Crystal Electron Density Research Articles

New ion radii for oxides and oxysalts, fluorides, chlorides and nitrides.

Ion radii are derived here from the characteristic (grand mean) bond lengths for (i) 135 ions bonded to oxygen in 459 configurations (on the basis of coordination number) using 177 143 bond lengths extracted from 30 805 ordered coordination polyhedra from 9210 crystal structures; and (ii) 76 ions bonded to nitrogen in 137 configurations using 4048 bond lengths extracted from 875 ordered coordination polyhedra from 434 crystal structures. There are two broad categories of use for ion radii: (1) those methods which use the relative sizes of cation and anion radii to predict local atomic arrangements; (2) those methods which compare the radii of different cations (or the radii of different anions) to predict local atomic arrangements. There is much uncertainty with regard to the relative sizes of cations and anions, giving rise to the common failure of type (1) methods, e.g. Pauling's first rule which purports to relate the coordination adopted by cations to the radius ratio of the constituent cation and anion. Conversely, type (2) methods, which involve comparing the sizes of different cations with each other (or different anions with each other), can give very accurate predictions of site occupancies, physical properties etc. Methods belonging to type (2) can equally well use the characteristic bond lengths themselves (from which the radii are derived) in place of radii to develop correlations and predict crystal properties. Extensive quantum-mechanical calculations of electron density in crystals in the literature indicate that the radii of both cations and anions are quite variable with local arrangement, suggesting significant problems with any use of ion radii. However, the dichotomy between the experimentally derived ion radii and the quantum-mechanical calculations of electron density in crystals is removed by the recognition that ion radii are proxy variables for characteristic bond lengths in type (2) relations.

X-ray constrained wavefunctions based on Hirshfeld atoms. II. Reproducibility of electron densities in crystals of α-oxalic acid dihydrate.

The Hirshfeld atom-based X-ray constrained wavefunction fitting (HA-XCW) procedure is tested for its reproducibility, and the information content of the fitted wavefunction is critically assessed. Fourteen different α-oxalic acid dihydrate data sets are used for this purpose, and the first joint fitting to 12 of these data sets is reported. There are systematic features in the electron density obtained from all data sets which agree with higher level benchmark calculations, but there are also many other strong systematic features which disagree with the reference calculations, most notably those associated with the electron density near the nuclei. To enhance reproducibility, three new protocols are described and tested to address the halting problem of XCW fitting, namely: an empirical power-function method, which is useful for estimating the accuracy of the structure factor uncertainties; an asymptotic extrapolation method based on ideas from density functional theory; and a `conservative method' whereby the smallest value of the regularization parameter is chosen from a series of data sets, or subsets.

Crystal structures.

A personal view is offered on various solved and open problems related to crystal structures: the present state of reconstructing the crystal electron density from X-ray diffraction data; characterization of atomic and molecular motion from a combination of atomic displacement parameters and quantum chemical calculations; Bragg diffraction and diffuse scattering: twins, but different; models of real (as opposed to ideal) crystal structures from diffuse scattering; exploiting unexplored neighbourhoods of crystallography to mathematics, physics and chemistry.

Analyzing Hydration Differences in Cocrystal Polymorphs: High-Resolution X-ray Investigation of Caffeine–Glutaric Acid Cocrystals

Two polymorphic forms of caffeine (CAF)–glutaric acid (GLU) cocrystals have been studied via high-resolution X-ray crystallography and Bader’s quantum theory of atoms in molecules (QTAIM). For both the monoclinic, 1, and triclinic, 2, systems the experimental charge density distributions of the 1:1 ratio of CAF and GLU polymorphs have been determined and compared. Previous studies have determined that 1 is less stable than 2, in relative humidity (RH) testing. A topological analysis of the electron density distribution (EDD) revealed little difference between the two polymorph internal systems. The packing densities (0.76 vs 0.74) and lattice energies (−101.1 vs −107.1 kJ mol–1) of 1 and 2, respectively, are nearly equivalent, implying that the differences in hygroscopicity between the two polymorphs are not due to crystal lattice porosity or stability. A topological analysis of the number and strength of hydrogen bonds for 1 and 2 revealed nine hydrogen bonds in both polymorphs. “Classical” (O–H···X) hydrogen bonds were similarly present in both polymorphs, stabilizing the cocrystals. However, the sum of the stability produced from the “nonclassical” (C–H···X) bonds is higher in 2: −27.6 vs −38.2 kJ mol–1 for 1 and 2, respectively. One of the nine hydrogen bonds in 1 and 2 varies from the others, caused by the torsional rotation of the aliphatic carbon chain in GLU. This bond is critical for packing stabilization, creating a parallelogram-like packing arrangement in 2 in comparison to ribboning in 1. A Hirshfeld surface analysis found that the percentages of O–H···X hydrogen bonds were nearly identical in 1 and 2 (23.9% vs 22.1%); however, the H···H contacts were higher in 2 (61.4% vs 65.8% for 1 and 2, respectively), suggesting that more hydrogen-based contacts require competitive displacement by water in the hydration of 2 in comparison to 1. Additionally, a stabilizing aromatic cycle stack between CAF molecules is present in 2 due to the varied parallelogram packing arrangement, which was absent in 1; this provided ∼11.3 kJ mol–1 of stability to the system of 2. The solid-state entropies and molecular dipole moments (MDMs) of 1 and 2 supported the relative stability of the individual polymorphs, with 1 having a higher entropy and dipole moment in comparison to 2 (123.2 vs 112.8 J K–1 mol–1 and 7.45 and 4.93 D for 1 and 2, respectively), implying that it has the potential to hydrate more rapidly. These findings are in good agreement with previous experimental RH stability studies, giving further insight into the information gained from thermally averaged ground-state crystal electron density data.

Experimental Charge Density Analysis and Electrostatic Properties of Crystalline 1,3-Bis(Dimethylamino)Squaraine and Its Dihydrate from Low Temperature (T = 18 and 20 K) XRD Data

Multipolar refinements of structural models fitting extensive sets of X-ray diffraction (XRD) data from single crystals of 1,3-bis(dimethylamino)squaraine [SQ, C8H12N2O2] and its dihydrate [SQDH, C8H12N2O2·2H2O], collected at very low T (18 ± 1 K for SQ, 20 ± 1 K for SQDH), led to an accurate description of their crystal electron density distributions. Atomic volumes and charges have been estimated from the experimental charge densities using the Quantum Theory of Atoms in Molecules (QTAIM) formalism. Our analysis confirms the common representation (in the literature and textbooks) of the squaraine central, four-membered squarylium ring as carrying two positive charges, a representation that has been recently questioned by some theoretical calculations: the integrated total charge on the C4 fragment is estimated as ca. +2.4e in SQ and +2.2e in SQDH. The topology of the experimental electron density for the SQ squaraine molecule is modified in the dihydrated crystal by interactions between the methyl groups and the H2O molecules in the crystal. Maps of the molecular electrostatic potential in the main molecular planes in both crystals clearly reveal the quadrupolar charge distribution of the squaraine molecules. Molecular quadrupole tensors, as calculated with the PAMoC package using both Stewart and QTAIM distributed multipole analysis (DMA), are the same within experimental error.

Experimental distribution of electron density in crystals of Ph3Sb(O2CCH=CH–CH=CH–CH3)2 complex: the selection of a reference point for the source function in the absence of a bond critical point between atoms

In this article, we report on the results of experimental and theoretical (DFT calculation of an isolated molecule) investigation of electron density in a triphenylantimony disorbate complex (triphenylantimony bis(hexa-2,4-dienoate), Ph3Sb(O2CCH=CH–CH=CH–CH3)2 (1)). A topological analysis of the electron density was carried out in the framework of the quantum theory of atoms in molecules (QTAIM), which allows to study the nature of chemical bonds and the molecular graph in Ph3Sb(O2CCH=CH–CH=CH–CH3)2 complex. The molecular graph is an important tool for determining the interacting atoms. However, the molecular graph of the triphenylantimony disorbate complex did not show the presence of the “expected” intramolecular interactions between the antimony atom and the carbonyl oxygen one. Such a situation can be caused by electron density low curvature between these atoms. It is extremely difficult case, and sometimes it is not possible to find all the “expected” bond paths and critical points (3,−1). Thus, the molecular graph for this class of compounds does not provide a definitive picture of the chemical bonding and should be complemented with other descriptors, such as а source function (SF) and non-covalent interaction (NCI) index. It was found that in some cases using the SF on the NCI isosurface allows to interpret intramolecular interactions in the absence of a bond critical point more correctly. In this article, presence of intramolecular interaction in the absence of a bond critical point between the antimony atom and the carboxylate oxygen one was shown. The carboxylate fragment always acts as a source of the electron density for the Sb…O(carbonyl) interactions, whereas the antimony atom can be both a source and a sink for it.

Belok/XSA Diffraction Beamline for Studying Crystalline Samples at Kurchatov Synchrotron Radiation Source

AbstractIntroduction of “Belok/XSA” beamline‐modernization is presented here with commissioning of a modern synchrotron beamline with unique capabilities for the structural analysis of single and polycrystalline samples. The beamline is designed to perform important tasks in modern crystallography, such as solving and refining the atomic structure using the single‐ and polycrystal method, studying the real (defective) structure, and analyzing the distribution of electron density in crystals, it is also possible to study rapidly occurring processes, for example, temperature phase transitions. Due to the speed of data accumulation and a registration over a large solid angle, the method allows to study large numbers of samples. This beamline has become extremely demanded in the chemical, biological, material science, and archaeological community in Russia.

Intermolecular Interactions in Ionic Crystals of Nucleobase Chlorides—Combining Topological Analysis of Electron Densities with Energies of Electrostatic Interactions

Understanding intermolecular interactions in crystals of molecular ions continues to be difficult. On the one hand, the analysis of interactions from the point of view of formal charges of molecules, similarly as it is commonly done for inorganic ionic crystals, should be performed. On the other hand, when various functional groups are present in the crystal, it becomes natural to look at the interactions from the point of view of hydrogen bonding, π…π stacking and many other kinds of non-covalent atom–atom bonding. Often, these two approaches seem to lead to conflicting conclusions. On the basis of experimental charge densities of cytosinium chloride, adeninium chloride hemihydrate, and guanine dichloride crystals, with the help of theoretical simulations, we have deeply analysed intermolecular interactions among protonated nucleobases, chloride anions and water molecules. Here, in the second paper of the series of the two (Kumar et al., 2018, IUCrJ 5, 449–469), we focus on applying the above two approaches to the large set of dimers identified in analysed crystals. To understand electrostatic interactions, we analysed electrostatic interaction energies (Ees) computed directly from molecular charge densities and contrasted them with energies computed only from net molecular charges, or from a sum of electric multipolar moments, to find the charge penetration contribution to Ees. To characterize non-covalent interactions we performed topological analyses of crystal electron densities and estimated their interaction energies (EEML) from properties of intermolecular bond critical points. We show that the overall crystal architecture of the studied compounds is governed by the tight packing principle and strong electrostatic attractions and repulsions between ions. Many ions are oriented to each other in a way to strengthen attractive electrostatic interactions or weaken strong repulsion, but not all of them. Numerous bond critical points and bond paths were found between ions, including nucleobase cations despite their overall repulsive interactions. It is clear there is no correlation between EEML and Ees. However, strong relation between EEML and the charge penetration component of Ees is observed. The relation holds regardless of interaction types or whether or not interacting molecules bear the same or opposite charges. Thus, a charge density-based approach for computing intermolecular interaction energies and the atom–atom approach to analyse non-covalent interactions do complement each other, even in ionic systems.

Validation of X-ray Wavefunction Refinement.

In this work, the quality of the electron density in crystals reconstructed by the multipolar model (MM) and by X-ray wavefunction refinement (XWR) is tested on a set of high-resolution X-ray diffraction data sets of four amino acids and six tripeptides. It results in the first thorough validation of XWR. Agreement statistics, figures of merit, residual- and deformation-density maps, as well as atomic displacement parameters are used to measure the quality of the reconstruction relative to the measured structure factors. Topological analysis of the reconstructed density is carried out to obtain atomic and bond-topological properties, which are subsequently compared to the values derived from benchmarking periodic DFT geometry optimizations. XWR is simultaneously in better agreement than the MM with both benchmarking theory and the measured diffraction pattern. In particular, the obvious problems with the description of polar bonds in the MM are significantly reduced by using XWR. Similarly, modeling of electron density in the vicinity of hydrogen atoms with XWR is visibly improved.

Structural study of polymorphism in methylprednisolone aceponate

The crystal structures of methylprednisolone aceponate were determined by X-ray diffraction analysis at temperatures 90 K and 150 K: space group P212121, a = 14.8592(2), b = 19.6844(5), c = 26.1626(4) Å, Z = 12; R = 0.0598 (T = 90 K); space group P212121, a = 6.57348(14), b = 14.8295(3), c = 26.2214(5) Å, Z = 4; R = 0.0518 (T = 150 K). Features of structural changes in the phase transition were revealed. The abrupt change in the unit cell parameters in the phase transition was shown by low-temperature X-ray powder. The methods of degree of invariance of crystal electron density and molecular Voronoi−Dirichlet polyhedra were used for the analysis of polymorphism in methylprednisolone aceponate. The atomic structure at 90 K have a translational pseudosymmetry of electron density η = 0.329(1). The decrease of number of intermolecular contacts in the high-temperature modification due to rupture of intermolecular non-valence contacts C/O was observed.

A history of experimental phasing in macromolecular crystallography.

It was just over a century ago that W. L. Bragg published a paper describing the first crystal structures to be determined using X-ray diffraction data. These structures were obtained from considerations of X-ray diffraction (Bragg equation), crystallography (crystal lattices and symmetry) and the scattering power of different atoms. Although W. H. Bragg proposed soon afterwards, in 1915, that the periodic electron density in crystals could be analysed using Fourier transforms, it took some decades before experimental phasing methods were developed. Many scientists contributed to this development and this paper presents the author's own perspective on this history. There will be other perspectives, so what follows is a history, rather than the history, of experimental phasing.

Experimental and theoretical investigation of topological and energy characteristics of electron density in crystals of SbV o-amidophenolate complexes

Experimental and theoretical investigations of electron density in the complexes [4,6-ditert-butyl-N-(2,6-diisopropylphenyl)-1,2-amidophenolato]tricyclohexylantimony(V) (1) and [4,6-di-tert-butyl-N-(2,6-dimethylphenyl)-1,2-amidophenolato]tri-4-fluorophenylantimony(V) (2) were performed based on high-resolution X-ray diffraction data and density functional theory calculations (B3LYP/DGDZVP). The nature of chemical bonding, the energy of intraand intermolecular interactions in the crystals, and the atomic charge distribution were studied.

Extracting Extremely Localized Molecular Orbitals from X-ray Diffraction Data

The accurate determination of electron densities in crystals from high-resolution X-ray diffraction data has become more and more important over the years. The existing techniques to accomplish this task can be subdivided into two great families: the multipole models and the wave function-based strategies. The former, which are the most widely used, are essentially linear scaling and allow an easy chemical interpretation of the obtained molecular charge densities, but they are also characterized by some drawbacks, such as the possible presence of unphysical negative regions in the resulting electron distributions. On the contrary, the latter always provide quantum mechanically rigorous electron densities, but they are more computationally expensive and, above all, the ease of chemical interpretation is almost completely lost. In this context, in order to combine the easy chemical interpretability of the multipole models with the quantum mechanical rigor of the wave-function based methods, we have recently extended the X-ray constrained wave function approach proposed by Jayatilaka in the framework of a quantum chemistry technique for the a priori determination of Extremely Localized Molecular Orbitals (ELMOs), namely we have developed a new strategy that allows to extract from X-ray diffraction data a single Slater determinant built up wit Molecular Orbitals strictly localized on small molecular fragments (e.g., atoms, bonds or functional groups). Preliminary tests have shown that the determination of X-ray constrained ELMOs is really straightforward. Furthermore, given the reliable transferability of the obtained Molecular Orbitals, we are constructing new ELMOs databases that can be used as alternative to the existing pseudo-atoms libraries for refining crystallographic structures and electron distributions of large systems. A detailed comparison between the new technique and the multipole models is also currently under investigation.

Electron Density Sharpening as a General Technique in Crystallographic Studies

Sharpening is a powerful method to restore the details from blurred electron density in crystals with high overall temperature factors (B-factors). This valuable technique is currently not optimally used because of the uncertainty in the scope of its application and ambiguities in practice. We performed an analysis of ~2000 crystal data sets deposited in the Protein Data Bank and show that sharpening improves the electron density map in many cases across all resolution ranges, often with dramatic enhancement for mid- and low-resolution structures. It is effective when used with either experimental or model phases without introducing additional bias. Our tests also provide a practical guide for optimal sharpening. We further show that anisotropic diffraction correction improves electron density in many cases but should be used with caution. Our study demonstrates that a routine practice of electron density sharpening may have a broad impact on the outcomes of structural biology studies.

Thermo-elastic and bonding features of the CdMgS2 crystal: an ab initio study

An investigation of the chalcopyrite and rock-salt phase of the CdMgS2 compound was carried out in terms of FP-LAPW calculations. We found that the electron density of crystals contains important information required for studying the phase transition in this compound. By means of the atoms-in-molecules (AIM) and electron localization function (ELF) formalisms, we found that the chalcopyrite semiconductor phase of the investigated compound undergoes a phase transition to a metallic rock salt. The closure of the band gap is reflected in an increasing electronic delocalization across a change in the nature of the bonding. Thermal effects were included via a quasi-harmonic non-empirical Debye model in order to study their linear expansion parameters. Calculations of the elastic constants of this crystal are possible in both phases, but, obviously, this does not imply that the rock-salt phase is the thermodynamically stable one for this crystal.

Understanding Thermodynamic Properties at the Molecular Level: Multiple Temperature Charge Density Study of Ribitol and Xylitol

X-ray diffraction data of high quality measured to high resolution on crystals of the two pentitol epimers ribitol (centric) and xylitol (acentric) at 101, 141, and 181 K and data on the two compounds previously recorded at 122 K have formed the basis for multipole refinements with the VALRAY system. Our analysis showed that it is possible to obtain a reliable crystal electron density for an acentric compound (xylitol) from X-ray diffraction data and that the thermal motion can be deconvoluted from the static density in this temperature range. The Bader-type topological analysis of the static electron densities revealed virtually identical intramolecular interactions as well as very similar hydrogen bond interactions of ribitol and xylitol; the only minor differences are found in the weaker intermolecular interactions. The high-level periodic DFT calculations are in accordance with the thermodynamic measurements that show that the two pentitols have identical sublimation energies. A rigid body normal coordinate analysis was performed on the atomic displacement parameters obtained at the four different temperatures. The translational and librational mean square deviations derived through this analysis were used in a quantum statistical approach to derive frequencies of the corresponding harmonic oscillators. The analysis showed a consistent vibrational model for all temperatures. The frequencies were subsequently used to calculate crystal entropies assuming an Einstein-type behavior. These calculations show that the crystal entropy of ribitol is 8 J K(-1) mol(-1) higher than the crystal entropy of xylitol, confirming that it is a difference in the entropy of the two compounds that causes the difference in their free energy. Our results presented in this Article show the potential to use X-ray diffraction data to obtain physicochemical properties of crystals.

One-body potentials including electron interactions, and electron densities in crystals

The electron density ρ(r) in the ground state of a crystal can, in principle, be determined by measuring the x-ray intensities at the Bragg reflections. The work presented here has two aims: (1) To propose a method to determine a fundamental one-body potential V(r) solely from a knowledge of the measured charge density ρ(r). (2) To present new results for corrections to Dirac-Slater exchange theory, solely as functionals of the electron density ρ(r). The methods use the density matrix perturbation theory of March and Murray, recasting this into an approximate form suggested by gradient expansions in the density. Contact is made with the work of Herman and his coworkers, who propose a correction to the Dirac-Slater exchange energy of the form . Our work suggests that a term In ρ is also present, though its magnitude is probably small compared with the term proposed by Herman and coworkers [1].

First principles studies of thermal decomposition of anhydrous zinc oxalate

The results of first principles calculations of band structure, density of states and electron density topology of ZnC2O4 crystal are presented. The calculations have been performed with WIEN2k FP LAPW ab initio package. The obtained SCF electron density has been used in calculations of Bader’s QTAIM (quantum theory of atoms in molecules) topological properties of the electron density in crystal. Additional calculations of bond orders (Pauling, Bader, Cioslowski and Mixon) and bond valences according to bond valence model have been done. The obtained results are analyzed from the point of view of the thermal decomposition process, and this analysis indicates, that most probably this compound should decompose to metal oxide, carbon oxide and carbon dioxide, in agreement with the experiment.

Theoretical studies of thermal decomposition of anhydrous cadmium and silver oxalates

The results of first principles calculations of band structure, density of states and electron density topology of CdC2O4 and Ag2C2O4 crystals are presented. The calculations have been performed with WIEN2k ab initio program, using highly precise full potential linearized augmented plane wave (FP LAPW) method within Density Functional Theory formalism. The obtained SCF electron density has been used in calculations of Bader’s AIM (atoms in molecules) topological properties of the electron density in crystal. The obtained results show important similarities in electronic structure and electron density topology of both compounds and allow supposing, that during the thermal decomposition process these compounds should behave similarly, which is in agreement with the experiment.

Modeling the interplay of inter- and intramolecular hydrogen bonding in conformational polymorphs

The predicted stability differences of the conformational polymorphs of oxalyl dihydrazide and ortho-acetamidobenzamide are unrealistically large when the modeling of intermolecular energies is solely based on the isolated-molecule charge density, neglecting charge density polarization. Ab initio calculated crystal electron densities showed qualitative differences depending on the spatial arrangement of molecules in the lattice with the greatest variations observed for polymorphs that differ in the extent of inter- and intramolecular hydrogen bonding. We show that accounting for induction dramatically alters the calculated stability order of the polymorphs and reduces their predicted stability differences to be in better agreement with experiment. Given the challenges in modeling conformational polymorphs with marked differences in hydrogen bonding geometries, we performed an extensive periodic density functional study with a range of exchange-correlation functionals using both atomic and plane wave basis sets. Although such electronic structure methods model the electrostatic and polarization contributions well, the underestimation of dispersion interactions by current exchange-correlation functionals limits their applicability. The use of an empirical dispersion-corrected density functional method consistently reduces the structural deviations between the experimental and energy minimized crystal structures and achieves plausible stability differences. Thus, we have established which types of models may give worthwhile relative energies for crystal structures and other condensed phases of flexible molecules with intra- and intermolecular hydrogen bonding capabilities, advancing the possibility of simulation studies on polymorphic pharmaceuticals.