, nine cluster centers are defined at different locations in a 2D space with coordinate values within [-3.0, 3.0]. Second, a random number (between 1 and 100) of points is generated around each cluster center within a distance of 0.5. Finally, additional points are generated, which are randomly distributed in the 2D space so that a total of 1000 points is obtained. This dataset simulates the situation where clusters of molecules exist in their descriptor space, and the number of members for each cluster is different, i.e., some regions are more densely populated than other regions. 

To remove momentum fluctuations from the problem, BCAH assume that in the creeping flow limit, in which a system of small mass interacts strongly with a thermally equilibrated solvent, the distribution of values for the momenta for fixed coordinate values stays very near a state of local equilibrium, in which... 

To calculate the drift velocity and diffusivity for such a process, we imagine a discrete sequence of coordinates values X = X (t ) sampled at times ti, t2, ri, separated by At, and then take the limit At 0. Such a sequence could be generated by integrating Eq. (2.244) over each timestep to obtain the change AX for that step. Such a sequence will be approximately Markovian... 

Each object can be projected in the space defined by the new variables the coordinate values obtained are called scores. 

Difference between coordinate values of cursor 1 and cursor 2. 

Point-to-point geometric similarity of two bodies (e.g., two mixing tanks) requires three-dimensional correspondence. Every point in the first body is defined by specific x, y, and z coordinate values. The corresponding point in the second body is defined by specific x, y, and z coordinate values. The correspondence is defined by the following equation ... 

Thus, we have detailed how to construct a molecular PES as a sum of energies from chemically intuitive functional forms that depend on internal coordinates and on atomic (and possibly bond-specific) properties. However, we have not paid much attention to the individual parameters appearing in those functional forms (force constants, equilibrium coordinate values, phase angles, etc.) other than pointing out the relationship of many of them to certain spectroscopically measurable quantities. Let us now look more closely at the Art and Science of the parameterization process. 

Returning to the issue of convergence, as noted above the structure of each snapshot in a simulation can be described in the space of the PCA eigenvectors, there being a coefficient for each vector that is a coordinate value just as an x coordinate in three-dimensional Cartesian space is the coefficient of the i Cartesian basis vector (1,0,0). If a simulation has converged, die distribution of coefficient values sampled for each PCA eigenvector should be normal,... 

Sidebar 10.3 outlines the useful analogy to normal-mode analysis of molecular vibrations, where the null modes correspond to overall translations or rotations of the coordinate system that lead to spurious alterations of coordinate values, but no real internal changes of interatomic distances. For this reason, the internal metric M( of (10.29) is the starting point for analyzing intrinsic state-related (as opposed to size-related) aspects of a given physical system of interest. 

Using these coordinate values we may now evaluate the matrix elements of Eq. (14.12) by substituting for the dipole moments the dipole matrix elements between the initial and final states. This procedure yields explicitly time dependent matrix elements VAB(r). It is particularly interesting to consider the (0,0) resonances, for two reasons. First, the (0,0) resonances have no further splitting due to the spin orbit interaction and are therefore good candidates for detailed experimental study. Second, since these resonances only involve the matrix... 

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