How to get maximum structure information from anisotropic displacement parameters obtained by three-dimensional electron diffraction: an experimental study on metal-organic frameworks

Three-dimensional electron diffraction (3D ED) has been used for ab initio structure determination of various types of nanocrystals, such as metal-organic frameworks (MOFs), zeolites, metal oxides and organic crystals. These crystals are often obtained as polycrystalline powders, which are too small for single-crystal X-ray diffraction (SCXRD). While it is now possible to obtain accurate atomic positions of nanocrystals by adopting kinematical refinement against 3D ED data, most new structures are refined with isotropic displacement parameters (U _eq), which limits the detection of possible structure disorders and atomic motions. Anisotropic displacement parameters (ADPs, U^ij ) obtained by anisotropic structure refinement, on the other hand, provide information about the average displacements of atoms from their mean positions in a crystal, which can provide insights with respect to displacive disorder and flexibility. Although ADPs have been obtained from some 3D ED studies of MOFs, they are seldom mentioned or discussed in detail. We report here a detailed study and interpretation of structure models refined anisotropically against 3D ED data. Three MOF samples with different structural complexity and symmetry, namely ZIF-EC1, MIL-140C and Ga(OH)(1,4-ndc) (1,4-ndcH₂ is naphthalene-1,4-dicarboxylic acid), were chosen for the studies. We compare the ADPs refined against individual data sets and how they are affected by different data-merging strategies. Based on our results and analysis, we propose strategies for obtaining accurate structure models with interpretable ADPs based on kinematical refinement against 3D ED data. The ADPs of the obtained structure models provide clear and unambiguous information about linker motions in the MOFs.

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.

On the calculation of the electrostatic potential, electric field and electric field gradient from the aspherical pseudoatom model. II. Evaluation of the properties in an infinite crystal

The previously reported exact potential and multipole moment (EP/MM) method for fast and precise evaluation of the intermolecular electrostatic interaction energies in molecular crystals using the pseudoatom representation of the electron density [Nguyen, Macchi & Volkov (2020), Acta Cryst. A76, 630-651] has been extended to the calculation of the electrostatic potential (ESP), electric field (EF) and electric field gradient (EFG) in an infinite crystal. The presented approach combines an efficient Ewald-type summation (ES) of atomic multipoles up to the hexadecapolar level in direct and reciprocal spaces with corrections for (i) the net polarization of the sample (the `surface term') due to a net dipole moment of the crystallographic unit cell (if present) and (ii) the short-range electron-density penetration effects. The rederived and reported closed-form expressions for all terms in the ES algorithm have been augmented by the expressions for the surface term available in the literature [Stenhammar, Trulsson & Linse (2011), J. Chem. Phys. 134, 224104] and the exact potential expressions reported in a previous study [Volkov, King, Coppens & Farrugia (2006), Acta Cryst. A62, 400-408]. The resulting algorithm, coded using Fortran in the XDPROP module of the software package XD, was tested on several small molecular crystal systems (formamide, benzene, L-dopa, paracetamol, amino acids etc.) and compared with a series of EP/MM-based direct-space summations (DS) performed within a certain number of unit cells generated along both the positive and negative crystallographic directions. The EP/MM-based ES technique allows for a noticeably more precise determination of the EF and EFG and significantly better precision of the evaluated ESP when compared with the DS calculations, even when the latter include contributions from an array of symmetry-equivalent atoms generated within four additional unit cells along each crystallographic direction. In terms of computational performance, the ES/EP/MM method is significantly faster than the DS calculations performed within the extended unit-cell limits but trails the DS calculations within the reduced summation ranges. Nonetheless, the described EP/MM-based ES algorithm is superior to the direct-space summations as it does not require the user to monitor continuously the convergence of the evaluated properties as a function of the summation limits and offers a better precision-performance balance.

Putting pressure on aromaticity along with in situ experimental electron density of a molecular crystal

When pressure is applied, the molecules inside a crystal undergo significant changes of their stereoelectronic properties. The most interesting are those enhancing the reactivity of systems that would be otherwise rather inert at ambient conditions. Before a reaction can occur, however, a molecule must be activated, which means destabilized. In aromatic compounds, molecular stability originates from the resonance between two electronic configurations. Here we show how the resonance energy can be decreased in molecular crystals on application of pressure. The focus is on syn-1,6:8,13-Biscarbonyl[14]annulene, an aromatic compound at ambient conditions that gradually localizes one of the resonant configurations on compression. This phenomenon is evident from the molecular geometries measured at several pressures and from the experimentally determined electron density distribution at 7.7 GPa; the observations presented in this work are validated by periodic DFT calculations.

Aromatic compounds display enhanced stability due to delocalized π-bonding. Here, the authors study an aromatic, molecular crystal under pressure using x-ray diffraction and calculations, and observe a reduction in π-bond delocalization under increasing pressure.

A fullerene-carbene adduct as a crystalline molecular rotor: remarkable behavior of a spherically-shaped rotator

A new fullerene structure was recently obtained from the reaction of a Lewis basic N-heterocyclic carbene (NHC) and the Lewis acidic C60. The molecular features of the zwitterionic adduct can be described as a molecular rotor with the fullerene cage acting as the rotator that spins about one distinct axis given by its C-C single bond linkage with the imidazolium heterocycle stator. A detailed structural analysis of the compound by means of single-crystal X-ray diffraction (XRD) revealed significant differences in the packing motifs of solvent-free and solvent-containing crystals. Variable temperature single-crystal XRD experiments (80 K ≤ T ≤ 480 K) carried out to investigate the rotational dynamics of the fullerene group in the higher quality solvent-free structure revealed atomic displacement parameters consistent with fast rotation of the highly symmetric fullerene in the solid state, whereas the imidazolium unit remains in a fixed position and therefore represents the stator. DFT and semiempirical calculations were applied to get insight into the profile of the rotational potential of the fullerene unit, particularly considering interactions with the neighboring molecules in the crystal lattice. The results indicate that the crystal environment leads to the presence of one lowest energy minimum that is connected to seven others that are slightly higher in energy through rotational barriers of approximately 1.5-2.5 kcal mol(-1).

Dynamics and thermodynamics of crystalline polymorphs: α-glycine, analysis of variable-temperature atomic displacement parameters

Multitemperature synchrotron diffraction data to 0.5 Å resolution in the temperature range 10-298 K and neutron data at 18 K of the α-glycine polymorph have been collected at the KEK photon factory (PF), SPring-8 and the Institut Laue Langevin (ILL) for the study of molecular motion in the crystal and of the associated thermodynamic functions. Atomic displacement parameters (ADPs) of non-H atoms are obtained from refinements based on nonspherical atomic scattering factors (invariom model) to minimize correlation between parameters describing thermal motion and valence electron density. The ADPs in the temperature range 50-298 K from SPring-8 connect smoothly with those from neutron diffraction at 18 K and 288-323 K. The combined ADPs from both sources covering the temperature range 18-323 K are used for a normal-mode analysis in the molecular mean field approximation. The lattice vibration frequencies from the ADP analysis and the internal vibration frequencies from an ONIOM (B3LYP/6-311+G(2d,p):PM3) calculation together with the Einstein, Debye, and Nernst-Lindemann models of heat capacity are used to calculate Cp, Hvib, and Svib values that are in good agreement with those from calorimetry.

Temperature-dependent analysis of thermal motion, disorder and structures of tris(ethylenediamine)zinc(II) sulfate and tris(ethylenediamine)copper(II) sulfate

The crystal structures of the title compounds have been determined in the temperature range 140–290 K for the zinc complex, and 190–270 K for the copper complex. The two structures are isostructural in the trigonal space group P{\bar{3}1c} with the sulfate anion severely disordered on a site with 32 (D 3) symmetry. This sulfate disorder leads to a disordered three-dimensional hydrogen-bond network, with the N—H atoms acting as donors and the sulfate O atoms as acceptors. The displacement parameters of the N and C atoms in both compounds contain disorder contributions in the out-of-ligand plane direction owing to ring puckering and/or disorder in hydrogen bonding. In the Zn compound the vibrational amplitudes in the bond directions are closely similar. Their differences show no significant deviations from rigid-bond behaviour. In the Cu compound, a (presumably) dynamic Jahn–Teller effect is identified from a temperature-independent contribution to the displacement ellipsoids of the N atom along the N—Cu bond. These conclusions derive from analyses of the atomic displacement parameters with the Hirshfeld test, with rigid-body models at different temperatures, and with a normal coordinate analysis. This analysis considers the atomic displacement parameters (ADPs) from all different temperatures simultaneously and provides a detailed description of both the thermal motion and the disorder in the cation. The Jahn–Teller radii of the Cu compound derived on the basis of the ADP analysis and from the bond distances in the statically distorted low-temperature phase [Lutz (2010). Acta Cryst. C66, m330–m335] are found to be the same.

Specific heat of molecular crystals from atomic mean square displacements with the Einstein, Debye, and Nernst-Lindemann models

Analysis of atomic displacement parameters (ADPs) from multitemperature diffraction data provides mean-field molecular translation and libration frequencies. These quantities have been combined with molecular deformation frequencies calculated ab initio, e.g. by DFT methods, to calculate the specific heat Cv of molecular crystals of naphthalene, anthracene, and hexamethylenetetramine. If the difference Cp - Cv is represented by the Nernst-Lindemann relation, Cp curves from diffraction experiments and ab initio calculations agree well with those based on calorimetry. Agreement is better if the Debye rather than the Einstein model is chosen to represent the contribution of the translational vibrations. Compressibilities estimated from the differences Cp - Cv are 2-5 times higher than those obtained from compressibility measurements at 298 K and Grüneisen constants derived from the temperature dependence of ADPs.

Molecular Motion in Crystalline Naphthalene: Analysis of Multi-Temperature X-Ray and Neutron Diffraction Data

Single crystals of h8-naphthalene have been examined by both X-ray and neutron diffraction over a range of temperatures from 5 to 295 K. The aim of this case study was to measure the anisotropic displacement parameters (ADPs) of carbons and hydrogens and to interpret them using the model of thermal motion proposed by Bürgi and Capelli (Acta Cryst. 2000, A56, 403). The traditional rigid-body analysis expresses the low-frequency motions in terms of molecular translations and librations only, whereas the Bürgi-Capelli treatment also includes the high-frequency internal modes. We show that a considerable improvement occurs by representing the internal modes by a single second-rank tensor and that a further improvement follows by including a Grüneisen parameter to account for volume thermal expansion. By applying the treatment to multi-temperature diffraction data, there is a considerable reduction in the ratio of number of adjustable parameters/number of independent observations.