Density procrystal

The topological characteristics of CPs on bonds gives a quantitative explanation of the known effect that the formation of a Ge crystal is accompanied by shifting the electron density towards the Ge-Ge bonding line. This is can be seen by comparing the parameters of the curvature of the electron density at the critical point (3,-1) with analogous parameters for a procrystal (a set of noninteracting spherical atoms placed at the same... 

Let us suppose that the atomic positional parameters r, and the thermal smearing factors 71 have been obtained by neutron diffraction. We can then calculate the X-ray scattering of the procrystal density with spherical-atom X-ray scattering factors, using... 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

The multipole description of the charge density makes it possible to identify a pseudomolecule in a crystal. The pseudomolecule is distinct from the molecules in the procrystal, composed of noninteracting molecules, in the same sense as... 

FIG. 11.11 Electron-density difference maps on Li2BeF4 calculated with all reflections < sin 6/1 = 0.9 A"1 (81 K). (a) Based on the neutral atom procrystal model, (b) based on the ionic model. Contour levels are drawn at intervals of 0.045 eA"3.1 Full lines for positive density, dashed lines for negative and zero density. The standard deviation, estimated from [2Lff2(F0)]1/2N, is 0.015 eA-3. Source Seiler and Dunitz (1986). 

In the following, this approach has been used to study the energy features of 3-NTO [29]. The kinetic, g(r), and potential, v(r), energy density maps have been calculated from the experimental electron densities with the WinXPRO program package [39] using the approach described above, as well as their difference with respect to atomic procrystals with no chemical bonds. Critical point characteristics have also been similarly analyzed. 

The first line lists the experimental result the second line- the atomic procrystal p is the electron density V2p is the Laplacian, g, v and he are the kinetic, potential and total electronic energies at the critical point R is the bond path length. 

The comparison of the properties at the critical points in the P-NTO electron density with those in the atomic procrystals (Table 6) reveals the contraction of the electron density towards the shared interaction lines during crystal/molecule formation. This contraction is accompanied by a significant enhancement of the local kinetic, potential and electronic energies (Table 7). 

Another study focusing on the comparison between theoretical and experimental densities is that of Tsirelson el al. on MgO.133 Here precise X-ray and high-energy transmission electron diffraction methods were used in the exploration of p and the electrostatic potential. The structure amplitudes were determined and their accuracy estimated using ab initio Hartree-Fock structure amplitudes. The model of electron density was adjusted to X-ray experimental structure amplitudes and those calculated by the Hartree-Fock model. The electrostatic potential, deformation density and V2p were calculated with this model. The CPs in both experimental and theoretical model electron densities were found and compared with those of procrystals from spherical atoms and ions. A disagreement concerning the type of CP at ( , 0) in the area of low,... 

Fig. 15. Difference density (DD) maps of nitrobenzene (65) with respect to a procrystal of spherically averaged atoms, on cuts vertical to the molecular plane through the midpoints of bonds, as shown in the left top diagram. Under each DD map, the difference between this map and the same map rotated by 90° is shown (AAp), indicating the n-bond ellipticity. Contour line values are 0.05n e/A , n=l,2,3,..

In early experimental studies predominantly in the 1970s and 1980s, so-called deformation electron density studies were typically conducted. Thus, the electron density is modeled using the approach described above, and the electron density of a promolecule (procrystal), constructed by placing spherical free atoms densities at all atomic centers, is subtracted from the fully modeled density. The difference then shows the redistribution of electron density that arises due to chemical bond formation. This approach readily shows electron density buildup in the center of most covalent... 

The standard procrystal model assumes that the electron distribution in the crystal is very nearly equal to a superposition of previously known rigid atomic density distributions, which are smeared by harmonic lattice vibrations. The structure factor F(h, k, 1) with integer Miller indices h, k, I for N atoms per unit cell, located at the positions then becomes... 

Procrystal electron density threshold for a migration pathway Bond valence site energy threshold for a migration pathway Absolute bond valence sum mismatch Bond valence sit energy of ion M Density functional theory... 

Fig. 12 Regions in the Li4GeS4 structure accessible to moving LU ions accrading to BVSE energy calculations top half of the unit cell) for AEbvse of (a) 0-95 eV, (b) 1.1 eV and (c) 1.35 eV and the procrystal analysis bottom half of the unit cell) showing paths with electron density isovalues of 0.0016 au, 0.0018 au, and 0.0024 au, respectively...

Given the correlation between bond valence and electron density, it appears tempting to compare also what electron density maps and maps of the BVSE predict as ion transport pathways. Hirshfeld surface analysis has been explored to characterize intermolecular interactions in molecular crystals [47,48]. This analysis is based on the procrystal, which is obtained from superposition of spherical atomic electron densities placed at the crystal structure positions, a quantity that can readily be calculated from the structure using software tools such as CrystalExplorer [49]. The approach was also explored as a tool to map out voids in porous crystals such as metal organic framework materials and zeolites [50]. 

Fig. 13 Correlation between AFbvse and the difference in procrystal electron density App , for interstitial sites and pathway bottlenecks in various lithium-ion conducting oxides relative to the respective minimum of the same quantity in the respective structure (equilibrium sites). Reinterpretation of data from the procrystal study by Fils0 et al. [51]...

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