Contemporary X-ray electron-density studies using synchrotron radiation

In data we trust: X-ray diffraction experiments for charge density investigations

A short commentary is given on the paper by Vosegaard et al. [Acta Cryst. (2023), 79, 380-391], which compares charge density models derived from four datasets measured on conventional diffractometers with the results of a high-quality dataset from SPring-8.

Comparative study of conventional and synchrotron X-ray electron densities on molecular crystals

Five different electron density datasets obtained from conventional and synchrotron single crystal X-ray diffraction experiments are compared. The general aim of the study is to investigate the quality of data for electron density analysis from current state-of-the-art conventional sources, and to see how the data perform in comparison with high-quality synchrotron data. A molecular crystal of melamine was selected as the test compound due to its ability to form excellent single crystals, the light atom content, and an advantageous suitability factor of 3.6 for electron density modeling. These features make melamine an optimal system for conventional X-ray diffractometers since the inherent advantages of synchrotron sources such as short wavelength and high intensity are less critical in this case. Data were obtained at 100 K from new in-house diffractometers Rigaku Synergy-S (Mo and Ag source, HyPix100 detector) and Stoe Stadivari (Mo source, EIGER2 1M CdTe detector), and an older Oxford Diffraction Supernova (Mo source, Atlas CCD detector). The synchrotron data were obtained at 25 K from BL02B1 beamline at SPring-8 in Japan (λ = 0.2480 Å, Pilatus3 X 1M CdTe detector). The five datasets were compared on general quality parameters such as resolution, ⟨I/σ⟩, redundancy and R factors, as well as the more model specific fractal dimension plot and residual density maps. Comparison of the extracted electron densities reveals that all datasets can provide reliable multipole models, which overall convey similar chemical information. However, the new laboratory X-ray diffractometers with advanced pixel detector technology clearly measure data with significantly less noise and much higher reliability giving densities of higher quality, compared to the older instrument. The synchrotron data have higher resolution and lower measurement temperature, and they allow for finer details to be modeled (e.g. hydrogen κ parameters).

Multipole electron densities and structural parameters from synchrotron powder X-ray diffraction data obtained with a MYTHEN detector system (OHGI)

Powder X-ray diffraction has some inherent advantages over traditional single-crystal X-ray diffraction in accurately determining electron densities and structural parameters due to the lower requirements for sample crystallinity, simpler corrections and measurement simultaneity. For some simple inorganic materials, it has been shown that these advantages can compensate for disadvantages such as peak overlap and error-prone background subtraction. Although it is challenging to extend powder X-ray diffraction-based electron-density studies to organic materials with significant peak overlap, previous results using a dedicated vacuum diffractometer with a large image-plate camera (AVID) demonstrated that it can be done. However, the vacuum setup with the off-line detector system was found to prohibit a widespread use. Fast microstrip detectors, which have been employed at a number of powder diffraction beamlines, have the potential to facilitate electron-density studies. Nevertheless, no electron-density studies even for materials with slight peak overlap have been performed with microstrip detectors. One of the most critical problems has been a difference in sensitivity between microstrip channels, which substantially defines the dynamic range of a detector. Recently, a robust approach to this problem has been developed and applied to a total scattering measurement system (OHGI) with 15 MYTHEN microstrip modules. In the present study, synchrotron powder X-ray diffraction data obtained with OHGI are evaulated in terms of multipole electron densities and structural parameters (atomic positions and displacement parameters). These results show that, even without a dedicated setup and perfect samples, electron-density modelling can be carried out on high-quality powder X-ray diffraction data. However, it was also found that the required prior information about the sample prohibits widespread use of the method. With the presently obtainable data quality, electron densities of molecular crystals in general are not reliably obtained from powder data, but it is an excellent, possibly superior, alternative to single-crystal measurements for small-unit-cell inorganic solids. If aspherical atomic scattering factors can be obtained from other means (multipole databases, theoretical calculations), then atomic positions (including for hydrogen) and anisotropic atomic displacement parameters (non-hydrogen atoms) of excellent accuracy can be refined from synchrotron powder X-ray diffraction data on organic crystals.

Insight into the Bonding and Aggregation of Alkyllithiums by Experimental Charge Density Studies and Energy Decomposition Analyses

In this Article, the organolithiums [((-)-sparteine)Li^tBu] (1), [(ABCO)Li^tBu]₂ (2), and [(ABCO)₂(LiⁱPr)₄] (3) are investigated by means of experimental and theoretical charge density determination to elucidate the nature of the Li-C and Li-N bonds. Furthermore, the valence shell charge concentrations (VSCCs) in the nonbonding region of the deprotonated C_α-atom will provide some insight on the localization of the carbanionic lone pair. Analysis of the electron density (ρ(r_BCP)), Laplacian (∇²ρ(r_BCP)), and the energy decomposition (EDA) confirmed that the Li-C/N bond exhibits astonishingly similar characteristics, to reveal an increasingly polar contact with decreasing aggregate size. This explains former observations on the incorporation of halide salts in organolithium reagents. Furthermore, it could be shown that the bonding properties of the ⁱPr group are similar to those of the ^tBu substituent. The accuracy of fit to all previously determined properties in organolithiums is remarkable.

Accurate high-resolution single-crystal diffraction data from a Pilatus3 X CdTe detector

Hybrid photon-counting detectors are widely established at third-generation synchrotron facilities and the specifications of the Pilatus3 X CdTe were quickly recognized as highly promising in charge-density investigations. This is mainly attributable to the detection efficiency in the high-energy X-ray regime, in combination with a dynamic range and noise level that should overcome the perpetual problem of detecting strong and weak data simultaneously. These benefits, however, come at the expense of a persistent problem for high diffracted beam flux, which is particularly problematic in single-crystal diffraction of materials with strong scattering power and sharp diffraction peaks. Here, an in-depth examination of data collected on an inorganic material, FeSb2, and an organic semiconductor, rubrene, revealed systematic differences in strong intensities for different incoming beam fluxes, and the implemented detector intensity corrections were found to be inadequate. Only significant beam attenuation for the collection of strong reflections was able to circumvent this systematic error. All data were collected on a bending-magnet beamline at a third-generation synchrotron radiation facility, so undulator and wiggler beamlines and fourth-generation synchrotrons will be even more prone to this error. On the other hand, the low background now allows for an accurate measurement of very weak intensities, and it is shown that it is possible to extract structure factors of exceptional quality using standard crystallographic software for data processing (SAINT-Plus, SADABS and SORTAV), although special attention has to be paid to the estimation of the background. This study resulted in electron-density models of substantially higher accuracy and precision compared with a previous investigation, thus for the first time fulfilling the promise of photon-counting detectors for very accurate structure factor measurements.

Multipole electron densities and atomic displacement parameters in urea from accurate powder X-ray diffraction

Electron density determination based on structure factors obtained through powder X-ray diffraction has so far been limited to high-symmetry inorganic solids. This limit is challenged by determining high-quality structure factors for crystalline urea using a bespoke vacuum diffractometer with imaging plates. This allows the collection of data of sufficient quality to model the electron density of a molecular system using the multipole method. The structure factors, refined parameters as well as chemical bonding features are compared with results from the high-quality synchrotron single-crystal study by Birkedal et al. [Acta Cryst. (2004), A60, 371-381] demonstrating that powder X-ray diffraction potentially provides a viable alternative for electron density determination in simple molecular crystals where high-quality single crystals are not available.

Quantum Crystallography: Current Developments and Future Perspectives

Crystallography and quantum mechanics have always been tightly connected because reliable quantum mechanical models are needed to determine crystal structures. Due to this natural synergy, nowadays accurate distributions of electrons in space can be obtained from diffraction and scattering experiments. In the original definition of quantum crystallography (QCr) given by Massa, Karle and Huang, direct extraction of wavefunctions or density matrices from measured intensities of reflections or, conversely, ad hoc quantum mechanical calculations to enhance the accuracy of the crystallographic refinement are implicated. Nevertheless, many other active and emerging research areas involving quantum mechanics and scattering experiments are not covered by the original definition although they enable to observe and explain quantum phenomena as accurately and successfully as the original strategies. Therefore, we give an overview over current research that is related to a broader notion of QCr, and discuss options how QCr can evolve to become a complete and independent domain of natural sciences. The goal of this paper is to initiate discussions around QCr, but not to find a final definition of the field.

Determination of d-Orbital Populations in a Cobalt(II) Single-Molecule Magnet Using Single-Crystal X-ray Diffraction

The tetrahedral cobalt(II) compound (Ph₄P)₂[Co(SPh)₄] was the first mononuclear transition-metal complex shown to exhibit slow relaxation of the magnetization in zero external magnetic field. Because the relative populations of the d orbitals play a vital role in dictating the magnitude of the magnetic anisotropy, the magnetic behavior of this complex is directly related to its electronic structure, yet the exact role of the soft, thiophenolate ligands in influencing the d-electron configuration has previously only been investigated via theoretical methods. To provide detailed experimental insight into the effect of this ligand field, the electron density distribution in this compound was determined from low-temperature, single-crystal X-ray diffraction data and subsequent multipole modeling. Topological analysis of the electron density indicates significant covalent contributions to the cobalt-sulfur bonds. The derived d-orbital populations further reveal a fully occupied d _z² orbital, minor d _xz orbital population, and nearly equal population of the d _xy, d _{x²- y²}, and d _yz orbitals. Notably, we find that an electrostatic interaction between Co(II) and one hydrogen atom from a thiophenolate group in the xz plane increases the energy of the d _{x²- y²} orbital, leading to the nearly equal population with d _xy and strong magnetic anisotropy.

X-ray electron density investigation of chemical bonding in van der Waals materials

Van der Waals (vdW) solids have attracted great attention ever since the discovery of graphene, with the essential feature being the weak chemical bonding across the vdW gap. The nature of these weak interactions is decisive for many extraordinary properties, but it is a strong challenge for current theory to accurately model long-range electron correlations. Here we use synchrotron X-ray diffraction data to precisely determine the electron density in the archetypal vdW solid, TiS₂, and compare the results with density functional theory calculations. Quantitative agreement is observed for the chemical bonding description in the covalent TiS₂ slabs, but significant differences are identified for the interactions across the gap, with experiment revealing more electron deformation than theory. The present data provide an experimental benchmark for testing theoretical models of weak chemical bonding.

Accurate charge densities from powder X-ray diffraction - a new version of the Aarhus vacuum imaging-plate diffractometer

In recent years powder X-ray diffraction has proven to be a valuable alternative to single-crystal X-ray diffraction for determining electron-density distributions in high-symmetry inorganic materials, including subtle deformation in the core electron density. This was made possible by performing diffraction measurements in vacuum using high-energy X-rays at a synchrotron-radiation facility. Here we present a new version of our custom-built in-vacuum powder diffractometer with the sample-to-detector distance increased by a factor of four. In practice this is found to give a reduction in instrumental peak broadening by approximately a factor of three and a large improvement in signal-to-background ratio compared to the previous instrument. Structure factors of silicon at room temperature are extracted using a combined multipole-Rietveld procedure and compared with ab initio calculations and the results from the previous diffractometer. Despite some remaining issues regarding peak asymmetry, the new diffractometer yields structure factors of comparable accuracy to the previous diffractometer at low angles and improved accuracy at high angles. The high quality of the structure factors is further assessed by modelling of core electron deformation with results in good agreement with previous investigations.

Experimental charge-density studies: data reduction and model quality: the more the better?

In this review, recent developments concerning data and model quality in experimental charge-density investigations from a personal view-point are described. Data quality is not only achieved by the high resolution, high I/σ(I) values, low merging R values and high multiplicity. The quality of the innermost reflections especially is crucial for mapping the density distribution of the outermost valence electrons and can be monitored by (I/σ)^asymptotic. New detector technologies seem to be promising improvements. Empirical corrections to correct for low-energy contamination of mirror-focused X-ray data and for resolution- and temperature-dependent errors caused by factors such as thermal diffuse scattering are described. Shashlik-like residual density patterns can indicate the need for an anharmonic description of the thermal motion of individual atoms. The physical reliability of the derived model must be thoroughly analysed. The derived probability density functions for the mean-squared atomic vibrational displacements especially should have only small negative values. The treatment of H atoms has been improved by methods to estimate anisotropic thermal motion. For very high resolution data, the polarization of the core density cannot be neglected. Several tools to detect systematic errors are described. A validation tool is presented that easily detects when the refinement of additional parameters yields a real improvement in the model or simply overfits the given data. In all investigated structures, it is proved that the multipole parameters of atoms with a comparable chemical environment should be constrained to be identical. The use of restraints could be a promising alternative.

Anharmonic motions versus dynamic disorder at the Mg ion from the charge densities in pyrope (Mg3Al2Si3O12) crystals at 30 K: six of one, half a dozen of the other

The possible occurrence of static/dynamic disorder at the Mg site in pyrope (Mg₃Al₂Si₃O₁₂), with or without anharmonic contribution to the thermal vibrations even at low temperatures, has been largely debated but conclusions were contrasting. Here a report is given on the experimental charge density distribution, ρ_EXP, of synthetic pyrope at T = 30 K, built through a Stewart multipolar expansion up to l = 5 and based on a very precise and accurate set of in-home measured single-crystal X-ray diffraction amplitudes with a maximum resolution of 0.44 Å. Local and integral topological properties of ρ_EXP are in substantial agreement with those of ρ_THEO, the corresponding DFT-grade quantum charge density of an ideal pyrope crystal, and those derived from synchrotron investigations of chemical bonding in olivines. Relevant thermal atomic displacements, probably anharmonic in nature, clearly affect the whole structure down to 30 K. No significant (> 2.5σ) residual Fourier peaks are detectable from the ρ_EXP distribution around Mg, after least-squares refinement of a multipole model with anharmonic thermal motion at the Mg site. Experimental findings were confirmed by a full analysis of normal vibration modes of the DFT-optimized structure of the perfect pyrope crystal. Mg undergoes wide displacements from its equilibrium position even at very low temperatures, as it is allocated in a ∼ 4.5 Å large dodecahedral cavity and involved in several soft phonon modes. Implications on the interplay among static/dynamic disorder of Mg and lattice vibrational degrees of freedom are discussed.

Low-temperature powder X-ray diffraction measurements in vacuum: analysis of the thermal displacement of copper

A serious limitation of the all-in-vacuum diffractometer reported by Straasø, Dippel, Becker & Als-Nielsen [J. Synchrotron Rad.(2014),21, 119–126] has so far been the inability to cool samples to near-cryogenic temperatures during measurement. The problem is solved by placing the sample in a jet of helium gas cooled by liquid nitrogen. The resulting temperature change is quantified by determining the change in unit-cell parameter and atomic displacement parameter of copper. The cooling proved successful, with a resulting temperature of ∼95 (3) K. The measured powder X-ray diffraction data are of superb quality and high resolution [up to sinθ/λ = 2.2 Å−1], permitting an extensive modelling of the thermal displacement. The anharmonic displacement of copper was modelled by a Gram–Charlier expansion of the temperature factor. As expected, the corresponding probability distribution function shows an increased probability away from neighbouring atoms and a decreased probability towards them.

Synchrotron powder diffraction of silicon: high-quality structure factors and electron density

Crystalline silicon is an ideal compound to test the current state of experimental structure factors and corresponding electron densities. High-quality structure factors have been measured on crystalline silicon with synchrotron powder X-ray diffraction. They are in excellent agreement with benchmark Pendellösung data having comparable accuracy and precision, but acquired in far less time and to a much higher resolution (sin θ/λ < 1.7 Å(-1)). The extended data range permits an experimental modelling of not only the valence electron density but also the core deformation in silicon, establishing an increase of the core density upon bond formation in crystalline silicon. Furthermore, a physically sound procedure for evaluating the standard deviation of powder-derived structure factors has been applied. Sampling statistics inherently account for contributions from photon counts as well as the limited number of diffracting particles, where especially the latter are particularly difficult to handle.

Modelling the experimental electron density: only the synergy of various approaches can tackle the new challenges

Electron density is a fundamental quantity that enables understanding of the chemical bonding in a molecule or in a solid and the chemical/physical property of a material. Because electrons have a charge and a spin, two kinds of electron densities are available. Moreover, because electron distribution can be described in momentum or in position space, charge and spin density have two definitions and they can be observed through Bragg (for the position space) or Compton (for the momentum space) diffraction experiments, using X-rays (charge density) or polarized neutrons (spin density). In recent years, we have witnessed many advances in this field, stimulated by the increased power of experimental techniques. However, an accurate modelling is still necessary to determine the desired functions from the acquired data. The improved accuracy of measurements and the possibility to combine information from different experimental techniques require even more flexibility of the models. In this short review, we analyse some of the most important topics that have emerged in the recent literature, especially the most thought-provoking at the recent IUCr general meeting in Montreal.

Charge density investigations on [2,2]-paracyclophane – in data we trust

Four datasets on [2,2]-paracyclophane were collected in-house and at the Advanced Photon Source at two different temperatures for charge density investigation. Global data quality indicators such as high resolution, high I/σ(I) values, low merging R values and high multiplicity were matched for all four datasets. The structural parameters did not show significant differences, but the synchrotron data depicted deficiencies in the topological analysis. In retrospect these deficiencies could be assigned to the low quality of the innermost data, which could have been identified by e.g. merging R values for only these reflections. In the multipole refinement these deficiencies could be monitored using DRK-plot and residual density analysis. In this particular example the differences in the topological parameters were relatively small but significant.

Crystallography and large research infrastructures, a perfect marriage

The role of large-scale neutron and synchrotron facilities in the development of crystallographic research is discussed.