Spherical-atom approximation

Early studies, which did not include many high-order reflections, revealed systematic differences between spherical-atom X-ray- and neutron-temperature factors (Coppens 1968). Though the spherical-atom approximation of the X-ray treatment is an important contributor to such discrepancies, differences in data-collection temperature (for studies at nonambient temperatures) and systematic errors due to other effects cannot be ignored. For instance, thermal diffuse scattering (TDS) is different for neutrons and X-rays. As the effect of TDS on the Bragg intensities can be mimicked by adjustment of the thermal parameters, systematic differences may occur. Furthermore, since neutron samples must be... 

A preliminary knowledge of the crystal structure is important prior to a detailed charge density analysis. Direct methods are commonly used to solve structures in the spherical atom approximation. The most popular code is the Shelx from Sheldrick [26] which provides excellent graphical tools for visualization. The refinement of the atom positional parameters and anisotropic temperature factors are carried out by applying the full-matrix least-squares method on a data corrected if found necessary, for absorption and diffuse scattering. Hydrogen atoms are either fixed at idealized positions or located using the difference Fourier technique. 

Numerical differences may originate from alternative approximations to the hardness kernel. The simple expression for the off-diagonal elements of the hardness matrix in EEM (Eq. (18)) stems from the fact that we use a spherical atom approximation and an atomic partitioning of the electron cloud. The two-center electron repulsion integral then reduces to the Coulombic form (4nfio)" e, ep/R p, which equals k/R,p in eV. 

The EEM formalism represents a comprehensive and internally consistent framework for the quantitative as well as qualitative understanding and computation of atom-in-a-molecule sensitivities. The method is direct, due to an adequate separation of the variables, allowed by a spherical-atom approximation. The potential for studying molecules, (ionic) solids and molecule-surface interactions has been fully demonstrated. There are several parameterizations possible, all of them relying on quantum-mechanical calculations for estimating atomic electronegativities and hardnesses. At present, the numerical results are conform with a Mulliken population analysis on STO-3G wavefunctions, but there is no reason why other more sophistieated approaches could not be used. Its simplicity forms a powerful tool for the experimental chemist, who is advised to include the environment into the models, avoiding isolated-atom approaches whenever possible. 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

In order to explain his experimental results, Rutherford designed a new picture of the atom. He proposed that the atom occupies a spherical volume approximately I0 8cm in radius and at the center of each atom there is a nucleus whose radius is about 10 u cm. He further proposed that this nucleus contains most of the mass of the atom, and that it also has a positive charge that is some multiple of the charge on the electron. The region of space outside the nucleus must be occupied by the electrons. We see from Figure 14-11 that Rutherford s picture requires that most of the volume of the atom be a region of very low density. 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

In support of this conclusion, Hanson et al. (1973) found, for a large data set on sucrose (with sin 0/k < 0.81 A-1), a mean difference between the phases ipx from a spherical-atom refinement and spherical-atom calculation with neutron parameters of 1.8°. A better approximation to the true phases is... 

The electronic polarisability of a spherical atom may be calculated in a number of simplified ways. In the oldest approximation, an atom is regarded as a conductive sphere of radius R, when the polarisability may be shown to be 4k 0R3, a quantity that is closely related to the actual volume of a molecule. In the more realistic semi-classical Bohr model of a hydrogen atom, the application of a field normal to the plane of the electron orbit, radius R, will produce a small shift, — x, in the orbit, as shown in Fig. 2.2. To a first approximation the distance of the orbit from the nucleus will still be R and the dipole moment p. induced in the atom will have magnitude ex. At equilibrium, the external field acting on the electron is balanced by the component of the Coulombic field from the positive nucleus in the field direction ... 

Here, an and a are the polarizabilities of the diatomic complex parallel and perpendicular to the internuclear separation, R12. A purely classical theory, which accounts for the electrostatic distortion of the local field by the proximity of a point dipole (the polarized collisional partner), suggests that )S(Ri2) 6ao/Ri2 with (Xq designating the permanent polarizability of an unperturbed atom. This expression is known to approximate the induced anisotropy of such diatoms fairly well. This anisotropy gives rise to the much studied pressure-induced depolarization of scattered light and to depolarized CILS spectra in general. The depolarization of light by dense systems of spherical atoms or molecules has been known as an experimental fact for a long time. It is, however, discordant with Smoluchowski s and Einstein s... 

Harrison has performed SIC-LSD calculations for the 3d series but without sphericalizing the spin-orbital densities. The SIC is calculated for Cartesian orbitals and then the central-field approximation is reintroduced by spherically averaging the contributions to the energy and to the potential arising from a given shell. As Harrison pointed out, the errors introduced by sphericalizing the orbital densities are present even for spherical atoms because of the non-linear orbital density dependence of the SIC (see also Ref. 186). His results for the d" s-d" s separation are shown in Fig. 7. The... 

The most important computational models in use today for proteins are based on a molecular mechanics description. They represent the protein as a collection of spherical particles (the atoms), approximately incompressible, connected together by springs, each one bearing a small electric charge [30, 44]. Solvent molecules can be described in the same way. To parameterize such a model for a large class of molecules like proteins takes several decades of researcher-years. Once in place, and despite its simplicity, a molecular mechanics model is a powerful tool to study the structure and stability of biomolecules. 

For spherical atoms, the polarizability in the Kirkwood approximation can be written as... 

From the density, it is now possible to construct the potential for the next iteration, and also to calculate total energies. During the self-consistency loop, we make a spherical approximation to the potential. This is because the computational cost for using the full potential is very high, and that the density converges rather badly in the corners of the unit cell. In order not to confuse matters with the ASA (Atomic Sphere Approximation) this is named the Spherical Cell Approximation (SC A) [65]. If we denote the volume of the Wigner-Seitz cell centered at R by QR, we have fiR = QWR — (4 /3), where w is the Wigner-Seitz radius. This means that the whole space is covered by spheres, just as in the ASA. We will soon see why this is practical, when we try to create the potential. 

An alternative to the spherical jellium approximation just described is to use the tried and tested methods of theoretical chemistry, namely the energy variational principle, to determine the most probable geometrical structure for atomic clusters. This is the basis of the Hiickel method, a rough outline of which is as follows. 

For valence states of nonhydride molecules, there is no reason to expect that the generalized pure precession approximation should be valid. In contrast, Rydberg orbitals, because of their large size and nonbonding, single-center, near-spherical, atomic-like character, are almost invariably well-described by the pure precession picture in terms of nl-complexes. 

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