Density projected

For the same reason, new density projection operators are redefined ... 

Fig. 10. Electron density projection along -strand direction—hydrogen-bonding direction (a-axis) horizontal, and intersheet direction (c-axis) vertical—and skeletal models of polyGln8 (Q8) and polyGln45 (Q45) assemblies. The unit cell for both peptides was monoclinic, with lattice constants a = 9.73 A, b = 7.14 A, c = 8.16 A, and y = 95.7° for Q8, and a = 9.66 A, b = 7.10 A, c = 8.33 A, and y = 94.0° for Q45. The side chains are nearly overlapped in the hydrogen-bonding direction. This difference in side chain conformation and disorder likely accounts for the differences in observed intensity between their diffraction patterns.

Fig. 11 One-dimensional electron density projection along the c-axis for a perylene dye intercalated in an Mg/Al LDH and a structural model based on this data. Reprinted with permission from [150]. Copyright Wiley...

The first step in the Fourier synthesis of the caffeine-pyrogallol complex was to choose an initial model so an electron density projection could be calculated. From the previous results it was determined that the unit cell was tetragonal measuring 23.26 X 23.26 X 6.99 A., and that eight caffeine-pyrogallol complex moieties reside in the unit cell. This information alone does not give any indication of the way in which the molecules are packed inside the unit cell, but the symmetry operations of the space group can be used to eliminate many models. For example, one of the symmetry operators in the unit cell is a fourfold axis. 

Additional data about the structure of black films are obtained by X-ray diffraction method. The first steps [336,338] have been performed with vertical foam films in a frame in a horizontal scanning diffractometer. Black films from decyltrimethyl ammonium decyl sulphate and NaBr solutions have been studied. The film thickness was calculated using a model of the mean electron density projection on the film normal. However, there was no indication whether the films were CBF or NBF. Platikanov et al. [339,340] used a new device for investigation of a horizontal black films from aqueous NaDoS solution (see Section 2.2.6). They found essentially different X-ray diffraction traces for the three types of black films CBF, NBF and stratified black films. This indicates their different structure. Precise X-ray reflectivity measurements with CBF and NBF films from NaDoS and NaCl aqueous solutions [341-343] provided more details about their structure. The data obtained for the thicknesses of the respective layers which detail the film structure are given below... 

Fig. 2.1. Electron density projection (projected parallel to c) for coesite (SiOj) obtained from x-ray data.

Consider an atom at some position in the unit cell. Its electron density projected onto one of the axes, say, the jc-axis, will have the shape of a Gaussian curve ... 

Our IBM 7040 least-squares programme was modified to cope with this situation. After several cycles of refinement of the gold atomic parameter, a c-axis difference electron-density projection was computed. On the resulting map it WM possible to locate the fluorine atoms there are only two independent fluorines, one in the general position x, y, z), 12(c), and the other in the special position (J, 0, 0), 6(o). Structure factors were calculated by use of the scattering factors of the International Tables for Au and F, that for gold being corrected for the real part of the anomalous dispersion effect B was taken as 2-0 A. ... 

Figure 16.23. Schematic structures of PEO conformation in V2O5 xerogel and their expected electron density projections along the c axis, (a) coil conformation, (b) zigzag conformation forming bilayer. [Adapted, by permission, from Liu Y J, Schindler J L, DeGroot D C, Kannewurf C R, Hirpo W, Kanatzidis M G, Chem. of Mat., 8, No.2, 1996, 525-34.]...

P electric power, conditional probability density, projection... 

The situation with regard to 2-phenylazulene resembles somewhat that of azulene itself. The structure was first thought to be disordered,70 as had been shown for azulene, and refined in projection to an R of 17 percent however, later calculations based on the ordered model71 reduced the R value to 14 percent. The two structures and the correspond ing electron density projections are shown in Figure 7. At present, it is not known which model is correct, but it is certain that at least one of these electron density maps is incorrect. 

FIGURE 6 The electron density projections corresponding to Figure 5 (a) ordered, (b) disordered. From Robertson et al. ... 

A faithful representation of the spin-density projection P(x) would be obtained by this method of frequency encoding only if the spectrum F(x - x ) were a delta spike, or a narrow liquid-state NMR line. 

Fig. 1. Electron-density projection in the direction of the b axis of D-erythro-psntuloss p-bromophenylosazone. Note that C-l is linked two bulky atoms only (C and N).

The three methanesulfonyl derivatives, sulfur, selenium, and tellurium dimethanethiosulfonates, are isomorphous. Structure analyses have been carried out, starting with tellurium dimethanethiosulfonate and using tellurium as a heavy atom in the first stages. So far, the structure of the tellurium compound has been published in detail 128), and an electron density projection along the short crystal axis of the trisulfide (98) has shown the unbranched sulfur chain structure. 

An electron density projection (110) of another representative of the tetragonal series, di-p-toluenesulfony 1 triselenide, shows the triselenide in the trans form, as contrasted with the cis form in cyanogen triselenide. On the basis of the isomorphism, the same unbranched, trans structures can be inferred for di-p-toluenesulfonyl trisulfide and selenium di-p-toluene-thiosulfonate, as well as for the two tetragonal benzenesulfonyl compounds. 

Fig. 7. Electron density projection along the twofold axis of irans-dichloro-dien-cobalt(IIJ) hexathionate monohydrate, showing the Irans-lranx form of the hexathionate ion. Reproduced, with permission, from Acta Chem. Scand. 13, 201 (1959).

An electron density projection (Fig. 7) along the c-axis (107) shows an unbranched structure for the sulfur chain of the hexathionate ion also in this salt, but here, the chain has the extended trans-trans form of the helixes of fibrous sulfur. 

As the collinear approach, the KU scheme uses a fixed quantization direction for the spin, but the projection of the magnetization is used to define the xc energy of a polarized system rather than the spin-density projection... 

Fig. 7 Modeling a high-resolution dimeric structure, (a) Estimate of the mean distance on the example of spin label K-221R1. The primary DEER trace (P(t)A (0)), the form factor and the distance distribution obtained by Tikhonov regularization (the L curve is shown in the inset) with the software Deer Analysis are presented, (b) Fits of primary experimental DEER data black lines) by simulated data red lines) corresponding to the final structure of the NhaA dimer and a distribution of spin label conformations modeled by a rotamer library, (c) C2 symmetry axis of the dimer created with the EPR constraints, (d) Comparison between the EPR stmcture and the electron density projection to the membrane plane obtained by cryo-EM on 2D crystals, (e) Comparison between the EPR structure and the dimer modeled on the refined cryo-EM data (PDB 3EI1). Adapted from [77]...

Fig. 11. Features of a difference electron density projection for TMA (3) H O PA. The difference electron density which represents the enclathrated picric acid molecules is contoured at levels of 1, 2, 3 e A In TMA H2O PA the molecules lie in the (III) planes and the hydrogen bonds do not lie in the plane of the projection but are directed out of this plane. For TMA H O the present diagram serves as a slightly distorted representation of the arrangement in the layers of the framework TMA and HjO molecules, as viewed along a. The hydrogen bonds have the following lengths, averaged over corresponding bonds in TMA H O % PA and the five independent layers in TMA HjO (the values in parentheses are the RMS deviations from the means) a 2.641 (15) A b 2.632 (52) c 2.874 (44) d 2.850 (52) e 2.592 (29). (Taken from Ref. )...

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