Model deformation density

A number of different atom-centered multipole models are available. We distinguish between valence-density models, in which the density functions represent all electrons in the valence shell, and deformation-density models, in which the aspherical functions describe the deviation from the IAM atomic density. In the former, the aspherical density is added to the unperturbed core density, as in the K-formalism, while in the latter, the aspherical density is superimposed on the isolated atom density, but the expansion and contraction of the valence density is not treated explicitly. 

In lower pressure environments, the wave profiles are dominated by the consequences of deformation of the samples to fill the voids. This irreversible crush-up process strongly controls the wave speeds, which have anomalously low values at low initial sample densities. Modeling of this problem is... 

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 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. 

Hirshfeld (1971) was among the first to introduce atom-centered deformation density functions into the least squares procedure. Hirshfeld s formalism is a deformation model, in which the leading term is the unperturbed IAM density, and the deformation functions are of the form cos" 0jk, where 9jk is the angle between the radius vector r7 and axis k of a set of (n + l)(n + 2)/2 polar axes on each atom /, as defined in Table 3.8 (Hirshfeld 1977). The atomic deformation on atom j is described as... 

The deformation density is defined as the difference between the total density and the density calculated with a reference model based on unbiased positional and thermal parameters. The deformation density is obtained by Fourier transform, like the residual density [Eq. (5.9)], but with Fca c from the reference state with which the experimental density is to be compared. 

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. 

The implications of a charge density least-squares refinement can be visualized by calculation of the deformation density corresponding to the least-squares fitted model. 

The static model deformation density corresponding to the multipole refinement results is given by... 

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.

Static deformation density maps can be compared directly with theoretical deformation densities. For tetrafluoroterephthalonitrile (l,4-dicyano-2,3,5,6-tetra-fluorobenzene) (Fig. 5.13), a comparison has been made between the results of a density-functional calculation (see chapter 9 for a discussion of the density-functional method), and a model density based on 98 K data with a resolution of (sin 0//)max = 1.15 A -1 (Hirshfeld 1992). The only significant discrepancy is in the region of the lone pairs of the fluorine and nitrogen atoms, where the model functions are clearly inadequate to represent the very sharp features of the density distribution. 

Hirshfeld (1984) found the electrostatic charge balance at the F nuclei, based on the experimental deformation density, to be several times more repulsive (i.e., anti-bonding) than that of the promolecule. Very sharp dipolar functions at the exocyclic C, N, and F atoms, oriented along the local bonds, were introduced in a new refinement in which the coefficients of the sharp functions were constrained to satisfy the electrostatic Hellmann-Feynman theorem (chapter 4). The electrostatic imbalance was corrected with negligible changes in the other parameters of the structure. The model deformation maps were virtually unaffected, except for the innermost contour around the nuclear sites. 

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°). 

Stewart s conclusion underscores the need for short-wavelength, low-temperature studies, if very high accuracy electrostatic properties are to be evaluated by Fourier summation. But, as pointed out by Hansen (1993), the convergence can be improved if the spherical atoms subtracted out are modified by the k values obtained with the multipole model. Failure to do this causes pronounced oscillations in the deformation density near the nuclei. For the binuclear manganese complex ( -dioxo)Mn(III)Mn(IV)(2,2 -bipyridyl)4, convergence of the electrostatic potential at the Mn nucleus is reached at 0.7 A" as checked by the inclusion of higher-order data (Frost-Jensen et al. 1995). 

The model deformation densities for all three compounds show a peak between the metal atom and the bonded sulfur, located closer to the sulfur atoms, in a region not covered by Fig. 10.7. A local density functional calculation of pyrite by Zeng and Holzwart (1994) gives a theoretical deformation density which closely reproduces the features of the experimental densities, including the Fe—S bond peak. No orbital populations are as yet available from this calculation. 

Many of the currently available studies of metal-metal bonding were completed before the multipole model and the topological analysis of the total density were fully developed. For this reason, the discussions reported below focus on the deformation density distributions, and their comparison with theoretical results, though a more quantitative analysis is now possible and would be of considerable interest. 

FIG. 11.3 Comparison of the ab-initio local density functional deformation density for Si with the experimental static model deformation density. Contour interval is 0.025 e A-3. Negative contours are dashed lines. Source Lu and Zunger (1992), Lu et al. (1993). 

Schneider, Hansen, and Kretschmer (SHK) (1981) have measured the 19 reflections with sin 0/A < 0.7 eA 3 with 0.03 A y-radiation. Deformation densities based on these reflections show a small accumulation of charge of height 0.19 eA 3 at 1/4 1/4 0 and equivalent positions, which is between nearest neighbors located along the [110] directions, as well as an accumulation of similar height, but somewhat more extended, in the voids between the atoms at 1/4 1/4 1/2. This seems fully compatible with a hybrid bonding model. 

In such models, the bonding is considered to be partially ionic with a charge transfer from A1 to the Ni 3d valence band. To explain the properties of /J NiAl at a more sophisticated level, Fox and Tabernor (1991) measured four low-angle structure factors by the HEED critical-voltage technique. The deformation density based on these four reflections shows a depletion of density around both the Ni and A1 atoms, and a buildup of about 0.13 eA-3 along the [111] direction halfway between Ni and A1 nearest neighbors. 

The density dependence of Vg in Kr was determined by field ionization of CH3I [62] and (0113)28 [63]. Whereas previous studies found a minimum in Vg at a density of 12 X 10 cm [66], the new study indicates that the minimum is at 14.4 x 10 cm (see Fig. 3). This is very close to the density of 14.1 x 10 cm at which the electron mobility reaches a maximum in krypton [67], a result that is consistent with the deformation potential model [68] which predicts the mobility maximum to occur at a density where Vg is a minimum. The use of (0113)28 permitted similar measurements of Vg in Xe because of its lower ionization potential. The results for Xe are also shown in Fig. 3 by the lower line. 

To negate this observation, the conviction that covalent interaction mandates an excess bonding density in all cases, prompted the formulation of aspherical atomic densities to reflect the requirements of bonding theory. By multipole expansion of atomic densities, based on real spherical harmonics, in line with traditional models of orbital hybridization, the mandated deformation densities are retrieved. Increased flexibility of the model by the introduction of scaling parameters further ensures the elimination of any discrepancies with the theory. However, it is debatable whether this exercise proofs anything other than the power of well-chosen parameters to improve the fit between incompatible data sets. 

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]).

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