Abstract. The various ways of presenting the topographical features of electron densities and difference densities are discussed and illustrated with graphical examples. A systematic terminology is introduced. Id2·"dNp(I,...,N;{R,,}), (2) where we assume a system of N electrons with the nuclear configuration specified by {R,.} and a numeral J refers to the combined space and spin coordinates (r;. s;) of electron j.
Highly accurate absolute measurements of the X-ray structure factors of silicon [Aldred & Hart, Proc. R. Soc. London, Ser. A, (1973), 332, 223-238] have been used to analyse a number of models for the electron distribution. Initially, the valence-electron distribution (with the neon core assumed to be unmodified from that of the isolated silicon atoms) was built up with a radial basis of the form r ~ exp (-Q') and non-sphericity was allowed for by inclusion of octupole and hexadecapole terms. Improved representation was achieved with related models in which deformations from the total isolated-atom electron density were refined instead. The exact shape of the deformation electron density in the region of the bond was sensitively dependent on the monopole deformation term. The anomalous-dispersion contributions (Af') to the scattering factors were refined and found to be in agreement with recent interferometric measurements, but not with recent calculations. The octupole density term is slightly sharper at 293-2 than at 92.2 K, and the structure factor for the 222 reflection is predicted to be larger at the higher temperature. These effects may be due to a failure of the convolution approximation or to uncertainties in the anharmonic corrections to the structure-factor data.
Some techniques for charge-density analysis proposed by R. F. Stewart have been compared by application to s-triazine, with X-ray data and neutron structural parameters. There is strong correlation between population coefficients for density basis functions which overlap heavily with each other, and in extreme cases individual populations are indeterminate. Multipole expansions centred on the nuclei are better defined than sets of orbital products which include two-centre terms. Trigonal octupole functions centred on the carbon and nitrogen nuclei give a marked improvement in the description of the scattering experiment provided the exponents which determine the radial variation of the density functions are optimized. The optimum exponent varies with the length of the multipole expansion. New algorithms developed for these studies give improved computational efficiency.
X-ray powder diffraction data for diamond have been re-analysed, in terms of an 'at rest' deformation-density model of monopole, octupole, hexadecapole and hexacontatettarapole terms of exponential form. The values of the parameters describing the third and fourth-order terms, and the refinement indices, are no different from those of an 'at rest' valence-density model. Unlike earlier analyses, however, the model is consistent with experimental measurements of the absolute scale, and with independently determined values of the Debye-Waller parameter. The predicted values of physical properties such as the gradient of the electric-field gradient at the carbon nucleus are shown to be strongly model-dependent. It is noted that the fit to the data, and in particular to the observed structure factor of the overlapping 333-511 reflection, is poor for all published charge-density analyses. Recent theoretical (crystal 'Hartree-Fock') calculations indicate, within the convolution and harmonic approximations, an experimental B value of 0-172 + 0-006 A2 and a scale factor of 1.004 _+ 0.002. These theoretical results are, however, in poor agreement with the experimental structure factors.
Neutron powder patterns from two samples have been analysed to provide information on the thermal motion in diamond. The resulting Debye-Waller factor of 0.14-0.17 /~2 lends support to the latticedynamic value of 0.149-0.150 ,~2. The small effect of extinction (less than 4%) is well described by the Becker and Coppens formula but poorly described by that of Zachariasen. The accuracy of the data necessitated a correction for thermal diffuse scattering.
The electron density distribution in melamine has been studied with X-ray diffraction data and neutron structural parameters. The at-rest valence density is represented as a set of nuclear-centred multipole density functions with Slater-type radial functions. Two series of analyses were compared, the first with the radial exponents fixed at the standard molecular values and the second with these exponents as variable parameters. Exponent refinement allows a marked improvement in the fit of the model to the data. The population coefficients of the multipole terms are better defined when the exponents are optimized. On chemically similar atoms the populations of the monopole terms are inversely related to the exponents. The carbon atom parameters agree to high precision. Exponents for the ring and amine nitrogens differ and small differences within each set are related to the hydrogen-bond and packing environment. The exponents and electron density near the nucleus are relatively low for hydrogens involved in hydrogen bonding. The most significant deformation functions in the multipole expansion have a symmetry compatible with nearest-neighbour geometry. Differences between populations are related to distortions from the idealized geometry or to hydrogen-bonding interactions. The inclusion of a hydrogen dipole deformation term with a large exponent results in internally consistent populations which are correlated with N-H stretching amplitudes in the structure. This suggests that the convolution approximation is invalid at this level of structure refinement. IntroductionThe free molecule of 2-4-6-triamino-s-triazine (melamine) has (5m2 symmetry. Each of the carbon and nitrogens has mm2 symmetry and the hydrogen atoms have symmetry m. In the crystal structure, which was first studied with X-ray data by Hughes (1941), there is one molecule in the asymmetric unit. The structure has since been more accurately determined with both X-ray and neutron data, by Varghese, O'Connell & Maslen (1977). A diagram of the structure is given in Fig. 1.The unit cell is roughly equi-dimensional. The lack of a short cell dimension is a considerable advantage for charge-density analysis. In each principal direction there are points close to the origin of reciprocal space, where the contribution of the valence-electron density to the scattering is maximal. The parameters which determine the charge density can be determined to far higher accuracy than those for a smaller structure with data of comparable quality.The symmetry of the free molecule is expected to persist approximately in the crystal. At this level of approximation the carbon and nitrogen atoms in the structure are equivalent in threes and the hydrogen atoms in sixes. This provides an internal check on the validity of the results. ... ~"I-(4)+'+ C(1) .N(8). ~: N(6) " -H(4) " . ,< H(4) Fig. 1. Diagram of the melamine structure based on the neutron parameters of Varghese, O'Connell & Maslen (1977). ELECTRON-DENSITY STUDIES. VThe atoms which are equivalent to first order are d...
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