K-expansion-contraction parameter

On the carbons, nitrogens and oxygens expansions up to octapole level were introduced, whereas the expansions were limited to quadrupole level for the hydrogen atoms. All atoms were given a k expansion/contraction parameter for the spherical monopole term, and all atoms except the hydrogens were given k parameters to expand or contract the non-spherical poles. The k and k values on 0(1) and 0(3), on N(l) and N(3), on C(l) and C(3) and on 0(21) and 0(22) were constrained to be equal. 

To preserve the shell structure of the spherical component of the valence density, the radial function of the bonded atom may be described by the isolated atom radial dependence, modified by the k expansion-contraction parameter. 

The nature of the charge density parameters to be added to those of the structure refinement follows from the charge density formalisms discussed in chapter 3. For the atom-centered multipole formalism as defined in Eq. (3.35), they are the valence shell populations, PLval, and the populations PUmp of the multipolar density functions on each of the atoms, and the k expansion-contraction parameters for... 

Here P and Plm are monopole and higher multipole populations / , are normalized Slater-type radial functions ylm are real spherical harmonic angular functions k and k" are the valence shell expansion /contraction parameters. Hartree-Fock electron densities are used for the spherically averaged core and valence shells. This atom centered multipole model may also be refined against the observed data using the XD program suite [18], where the additional variables are the population and expansion/contraction parameters. If only the monopole is considered, this reduces to a spherical atom model with charge transfer and expansion/contraction of the valence shell. This is commonly referred to as a kappa refinement [19]. 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

The radial deformation of the valence density is accounted for by the expansion-contraction variables (k and k ). The ED parameters P, Pim , k, and k are optimized, along with conventional crystallographic variables (Ra and Ua for each atom), in an LS refinement against a set of measured structure factor amplitudes. The use of individual atomic coordinate systems provides a convenient way to constrain multipole populations according to chemical and local symmetries. Superposition of pseudoatoms (15) yields an efficient and relatively simple analytic representation of the molecular and crystalline ED. Density-related properties, such as electric moments electrostatic potential and energy, can readily be obtained from the pseudoatomic properties [53]. 

A simple modification of the IAM model, referred to as the K-formalism, makes it possible to allow for charge transfer between atoms. By separating the scattering of the valence electrons from that of the inner shells, it becomes possible to adjust the population and radial dependence of the valence shell. In practice, two charge-density variables, P , the valence shell population parameter, and k, a parameter which allows expansion and contraction of the valence shell, are added to the conventional parameters of structure analysis (Coppens et al. 1979). For consistency, Pv and k must be introduced simultaneously, as a change in the number of electrons affects the electron-electron repulsions, and therefore the radial dependence of the electron distribution (Coulson 1961). 

Alternatively, the deformation ED can be described by a series of multipoles, that is, spherical harmonic density functions, the parameters of which can be refined by the LS technique. The core ED is described as spherically symmetrical, using the so-called k formalism. The k parameter expresses the isotropic expansion ( < > 1) or contraction (/c < 1) of a valence shell as a whole. The higher-order multipoles describe the deviations of the ED from spherical symmetry. 

N, is the normalization factor, n, and are parameters depending on the atomic type. Pi are the multipolar population parameters and k and k are the contraction-expansion coefficients [11] for, respectively, spherical and multipolar valence densities. We have chosen orthogonal reference axes which respect the tetrahedral (23) T point group for Si and A1 atoms of the scolecite in order to reduce the number of multipolar parameters only the cubic harmonic multipoles (one octupole / = 3 and two hexadecapoles / = 4) have been refined for these two atoms. The pseudo-atom expansion was extended to the octupoles (/ = 3) for 0 including oxygen of water, and to the dipoles (/ = 1) for H. The best radial functions of Si and A1 atoms were obtained by inspection of the residual maps [12], ( / = 4,4,4,4 (1 = 1-4)) s were taken from Clement and Raimondi [13] i si = 3.05 bohr, = 2.72 bohr. For 0 atoms, = 4.5 bohr and the multipole exponents were respectively n = 2, 3, 4 up to the octupole level. 

Where a is the longitudinal stress, e is corresponding strain, and E is called Young s modulus (or the modulus of elasticity). Similarly, in shear deformation, the modulus is called the shear modulus or the modulus of rigidity (G). When a hydrostatic force is applied, a third elastic modulus is used the modulus of compressibility or bulk modulus (K). It is defined as the ratio of hydrostatic pressure to volume strain. A deformation (elongation or compression) caused by an axial force is always associated with an opposite deformation (contraction or expansion) in the lateral direction. The ratio of the lateral strain to the longitudinal strain is the fourth elastic constant called Poisson s ratio (v). For a small deformation, elastic parameters can be correlated in the following way ... 

The predictions of eq. (24) were tested with glass beads (0.456 mm and 1.08 mm) and lead shots (2.18 mm) for liquid superficial velocities of water and water solutions of polyethylene glycol in the 0.4 - 38 cm/s range [23]. The k parameter was estimated by using the spherical cap assumption and from previous research studies. It was proved that the sign of the numerator of eq. (24) correctly predicts the initial contraction or expansion of the bed. 

More promising is to describe the deformation electron density by a series of spherical harmonic density functions (multipoles), which can be included into least-squares refinement. The inner (core) electron shells of an atom are presumed and the k parameter, which describes the isotropic expansion (ic <1) or contraction (/c > 1) of the valence shell as a whole. Multipole parameters of higher orders describe deviations of the electron density from spherical symmetry. They can be related to the products of atomic... 

LEED optimization [150] with a total number of 26 independent parameters leads to a structural model for the second phase (Fig. 3.23d). The oxygen atoms in the bridge sites reside 0.12-0.18 A above the top vanadium layer, while oxygen atoms in the 4 hollow sites are 0.5 A above the top layer. The structural discrepancy determined by different methods corresponded to (1) the ab initio data are for T = 0 K, whereas the LEED was taken at room temperature (2) the involvement of the (1 x 4) and (1x6) superstructures may influence the LEED measurements. Nevertheless, both the LEED and DFT optimization revealed an expansion of the flrst-layer spacing (+4.2 %) and a contraction (—3.2 %) of the second with respect to the clean V(OOl) surface that contracts by —7 to —8.5 % (LEED) or -15 % (DET). 

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