Density- and wavefunction-normalized Cartesian spherical harmonics forl≤ 20

The widely used pseudoatom formalism [Stewart (1976).Acta Cryst. A32, 565–574; Hansen & Coppens (1978).Acta Cryst.A34, 909–921] in experimental X-ray charge-density studies makes use of real spherical harmonics when describing the angular component of aspherical deformations of the atomic electron density in molecules and crystals. The analytical form of the density-normalized Cartesian spherical harmonic functions for up tol≤ 7 and the corresponding normalization coefficients were reported previously by Paturle & Coppens [Acta Cryst.(1988), A44, 6–7]. It was shown that the analytical form for normalization coefficients is available primarily forl≤ 4 [Hansen & Coppens, 1978; Paturle & Coppens, 1988; Coppens (1992).International Tables for Crystallography, Vol. B,Reciprocal space, 1st ed., edited by U. Shmueli, ch. 1.2. Dordrecht: Kluwer Academic Publishers; Coppens (1997).X-ray Charge Densities and Chemical Bonding. New York: Oxford University Press]. Only in very special cases it is possible to derive an analytical representation of the normalization coefficients for 4 <l≤ 7 (Paturle & Coppens, 1988). In most cases forl> 4 the density normalization coefficients were calculated numerically to within seven significant figures. In this study we review the literature on the density-normalized spherical harmonics, clarify the existing notations, use the Paturle–Coppens (Paturle & Coppens, 1988) method in the WolframMathematicasoftware to derive the Cartesian spherical harmonics forl≤ 20 and determine the density normalization coefficients to 35 significant figures, and computer-generate a Fortran90 code. The article primarily targets researchers who work in the field of experimental X-ray electron density, but may be of some use to all who are interested in Cartesian spherical harmonics.

Chen's Lattice Inversion Embedded-Atom Method for Nial and Ni3Al Alloy

We explored a new type alloy EAM potential (CLI-EAM) that the value of atomic electron density and pair potential between distinct atoms are obtained by Chen’s lattice inversion based on first-principles calculations. The alloy CLI-EAM potential acquired from NiAl alloy can also apply in Ni3Al successfully and the results of basic properties agreed with the experiments. The results of formation energy of point defects of NiAl and Ni3Al alloy indicate that the structural defects are anti-site defects of Al when enrichments of Al atoms.

Chen's Lattice Inversion Embedded-Atom Method for FCC Metal

We explored a new type EAM potential (CLI-EAM) that the value of atomic electron density and pair potential functions are obtained by Chen’s lattice inversion based on first-principles calculations. This EAM potential is applied to Cu, Ag, Cu and Pt metals successfully and the results of basic properties agreed with the experiments. For the same metal, the cohesive energy of fcc structures are the lower than bcc structures.

Optimal local axes and symmetry assignment for charge-density refinement

In the perspective of building an expanded electron-density library based on multipolar modelling for common chemical atom types, new consistent local axes systems are proposed. Optimal symmetry constraints can consequently be applied to atoms and a large number of multipole populations can be fixed to a zero value. The introduction of symmetry constraints in the multipolar refinement of small compounds allows a reduction in the number of multipolar parameters stored in the library and required for the description of the atomic electron density. In a refinement where the symmetry constraints are not applied, the use of optimal axes enables the deviations from the local pseudo-symmetry to be highlighted. The application of symmetry constraints or restraints on multipoles is more effective when the axes are in accordance with the local geometry of the atom and can lead to improved crystallographicRfreeresiduals. The new local axes systems, based on two or three atom neighbours, can also be usefully applied to the description of transition metal complexes.

Refinement of proteins at subatomic resolution withMOPRO

Crystallography at subatomic resolution permits the observation and measurement of the non-spherical character of the atomic electron density. Charge density studies are being performed on molecules of increasing size. TheMOPROleast-squares refinement software has thus been developed, by extensive modifications of the programMOLLY, for protein and supramolecular chemistry applications. The computation times are long because of the large number of reflections and the complexity of the multipolar model of the atomic electron density; the structure factor and derivative calculations have thus been parallelized. Stereochemical and dynamical restraints as well as the conjugate gradient algorithm have been implemented. A large number of the normal matrix off-diagonal terms turn out to be very small and the block diagonal approximation is thus particularly efficient in the case of large structures at very high resolution.

Chemical Bonding and the X-ray Scattering Formalism

The assumption that the atomic electron density is well described by the spherically averaged density of the isolated atom has been the basis of X-ray structure analysis since its inception. The independent-atom model (IAM) is indeed a very good approximation for the heavier atoms, for which the valence shell is a minor part of the total density, but is much less successful for the lighter atoms. The lightest atom, hydrogen, has no inner shells of electrons, so that the effect of bonding is relatively pronounced. Because of the overlap density in covalent X—H bonds (X = C, N, O), the mean of the hydrogen electron distribution is significantly displaced inwards into the bond. When a spherical IAM hydrogen scattering factor is used in a least-squares adjustment of the atomic “position,” the result will be biased because the centroid of the density associated with the H atom is shifted in the direction of the bond. The result is an apparent shortening of X—H bonds which is far beyond the precision of X-ray structure determination (Hanson et al. 1973). For sucrose, for example, the differences between X-ray and neutron bond lengths are 0.13 (1) Å averaged over 14 C—H bonds, and 0.18 (3) A averaged over eight O—H bonds (Hanson et al. 1973). The observed discrepancy between X-ray results and spectroscopic values was first explained in terms of the electron distribution in the 1950s by Cochran (1956) and Tomii (1958). That the bond density is also of significance for heavier atoms is evident from the occurrence of the spherical-atom forbidden (222) reflection of diamond and silicon, even at low temperatures where anharmonic thermal effects (see chapter 2) are negligible. The historical importance of the nonzero intensity of the diamond (222) reflection is illustrated by the following comment made by W. H. Bragg, in 1921: . . . Another point of interest is the existence of a small (222) reflection (in diamond). This has been looked for previously but without success. . . .