Azulene revisited: solid-state structure, invariom modeling and lattice-energy minimization of a classical example of disorder.

The molecular and solid-state structure of azulene both raise fundamental questions. Therefore, the disordered crystal structure of azulene was re-refined with invariom non-spherical atomic scattering factors from new single-crystal X-ray diffraction data with a resolution of d = 0.45 Å. An unconstrained refinement results in a molecular geometry with Cs symmetry. Refinements constrained to fulfill C2v symmetry, as observed in the gas phase and in high-level ab initio calculations, lead to similar figures of merit and residual densities as unconstrained ones. Such models are consistent with the structures from microwave spectroscopy and electron diffraction, albeit they are not the same. It is shown that for the disorder present in azulene, the invariom model describes valence electron density as successfully as it does for non-disordered structures, although the disorder still leads to high correlations mainly between positional parameters. Lattice-energy minimizations on a variety of ordered model structures using dispersion-corrected DFT calculations reveal that the local deviations from the average structure are small. Despite the molecular dipole moment there is no significant molecular ordering in any spatial direction. A superposition of all ordered model structures leads to a calculated average structure, which explains not only the experimental determined atomic coordinates, but also the apparently unusual experimental anisotropic displacement parameters.

Properties of atoms under pressure: bonded interactions of the atoms in three perovskites.

The crystal structures for the three perovskites, CaSnO(3), YAlO(3), and LaAlO(3), were geometry optimized at the density functional theory level for a wide range of simulated isotropic pressures up to 80 GPa. The connections between the geometry optimized bond lengths, R(M-O), the values of the electron density, ρ(r(c)), the local kinetic, G(r(c)), potential, V(r(c)), energy densities, H(r(c)), and the Laplacian, ∇(2)(r(c)), at the bond critical points, r(c), for the M-O nonequivalent bonded interactions were examined. With increasing pressure, ρ(r(c)) increases along four distinct trends when plotted in terms of the Al-O, Ca-O, Sn-O, Y-O, and La-O bond lengths, but when the bond lengths were plotted in terms of ρ(r(c))/r where r is the periodic table row number of the M atoms, the data scatter along a single trend modeled by the power law regression expression R(M-O) = 1.41(ρ(r(c))/r)(-0.21), an expression that is comparable with that obtained for the bonded interactions for a large number of silicate and oxides crystals, R(M-O) = 1.46(ρ(r(c))/r)(-0.19) and that obtained for a relatively large number of hydroxyacid molecules R(M-O) = 1.39(s/r)(-0.22) where s is the Pauling bond strength of a bonded interaction. The similarity of the expressions determined for the perovskites, silicate and oxides crystals, and hydroxyacid molecules suggest that the bonded interactions in molecules and crystal are not only similar and comparable. The close correspondence of the expressions for the perovskites, the silicate and oxide crystals, and the molecules indicates that Pauling bond strength and ρ(r(c)) are comparable measures of the bonded interactions, the larger the accumulation of the electron density between the bonded atoms the larger the value of s, the shorter the bond lengths. It also indicates that the bonded interactions that govern the bond length variations behave as if largely short ranged. Like ρ(r(c))/r, the values of G(r(c))/r, V(r(c))/r, ∇(2)(r(c))/r likewise correlate in terms of R(M-O) in a single trend. With increasing pressure, the value of V(r(c)) decreases at a faster rate than G(r(c)) increases consistent with the observation that ρ(r(c)) increases with increasing pressure thereby stabilizing the structures at high pressures. As evinced by the well-developed power law trends between R(M-O) and the bond critical point properties, the bulk of the bonded interactions for the perovskites are concluded to change progressively from closed-shell to intermediate polar covalent interactions with increasing pressure. A well-developed trend between the ratios ∣V(r(c))∣ /G(r(c)) and H(r(c))/ρ(r(c)) is consistent with this conclusion. The employment of a positive value for the Laplacian alone in distinguishing between closed shell and polar covalent bonded interactions is unsatisfactory when 2G(r(c)) > ∣V(r(c))∣ > G(r(c)).

Can the interaction density be measured? The example of the non-standard amino acid sarcosine.

The experimental charge density rho(r) of the non-standard amino acid sarcosine has been determined based on an extensive and complete data set measured at 100 K to high resolution (sin theta/lambda = 1.18 A(-1)) by single-crystal X-ray diffraction. Anisotropic thermal motion of the H atoms, obtained from TLS + ONIOM cluster methods, was included in the structural model. Based on the multipole-model geometry, the theoretical Hartree-Fock interaction density of a molecule in the crystal has been calculated with CRYSTAL98. It manifests itself in local rearrangements of rho(r) and can be reproduced with a multipole projection via simulated structure factors. An attempt has also been made to obtain the interaction density from a combination of experimental and theoretical charge densities using either a whole-molecular calculation or the invariom database. Agreement with the periodic Hartree-Fock interaction density is qualitative. It is shown that invarioms reproduce the features of the theoretical multipole-projected whole-molecular electron density, and can be used to approximate it.

Redetermination, invariom-model and multipole refinement of L-ornithine hydrochloride.

The structure of L-ornithine hydrochloride, C(5)H(13)N(2)O2+Cl(-), has been redetermined at 100 K by single-crystal X-ray diffraction within a project that aims to generate accurate bond-distance restraints for the invariom refinement of proteins. The high-resolution data were subject to an invariom and a multipole refinement, and the resulting electron densities on a grid were compared. Improvements in the conventional R factor obtained by multipole modelling were smaller than in other structures containing solely the elements CHNO owing to Cl core scattering. Cruickshank's diffraction-component precision index and Stevens & Coppens suitability factor are discussed.

Theoretical electron density distributions for Fe- and Cu-sulfide earth materials: a connection between bond length, bond critical point properties, local energy densities, and bonded interactions.

Bond critical point and local energy density properties together with net atomic charges were calculated for theoretical electron density distributions, rho(r), generated for a variety of Fe and Cu metal-sulfide materials with high- and low-spin Fe atoms in octahedral coordination and high-spin Fe atoms in tetrahedral coordination. The electron density, rho(rc), the Laplacian, triangle down2rho(rc), the local kinetic energy, G(rc), and the oxidation state of Fe increase as the local potential energy density, V(rc), the Fe-S bond lengths, and the coordination numbers of the Fe atoms decrease. The properties of the bonded interactions for the octahedrally coordinated low-spin Fe atoms for pyrite and marcasite are distinct from those for high-spin Fe atoms for troilite, smythite, and greigite. The Fe-S bond lengths are shorter and the values of rho(rc) and triangle down2rho(rc) are larger for pyrite and marcasite, indicating that the accumulation and local concentration of rho(r) in the internuclear region are greater than those involving the longer, high-spin Fe-S bonded interactions. The net atomic charges and the bonded radii calculated for the Fe and S atoms in pyrite and marcasite are also smaller than those for sulfides with high-spin octahedrally coordinated Fe atoms. Collectively, the Fe-S interactions are indicated to be intermediate in character with the low-spin Fe-S interactions having greater shared character than the high-spin interactions. The bond lengths observed for chalcopyrite together with the calculated bond critical point properties are consistent with the formula Cu+Fe3+S2. The bond length is shorter and the rho(rc) value is larger for the FeS4 tetrahedron displayed by metastable greigite than those displayed by chalcopyrite and cubanite, consistent with a proposal that the Fe atom in greigite is tetravalent. S-S bond paths exist between each of the surface S atoms of adjacent slabs of FeS6 octahedra comprising the layer sulfide smythite, suggesting that the neutral Fe3S4 slabs are linked together and stabilized by the pathways of electron density comprising S-S bonded interactions. Such interactions not only exist between the S atoms for adjacent S8 rings in native sulfur, but their bond critical point properties are similar to those displayed by the metal sulfides.

Bond length and local energy density property connections for non-transition-metal oxide-bonded interactions.

For a variety of molecules and earth materials, the theoretical local kinetic energy density, G(r(c)), increases and the local potential energy density, V(r(c)), decreases as the M-O bond lengths (M = first- and second-row metal atoms bonded to O) decrease and the electron density, rho(r(c)), accumulates at the bond critical points, r(c). Despite the claim that the local kinetic energy density per electronic charge, G(r(c))/rho(r(c)), classifies bonded interactions as shared interactions when less than unity and closed-shell when greater, the ratio was found to increase from 0.5 to 2.5 au as the local electronic energy density, H(r(c)) = G(r(c)) + V(r(c)), decreases and becomes progressively more negative. The ratio appears to be a measure of the character of a given M-O bonded interaction, the greater the ratio, the larger the value of rho(r(c)), the smaller the coordination number of the M atom and the more shared the bonded interaction. H(r(c))/rho(r(c)) versus G(r(c))/rho(r(c)) scatter diagrams categorize the M-O bonded interactions into domains with the local electronic energy density per electron charge, H(r(c))/rho(r(c)), tending to decrease as the electronegativity differences for the bonded pairs of atoms decrease. The values of G(r(c)) and V(r(c)), estimated with a gradient-corrected electron gas theory expression and the local virial theorem, are in good agreement with theoretical values, particularly for the bonded interactions involving second-row M atoms. The agreement is poorer for shared C-O and N-O bonded interactions.

Si-O bonded interactions in silicate crystals and molecules: a comparison.

Bond critical point, local kinetic energy density, G(rc), and local potential energy density, V(rc), properties of the electron density distributions, rho(r), calculated for silicates such as quartz and gas-phase molecules such as disiloxane are similar, indicating that the forces that govern the Si-O bonded interactions in silica are short-ranged and molecular-like. Using the G(rc)/rho(rc) ratio as a measure of bond character, the ratio increases as the Si-O bond length, the local electronic energy density, H(rc)= G(rc) + V(rc), and the coordination number of the Si atom decrease and as the accumulation of the electron density at the bond critical point, rho(rc), and the Laplacian, inverted Delta2 rho(rc), increase. The G(rc)/rho(rc) and H(rc)/rho(rc) ratios categorize the bonded interaction as observed for other second row atom M-O bonds into discrete categories with the covalent character of each of the M-O bonds increasing with the H(rc)/rho(rc) ratio. The character of the bond is examined in terms of the large net atomic charges conferred on the Si atoms comprising disiloxane, stishovite, quartz, and forsterite and the domains of localized electron density along the Si-O bond vectors and on the reflex side of the Si-O-Si angle together with the close similarity of the Si-O bonded interactions observed for a variety of hydroxyacid silicate molecules and a large number of silicate crystals. The bond critical point and local energy density properties of the electron density distribution indicate that the bond is an intermediate interaction between Al-O and P-O bonded interactions rather than being a closed-shell or a shared interaction.

Introduction and validation of an invariom database for amino-acid, peptide and protein molecules.

A database of invarioms for structural refinement of amino-acid, oligopeptide and protein molecules is presented. The spherical scattering factors of the independent atom or promolecule model are replaced by ;individual' aspherical scattering factors that take into account the chemical environment of a bonded atom. All amino acids were analysed in terms of their invariom fragments. In order to generate 73 database entries that cover this class of compounds, 37 model compounds were geometry-optimized and theoretical structure factors were calculated. Multipole refinements were then performed on these theoretical structure factors to yield the invariom database. Validation of this database on an extensive number of experimental small-molecule crystal structures of varying quality and resolution shows that invariom modelling improves various figures of merit. Differences in figures of merit between invariom and promolecule models give insight into the importance of disorder for future protein-invariom refinements. The suitability of structural data for application of invarioms can be predicted by Cruickshank's diffraction-component precision index [Cruickshank (1999), Acta Cryst. D55, 583-601].

Invarioms for improved absolute structure determination of light-atom crystal structures.

The determination of molecular absolute configuration from an X-ray analysis for structures that contain only light elements is challenging owing to the weak anomalous dispersion signal. The achievable precision of the Flack x parameter for such structures is therefore limited, especially when the independent-atom model is employed. Invariom modelling can improve this situation. Invarioms are theoretically predicted pseudoatoms within the Hansen & Coppens multipole formalism. They are transferable from one molecule to another and provide generalized aspherical atomic form factors. It is shown that, by application of the invariom approach, the precision and standard uncertainty of the Flack x parameter and therefore the reliability of deducing molecular chirality in an absolute structure determination can be improved.

Phase measurement for accurate mapping of chemical bonds in acentric space groups.

Although the electron density is fundamental to the study of chemical bonding and density-functional theory, it cannot be accurately mapped experimentally for the important class of crystals lacking inversion symmetry, since structure factor phase information is normally inaccessible. We report the combination of x-ray and electron diffraction experiments for the determination of the electron density in acentric AlN, using multiple-scattering effects in convergent-beam electron diffraction to obtain sensitivity to structure factor phases, and describe a new error metric and weighting scheme for multipole refinement using combined measurements of structure factor magnitudes and phases.