Deformation density dynamic

Figure 2. L-alanine. Dynamic deformation density in the COO plane, (a) Model dynamic deformation density A Modei. (b) MaxEnt dynamic deformation density (Agj, (x)) map obtained with a non-uniform prior of spherical-valence shells. Map size 6.0A x 6.0A Contour levels from -1.0 to 1.0 eA 3, step 0.075 e A-f...

Figure 3. L-Alanine. Dynamic deformation density in the COO- plane, (a) - m(x). 

The MaxEnt deformation density in the COO- plane is shown in Figure 6(a). The deformation map shows correct qualitative features differences between the single C-C bond and the C-0 bonds are clearly visible, and so are the lone-pair maxima on the oxygen atoms. If compared to the conventional dynamic deformation density... 

Figure 6. l-Alanine. Fit to noisy data. Calculation A. MaxEnt deformation density and error map in the COO- plane Map size, orientation and contouring levels as in Figure 2. (a) MaxEnt dynamic deformation density A uP. (b) Error map qME - Model. 

When observed structure factors are used, the thermally averaged deformation density, often labeled the dynamic deformation density, is obtained. An attractive alternative is to replace the observed structure factors in Eq. (5.8) by those calculated with the multipole model. The resulting dynamic model deformation map is model dependent, but any noise not fitted by the muitipole functions will be eliminated. It is also possible to plot the model density directly using the model functions and the experimental charge density parameters. In that case, thermal motion can be eliminated (subject to the approximations of the thermal motion formalism ), and an image of the static model deformation density is obtained, as discussed further in section 5.2.4. 

Figure 5.12 shows both the dynamic and the static model deformation densities in the plane of the oxalic acid molecule, based on the data set also used for Fig. 5.2. The increase in peak height, due to higher resolution, and reduction in background noise relative to the earlier maps is evident. The model acts as a noise filter because the noise is generally not fitted by the model functions during the minimalization procedure.

The effect of phase differences on dynamic deformation density may be estimated as follows (Souhassou et al. 1991). The amplitudes of the deformation density Fourier series may be written as (Fig. 5.9)... 

Fig. 4. The phosphazene ring (a) island delocalization model predicting nodes in TT-density at the phosphorus atoms (b) dynamic deformation density (at 0.1 eA 3) in the plane of the ring (c) theoretical deformation density (at 0.05 eA-3) of cyclic phosphazene was used as a model (reproduced with permission from Cameron et al. [43]).

However, these density maps which give a dynamic deformation density do not readily lead to numbers describing charges and electrostatic properties on the other... 

Once the multipole analysis of the X-ray data is done, it provides an analytical description of the electron density that can be used to calculate electrostatic properties (static model density, topology of the density, dipole moments, electrostatic potential, net charges, d orbital populations, etc.). It also allows the calculation of accurate structure factors phases which enables the calculation of experimental dynamic deformation density maps [16] ... 

For dynamic deformation maps, < >m differs from s when the crystal is acentric. Neglecting this phase difference can underestimate the deformation density of a covalent bond by 0.21 A-3 which represents something like one-third to one-half of the deformation density [76]. 

Figure 13. Dynamic deformation density on a pyrrole plane (a), in the (x,y) four nitrogen plane (b) and in the (y,z) N-Fe-O (THF) plane (c) in TPPFedHFte corresponding theoretical density in the (x,y) plane for the 5E2g state of PFe(H20)2 [33] d). Contours as in Figure 6 (reproduced from [34]).

FIGURE 9.17 (cont d). Examples of deformation density maps, (b) Dynamic deformation-density map in which motion of the atoms has not been corrected for. ... 

It is also evident that this phenomenological approach to transport processes leads to the conclusion that fluids should behave in the fashion that we have called Newtonian, which does not account for the occurrence of non-Newtonian behavior, which is quite common. This is because the phenomenological laws inherently assume that the molecular transport coefficients depend only upon the thermodyamic state of the material (i.e., temperature, pressure, and density) but not upon its dynamic state, i.e., the state of stress or deformation. This assumption is not valid for fluids of complex structure, e.g., non-Newtonian fluids, as we shall illustrate in subsequent chapters. 

The above model assumes that both components are dynamically symmetric, that they have same viscosities and densities, and that the deformations of the phase matrix is much slower than the internal rheological time [164], However, for a large class of systems, such as polymer solutions, colloidal suspension, and so on, these assumptions are not valid. To describe the phase separation in dynamically asymmetric mixtures, the model should treat the motion of each component separately ( two-fluid models [98]). Let Vi (r, t) and v2(r, t) be the velocities of components 1 and 2, respectively. Then, the basic equations for a viscoelastic model are [164—166]... 

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