Deformation densities

Thanks to the particular choice made for the NUP, taken equal to a superposition of spherical atoms, it is for the first time possible within the present approach to compute MaxEnt deformation maps in a straightforward manner. Once the Lagrange multipliers X have been obtained, the deformation density is simply... 

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. 

Sections of the density from one of these fits, which we will refer to as calculation B, are shown in Figure 7 the MaxEnt deformation density in the COO- plane is shown in Figure 7(a) Figure 7(b) is the difference between the MaxEnt valence density and the reference density in the same plane. The lower noise content of the data is clearly visible, when the map is compared with the one for calculation A in particular, the lone pairs on the oxygen atoms are better defined. The rms deviation from the reference is as low as 0.023 e A 3. 

Figure 3. Static deformation density in Si-O-Si bridge planes Si,-0,-Si2 and Si,-O,o-Sij. Contours as in Figure 1.

Figure 6.7 shows the calculated electron density distributions for the H2 and N2 molecules in their equilibrium geometry together with the standard deformation densities. There is clearly a buildup of electron density in the bonding region in both molecules. In the N2 molecule there is also an increase in the electron density in the lone pair region and a de-... 

Figure 6.8 Deformation densities for the F2 molecule, (a) Standard deformation density. Note that there is no charge buildup-in the bonding region between the nuclei, (b) Modified deformation density molecular density minus the density of atoms in the (ls)2(2s)2(2px)2(2py)2( 2pe)1 reference state showing a buildup of charge in the bonding region. (Reproduced with permission from P. Coppers [1997].)...

Clearly the form of a deformation density depends crucially on the definition of the reference state used in its calculation. A deformation density is therefore meaningful only in terms of its reference state, which must be taken into account in its interpretation. As we will see shortly, the theory of AIM provides information on bonding directly from the total molecular electron density, thereby avoiding a reference density and its associated problems. But first we discuss experimentally obtained electron densities. 

In the next section we will see how the theory of AIM enables us to obtain chemical insight directly from the experimental density as determined by experiment or by calculation, thereby avoiding the need for deformation densities. 

Figure 6.9 (a) Standard deformation density of tetrafluoroterephthalonitrile in the molecular plane. Contour interval is 0.1 e A-3, terminated at 1.5 e A 3. (b) Molecular diagram with a box around the fragment shown in the deformation map (a). (Reproduced with permission from F. L. Hirshfeld, Acta Crystallogr., B40, 613, 1984.)... 

G. Will, Electron Deformation Density in Titanium Diboride Chemical Bonding in TiB2, Jour. Sol. St. Chem., 177, 628 (2004). 

For the 2-cyanoguanidine molecule6 the static deformation density has been mapped by least-square refinement against low-temperature X-ray data in order to explain the fact that the C—N bonds around the C atom are almost identical and the fact that a large negative charge (—0.2 e) is on the N(3) atom. Hence one must take all the resonance forms (2) into consideration. 

Figure 1,8 Electron density plots along (110) plane of BeO (A) effective total electron density (pseudoatom approximation) (B) total electron density of lAM (C) deformation density (pseudoatom-IAM). From Downs (1992). Reprinted with permission of Springer-Verlag, New York.

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. 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

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... 

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