On QM/MM and MO/MO cluster calculations of all-atom anisotropic displacement parameters for molecules in crystal structures

Linking solid-state phenomena via energy differences in `archetype crystal structures'

Categorization underlies understanding. Conceptualizing solid-state structures of organic molecules with `archetype crystal structures' bridges established categories of disorder, polymorphism and solid solutions and is herein extended to special position and high-Z' structures. The concept was developed in the context of disorder modelling [Dittrich, B. (2021). IUCrJ, 8, 305-318] and relies on adding quantum chemical energy differences between disorder components to other criteria as an explanation as to why disorder - and disappearing disorder - occurs in an average structure. Part of the concept is that disorder, as probed by diffraction, affects entire molecules, rather than just the parts of a molecule with differing conformations, and the finding that an R·T energy difference between disorder archetypes is usually not exceeded. An illustrative example combining disorder and special positions is the crystal structure of oestradiol hemihydrate analysed here, where its space-group/subgroup relationship is required to explain its disorder of hydrogen-bonded hydrogen atoms. In addition, we show how high-Z' structures can also be analysed energetically and understood via archetypes: high-Z' structures occur when an energy gain from combining different rather than overall alike conformations in a crystal significantly exceeds R·T, and this finding is discussed in the context of earlier explanations in the literature. Twinning is not related to archetype structures since it involves macroscopic domains of the same crystal structure. Archetype crystal structures are distinguished from crystal structure prediction trial structures in that an experimental reference structure is required for them. Categorization into archetype structures also has practical relevance, leading to a new practice of disorder modelling in experimental least-squares refinement alluded to in the above-mentioned publication.

lamaGOET: an interface for quantum crystallography

In quantum crystallography, theoretical calculations and crystallographic refinements are closely intertwined. This means that the employed software must be able to perform both quantum-mechanical calculations and crystallographic least-squares refinements. So far, the program Tonto is the only one able to do that. The lamaGOET interface described herein deals with this issue since it interfaces dedicated quantum-chemical software (the widely used Gaussian package and the specialized ELMOdb program) with the refinement capabilities of Tonto. Three different flavours of quantum-crystallographic refinements of the dipetide glycyl-l-threonine dihydrate are presented to showcase the capabilities of lamaGOET: Hirshfeld atom refinement (HAR), HAR-ELMO, namely HAR coupled with extremely localized molecular orbitals, and X-ray constrained wavefunction fitting.

On modelling disordered crystal structures through restraints from molecule-in-cluster computations, and distinguishing static and dynamic disorder

Distinguishing disorder into static and dynamic based on multi-temperature X-ray or neutron diffraction experiments is the current state of the art, but is only descriptive, not predictive. Here, several disordered structures are revisited from the Cambridge Crystallographic Data Center 'drug subset', the Cambridge Structural Database and own earlier work, where experimental intensities of Bragg diffraction data were available. Using the molecule-in-cluster approach, structures with distinguishable conformations were optimized separately, as extracted from available or generated disorder models of the respective disordered crystal structures. Re-combining these 'archetype structures' by restraining positional and constraining displacement parameters for conventional least-squares refinement, based on the optimized geometries, then often achieves a superior fit to the experimental diffraction data compared with relying on experimental information alone. It also simplifies and standardizes disorder refinement. Ten example structures were analysed. It is observed that energy differences between separate disorder conformations are usually within a small energy window of RT (T = crystallization temperature). Further computations classify disorder into static or dynamic, using single experiments performed at one single temperature, and this was achieved for propionamide.

Studying the hydrogen atom position in the strong-short intermolecular hydrogen bond of pure and 5-substituted 9-hydroxyphenalenones by invariom refinement and ONIOM cluster computations

The solid-state structures of three H-bonded enol forms of 5-substituted 9-hydroxyphenalenones were investigated to accurately determine the H atom positions of the intramolecular hydrogen bond. For this purpose, single-crystal X-ray diffraction (SC-XRD) data were evaluated by invariom-model refinement. In addition, QM/MM computations of central molecules in their crystal environment show that results of an earlier standard independent atom model refinement, which pointed to the presence of a resonance-assisted hydrogen bond in unsubstituted 9-hydroxyphenalone, are misleading: in all our three and the earlier solid-state structures the lowest energy form is that of an asymmetric hydrogen bond (CS form). Apparent differences of results from SC-XRD and other analytical methods are explained.

Can we trust the experiment? Anisotropic displacement parameters in 1-(halomethyl)-3-nitrobenzene (halogen = Cl or Br)

1-(Chloromethyl)-3-nitrobenzene, C₇H₆NClO₂, and 1-(bromomethyl)-3-nitrobenzene, C₇H₆NBrO₂, were chosen as test compounds for benchmarking anisotropic displacement parameters (ADPs) calculated from first principles in the harmonic approximation. Crystals of these compounds are isomorphous, and theory predicted similar ADPs for both. In-house diffraction experiments with Mo Kα radiation were in apparent contradiction to this theoretical result, with experimentally observed ADPs significantly larger for the bromo derivative. In contrast, the experimental and theoretical ADPs for the lighter congener matched reasonably well. As all usual quality indicators for both sets of experimental data were satisfactory, complementary diffraction experiments were performed at a synchrotron beamline with shorter wavelength. Refinements based on these intensity data gave very similar ADPs for both compounds and were thus in agreement with the earlier in-house results for the chloro derivative and the predictions of theory. We speculate that strong absorption by the heavy halogen may be the reason for the observed discrepancy.

Fast energy minimization of the CCDC drug-subset structures by molecule-in-cluster computations allows independent structure validation and model completion

Optimizing structures with computations on clusters of molecules permits generation of structure-specific restraints for refinement and structure validation.

Disappearing disorder

Disorder in crystal structures can disappear, depending on the circumstances, as shown by multi-temperature measurements, aspherical-atom refinement and computational analyses.

The Color of the Elements: A Combined Experimental and Theoretical Electron Density Study of ScB2 C2

The chemical or physical control parameters for the onset of superconductivity in MB₂ C₂ hetero-graphene materials are unclear. This is mainly due to the almost ubiquitous positional B/C disorder, rendering the description of real structures of borocarbides into one of the most challenging problems in materials science. We will show that high-resolution X-ray diffraction data provides all the essential information to decode even complex coloring problems due to B/C disorder. Electron density studies and subsequent analyses of the fine structure of the Laplacian of the electron density resolves the local electronic structure of ScB₂ C₂ at sub-atomic resolution and allows for an unequivocal identification of all atoms involved in the coloring scenario. This information could finally be used to identify the electron deficient character of the B/C layers in ScB₂ C₂ and to synthesize the first bimetallic hetero-metallocene with lithium and scandium atoms embedded in the pentagonal and heptagonal voids, respectively.

A new tool for validating theoretically derived anisotropic displacement parameters with experiment: directionality of prolate displacement ellipsoids

How well do anisotropic displacement parameters from theory match experiment? The orientation of prolate ellipsoids contributes to the answer!

On avoiding negative electron density in Gram-Charlier refinements of anharmonic motion: the example of glutathione

The structure of glutathione, γ-l-Glutamyl-l-cysteinyl-glycine (C10H17N3O6S), was studied by multi-temperature single-crystal X-ray diffraction. Residual density maps from conventional independent atom model refinement gave indication of anharmonic motion in the molecule. This was further investigated by invariom refinement with anisotropic displacement parameters for all atoms, which described asphericity due to chemical bonding and lone pairs; afterwards only the residual-density signal of anharmonic motion remained. Treating anharmonicity with third-order Gram-Charlier displacement parameters led to regions with unphysical negative electron density. In contrast, a maximum entropy method (MEM) determination of the electron density successfully takes the features into account. Respective difference electron density plots (MEM minus prior and [Invariom+GC] minus invariom) agree well with each other. Challenges in treating and understanding the phenomenon are discussed. A procedure is proposed how unphysical negative electron density can be avoided. It is closely related to the free lunch algorithm.

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.

Accurate Bond Lengths to Hydrogen Atoms from Single-Crystal X-ray Diffraction by Including Estimated Hydrogen ADPs and Comparison to Neutron and QM/MM Benchmarks

Amino acid structures are an ideal test set for method-development studies in crystallography. High-resolution X-ray diffraction data for eight previously studied genetically encoding amino acids are provided, complemented by a non-standard amino acid. Structures were re-investigated to study a widely applicable treatment that permits accurate X-H bond lengths to hydrogen atoms to be obtained: this treatment combines refinement of positional hydrogen-atom parameters with aspherical scattering factors with constrained "TLS+INV" estimated hydrogen anisotropic displacement parameters (H-ADPs). Tabulated invariom scattering factors allow rapid modeling without further computations, and unconstrained Hirshfeld atom refinement provides a computationally demanding alternative when database entries are missing. Both should incorporate estimated H-ADPs, as free refinement frequently leads to over-parameterization and non-positive definite H-ADPs irrespective of the aspherical scattering model used. Using estimated H-ADPs, both methods yield accurate and precise X-H distances in best quantitative agreement with neutron diffraction data (available for five of the test-set molecules). This work thus solves the last remaining problem to obtain such results more frequently. Density functional theoretical QM/MM computations are able to play the role of an alternative benchmark to neutron diffraction.

Dynamic quantum crystallography: lattice-dynamical models refined against diffraction data. I. Theory

This study demonstrates and tests the refinement of a lattice-dynamical model derived from periodic ab initio calculations at the Γ point against elastic diffraction data (X-ray or neutron). Refinement of only a handful of parameters is sufficient to obtain a similar agreement with the data as the conventional crystallographic model using anisotropic displacement parameters. By refinement against X-ray data, H displacement parameters are obtained which compare favourably with those from neutron diffraction experiments. The approach opens the door for evaluating thermodynamic properties, and for refinement against multi-temperature data, against inelastic diffraction data, spectroscopic information and thermal diffuse scattering data.

Anisotropic thermal motion in transition-metal carbonyls from experiments and ab initio theory

This proof-of-concept study extends the ab initio computation of anisotropic displacement parameters to complexes with transition metal centres.

Estimating temperature-dependent anisotropic hydrogen displacements with the invariom database and a new segmented rigid-body analysis program

Invariom partitioning and notation are used to estimate anisotropic hydrogen displacements for incorporation in crystallographic refinement models. Optimized structures of the generalized invariom database and their frequency computations provide the information required: frequencies are converted to internal atomic displacements and combined with the results of a TLS (translation-libration-screw) fit of experimental non-hydrogen anisotropic displacement parameters to estimate those of H atoms. Comparison with TLS+ONIOM and neutron diffraction results for four example structures where high-resolution X-ray and neutron data are available show that electron density transferability rules established in the invariom approach are also suitable for streamlining the transfer of atomic vibrations. A new segmented-body TLS analysis program called APD-Toolkit has been coded to overcome technical limitations of the established program THMA. The influence of incorporating hydrogen anisotropic displacement parameters on conventional refinement is assessed.

Anisotropic displacement parameters from dispersion-corrected DFT methods and their experimental validation by temperature-dependent X-ray diffraction

How reliably can anisotropic displacement parameters be derived from theory? Experiments and computations on pentachloropyridine shed new light on this question.

Contributions of charge-density research to medicinal chemistry

This article reviews efforts in accurate experimental charge-density studies with relevance to medicinal chemistry. Initially, classical charge-density studies that measure electron density distribution via least-squares refinement of aspherical-atom population parameters are summarized. Next, interaction density is discussed as an idealized situation resembling drug-receptor interactions. Scattering-factor databases play an increasing role in charge-density research, and they can be applied both to small-molecule and macromolecular structures in refinement and analysis; software development facilitates their use. Therefore combining both of these complementary branches of X-ray crystallography is recommended, and examples are given where such a combination already proved useful. On the side of the experiment, new pixel detectors are allowing rapid measurements, thereby enabling both high-throughput small-molecule studies and macromolecular structure determination to higher resolutions. Currently, the most ambitious studies compute intermolecular interaction energies of drug-receptor complexes, and it is recommended that future studies benefit from recent method developments. Selected new developments in theoretical charge-density studies are discussed with emphasis on its symbiotic relation to crystallography.

On the temperature dependence of H-U(iso) in the riding hydrogen model

The temperature dependence of hydrogen U_iso and parent U_eq in the riding hydrogen model is investigated by neutron diffraction, aspherical-atom refinements and QM/MM and MO/MO cluster calculations. Fixed values of 1.2 or 1.5 appear to be underestimated, especially at temperatures below 100 K.

The temperature dependence of H-U_iso in N-acetyl-l-4-hydroxyproline monohydrate is investigated. Imposing a constant temperature-independent multiplier of 1.2 or 1.5 for the riding hydrogen model is found to be inaccurate, and severely underestimates H-U_iso below 100 K. Neutron diffraction data at temperatures of 9, 150, 200 and 250 K provide benchmark results for this study. X-ray diffraction data to high resolution, collected at temperatures of 9, 30, 50, 75, 100, 150, 200 and 250 K (synchrotron and home source), reproduce neutron results only when evaluated by aspherical-atom refinement models, since these take into account bonding and lone-pair electron density; both invariom and Hirshfeld-atom refinement models enable a more precise determination of the magnitude of H-atom displacements than independent-atom model refinements. Experimental efforts are complemented by computing displacement parameters following the TLS+ONIOM approach. A satisfactory agreement between all approaches is found.

CLUSTERGEN: a program for molecular cluster generation from crystallographic data

A new program,CLUSTERGEN, for molecular cluster generation is introduced.CLUSTERGENprovides the quantum mechanics/molecular mechanics (QM/MM) input files for program packages such asADF[Baerendset al.(2012). Vrije Universiteit, Amsterdam, The Netherlands] andGAUSSIAN[Frischet al.(2009). Gaussian Inc., Pittsburgh, Pennsylvania, USA]. Additionally, it prints out a standardCRYSTAL[Dovesiet al.(2009). University of Turin, Italy] input and, in general, facilitates file-format manipulation. TheCLUSTERGENprogram is supported by an extensive manual and a user-friendly graphical interface. The code is freely available and carefully commented, which makes it easily modifiable. Exemplary applications ofCLUSTERGENconcerning QM/MM calculations and derivation of nucleus-independent chemical shift indices are demonstrated.