Dylan Jayatilaka

Directional anisotropy in electron density provides key information about chemical bonding that is not readily accessible from conventional electron-density maps. Here, a model-independent framework is presented for decomposing experimental structure factors into angular components using spherical harmonics. Reciprocal-space projection onto spherical harmonics followed by standard Fourier synthesis yields angularly filtered density maps. The ℓ = 0 component captures the isotropic part of the density, while the ℓ = 1 components resemble p _x , p _y and p _z –like dipolar functions that highlight directional electronic structure. Applications to high-resolution datasets, including urea, the Gly–Ala dipeptide and a 0.97 Å β -lactamase structure, reveal chemically interpretable dipolar features associated with carbonyl and amide bonds, N–H interactions and aromatic π systems. Quantitative analysis using bond-centred sampling demonstrates stable dipolar signatures that remain detectable under moderate resolution truncation. These results establish spherical-harmonic angular decomposition as a practical framework for extracting directional electronic information from crystallographic electron-density maps. Synopsis: Angular decomposition of experimental structure factors reveals dipolar anisotropy and directional electron-density features that are directly meaningful for chemical interpretation.

In X-ray constrained wavefunction (XCW) fitting, external information, such as electron correlation and polarization, is included into a single-determinantal isolated-molecule wavefunction. In a first step, we show that the extraction of these two physical effects by XCW fitting is complete and accurate by comparing to theoretical reference calculations. In a second step, we show that fitting to data from single-crystal x-ray diffraction measurements provides the same results qualitatively and how the physical effects can be separated, although always inherently convolved in the experiment. We further demonstrate that exchange–correlation potentials are systematically affected by XCW fitting in a physically meaningful way, which could be exploited for method development in quantum chemistry, subject to some remaining challenges that we also outline.

The X-ray constrained wavefunction (XCW) procedure for obtaining an experimentally reconstructed wavefunction from X-ray diffraction data is reviewed. The two-center probability distribution model used to perform nuclear-position averaging in the original paper [Grimwood & Jayatilaka (2001).Acta Cryst.A57, 87–100] is carefully distinguished from the newer one-center probability distribution model. In the one-center model, Hirshfeld atoms are used, and the Hirshfeld atom based X-ray constrained wavefunction (HA-XCW) procedure is described for the first time, as well as its efficient implementation. In this context, the definition of the related X-ray wavefunction refinement (XWR) method is refined. The key halting problem for the XCW method – the procedure by which one determines when overfitting has occurred – is named and work on it reviewed.

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.

Density functional theory (DFT) is currently experiencing a golden age. The past two decades witnessed remarkable advances in the general applicability of density functionals in the top rungs of Jacob’s Ladder. Nevertheless, Jacob’s Ladder may have reached its highest rung in terms of dependencies on occupied (rung four) and unoccupied orbitals (rung five). Moreover, the fifth rung is associated with a computational cost far greater than the lower rungs. Another limitation is that each rung includes dozens of different functionals, and at present, there is no clear pathway for systematic improvements within each rung of the ladder. This highlight provides an overview of the exchange–correlation (XC) hole and how it could be used in developing new density functionals. We begin with a brief overview of the current status and challenges in developing better density functionals, followed by the intimate relationship between the XC functional and hole. We present a conceptually simple and computationally economical method for calculating the XC hole and how this method could offer new directions in developing better exchange–correlation functionals.

CrystalExplorer is a native cross-platform program supported on Windows, MacOS and Linux with the primary function of visualization and investigation of molecular crystal structures, especially through the decorated Hirshfeld surface and its corresponding two-dimensional fingerprint, and through the visualization of void spaces in the crystal via isosurfaces of the promolecule electron density. Over the past decade, significant changes and enhancements have been incorporated into the program, such as the capacity to accurately and quickly calculate and visualize quantitative intermolecular interactions and, perhaps most importantly, the ability to interface with the Gaussian and NWChem programs to calculate quantum-mechanical properties of molecules. The current version, CrystalExplorer21, incorporates these and other changes, and the software can be downloaded and used free of charge for academic research.

Although hydrogen bonding is one of the most important motifs in chemistry and biology, H-atom parameters are especially problematic to refine against X-ray diffraction data. New developments in quantum crystallography offer a remedy. This article reports how hydrogen bonds are treated in three different quantum-crystallographic methods: Hirshfeld atom refinement (HAR), HAR coupled to extremely localized molecular orbitals and X-ray wavefunction refinement. Three different compound classes that form strong intra- or intermolecular hydrogen bonds are used as test cases: hydrogen maleates, the tripeptide L-alanyl-glycyl-L-alanine co-crystallized with water, and xylitol. The differences in the quantum-mechanical electron densities underlying all the used methods are analysed, as well as how these differences impact on the refinement results.

The capability of X-ray constrained wavefunction (XCW) fitting to introduce relativistic effects into a non-relativistic wavefunction is tested. It is quantified how much of the reference relativistic effects can be absorbed in the non-relativistic XCW calculation when fitted against relativistic structure factors of a model HgH₂molecule. Scaling of the structure-factor sets to improve the agreement statistics is found to introduce a significant systematic error into the XCW fitting of relativistic effects.

The relationship between the structure and the properties of a drug or material is a key concept of chemistry. Knowledge of the three-dimensional structure is considered to be of such importance that almost every report of a new chemical compound is accompanied by an X-ray crystal structure – at least since the 1970s when diffraction equipment became widely available. Crystallographic software of that time was restricted to very limited computing power, and therefore drastic simplifications had to be made. It is these simplifications that make the determination of the correct structure, especially when it comes to hydrogen atoms, virtually impossible. We have devised a robust and fast system where modern chemical structure models replace the old assumptions, leading to correct structures from the model refinement against standard in-house diffraction data using no more than widely available software and desktop computing power. We call this system NoSpherA2 (Non-Spherical Atoms in Olex2). We explain the theoretical background of this technique and demonstrate the far-reaching effects that the improved structure quality that is now routinely available can have on the interpretation of chemical problems exemplified by five selected examples.