Valency radial charge density

Figure 15. Radial charge density plot for the resonant p-type virtual orbital for dilation angles 9 = 0.0 and 6 = 90pt (0.42 radians) in e-Be scattering. The role of optimal theta in the accumulation of electron density near the nucleus is clearly seen. In the inset, the maximum is seen to occur at rmaz — 2.5 a.u., very close to that for the rmax of the outer valence 2s orbital, seen in fig. 14- Though a cursory look at the nodal pattern identifies this as a 4P orbital, the dominant contribution to the charge density distribution is mainly of 2p-iype.

Figure SI. Radial charge density plot for the resonant FD amplitude in e-Ca scattering. The number of radial nodes for the resonant FD amplitude identifies it as a Sp-iype orbital with a predominantly dp-type character. The role of optimal theta (9opt = 0.S9 radians) in accumulation of electron density near the nucleus is evident. The electron density for the temporarily captured electron accumulates at rmat = IS a.u. indicating that the impinging electron stays far away even from the outermost valence electrons of the target.

Ionic radii can be defined either by the maximum of the radial charge density, r, or the expectation value, (r , of an outer valence orbital. The DF... 

Figure 1 Radial charge density of the valence orhitals of Gd+ (a) (This figure was pubhshed in Ref 8, Copyright Elsevier (2003)) and Pr + (h). (Redrawn fi om B. G. Wyhoume, Ref. 9)...

Selected band results for the cerium pnictides. The angular momentum decomposition is a single-site decomposition of the wave function within the mufiln tins, not the LCAO decomposition. The radial charge density of the 4f-orbital at the muffine tin boundary and the valence band width (Ep - [,) are tabulated here as they enter into the discussion of relative peak heights (section 3.7). Units are atomic units except the band width which is given in electron... 

The differences between the actinide and lanthanide metals can be rationalized by a consideration of the differences between the 4f- and 5f-electron shells [25]. In the 4f series, all the 4f electrons (added after cerium) are buried in the interior of the electron cloud. The 4f electrons are thus confined to the core of the atom, and experience relatively little interaction with electrons in the 5d shell. The maxima in the radial charge density occur well inside the usual interatomic distances in solids, and consequently the 4f electron properties of the free atoms are retained in the metallic as well as ionic lanthanide solids. Cerium is the only 4f metal that does not conform to this generalization, presumably because its 4f-electron shell is not yet fully stabilized. The actinide 5f electrons behave quite differently. For the early members of the actinide series, the Sf electrons have a greater radial distribution than do their 4f homologs. The first few 5f electrons are not confined to the core of the atom, and they can therefore interact or mix with the other valence electrons to affect interatomic interactions in the solid state. Beyond plutonium, all the 5f electrons are localized within the atomic core, and the resemblance between the f-block elements becomes closer. Americium is the first actinide metal whose crystal structure resembles that of the lanthanide metals. In the transcurium metals, the resemblance to the lanthanide metals becomes increasingly stronger. The room-temperature crystal structure for the elements for Am to Cf is dhep, just as it is in the light lanthanides. 

Weinberger et al. (77) computed for HfC and TaC as well as for HfN and B1 TaN (78) the following quantities by means of the self-consistent, relativistic KKR method (a) partial / like charges in a special atomic sphere P for selected electron states (b) partial / like charges in a special atomic sphere P corresponding to the occupied region of the valence bands (c) / -like radial charge densities in a special atomic sphere P. 

Extra radial flexibility has been proved necessary in order to model the valence charge density of metal atoms, in minerals [6,11], and coordination complexes [5], and similar evidence of the inability of single-exponential deformation functions to account for all the information present in the observations have also been found in studies of organic [12, 13] and inorganic [14] molecular crystals. 

A simple modification of the IAM model, referred to as the K-formalism, makes it possible to allow for charge transfer between atoms. By separating the scattering of the valence electrons from that of the inner shells, it becomes possible to adjust the population and radial dependence of the valence shell. In practice, two charge-density variables, P , the valence shell population parameter, and k, a parameter which allows expansion and contraction of the valence shell, are added to the conventional parameters of structure analysis (Coppens et al. 1979). For consistency, Pv and k must be introduced simultaneously, as a change in the number of electrons affects the electron-electron repulsions, and therefore the radial dependence of the electron distribution (Coulson 1961). 

As an application to atomic systems, the efifective Slater atomic model will be considered (Slater, 1930 dementi Raimondi, 1963 dementini et al., 1967) that is employing the Slater effective charge and the valence electrons through the potential (4.312) and the radial electronic density for a given quantum (shell) number n and the orbital exponent related with Eqs. (4.317) and (4.131), here as... 

Figure 5.8b. Pseudo valence charge density contour plots of the (a) RuO2(110) surface in comparison with (b) the RuO2(001) surface cut through the cus-Ru atoms. These plots are defined as the difference between the total valence electron density and a linear superposition of radially symmetric atomic charge densities. Contours of constant charge density are separated by 0.15 eV/A. Electron depletion and accumulation are marked by dashed and solid lines, respectively. In addition, regions of electron accumulation are shadowed...

Here P and Plm are monopole and higher multipole populations / , are normalized Slater-type radial functions ylm are real spherical harmonic angular functions k and k" are the valence shell expansion /contraction parameters. Hartree-Fock electron densities are used for the spherically averaged core and valence shells. This atom centered multipole model may also be refined against the observed data using the XD program suite [18], where the additional variables are the population and expansion/contraction parameters. If only the monopole is considered, this reduces to a spherical atom model with charge transfer and expansion/contraction of the valence shell. This is commonly referred to as a kappa refinement [19]. 

The Sn 5 s and 5p radial functions, from a nonrelativistic calculation for the free 5sz5pz atom, are plotted in Fig. 7. Roughly 8% of the 5s charge extends outside the Wigner-Seitz radius, rws, for / —Sn the 5s orbital, with much of its density in a region in which Zen is about equal to the valence, is actually somewhat in the interior of the atom. It is not unlike the d orbitals of transition metals, which, as earlier noted, maintain much of their atomic quality in a metal. Thus it is quite plausible that the valence s character in Sn is much like the free atom 5 s, except for a renormalization within the Wigner-Seitz cell. The much more extended 5p component, on the other hand, is not subject to simple renormalization the p character near the bottom of the band takes on a form more like the dot-dash curve of Fig. 7. It nevertheless appears useful to account for charge terms of a pseudo P component and a renormalized s. 

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