Atomic asphericity

I Spherical, Harmonic II Spherical Anharmonic III Aspherical Harmonic6 IV All atoms Aspherical, Anharmonic ... 

One parameter in Cp-C=N for which the X-ray (Table 8) and gas-phase results do disagree very significantly" is the Ci—C4 single bond length. The numerical situation is illustrated in Figure 4 and is a clear example of the effect of atomic asphericity on X-ray bond lengths there is a clear and substantial movement of C4 towards the triple-bond density, while N5 remains static due to the balancing effect of the N lone pair. 

Hansen, N.K. and Coppens, P. (1978) Testing aspherical atoms refinements on small-molecule data... 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

The Multipole Description of the Charge Density of Aspherical Atoms... 

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 aspherical features of the density are described by the summation added to the /c-expression. The summation includes an additional monopole, which may be omitted for first- and second-row atoms, but is necessary to describe the outer s-electron shell of transition metal atoms, which is much more diffuse than the outermost d shell. 

Substitution of the atomic density expression of Eq. (3.35) gives the aspherical atom scattering factor of atom j as... 

The results show only a modest variation when the van der Waals radii are changed within reasonable bounds (Table 6.2). As the data were not refined with the aspherical atom formalism, the scale of the observed structure factors may be biased, an effect estimated on the basis of other studies (Stevens and Coppens 1975) to correspond to a maximal lowering of the scale by 2%. Values corrected for this effect are listed in the last two columns of Table 6.2. Since neutral TTF and TCNQ have, respectively, 72 and 52 valence electrons, the results imply a charge transfer close to 0.60 e. 

Only the spherical and dipolar density terms contribute to the integral on the right. Assuming, for simplicity, that the deformation is represented by the valence-shell distortion (i.e., the second monopole in the aspherical atom expansion is not used), we have, with density functions p normalized to 1, for each atom ... 

For the static density, the zero term in the potential can be expressed in terms of the multipole coefficients of the aspherical-atom formalism. Substituting for atom j at... 

In low-spin transition metal complexes, the preferential occupancy of the d orbitals in the crystal field tends to create excess density in the voids between the bonds, which means that anharmonicity tends to reinforce the electron density asphericity. We will discuss, in the following sections, to what extent the two effects can be separated by combined use of aspherical atom and anharmonic thermal motion formalisms. 

The valence M-shell of silicon is found to expand by about 6% (km — 0.9382), in agreement with the results of a much earlier aspherical atom refinement, which gave k = 0.956 (9) (Hansen and Coppens 1978). Lu, Zunger and Deutsch... 

The introduction of standard aspherical-atom scattering factors leads to a very significant improvement in Hirshfeld s rigid bond test. The results are a beautiful confirmation of Hirshfeld s (1992) statement that an accurate set of nuclear coordinates (and thermal parameters ) and a detailed map of the electron density can be obtained, via X-ray diffraction, only jointly and simultaneously, never separately or independently . 

The STO-3G basis set has two obvious shortcomings The first is that all basis functions are either themselves spherical or come in sets which, taken together, describe a sphere. This means that atoms with spherical molecular environments or nearly spherical molecular environments will be better described than atoms with aspherical molecular environments . This suggests that comparisons among different molecules will be biased in favor of those incorporating the most spherical atoms. The second shortcoming follows from the fact that basis functions are atom centered. This restricts their flexibility to describe electron distributions between nuclei ( bonds ). 

Split-Valence Basis Set. A Basis Set in which the Core is represented by a single set of Basis Functions (a Minimal Basis Set) and the Valence is represented by two or more sets of Basis Functions. This allows for description of aspherical atomic environments in molecules. 3-2IG, 6-3IG and 6-3IIG are split-valence basis sets. 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

Hansen NK, Coppens P (1978) Electron population analysis of accurate diffraction data. 6. Testing aspherical atom refinements on small-molecule data sets. Acta Crystallogr A 34 909-921... 

Fig. I. Schematic representation of the phenomena of magnetostriction. The surrounding atoms, schematised as positive charges, are displaced from their initial symmetrical position (open circles) to their final strained positions (black circles) due to the electrostatic interactions with the aspherical electron distribution.

Molecular geometry, namely, each atomic position ry in the unit cell, is conventionally determined by the least-squares fitting method based on the residue between IF qI and FC. A simple least-squares technique used in the conventional crystal structure analysis gives a slightly biased atomic position, because of the aspherical electron-density distribution caused by the chemical environment (15, 18). To obtain the unbiased deformation density, the atomic positions are commonly de-... 

In 1973, Iwata and Saito determined the electron-density distribution in crystals of [Co(NH3)6]fCo(CN)6l (37). This was the first determination of electron density in transition metal complexes. In the past decade, electron-density distributions in crystals of more than 20 transition metal complexes have been examined. Some selected references are tabulated in Table I. In most of the observed electron densities, aspherical distributions of 3d electron densities have been clearly detected in the vicinities of the metal nuclei. First we shall discuss the distributions of 3d electron density in the transition metal complexes. Other features, such as effective charge on transition metal atoms and charge redistribution on chemical bond formation, will be discussed in the following sections. 

Most coordination compounds have octahedral, square-planar, or tetrahedral structures. Asphericities of 3d electron-density distributions around transition metal atoms placed in an octahedral and tetrahedral environment differ distinctly from each other. Thus, they will be dealt with separately. 

It is reasonable to compare the observed deformation density of Cr042- with the theoretical one for Mn04", since the electronic structures of both systems can be regarded as identical except for the different oxidation numbers of the central transition metal atoms. Figure 7 shows the theoretical deformation density around the Mn nucleus on the same section as Fig. 6. Aspherical electron-density distributions around the metal atoms closely resemble each other. The theoretical deformation density presents pronounced positive peaks around the Mn nucleus at 0.30 A with peak height of 2 e A-3 on the extension of the 0—Mn bonds. Positive peak height and the distribution of nega-... 

Rees and Mitschler have estimated the 3d electron populations in t2g and eg orbitals for both limited electronic configurations of (3g05(4s)1 and (3d)6 (63). The electron population in the 4s orbital was not refined because a diffuse distribution of 4s electrons affects only a small number of structure amplitudes with very low reflection angles. The estimated electronic configurations of a Cr atom are 3d(t2g)3H3d(eg)12(4s) and 3d(t2g)453d(eg) 5, respectively, being consistent with the weak asphericity in 3d electron distribution. 

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