Shell density

This treatment may be extended to spheres core-shell structure. If the core density is p 0 to fp the shell density is p2 in the range o density of the surrounding medium is Pq, th of the structure factor becomes... 

Table 4.15 summarizes optimized bond lengths and NBO Lewis-like structures for 15 saturated normal-valent H MMH compounds (M = W-Pt) as well as corresponding hydrides of hypovalent Ta for comparison. The accuracy of the localized Lewis-like description (as measured by %pf) is found to be reasonably high both for normal-valent and for hypovalent species, typically >98% of the valence-shell density and >99.5% of the total density. 

In this nonvariational approach for the first term represents the potential of the exchange-correlation hole which has long range — 1/r asymptotics. We recognize the previously introduced splitup into the screening and screening response part of Eq. (69). As discussed in the section on the atomic shell structure the correct properties of the atomic sheU structure in v arise from a steplike behavior of the functional derivative of the pair-correlation function. However the WDA pair-correlation function does not exhibit this step structure in atoms and decays too smoothly [94]. A related deficiency is that the intershell contributions to E c are overestimated. Both deficiencies arise from the fact that it is very difficult to represent the atomic shell structure in terms of the smooth function p. Substantial improvement can be obtained however from a WDA scheme dependent on atomic shell densities [92,93]. In this way the overestimated intershell contributions are much reduced. Although this orbital-depen-... 

For a given value of n, the functions httk are identical to a sum of spherical harmonics with l = n, n — 2, n — 4,..., (0,1) for n > 1. The relationships are summarized in Table 3.8. For n = 0,1, the Hirshfeld functions are identical to the spherical harmonics with / = 0, 1, but, starting with the n = 2 functions, lower-order spherical harmonics are included for each n value. Unlike the spherical harmonics, the hnl functions are therefore not mutually orthogonal. As the radial functions in Eq. (3.48) contain the factor r", quite diffuse s, p, and d functions are included in the n = 2, 3, and 4 sets. For n <4 there are 35 deformation functions on each atom, compared with 25 valence-shell density functions with / < 4 in the multipole expansion of Eq. (3.35). 

Substitution of Eq. (10.18) in Eq. (10.20), with the value of n(2) in Eq. (10.18) equal to 4 for 3d valence-shell density functions of first-row transition metal atoms (chapter 3), gives1... 

The present contribution considers general electronic states of solvated molecules and is not limited to closed shell molecular compounds. For closed shell molecular systems, methods utilizing closed shell coupled-cluster electronic structure and closed shell density density functional theory for the electronic structure of the solvated system have appeared in the literature [54-67],... 

This method requires only a crude structural model as a starting model. In this analysis, the starting model was a homogeneous spherical shell density for the carbon cage. As for the temperature factors of all atoms, an isotropic harmonic model was used an isotropic Gaussian distribution is presumed for a La atom in the starting model. Then, the radius of the C82 sphere was refined as structural parameter in the Rietveld refinement. 

M. Odelius, D. Laikov, and J. Butter (2003) Excited state geometries within time-dependent and restricted open-shell density functional theories. J. Mol. 

This chart consists of contour diagrams of the electron densities characteristic of the shell model of the molecules H, Lit, Bj, C2l N2i Ot, end F. Both the total molecular density and the constituent shell densities are displcyed at the experimental intemuclear. distance of eoch molecule. (He, Be, and Ne which ore members of this homonueleor series ore not bound in their ground state and therefore not displayed.)... 

R Workspace Must be large enough for a stack of number of.shells density matrices. 

Pay represents an element of the closed-shell density matrix defined as ... 

Here, denotes an element of the (closed-shell) density matrix given by... 

Geometries within Time-Dependent and Restricted Open-Shell Density Functional Theories. 

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