Density radial function

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

The Fourier transform of the spherical atomic density is particularly simple. One can select S to lie along the z axis of the spherical polar coordinate system (Fig. 1.4), in which case S-r = Sr cos. If pj(r) is the radial density function of the spherically symmetric atom,... 

A hydrogen atom is described by the Gaussian radial density function... 

A variety of empirical rules exist for choosing the exponent(s) for a set of polarization functions. If only a single set is desired, one possible choice is to make the maximum in tlie radial density function, equal to that for the existing valence set (e.g., the 3d functions that best overlap the 2p functions for a first-row atom - note that the radial density is used instead of the actual overlap integral because the latter, by symmetry, must be zero). 

Fig. 15. The radial density function in the steady-state of eqn. (95) for the exchange...

S. L. Strong, and B. L. Averbach Radial Density Functions for Liquid Mercury and Lead. Phys. Rev. 138a, 1336 (1965). 

Here the first term in the D(r) function is the average radial density function of the material which has the form of a parabola with po equal to the average density in electron units, and the second term oscillates about the parabola indicating distances of high and low atomic (electronic) density. The G(r) function subtracts off the parabola so that the oscillation is about zero. Several representative RDFs of minerals are shown in Figure 22. There are several other types of radial distribution functions used in the literature, as well as the analog in EXAFS analysis below. For a complete review of the various types of analysis see Klug and Alexander (1974) and Warren (1969). 

Fig. 2.2 Radial density functions for = 2 for the hydrogen atom. These functions give the relative electron density (e pm ) as a function of distance from the nucleus. They were prepared by squaring the wave functions given in Fig. 2.1.

For Eq. (11) S is the Bragg vector S = 2ttH, IT is the row vector (htk,l) and the scalar S - S = 4ir sin 0/A. The index / covers the N atoms in the unit cell. The atomic scattering factor f (S) is the Fourier-Bessel transform of the electronic, radial density function of the isolated atom. This density function is usually derived from a spin-restricted Hartree-Fock wave function for the atom in its ground state. The structure fac-... 

Fig. 15. The radial density function in the steady-state of eqn. (95) for the exchange interaction J134] with A = 10 s, L = 0.14nm and D = 10" s (-------),...

There is a continuing interest in exploring possible relationships between the shell structures of atoms and their electronic density distributions [31-39]. In this respect, considerable attention has focused upon the radial density function, D(r) = 4nr p(r), which goes through a series of maxima and minima with increasing radial distance from the nucleus [6,31-36,40], [p(r) is the electronic density function since atomic charge distributions are spherically symmetric... 

Note that the spatial orbitals of neon, Eqs. (5.428)-(5.435), are individually radial normalized to one such that the resulting radial density function. 

The resulting radial density function as well the BE and actual Markovian ELFs are depicted in the Figure 5.7. (a)-(d) for all above considered levels of orbital stmcture of Ne. First of all, for all ELFs a clear maximum and minimum corresponding to regions within and between shells are remarked, respectively. 

A natural goal of simulation would be the computation of the relative probabilities of these various states. A more elementary task is to compute the radial distribution which gives the distribution of distance between atom pairs observed. The radial density function may be approximated from a histogram of all pan-distances observed in a long simulation. (There are 21 at each step, so the amount of data is helpfully increased, reducing the sampling error .) This distribution is displayed in Fig. 3.5. The peaks of the radial distribution function are correlated with the various interatomic distances that appear in the cluster configurations shown in Fig. 3.4. 

Although mass and size are fundamental parameters in particle technology, the major difference between particle aggregates and compact particles is their internal stmcture. Structure refers to the spatial distribution of primary particles. This distribution can be described by the radial density function ... 

Electron density function. In order to carry out a self-consistent eneigy band calculation, it is necessary to calculate the electron density function associated with a symmetrized relativistic APW function. For the Bloch state k. A), the spherically averaged radial density function within the j-type APW sphere is given by... 

The radial distribution function D(r) represents the probability of finding an electron between the distances r and r + dr from the nucleus, regardless of directionThis radial density function reveals the atomic shell structure when plotted as function of r. Its integration over r gives the total number of electrons of the system... 

Fig. 9.2 Comparison of the Pq, Pi and P2 radial density functions forCf +. Density differences have been scaled by a factor 10,000...

We first illustrate the difference of the radial density functions D r) defined as (see also expression (9.34))... 

Fig. 1.2 Radial density function for liquid gold. W is the ratio of the racial distribution function density, p r), to the bulk density, />o- The radial density for crystallinegold is indicated by the vertical lines at the bottom of the figure (Vineycrd ...

By choosing a suitable coordinate system in which r is along the z axis, the marginal radial density function can be extracted as... 

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