Structure Model - Spherical Atom

In normal, routine structure determination experiments, the atomic scattering factors (fj) used are derived from spherically averaged ground state electronic configurations of neutral atoms. The positions of these scattering centers convoluted with thermal motion are then used to calculate structure factors (FH), which are compared with observed structure factors derived from the observed Bragg intensities [3]. 

Treating the positional parameters and the elements of the thermal displacement tensor as variables, a best fit of the observed to the calculated structure factors in a least squares sense is determined. As this is a non-linear procedure, it is essential to over determine the problem. For a routine structure 

Given the quality of data described above, it is possible to go beyond the neutral spherical atom model, and to determine the redistribution of valence electrons due to chemical bonding. In other words, we can develop a description of the electron density distribution that includes charge transfer and non-spherical atoms. 

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Iversen et al, in their study of crystalline beryllium [32], were the first to make use of NUP distributions calculated by superposition of thermally-smeared spherical atoms. More recently, a superposition of thermally-smeared spherical atoms was used as NUP in model studies on noise-free structure factor amplitudes for crystalline silicon and beryllium by de Vries et al. [38]. The artefacts present in the densities computed with a uniform prior-prejudice distributions have been shown to disappear upon introduction of the NUP. No quantitative measure of the residual errors were given. 

The core and valence monopole populations used for the MaxEnt calculation were the ones of the reference density (electrons in the asymmetric unit iw = 12.44 and nvalence = 35.56). The phases and amplitudes for this spherical-atom structure, union of the core fragment and the NUP, are already very close to those of the full multipolar model density to estimate the initial phase error, we computed the phase statistics recently described in a multipolar charge density study on 0.5 A noise-free data [56],... 

For a number of 1907 acentric reflexions up to 0.463 A resolution, the mean and rms phase angle differences between the noise-free structure factors for the full multipolar model density and the structure factors for the spherical-atom structure (in parentheses we give the figures for 509 acentric reflexions up to 0.700A resolution only) were (Acp) = 1.012(2.152)°, rms(A( >) = 2.986(5.432)° while... 

A preliminary least-squares refinement with the conventional, spherical-atom model indicated no disorder in the low-temperature structure, unlike what had been observed in a previous room-temperature study [4], which showed disorder in the butylic chain at Cl. The intensities were then analysed with various multipole models [12], using the VALRAY [13] set of programs, modified to allow the treatment of a structure as large as LR-B/081 the original maximum number of atoms and variables have been increased from 50 to 70 and from 349 to 1200, respectively. The final multipole model adopted to analyse the X-ray diffraction data is described here. 

The effect of pressure on the ground-state electronic and structural properties of atoms and molecules have been widely studied through quantum confinement models [53,69,70] whereby an atom (molecule) is enclosed within, e.g., a spherical cage of radius R with infinitely hard walls. In this class of models, the ground-state energy evolution as a function of confinement radius renders the pressure exerted by the electronic density on the wall as —dEldV. For atoms confined within hard walls, as in this case, pressure may also be obtained through the Virial theorem [69] ... 

Mallinson et al. (1988) have performed an analysis of a set of static theoretical structure factors based on a wave function of the octahedral, high-spin hexa-aquairon(II) ion by Newton and coworkers (Jafri et al. 1980, Logan et al. 1984). To simulate the crystal field, the occupancy of the orbitals was modified to represent a low-spin complex with preferential occupancy of the t2g orbitals, rather than the more even distribution found in the high-spin complex. The complex ion (Fig. 10.14) was centered at the corners of a cubic unit cell with a = 10.000 A and space group Pm3. Refinement of the 1375 static structure factors (sin 8/X < 1.2 A 1) gave an agreement factor of R = 4.35% for the spherical-atom model with variable positional parameters (Table 10.12). Addition of three anharmonic thermal... 

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. 

The total contribution from a specific complex can be calculated by summing over all of its intramolecular distances. In order to make a complete structural model of the solution all other contributions would also have to be included. Distances from the complex to surrounding atoms are often approximated by assuming it to occupy a spherical hole in an evenly distributed electron density. Other interactions can also be approximated by a combination of discrete interactions for short distances and a continuous atomic distribution for long distances. 

A metallic crystal can be pictured as containing spherical atoms packed together and bonded to each other equally in all directions. We can model such a structure by packing uniform, hard spheres in a manner that most efficiently uses the available space. Such an arrangement is called closest packing (see Fig. 16.13). The spheres are packed in layers in which each sphere is surrounded by six others. In the second layer the spheres do not lie direotlv over those in the first layer. Instead, each one occupies an indentation (or dimple) formed by three spheres in the first layer. In the third layer the spheres can occupy the dimples of the second layer in two possible ways. They can occupy positions so that each sphere in the third layer lies directly over a sphere in the first layer (the aba arrangement), or they can occupy positions... 

The electrons are negatively charged particles. The mass of an electron is about 2000 times smaller than that of an proton or neutron at 0.00055 amu. Electrons circle so fast that it cannot be determined where electrons are at any point in time, rather, we talk about the probability of finding an electron at a point in space relative to a nucleus at any point in time. The image depicts the old Bohr model of the atom, in which the electrons inhabit discrete "orbitals" around the nucleus much like planets orbit the sun. This model is outdated. Current models of the atomic structure hold that electrons occupy fuzzy clouds around the nucleus of specific shapes, some spherical, some dumbbell shaped, some with even more complex shapes. Even though the simpler Bohr model of atomic structure has been superseded, we still refer to these electron clouds as "orbitals". The number of electrons and the nature of the orbitals they occupy basically determines the chemical properties and reactivity of all atoms and molecules. 

The conventional structure factor formalism utilized in standard structure determinations invokes the concept of the promolecule the superposition of isolated (spherical) atomic densities derived, for example, via the Hartree-Fock procedure [45], While this model mimics the dominant topological features of the ED (local maxima at the nuclear positions) reasonably well, it completely neglects density deformations due to bonding. Unfortunately, this omission leads to biases in estimates of the structural [46,47] and thermal parameters [48]. 

Direct access to structure determination, i.e., atomic positions is obtained via the independent atom model (IAM, also denoted as the isolated-spherical-atom model). Here, the electron density is written as a sum of free atomic densities... 

Most pure metals adopt one of three crystal structures, Al, copper structure, (cubic close-packed), A2, tungsten structure, (body-centred cubic) or A3, magnesium structure, (hexagonal close-packed), (Chapter 1). If it is assumed that the structures of metals are made up of touching spherical atoms, (the model described in the previous section), it is quite easy, knowing the structure type and the size of the unit cell, to work out their radii, which are called metallic radii. The relationships between the lattice parameters, a, for cubic crystals, a, c, for hexagonal crystals, and the radius of the component atoms, r, for the three common metallic structures, are given below. 

Clearly, the properties of the electron posed problems about the inner structure of atoms. If everyday matter is electrically neutral, the atoms that make it up must be neutral also. But if atoms contain negatively charged electrons, what positive charges balance them And if an electron has such an incredibly tiny mass, what accounts for an atom s much larger mass To address these issues, Thomson proposed a model of a spherical atom composed of diffuse, positively charged matter, in which electrons were embedded like raisins in a plum pudding. ... 

Another study focusing on the comparison between theoretical and experimental densities is that of Tsirelson el al. on MgO.133 Here precise X-ray and high-energy transmission electron diffraction methods were used in the exploration of p and the electrostatic potential. The structure amplitudes were determined and their accuracy estimated using ab initio Hartree-Fock structure amplitudes. The model of electron density was adjusted to X-ray experimental structure amplitudes and those calculated by the Hartree-Fock model. The electrostatic potential, deformation density and V2p were calculated with this model. The CPs in both experimental and theoretical model electron densities were found and compared with those of procrystals from spherical atoms and ions. A disagreement concerning the type of CP at ( , 0) in the area of low,... 

In view of this result it is unlikely that any occupied MO will be degenerate with any unoccupied MO, in particular the familiar spherically constrained model of atomic structure in which (e.g.) all 3d AOs, whether occupied or not, have the same energy and radial function is likely to violate the aufbau principle. 

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