Spherical atom

In metals the situation is quite the opposite. The spherical atoms move easily from liquid to solid and the interface moves quickly in response to very small undercoolings. Latent heat is generated rapidly and the interface is warmed up almost to T, . The solidification of metals therefore tends to be heat-flow controlled rather than interface controlled. 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

This expression describes how the energy converges as we add successive s functions, p functions, d functions, f functions, and so on, to spherical atoms. 

When we deal with any spherical atomic ion in a vacuum, we may regard it as a charge sphere of radius a bearing a charge + or —q. We shall find that the correct expression for the total energy in the field is obtained by integrating (3) over all space outside the sphere. The electrical capacity of any spherical conductor is equal to its radius. The work to place a charge + or — q on this sphere is... 

Strictly speaking, the size of an atom is a rather nebulous concept The electron cloud surrounding the nucleus does not have a sharp boundary. However, a quantity called the atomic radius can be defined and measured, assuming a spherical atom. Ordinarily, the atomic radius is taken to be one half the distance of closest approach between atoms in an elemental substance (Figure 6.12). 

Packing efficiency is defined as the percent of the total volume of a solid occupied by (spherical) atoms. The formula is... 

Force fields split naturally into two main classes all-atom force fields and united atom force fields. In the former, each atom in the system is represented explicitly by potential functions. In the latter, hydrogens attached to heavy atoms (such as carbon) are removed. In their place single united (or extended) atom potentials are used. In this type of force field a CH2 group would appear as a single spherical atom. United atom sites have the advantage of greatly reducing the number of interaction sites in the molecule, but in certain cases can seriously limit the accuracy of the force field. United atom force fields are most usually required for the most computationally expensive tasks, such as the simulation of bulk liquid crystal phases via molecular dynamics or Monte Carlo methods (see Sect. 5.1). 

Up to now, we have described the crystalline arrays favored by spherical objects such as atoms, but most molecules are far from spherical. Stacks of produce illustrate that nonspherical objects require more elaborate arrays to achieve maximal stability. Compare a stack of bananas with a stack of oranges. Just as the stacking pattern for bananas is less S3TTimetrical than that for oranges, the stmctural patterns for most molecular crystals are less S3TTimetrical than those for crystals of spherical atoms, reflecting the lower s Tnmetry of the molecules that make up molecular crystals. 

The two computational methods, CMS-Xa and LCAO B-spline DPT, for now provide consistent, comparable results [57] with little to choose between them in comparison with experiment in those cases presented here (Sections I.D. 1. a and I. D.a.2). The B-spline method holds the upper hand aesthetically by its avoidance of a model potential semiempirically partitioned into spherical atomic regions. More importantly it olfers greater scope for future development, particularly as the inevitable increases in available computing power open new doors. 

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. 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

Thanks to the particular choice made for the NUP, taken equal to a superposition of spherical atoms, it is for the first time possible within the present approach to compute MaxEnt deformation maps in a straightforward manner. Once the Lagrange multipliers X have been obtained, the deformation density is simply... 

When the reconstruction of the density is carried out by modulation of a prior prejudice of spherical atoms, only the deformation features have to be accommodated this can be accomplished relatively easily, and the Lagrange multipliers are usually below 0.01 in modulus, or even smaller for valence-only runs. No aliasing problems occur in the synthesis of (x). 

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. 

Coppens, P., Guru, T.N., Leung, P., Stevens, E.D., Becker, P. and Yang, Y. (1979) Atomic net charges and molecular dipole moments from spherical-atom X-ray refinement and the relation between atomic charge and shape, Acta Cryst. A, 35, 63-72. 

The four maxima and the saddle point are critical points in the function L(r) analogous to the maxima and saddle points in p(r) discussed in Chapter 6. Every point on the sphere of maximum charge concentration of a spherical atom is a maximum in only one direction, namely, the radial direction. In any direction in a plane tangent to the sphere, the function L does not change therefore the corresponding curvatures are zero. When an atom is part of... 

The orbitals containing the bonding electrons are hybrids formed by the addition of the wave functions of the s-, p-, d-, and f- types (the additions are subject to the normalization and orthogonalization conditions). Formation of the hybrid orbitals occurs in selected symmetric directions and causes the hybrids to extend like arms on the otherwise spherical atoms. These arms overlap with similar arms on other atoms. The greater the overlap, the stronger the bonds (Pauling, 1963). 

General properties and definitions of polarizabilities can be introduced without invoking the complete DFT formalism by considering first an elementary model the dipole of an isolated, spherical atom induced by a uniform electric field. The variation of the electronic density is represented by a simple scalar the induced atomic dipole moment. This coarse-grained (CG) model of the electronic density permits to derive a useful explicit energy functional where the functional derivatives are formulated in terms of polarizabilities and dipole hardnesses. 

Liebig, too, saw the aim of chemistry as the search not only to consolidate the truth of chemical proportions but to study the causes of the regularity and constancy of these proportions. Liebig took the cause of chemical action to lie in Newtonian-type atoms and forces of the Berzelian variety, that is, spherical atoms and electrical affinity forces. 31 This is a problem-solving tradition focused on atoms and powers, or mechanism and materialism.32... 

Consider first two simple spherical atoms, say argon atoms. The pair interaction potential has the general form depicted in Fig. 9.6. Note that for/ < the potential... 

The topological characteristics of CPs on bonds gives a quantitative explanation of the known effect that the formation of a Ge crystal is accompanied by shifting the electron density towards the Ge-Ge bonding line. This is can be seen by comparing the parameters of the curvature of the electron density at the critical point (3,-1) with analogous parameters for a procrystal (a set of noninteracting spherical atoms placed at the same... 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

Fig. 1. Assuming spherical atoms, the surface area of atom A is the amount of surface area not contained in other atoms. The depictions are a 2D analogy in which an atom s exposed surface area is represented by its exposed perimeter. Each lower diagram depicts the exposed perimeter of atom A in each upper diagram.

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,... 

FIG. 1.5 Spherical atom scattering factors for the isoelectronic F and Na ions. 

That the bond density is also of significance for heavier atoms is evident from the occurrence of the spherical-atom forbidden (222) reflection of diamond and silicon, even at low temperatures where anharmonic thermal effects (see chapter 2) are negligible. The historical importance of the nonzero intensity of the diamond (222) reflection is illustrated by the following comment made by W. H. Bragg, in 1921 ... 

The scale factor can be measured experimentally by a number of techniques, using either single crystal or powder samples (Stevens and Coppens 1975). Measurement for a number of crystals, including orthorhombic sulfur (S8) and x-deutero-glycylglycine, and comparison with least-squares values, indicate that scale factors from spherical-atom refinements are subject to a positive bias of... 

That the positive bias in the scale factors correlates with an increase in thermal parameters is evident from comparison of X-ray and neutron results (Coppens 1968). The apparent increase in thermal parameters of some of the atoms may be interpreted as the response of the spherical-atom model to the existence of overlap density. Because of the positive correlation between the temperature parameters and k, this increase is accompanied by a positive bias in k. 

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... 

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