Core electron density

None of the studies mentioned in Section 2.2 has explicitly addressed the main issue of the redistribution of core electron densities under MaxEnt requirements in the absence of high-resolution observations. This is indeed the key to explaining the unsatisfactory features encountered so far in the applications of the method to charge density studies. 

The Pauli operator of equations 2 to 5 has serious stability problems so that it should not, at least in principle, be used beyond first order perturbation theory (20). These problems are circumvented in the QR approach where the frozen core approximation (21) is used to exclude the highly relativistic core electrons from the variational treatment in molecular calculations. Thus, the core electronic density along with the respective potential are extracted from fully relativistic atomic Dirac-Slater calculations, and the core orbitals are kept frozen in subsequent molecular calculations. 

First, valence and core electrons are formally identical however, the separation of valence and core electron density is dictated by the standard view of atomic physics. Thus for an ion at j, coordinates r-hi are assigned to the z z electrons designated as core electrons, the understanding being that the states of the system are such that < r , >1/2 is a small quantity (< a0). 

One not so obvious problem with the shape-consistent REP formalism (or any nodeless pseudoorbital approach) is that some molecular properties are determined primarily by the electron density in the core region (some molecular moments, Breit corrections, etc.) and cannot be computed directly from the valence-only wave function. For Phillips-Kleinman (21) types of wave functions, Daasch et al. (52) have shown that the core electron density can be approximated quite accurately by adding in the atomic core orbitals and then Schmidt orthogonalizing the valence orbitals to the core. This new set of orbitals (core plus orthogonalized valence) is a reasonable approximation to the all-electron set and can be used to compute the desired properties. This will not work for the shape-consistent case because / from Eq. (18) cannot be accurately described in terms of the core orbitals alone. On the other hand, it is clear from that equation that the corelike portion of the valence orbitals could be reintroduced by adding in fy (53),... 

Equation (10) shows that the isomer shift IS is a direct measure of the total electronic density at the probe nucleus. This density derives almost exclusively from 5-type orbitals, which have non-zero electron densities at the nucleus. Band electrons, which have non-zero occurrence probabilities at the nucleus and 5-type conduction electrons in metals may also contribute, but to a lesser extent. Figure 3 shows the linear correlation that is observed between the experimental values of Sb Mossbauer isomer shift and the calculated values of the valence electron density at the nucleus p (0). The total electron density at the nucleus p C ) (Eq. 10) is the sum of the valence electron density p (0) and the core electron density p (0), which is assumed to be constant. This density is not only determined by the 5-electrons themselves but also by the screening by other outer electrons p-, d-, or /-electrons) and consequently by the ionicity or covalency and length of the chemical bonds. IS is thus a probe of the formal oxidation state of the isotope under investigation and of the crystal field around it (high- and low-spin Fe may be differentiated). The variation of IS with temperature can be used to determine the Debye temperature of a compound (see Eq. (13)). 

Cullity, B. D. Elements ofX-ray Diffraction, 2nd ed., Addison-Wesley Reading, Massachusetts, 1978. The effective nuclear charge is defined as the actual nuclear charge felt by a particular valence electron. It is expressed as Zgff = Z —c, where Z is the nuclear charge for the atom and a is the screening constant. This latter term corresponds to the number of core electrons, and the effectiveness of the orbitals to shield core electron density. 

In the pseudopotential method, core states are omitted from explicit consideration, a plane-wave basis is used, and no shape approximations are made to the potentials. This method works well for complex solids of arbitrary structure (i.e., not necessarily close-packed) so long as an adequate division exists between localized core states and delocalized valence states and the properties to be studied do not depend upon the details of the core electron densities. For materials such as ZnO, and presumably other transition-metal oxides, the 3d orbitals are difficult to accommodate since they are neither completely localized nor delocalized. For example, Chelikowsky (1977) obtained accurate results for the O 2s and O 2p part of the ZnO band structure but treated the Zn 3d orbitals as a core, thus ignoring the Zn 3d participation at the top of the valence region found in MS-SCF-Aa cluster calculations (Tossell, 1977) and, subsequently, in energy-dependent photoemission experiments (Disziulis et al., 1988). 

A general equation can be derived that describes the variation in direction of the valence electron density about the nucleus. The distortion from sphericity caused by valence electrons and lone-pair electrons is approximated by this equation, which includes a population parameter, a radial size function, and a spherical harmonic function, equivalent to various lobes (multipoles). In the analysis the core electron density of each atom is assigned a fixed quantity. For example, carbon has 2 core electrons and 4 valence electrons. Hydrogen has no core electrons but 1 valence electron. Experimental X-ray diffraction data are used to deri e the parameters that correspond to this function. The model is now more complicated, but gives a better representation of the true electron density (or so we would like to think). This method is useful for showing lone pair directionalities, and bent bonds in strained molecules. Since a larger number of diffraction data are included, the geometry of the molecular structure is probably better determined. 

The other two types of radiation that can diffract fi om crystals are neutron and electron beams. Unlike x-rays, neutrons are scattered on the nuclei, while electrons, which have electric charge, interact with the electrostatic potential. Nuclei, their electronic shells (i.e. core electron density), and electrostatic potentials, are all distributed similarly in the same crystal and their distribution is established by the crystal structure of the material. Thus, assuming a constant wavelength, the differences in the diffraction patterns when using various kinds of radiation are mainly in the intensities of the diffracted beams. The latter occurs because various types of radiation interact in their own way with different scattering centers. The x-rays are the simplest, most accessible and by far the most commonly used waves in powder diffraction. 

In an effort to emphasize the valence structure of chemical bonds, valence electron density maps have been constructed.9 In these studies the core electron density (the spin restricted Hartree-Fock Is orbital product for a first row atom) is assumed invariant to chemical bonding and is the basis of the scattering factor that is incorporated in Eq. (11). 

Some workers prefer to subtract the dominant, but chemically uninteresting, core electron density and model the total valence distribution in full... 

Case II n b Taken as Valence Electron Density with Small Admixture of Core Electron Density... 

X-ray contrast variation has been best applied to the metalloprotein ferritin, which consists of a mineral iron core (electron density - 1000 e nm ) surrounded by a spherical protein shell (410 enm ) of 24 regularly-arranged subunits [149,158-160]. The protein shell was matched out in 53% sucrose (w/w) or 0.66 g sucrose/ml solution, leaving the observed scattering to be caused only by the iron core (Fig. 17). Control experiments were performed on apoferritin which lacks the iron core [149,160]. Thus the Rq for native ferritin is 3.7 nm, that of apoferritin is 5.6 nm and that of the iron core is 2.9 nm [158-160]. The outer and inner radii of the protein of ferritin were found to be 6.3 and 3.55 nm [160]. The radius of the iron core was found to be 3.66 nm and corresponded well to the scattering from a uniform sphere [149], rather than to other models based on a collection of spherical micellar domains that were proposed from electron microscopy studies (Fig. 17). 

Still another idea is introduced in the contributions by Cooper and Allan.They removed the core electron density dominance problem by using momentum transformations. A similar expression exists for the electron density in momentum space ... 

Cooper and Allan ° have used momentum density in several studies. A problem remains in obtaining the momentum space densities because most calculations are performed with position space wave functions. In a sense, working in momentum space is yet another way to reduce the overweighting of the core electron density. Most of the following discussions on, e.g., molecular alignment and quantum similarity indices, remain valid when we... 

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