Multipole parameters

Analysis of the multipole parameters show that the probability of their statistical significance is at least 70%. The obtained value k = 0.922 indicates about 8% decompression of the spherical part of the valence shell in a crystal. This value is in accordance with earlier obtained results (accuracy about 4.5 - 10 %). 

The number of multipole parameters is reduced by the requirements of symmetry. As discussed in chapter 3, the only allowed multipolar functions are those having the symmetry of the site, which are invariant under the local symmetry operations. For example, only / = even multipoles can have nonzero populations on a centrosymmetric site, while for sites with axial symmetry the dipoles must be oriented along the symmetry axis. For a highly symmetric site having 6 mm symmetry, the lowest allowed / 0 is d66+ all lower multipoles being forbidden by the symmetry. The index-picking rules listed in appendix D give the information required for selection of the allowed parameters. 

The accuracy of the electrostatic moments based on the multipole parameters is a function of the errors in both the population coefficients Tvai and the atomic parameters Pimp- Let M represent the m x m variance-covariance matrix for these parameters, as in chapter 4. Let D be the derivative matrix with elements... 

While the calculation of the electrostatic functions from the multipole parameters parallels that of the calculation of the atomic electrostatic moments, there is an... 

The EFG data from the multipole parameters are, in principle, for the static crystal while the spectroscopic data are affected by vibrations. There may therefore by a systematic difference between the two sets of values, which is evident for a number of hydrogen-bonded hydroxyl groups and water molecules studied by Tegenfeldt and Hermansson (1985), but is not apparent in the data in Table 8.3. The EFG values for H atoms in hydrogen-bonds is further discussed in chapter 12. 

Two advanced techniques have been proposed and applied to some crystal structures (Section IV,C), in which aspherical distributions of valence electrons around an atom are directly taken into account in the least-squares calculations. Aspherical atomic form factors are introduced in the least-squares refinement in the first method (29, 38, 80) and multipole parameters describing the aspherical valence distributions are used in the second method (31, 34, 46). 

To take in account the nonspherical shape of the valence electron distribution, the K model has been improved by the addition of multipole parameters [lib]. Then, the pseudoatomic density is written (Molly program),... 

Because electron density is a local property, electron density studies of the peptide-like molecules show that the nonspherical part of the deformation density (i.e the P]m parameters of Eq. 8) remain essentially the same for a given atom in the same environment (the peptide residue, a phenyl ring, a methyl group...) [29], The same observation was made for porphyrin ligands [30] and by Brock, Dunitz, and Hirshfeld [37] for naphthalene and anthracene type molecules. All these observations suggest that the multipole parameters are highly transferrable from one atom to a chemically similar atom in different molecules and crystals. A key question is is it possible to determine for each chemical type of a given atom a small set of pseudoatom multipole parameters, and can such parameters be used to calculate electrostatic properties of new molecules To answer this question [29], two accurate but low resolution X-ray data sets (sin 0/Xmax = 0.65 A-1) were... 

It confirms the possibility of transferring multipole parameters. Having new xyz Uij parameters, new structure factors phases ([Pg.280]

As described by Stevens [35] and Holladay, Leung, and Coppens [36], d orbital occupancies of the metal atom can be derived from the multipole parameters assuming that the overlap density and the asphericity of any 4p orbital density are small. For first row transition metals, the d orbital expansion and the overlap density between d orbitals and the ligands are small then, the asphericity of the electron density of the metal atom is mainly due to the d orbital occupancies. From Eq. 8, we can write ... 

The above description of the excited states in terms of excitation amplitudes is frame and basis set dependent. A more convenient description is in terms of state multipoles. It can be generalised to excited states of different orbital angular momentum and provides more physical insight into the dynamics of the excitation process and the subsequent nature of the excited ensemble. The angular distribution and polarisation of the emitted photons are closely related to the multipole parameters (Blum, 1981). The representation in terms of state multipoles exploits the inherent symmetry of the excited state, leads to simple transformations under coordinate rotations, and allows for easy separation of the dynamical and geometric factors associated with the radiation decay. 

In practice, site multipole series for the electric potential are truncated, usually at quadrupoles or less. If higher terms in the series are not actually calculated, the degree of convergence is uncertain. The PD method, discussed in the next section, avoids this problem by optimizing for the best possible model of the electric potential with a given number of multipole terms. Thus in the PD method convergence is always the best possible for a given set of multipole parameters and potential points. 

Lamzin et mention that the electron density multipole parameters... 

Independent treatment of the multipole parameters of the two independent molecules in the crystal structure of II did not result in any significant difference in the population of corresponding multipole functions. Further experimental details are summarized in Table 1. 

Index I denotes the rank of a multipole. Parameters of the electrostatic field of point charges (/=0) are presented as... 

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... 

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