Valence density formalism

Combining the angular and radial functions discussed above leads to a valence-density formalism in which the density of each of the atoms is described as (Hansen and Coppens 1978)... 

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. 

The nature of the charge density parameters to be added to those of the structure refinement follows from the charge density formalisms discussed in chapter 3. For the atom-centered multipole formalism as defined in Eq. (3.35), they are the valence shell populations, PLval, and the populations PUmp of the multipolar density functions on each of the atoms, and the k expansion-contraction parameters for... 

The models used in this review are due to Coppens and his co-workers [//]. First, the K formalism [ 11a) permits an estimation of the net charge of the atom and allows for the expansion contraction of the perturbated valence density for each atom, the density is described as,... 

Having secured a set of n values for phosphorus, the pseudoatom model was fitted to the four simulated data sets to test the effectiveness of the pseudoatoms model s formal deconvolution of multipolar valence density features from thermal vibrations smearing. Results are illustrated in Figure 4 as maps of the model static deformation densities ... 

The valence DOS has been computed for Ni and Ag clusters within the CNDO formalism. Blyholder [54] examined the Nis and M13 clusters. In both cases of s- and p-orbitals are occupied and lie well below the d-orbitals. Most of the intensity is near the middle of the d-orbitals with a fall-off in intensity as the HOMO is approached. Density of states for Agv, Agio, Agi3, and Agig clusters shows a strong d-component cc. 3.5 eV wide. The... 

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]. 

From the inspection of the data in Table 2.4, it is clear that NO changes its original molecular character after adsorption. In general, coordination of nitric oxide leads to a pronounced redistribution of the electron and spin densities, accompanied by modification of the N-0 bond order and its polarization. Thus, in the case of the (MNO 7 10 and ZnNO 11 species, slender shortening of the N-0 bond is observed, whereas for the MNO 6 and CuNO 11 complexes it is distinctly elongated. Interestingly, polarization of the bound nitric oxide assumes its extreme values in the complexes of the same formal electron count ( NiNO 10 and CuNO 10) exhibiting however different valence. 

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

Table 3.7. Valence-shell NBOs of small cations in equilibrium geometry (cf Tables 3.4 and 3.6), with percentage Lewis density (%p(L)) for each species (an asterisk denotes a formal non-Lewis orbital)...

There is a formal similarity between adsorption and reactions such as metal deposition which gives rise to the concept of electrosorption valence. Consider the deposition of a metal ion of charge number on an electrode of the same material. If the electrode potential 4> is kept constant, the current density j is ... 

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). 

By virtue of the oxophilicity of aluminium, many compounds exist in which at least one of the metal s formal valencies is occupied by oxygen or, alternatively, where the coordination state of the metal is raised to 4, 5 or 6 by the donation of electron density from oxygen. This review presents aluminium oxides first and thereafter aluminium hydroxides and organooxides. Although the discussion concentrates on the sohd-state structural properties of such systems, solution structural, theoretical and reactivity studies are also presented. 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

In atomic clusters, oscillations of valence electrons are generated by time-dependent variations of the electronic T-even density p f, t) only. So, one may neglect in the SRPA formalism all T-odd densities and their variations Pk t)- This makes SRPA equations especially simple. In particular, the density variation (9) is reduced to... 

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