Density deformation functions

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The dipolar terms contribute to the electric field. With the density deformation functions of the multipole model (chapter 3) and Eq. (8.36), one obtains... 

A main source of model bias lies in the choice of exponents in the single-exponential-type functions r exp (-ar) that are commonly used as the radial parts of the deformation functions this choice is often more of an art than a science [4]. Very little is known about the optimal values to be used for elements other than those of the first two rows. Selection of the best value for the exponents n is usually carried out by systematically varying exponents and monitoring the effects on the R indices and/or residual densities [8, 9]. The procedure can in some cases be unsatisfactory, as is the case when very diffuse functions centred on one atom are used to model most of the density in the bond, and even some of the density on neighbouring atoms [10]. 

Extra radial flexibility has been proved necessary in order to model the valence charge density of metal atoms, in minerals [6,11], and coordination complexes [5], and similar evidence of the inability of single-exponential deformation functions to account for all the information present in the observations have also been found in studies of organic [12, 13] and inorganic [14] molecular crystals. 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

Hirshfeld (1971) was among the first to introduce atom-centered deformation density functions into the least squares procedure. Hirshfeld s formalism is a deformation model, in which the leading term is the unperturbed IAM density, and the deformation functions are of the form cos" 0jk, where 9jk is the angle between the radius vector r7 and axis k of a set of (n + l)(n + 2)/2 polar axes on each atom /, as defined in Table 3.8 (Hirshfeld 1977). The atomic deformation on atom j is described as... 

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). 

The appearance of a deformation density depends crucially on the definition of the reference state used in its calculation. This has occasionally been interpreted as an ambiguity and an argument against the use of the deformation density as an analytical tool. More precisely, a deformation density is meaningful only in terms of its reference state, which must be taken into account in the interpretation. The different deformation functions are complementary, and when used properly, they provide detailed understanding of the steps in the bond formation process. 

The density distribution function, gla (y), that accounts for the break-down of secondary aggregates attributed to non-linear rubber phase deformation. 

As the resolution of the Bragg reflection data is improved, it becomes possible to obtain information on the more minute details of electron density in a molecule. At high enough resolution information can be obtained on the redistribution of electron density (deformation density) around atoms when they combine to form a molecule. Electrons in molecules ma -form bonds or exist as lone pairs, thereby distorting the electron density around each atom and requiring a more complicated function to describe this overall electron density than normally used, in which it is treated as if it were spherically symmetrical (deformed to an ellipsoid in order to account for anisotropic displacements). This assumption is inherent in the use of spherically-symmetrical scattering factors although the elec-... 

Fig. 14. Shear viscosity, Tj, and extensional viscosity, Tj as a function of deformation rate of a low density polyethylene (LDPE) at 150°C (111). To convert...

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