Multipolar refinement

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

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. 

X-ray structure factors from the theoretical density. Using the theoretical structure factors, one can carry out multipolar refinement in parallel to experimental X-ray data and make in depth comparison of the topological properties from the two sets. 

In this review, we describe the basic steps in theoretical (molecular, nonperiodic) calculations to study the H-H bonding interactions. For the analysis of electron density obtained by high-resolution low-temperature X-ray diffraction experiments coupled with multipolar refinement [125], the reader is referred to the literature [65, 66]. 

Figure 14. Examples of several intramolecular H H bond path found by RKC in tetraphenyl borates by high resolution X-ray crystallographic experiments followed by multipolar refinement. CP mark the positions of the H H BCPs. See Refs. [50,51] for details (reproduced with permission from Ref [50]).

The limitations of the multipolar refinement are the exceptional demands it places on the X-ray data (in terms of quantity and quality) and the large number of additional parameters per atom that must be included in the least-squares refinement procedures. Nevertheless, charge density studies are becoming more and more popular due to the new experimental possibilities offered by the new area detectors and the more intense sources as provided by the synchrotrons. 

In an electron density (ED) study using multipolar refinement, Bach et al7 have analyzed the intermolecular interactions of the type C-F -O and C-F---F-C associated with pentafluorobenzoic acid at 110 K. These studies highlight the discrepancies in the experimental and theoretical values of the Laplacian at the bond critical points of C-F and examined the energetic disadvantage of the F - F interaction. Bianchi et al have shown that in the ( )-l,2-bis(4-pyridyl)ethylene... 

Scolecite gave the opportunity to relate the electron density features of Si-O-Si and Si-O-AI bonds to the atomic environment and to the bonding geometry. After the multipolar density refinement against Ag Ka high resolution X-ray diffraction data, a kappa refinement was carried out to derive the atomic net charges in this compound. Several least-squares fit have been tested. The hat matrix method which is presented in this paper, has been particularly efficient in the estimation of reliable atomic net charges in scolecite. 

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... 

As anticipated, the multipolar model is not the only technique available to refine electron density from a set of measured X-ray diffracted intensities. Alternative methods are possible, for example the direct refinement of reduced density matrix elements [73, 74] or even a wave function constrained to X-ray structure factor (XRCW) [75, 76]. Of course, in all these models an increasing amount of physical information is used from theoretical chemistry methods and of course one should carefully consider how experimental is the information obtained. 

Fig. 3.27. Left Spectral moments 71 of the rototranslational bands of several molecular pairs, as function of temperature. Various measurements ( , o, etc.) are compared with theoretical data based either on the fundamental theory (H2-H2, H2-He) or on refined multipolar induction models after [58]. Right Same as at left, except the spectral moment 70 is shown.

Souhassou [60] has performed a topological analysis of the multipolar electron density resulting from a Hansen-Coppens refinement against De Titta data. 

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. 

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