Aspherical 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. 

The aspherical density formalism of Hirshfeld is a deformation model with angular functions which are a sum over spherical harmonics. It will be described in more detail in section 3.2.6. All three models have been applied extensively in charge density studies (for a comparison, see Lecomte 1991). 

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... 

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... 

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. 

For vanadium, the ratios are smaller, and the dynamic density maps do not show a distinct maximum in the cube direction. The difference is attributed to anharmonicity of the thermal motion. Thermal displacement amplitudes are larger in V than in Cr, as indicated by the values of the isotropic temperature factors, which are 0.007 58 and 0.00407 A2 respectively. As in silicon, the anharmonic displacements are larger in the directions away from the nearest neighbors, and therefore tend to cancel the asphericity of the electron density due to bonding effects. 

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 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. 

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-... 

A (298 K), respectively. Consequently, the weak asphericity observed in the deformation density may be genuine and suggest a small difference of 3d electron occupancies in t2g and eg orbitals. 

Observed deformation densities are shown in Fig. 17a-d. The aspherical distribution of 3d electrons can be clearly seen in the vicinity of the Fe nucleus. Two positive peaks with 0.5 e A 3 are located at... 

The electronic structure will give rise to excess electron density in the directions of three nonbonding orbitals and deficiency in the directions of two antibonding orbitals. This simple expectation reasonably reproduces the observed asphericity of 3d electron distribution in the vicinity of the Fe atom. The magnitude of its asphericity is, however, small as in Cr(CO)6, in contrast to that in IColNO ]3" (Fig. 1). Likewise the 3d orbitals in Cr(CO)6, antibonding dxz and dyz orbitals, will... 

---