Multipole population

Examination of the multipole populations gives no indication of the discrepancy observed in the model maps, all populations from parallel refinements agreeing to within two esd s (Table 5). The one striking exception is the monopole population (P,) for carbon. This must be a simple difference in the partitioning of the charge density between atom centers in the model as there is no discernible difference in the model maps around the carbon position. 

TABLE 4.2 Local-Symmetry-Allowed Multipole Population Parameters in S4N4... 

D from the multipole population parameters, and a solution value of 11.6 D. With a nonuniform prior, a more acceptable, but still low, MEM value of 7.8 D is obtained. While this physical criterion shows the nonuniform prior to be preferable, the validity of the MEM enhancement in charge density studies remains to be assessed. 

The Relation Between the Occupancies of Transition-Metal Valence Orbitals and the Multipole Population Parameters... 

The d-orbital occupancies are derived from the experimental multipole populations by the inverse expression (Holladay et al. 1983)... 

The Matrix M 1 Relating c/-Orbital Occupancies / to Multipole Populations Ptmp (Eq. 10.9)... 

Holladay A, Leung PC, Coppens P (1983) Generalized relations between d-orbital occupancies of transition-metal atoms and electron-density multipole population parameters from X-ray diffraction data. Acta Crystallogr A 39 377-387... 

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]. 

As we are dealing with spherical harmonics, and as we are trying to model the aspherical atomic electron density, the orientation of the local atom centered coordinate system is, in principle, arbitrary, appropriate linear combinations always giving the same result. However, in practice it is helpful to choose a local coordinate system such that the multipoles are oriented in rational directions, and thus the most important multipole populations will lie in directions that would be expected to represent chemical bonds or lone pairs [2,20], e.g. for an sp2 hybridized atom, defining one bond as the x direction, the trigonal plane as the xy plane, and z perpendicular defines three lobes of the 33+... 

The redistribution of the valence electron density due to chemical bonding may be obtained from summing the multipole populations or Fourier transforming appropriately calculated structure factors, having removed the contribution from neutral spherical atoms, to produce a so-called deformation density map [2], This function was introduced by Roux et al. [23] and has been widely used since then. The deformation electron density represents the difference between the electron density of the system, p(r), and the electron... 

A. d Orbitals Occupancies from Multipole Population Parameters.282... 

In other words the multipole populations and the orbital populations are related through the matrix equation ... 

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 temperature dependent X—X+N study (100, 135, 170, and 205 K) on naphthalene [66] addresses the problem of thermal de-convolution, that is, the efficiency of the pseudoatom model to decouple density deformations due to chemical bonding from those due to nuclear motion. The authors analyze the self-consistency of multipole populations, extracted from different temperature XRD data, in terms of statistical distances d ) in the parameter space of the same refinement model ... 

A recent detailed X-ray diffraction study of the experimental electron density distribution in Fe(TTP)OCH3 has allowed the evaluation of the electron occupancies of the five 3d orbitals of the iron(III) ion (Lecomte, Chadwick, Coppens Stevens, 1983). A multipole population analysis gave the results presented in Table 3.11 and indicated a net charge of +2.5(2) on the iron ion, in good agreement with the expected oxidation... 

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