Quantum crystallography

Current developments and trends in quantum crystallography

Quantum crystallography is an emerging research field of science that has its origin in the early days of quantum physics and modern crystallography when it was almost immediately envisaged that X-ray radiation could be somehow exploited to determine the electron distribution of atoms and molecules. Today it can be seen as a composite research area at the intersection of crystallography, quantum chemistry, solid-state physics, applied mathematics and computer science, with the goal of investigating quantum problems, phenomena and features of the crystalline state. In this article, the state-of-the-art of quantum crystallography will be described by presenting developments and applications of novel techniques that have been introduced in the last 15 years. The focus will be on advances in the framework of multipole model strategies, wavefunction-/density matrix-based approaches and quantum chemical topological techniques. Finally, possible future improvements and expansions in the field will be discussed, also considering new emerging experimental and computational technologies.

Exploring crystal structure-physical property relationships with pressure

From its conception, X-ray crystallography has provided a unique understanding of the structure, bonding and electronic state of materials, which, in turn, unlocks a means of examining the properties and function of crystalline systems. Using state-of-the-art single-crystal X-ray diffraction, along with UV-Vis spectroscopy and DFT calculations, Zwolenik et al. [(2024). IUCrJ, 11, 519-527] have provided a comprehensive study of the structure-optical property relationship of 1,3-diacetylpyrene with methodologies that are increasingly accessible to non-specialist laboratories.

Analytic Construction of One-Electron Reduced Density Matrices from Electron Densities within Finite Basis Sets

We show how to construct analytically all one-electron reduced density matrices (1-RDMs) compatible with a given electron density within a finite basis set, provided that the density is specified as a symmetric quadratic form in terms of the basis functions. Contrary to the current belief, exact linear dependencies in the basis function products assist, rather than hinder, such constructions. By applying the N-representability conditions to the analytically reconstructed 1-RDMs, one can perform a constrained search over physically acceptable 1-RDMs that yield a given finite-basis-set density. The discussion is illustrated with worked-out examples.

Critical assessment of the x-ray restrained wave function approach: Advantages, drawbacks, and perspectives for density functional theory and periodic ab initio calculations

The x-ray restrained wave function (XRW) method is a quantum crystallographic technique to extract wave functions compatible with experimental x-ray diffraction data. The approach looks for wave functions that minimize the energies of the investigated systems and also reproduce sets of x-ray structure factors. Given the strict relationship between x-ray structure factors and electron distributions, the strategy practically allows determining wave functions that correspond to given (usually experimental) electron densities. In this work, the capabilities of the XRW approach were further tested. The aim was to evaluate whether the XRW technique could serve as a tool for suggesting new exchange-correlation functionals for density functional theory or refining existing ones. Additionally, the ability of the method to address the influences of the crystalline environment was also assessed. The outcomes of XRW computations were thus compared to those of traditional gas-phase, embedding quantum mechanics/molecular mechanics, and fully periodic calculations. The results revealed that, irrespective of the initial conditions, the XRW computations practically yield a consensus electron density, in contrast to the currently employed density functional approximations (DFAs), which tend to give a too large range of electron distributions. This is encouraging in view of exploiting the XRW technique to develop improved functionals. Conversely, the calculations also emphasized that the XRW method is limited in its ability to effectively address the influences of the crystalline environment. This underscores the need for a periodic XRW technique, which would allow further untangling the shortcomings of DFAs from those inherent to the XRW approach.

Dynamic Covalent Bonds in the Ebselen Class of Antioxidants Probed by X-ray Quantum Crystallography

Dynamic bonds are essential structural ingredients of dynamic covalent chemistry that involve reversible cleavage and formation of bonds. Herein, we explore the electronic characteristics of Se-N bonds in the organo-selenium antioxidant ebselen and its derivatives for their propensity to function as dynamic covalent bonds by employing high-resolution X-ray quantum crystallography and complementary computational studies. An analysis of the experimentally reconstructed X-ray wavefunctions reveals the salient electronic features of the Se-N bonds with very low electron density localized at the bonding region and a positive Laplacian value at the bond critical point. Bond orders and percentage covalency and ionicity estimated from the X-ray wavefunctions, along with localized orbital locator (LOL) and electron localization function (ELF) analyses show that the Se-N bond is unique in its closed shell-like features, despite being a covalent bond. Time-dependent DFT calculations simulate the cleavage of Se-N bonds in ebselen in the excited state, further substantiating their nature as dynamic bonds.

Influence of N-protonation on electronic properties of acridine derivatives by quantum crystallography

Applications of 9-aminoacridine (9aa) and its derivatives span fields such as chemistry, biology, and medicine, including anticancer and antimicrobial activities. Protonation of such molecules can alter their bioavailability as weakly basic drugs like aminoacridines exhibit reduced solubility at high pH levels potentially limiting their effectiveness in patients with elevated gastric pH. In this study, we analyse the influence of protonation on the electronic characteristics of the molecular organic crystals of 9-aminoacridine. The application of quantum crystallography, including aspherical atom refinement, has enriched the depiction of electron density in the studied systems and non-covalent interactions, providing more details than previous studies. Our experimental results, combined with a topological analysis of the electron density and its Laplacian, provided detailed descriptions of how protonation changes the electron density distribution around the amine group and water molecule, concurrently decreasing the electron density at bond critical points of N/O-H bonds. Protonation also alters the molecular architecture of the systems under investigation. This is reflected in different proportions of the N⋯H and O⋯H intermolecular contacts for the neutral and protonated forms. Periodic DFT calculations of the cohesive energies of the crystal lattice, as well as computed interaction energies between molecules in the crystal, confirm that protonation stabilises the crystal structure due to a positive synergy between strong halogen and hydrogen bonds. Our findings highlight the potential of quantum crystallography in predicting crystal structure properties and point to its possible applications in developing new formulations for poorly soluble drugs.

The electron density: a fidelity witness for quantum computation

There is currently no combination of quantum hardware and algorithms that can provide an advantage over conventional calculations of molecules or materials. However, if or when such a point is reached, new strategies will be needed to verify predictions made using quantum devices. We propose that the electron density, obtained through experimental or computational means, can serve as a robust benchmark for validating the accuracy of quantum computation of chemistry. An initial exploration into topological features of electron densities, facilitated by quantum computation, is presented here as a proof of concept. Additionally, we examine the effects of constraining and symmetrizing measured one-particle reduced density matrices on noise-driven errors in the electron density distribution. We emphasize the potential benefits and future need for high-quality electron densities derived from diffraction experiments for validating classically intractable quantum computations of materials.

Development of Stable Carbanionic Substituents

During the past decade, carbon (C-H) acids depicted as 'Tf2 CHR' (Tf=CF3 SO2 ) have attracted considerable attention as a new class of superacidic molecules, which show stronger acidity than sulfuric acid molecules. In recent years, the author has developed a synthetic methodology for such strong acids and has opened the door to chemistry of highly stabilised carbanions [Tf2 CR]- , which are the conjugate bases of the carbon acids. These carbanion-containing salts are stable and easy-to-handle species. Our efforts have revealed that the ionic but lipophilic characters of this type of carbanion can be used as a unique 'substituent' for increasing both the water solubility and the lipophilicity of organic compounds. This Personal Account provides an overview of our [Tf2 CR]- chemistry, including its synthesis, structure, reactivity, and applications.

Elucidating the nature of chemical bonds in a coordination compound through quantum crystallographic techniques

Investigations simultaneously involving multiple techniques of quantum crystallography could be very useful to prove the consistency of obtained results or to highlight different facets of the same scientific phenomenon or problem. Pinto et al. [Acta Cryst. (2023), B79, 282-296] exploit three different quantum crystallographic techniques (Hansen & Coppens multipole model refinement, QTAIM analysis of the electron density, and Hirshfeld atom refinement) to characterize the nature of chemical bonds and of intra/intermolecular interactions in an organometallic compound.

Extracting Quantitative Information at Quantum Mechanical Level from Noncovalent Interaction Index Analyses

The noncovalent interaction (NCI) index is nowadays a well-known strategy to detect NCIs in molecular systems. Even though it initially provided only qualitative descriptions, the technique has been recently extended to also extract quantitative information. To accomplish this task, integrals of powers of the electron distribution were considered, with the requirement that the overall electron density can be clearly decomposed as sum of distinct fragment contributions to enable the definition of the (noncovalent) integration region. So far, this was done by only exploiting approximate promolecular electron densities, which are given by the sum of spherically averaged atomic electron distributions and thus represent too crude approximations. Therefore, to obtain more quantum mechanically (QM) rigorous results from NCI index analyses, in this work, we propose to use electron densities obtained through the transfer of extremely localized molecular orbitals (ELMOs) or through the recently developed QM/ELMO embedding technique. Although still approximate, the electron distributions resulting from the abovementioned methods are fully QM and, above all, are again partitionable into subunit contributions, which makes them completely suitable for the NCI integral approach. Therefore, we benchmarked the integrals resulting from NCI index analyses (both those based on the promolecular densities and those based on ELMO electron distributions) against interaction energies computed at a high quantum chemical level (in particular, at the coupled cluster level). The performed test calculations have indicated that the NCI integrals based on ELMO electron densities outperform the promolecular ones. Furthermore, it was observed that the novel quantitative NCI-(QM/)ELMO approach can be also profitably exploited both to characterize and evaluate the strength of specific interactions between ligand subunits and protein residues in protein-ligand complexes and to follow the evolution of NCIs along trajectories of molecular dynamics simulations. Although further methodological improvements are still possible, the new quantitative ELMO-based technique could be already exploited in situations in which fast and reliable assessments of NCIs are crucial, such as in computational high-throughput screenings for drug discovery.

Introduction of a weighting scheme for the X-ray restrained wavefunction approach: advantages and drawbacks

In a quite recent study [Genoni et al. (2017). IUCrJ, 4, 136-146], it was observed that the X-ray restrained wavefunction (XRW) approach allows a more efficient and larger capture of electron correlation effects on the electron density if high-angle reflections are not considered in the calculations. This is due to the occurrence of two concomitant effects when one uses theoretical X-ray diffraction data corresponding to a single-molecule electron density in a large unit cell: (i) the high-angle reflections are generally much more numerous than the low- and medium-angle ones, and (ii) they are already very well described at unrestrained level. Nevertheless, since high-angle data also contain important information that should not be disregarded, it is not advisable to neglect them completely. For this reason, based on the results of the previous investigation, this work introduces a weighting scheme for XRW calculations to up-weight the contribution of low- and medium-angle reflections, and, at the same time, to reasonably down-weight the importance of the high-angle data. The proposed strategy was tested through XRW computations with both theoretical and experimental structure-factor amplitudes. The tests have shown that the new weighting scheme works optimally if it is applied with theoretically generated X-ray diffraction data, while it is not advantageous when traditional experimental X-ray diffraction data (even of very high resolution) are employed. This also led to the conclusion that the use of a specific external parameter λ_J for each resolution range might not be a suitable strategy to adopt in XRW calculations exploiting experimental X-ray data as restraints.

Real-Space Interpretation of Interatomic Charge Transfer and Electron Exchange Effects by Combining Static and Kinetic Potentials and Associated Vector Fields

Intricate behaviour of one-electron potentials from the Euler equation for electron density and corresponding gradient force fields in crystals was studied. Channels of locally enhanced kinetic potential and corresponding saddle Lagrange points were found between chemically bonded atoms. Superposition of electrostatic ϕ e s r and kinetic ϕ k r potentials and electron density ρ r allowed partitioning any molecules and crystals into atomic ρ - and potential-based ϕ -basins; ϕ k -basins explicitly account for the electron exchange effect, which is missed for ϕ e s -ones. Phenomena of interatomic charge transfer and related electron exchange were explained in terms of space gaps between zero-flux surfaces of ρ - and ϕ -basins. The gap between ϕ e s - and ρ -basins represents the charge transfer, while the gap between ϕ k - and ρ -basins is a real-space manifestation of sharing the transferred electrons caused by the static exchange and kinetic effects as a response against the electron transfer. The regularity describing relative positions of ρ -, ϕ e s -, and ϕ k - basin boundaries between interacting atoms was proposed. The position of ϕ k -boundary between ϕ e s - and ρ -ones within an electron occupier atom determines the extent of transferred electron sharing. The stronger an H⋅⋅⋅O hydrogen bond is, the deeper hydrogen atom's ϕ k -basin penetrates oxygen atom's ρ -basin, while for covalent bonds a ϕ k -boundary closely approaches a ϕ e s -one indicating almost complete sharing of the transferred electrons. In the case of ionic bonds, the same region corresponds to electron pairing within the ρ -basin of an electron occupier atom.

On the termination of the X-ray constrained wavefunction procedure: reformulation of the method for an unequivocal determination of λ

The X-ray constrained/restrained wavefunction (XCW/XRW) approach of quantum crystallography is revisited by introducing the stationary condition of the Jayatilaka functional with respect to the Lagrange multiplier λ. The theoretical derivation has unequivocally shown that the right value of λ is a maximum stationary point of the functional to optimize, thus enabling the solution of the longstanding problem of establishing the point at which to halt the XCW/XRW procedure. Based on the new finding, a reformulation of the X-ray constrained wavefunction algorithm is proposed and its implementation is envisaged. In addition to relying on more solid mathematical grounds, the new variant of the method will be intrinsically more physically meaningful, allowing a straightforward evaluation of the highest level of confidence with which the experimental X-ray diffraction data can be possibly reproduced.

Remarks on X-ray constrained/restrained wavefunction fitting

X-ray constrained/restrained wavefunctions (XCWs/XRWs) result from a combination of theory and experiment and are therefore affected by experimental errors and model uncertainties. The present XCW/XRW procedure does not take this into account, thus limiting the meaning and significance of the obtained wavefunctions.

X-ray constrained wavefunctions based on Hirshfeld atoms. I. Method and review

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.

π-Hole bonding in a new co-crystal hydrate of gallic acid and pyrazine: static and dynamic charge density analysis

A new cocrystal hydrate of gallic acid with pyrazine (4GA, Py, 4H₂O; GA₄PyW₄) was obtained and characterized by single crystal X-ray diffraction. In addition to structure determination, experimental charge density analysis was carried out in terms of Multipole Modelling (MP), X-ray wavefunction refinement (XWR) and maximum entropy method (MEM). As a part of XWR, the structural refinement via Hirshfeld atom refinement was carried out and resulted in O-H bond lengths close to values from neutron diffraction. A systematic comparison of molecular conformations and aromatic interactions in this new cocrystal hydrate was performed with other existing polymorphs of gallic acid. In GA₄PyW₄, the two symmetry-independent gallic acid molecules have a syn COOH orientation and form the common (COOH)₂ dimeric synthon. The carboxyl C atom displays the characteristics of π-holes with electropositive regions above and below the molecular plane and engages in acceptor-donor interactions with oxygen atoms of acidic O-H groups and phenol groups of neighbouring gallic acid molecules. The signature of the π-hole was identified from experimental charge density analysis, both in static density maps in MP and XWR as well as dynamic density in MEM, but it cannot be pinned down to a specific atom-atom interaction. This study presents the first comparison between an XWR and a MEM experimental electron-density determination.

Aurophilic Interactions Studied by Quantum Crystallography

This is the first use of a wave-function-based crystallographic method to characterize aurophilic interactions from X-ray diffraction data. Theoretical calculations previously suggested the importance of electron correlation and dispersion forces, but no influence of relativistic corrections to the Au...Au interaction energy was found. In this study, we confirm the importance of relativistic corrections in the characterization of aurophilic interactions in addition to electron correlation and dispersion.

Hirshfeld atom refinement was used to characterize aurophilic interactions from X-ray diffraction data. An intermediate closed-shell type of aurophilic interaction with some features of covalency was identified when both electron correlation and relativistic corrections were applied. Relativistic correction changes the electron density distribution more than electron correlation. Relativistic effects strongly dominate the metal core region also in the direction of the noncovalent interactions and all of the valence and bonding regions with regard to the Au···Au interaction.

Material research from the viewpoint of functional motifs

As early as 2001, the need for the ‘functional motif theory’ was pointed out to assist the rational design of functional materials. The properties of materials are determined by their functional motifs and by how they are arranged in the materials. Uncovering the functional motifs and their arrangements is crucial in understanding the properties of materials and rationally designing new materials of desired properties. The functional motifs of materials are the critical microstructural units (e.g. constituent components and building blocks) that play a decisive role in generating certain material functions, and could not be replaced with other structural units without losing or significantly suppressing the relevant functions. The role of functional motifs and their arrangements in materials with representative examples was presented. These examples could be classified into six types of material microscopic structures on a length scale smaller than ∼10 nm with maximum subatomic resolution, i.e. the crystal, magnetic, aperiodic, defect, local, and electronic structures. The method of functional motif analysis could be employed in the function-oriented design of materials, as elucidated by taking infrared nonlinear optical materials as an example. Machine learning is more efficient in predicting material properties and screening materials with high efficiency than high-throughput experimentation and high-throughput calculations. In extracting the functional motifs and finding their quantitative relationships, developing sufficiently reliable databases for material structures and properties is imperative.

H-Atom Assignment and Sb-O Bonding of [Mes3SbOH][O3SPh] Confirmed by Neutron Diffraction, Multipole Modeling, and Hirshfeld Atom Refinement

Neutron wavelength-resolved Laue diffraction experiments permit accurate refinement of the H-atom positions and anisotropic displacement parameters of [Mes₃SbOH][O₃SPh]. A multipole-based charge density refinement and a topological analysis of the refined electron density were also performed. Hirshfeld atom refinement (HAR) recovers the neutron-determined H-atom parameters, and the quantum-mechanical electron density used in HAR recovers the electron density topology from the refined multipole model. These results confirm that [Mes₃SbOH][O₃SPh] does indeed feature a hydroxystibonium cation with a nominal Sb-O single bond and not a stibine oxide with an Sb=O/Sb⁺-O^- bond.

Vanishing of the atomic form factor derivatives in non-spherical structural refinement - a key approximation scrutinized in the case of Hirshfeld atom refinement

When calculating derivatives of structure factors, there is one particular term (the derivatives of the atomic form factors) that will always be zero in the case of tabulated spherical atomic form factors. What happens if the form factors are non-spherical? The assumption that this particular term is very close to zero is generally made in non-spherical refinements (for example, implementations of Hirshfeld atom refinement or transferable aspherical atom models), unless the form factors are refinable parameters (for example multipole modelling). To evaluate this general approximation for one specific method, a numerical differentiation was implemented within the NoSpherA2 framework to calculate the derivatives of the structure factors in a Hirshfeld atom refinement directly as accurately as possible, thus bypassing the approximation altogether. Comparing wR₂ factors and atomic parameters, along with their uncertainties from the approximate and numerically differentiating refinements, it turns out that the impact of this approximation on the final crystallographic model is indeed negligible.

Estimating electron density at the bond critical point through atomic Hirshfeld surfaces

Interatomic surfaces often carry information related to the electron distribution in a molecule or crystal, not only being a visual aid but also enabling quantitative analyses. Under certain conditions, atomic Hirshfeld surfaces present a high resemblance to the interatomic surfaces obtained through the quantum theory of atoms in molecules (QTAIM), with the advantage of being easily calculated, even for crystal structures determined at low resolutions (i.e. when a charge-density refinement is not performed). Here an empirical relation between the curvedness property of the Hirshfeld surfaces and the electron density at the bond critical point for certain types of covalent and coordination interactions involving carbon atoms has been obtained. The exponential function was tested to estimate the electron density in different crystalline systems, and the highest deviation from reference values obtained through QTAIM was just 16%. Additionally, fine details of this fit may be salient to the difference in electronegativity of the atoms involved in the bond.

Theoretically derived thermodynamic properties can be improved by the refinement of low-frequency modes against X-ray diffraction data

Herein, a framework for the estimation of the thermodynamic properties of molecular crystals via the refinement of frequencies from density functional theory calculations against X-ray diffraction data is presented. The framework provides an efficient approach to including the contribution of acoustic modes in the thermodynamic properties. The obtained heat capacities for urea, the α- and β-glycine polymorphs, benzoic acid, and 4'-hydroxyacetophenone are in good agreement with those from adiabatic calorimetry.

Initial Maximum Overlap Method for Large Systems by the Quantum Mechanics/Extremely Localized Molecular Orbital Embedding Technique

Quantum chemistry offers a large variety of methods to treat excited states. Many of them are based on a multireference wave function ansatz and are therefore characterized by an intrinsic complexity and high computational costs. To overcome these drawbacks and also some limitations of simpler single-reference approaches (e.g., configuration interaction singles and time-dependent density functional theory), the single-determinant Δself-consistent field-initial maximum overlap method (ΔSCF-IMOM) has been proposed. This strategy substitutes the aufbau principle with a criterion that occupies molecular orbitals at successive SCF iterations on the basis of their maximum overlap with a proper set of guess orbitals for the target excited state. In this way, it prevents the SCF process to collapse to the ground state wave function and provides excited state single Slater determinant solutions to the SCF equations. Here, we propose to extend the applicability of the IMOM to the treatment of localized excited states of large systems. To accomplish this task, we coupled it with the QM/ELMO (quantum mechanics/extremely localized molecular orbitals) strategy, a quantum mechanical embedding method in which the most chemically relevant part of the system is treated with traditional quantum chemical approaches, while the rest is described by extremely localized molecular orbitals transferred from recently constructed libraries or proper model molecules. After presenting the theoretical foundations of the new IMOM/ELMO technique, in this paper, we will show and discuss the results of preliminary test calculations carried out on both model systems (i.e., decanoic acid, decene, decapentaene, and solvated acrolein) and a system of biological interest (flavin mononucleotide in the flavodoxin protein). We observed that, for localized excited states, the new IMOM/ELMO method provides reliable results, and it reproduces the outcomes of fully IMOM calculations within the chemical accuracy threshold (i.e., 0.043 eV) by including only a limited number of atoms in the QM region. Furthermore, the first application of our embedding technique to a larger biological system gave completely plausible results in line with those obtained through more traditional quantum mechanical methods, thus opening the possibility of using the new approach in future investigations of photobiology problems.

Charge density studies of multicentre two-electron bonding of an anion radical at non-ambient temperature and pressure

The variation of charge density of two-electron multicentre bonding (pancake bonding) between semi-quinone radicals with pressure and temperature was studied on a salt of 5,6-di-chloro-2,3-di-cyano-semi-quinone radical anion (DDQ) with 4-cyano-N-methyl-pyridinium cation (4-CN) using the Transferable Aspheric Atom Model (TAAM) refinement. The pancake-bonded radical dimers are stacked by non-bonding π-interactions. With rising pressure, the covalent character of interactions between radicals increases, and above 2.55 GPa, the electron density indicates multicentric covalent interactions throughout the stack. The experimental charge densities were verified and corroborated by periodic DFT computations. The TAAM approach has been tested and validated for atomic resolution data measured at ambient pressure; this work shows this approach can also be applied to diffraction data obtained at pressures up to several gigapascals.

Three-Layer Multiscale Approach Based on Extremely Localized Molecular Orbitals to Investigate Enzyme Reactions

Quantum mechanics/molecular mechanics (QM/MM) calculations are widely used embedding techniques to computationally investigate enzyme reactions. In most QM/MM computations, the quantum mechanical region is treated through density functional theory (DFT), which offers the best compromise between chemical accuracy and computational cost. Nevertheless, to obtain more accurate results, one should resort to wave function-based methods, which however lead to a much larger computational cost already for relatively small QM subsystems. To overcome this drawback, we propose the coupling of our QM/ELMO (quantum mechanics/extremely localized molecular orbital) approach with molecular mechanics, thus introducing the three-layer QM/ELMO/MM technique. The QM/ELMO strategy is an embedding method in which the chemically relevant part of the system is treated at the quantum mechanical level, while the rest is described through frozen ELMOs. Since the QM/ELMO method reproduces results of fully QM computations within chemical accuracy and with a much lower computational effort, it can be considered a suitable strategy to extend the range of applicability and accuracy of the QM/MM scheme. In this paper, other than briefly presenting the theoretical bases of the QM/ELMO/MM technique, we will also discuss its validation on the well-tested deprotonation of acetyl coenzyme A by aspartate in citrate synthase.

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.

The advanced treatment of hydrogen bonding in quantum crystallography

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.

Orbital-Free Quantum Crystallographic View on Noncovalent Bonding: Insights into Hydrogen Bonds, π⋅⋅⋅π and Reverse Electron Lone Pair⋅⋅⋅π Interactions

A detailed analysis of a complete set of the local potentials that appear in the Euler equation for electron density is carried out for noncovalent interactions in the crystal of a uracil derivative using experimental X-ray charge density. The interplay between the quantum theory of atoms in molecules and crystals and the local potentials and corresponding inner-crystal electronic forces of electrostatic and kinetic origin is explored. Partitioning of crystal space into atomic basins and atomic-like potential basins led us to the definite description of interatomic interaction and charge transfer. Novel physically grounded bonding descriptors derived within the orbital-free quantum crystallography provided the detailed examination of π-stacking and intricate C=O⋅⋅⋅π interactions and nonclassical hydrogen bonds present in the crystal. The donor-acceptor character of these interactions is revealed by analysis of Pauli and von Weizsäcker potentials together with well-known functions, e. g., deformation electron density and electron localization function. In this way, our analysis throws light on aspects of these closed-shell interactions hitherto hidden from the description.

Similarities and Differences between Crystal and Enzyme Environmental Effects on the Electron Density of Drug Molecules

The crystal interaction density is generally assumed to be a suitable measure of the polarization of a low-molecular weight ligand inside an enzyme, but this approximation has seldomly been tested and has never been quantified before. In this study, we compare the crystal interaction density and the interaction electrostatic potential for a model compound of loxistatin acid (E64c) with those inside cathepsin B, in solution, and in vacuum. We apply QM/MM calculations and experimental quantum crystallography to show that the crystal interaction density is indeed very similar to the enzyme interaction density. Less than 0.1 e are shifted between these two environments in total. However, this difference has non-negligible consequences for derived properties.

Crystallographic and Computational Electron Density of dx2-y2 Orbitals of Azo-Schiff Base Metal Complexes Using Conventional Programs

The crystal structures of two azobenzene derivative Schiff base metal complexes (new C₄₄H₄₀CuN₆O₂ of P-1 and known C₄₄H₃₈MnN₆O₇ of P2₁/c abbreviated as Cu and Mn, respectively) were (re-)determined experimentally using conventional X-ray analysis to obtain electron density using a PLATON program. Cu affords a four-coordinated square planar geometry, while Mn affords a hexa-coordinated distorted octahedral geometry whose apical sites are occupied by an acetate ion and water ligands, which are associated with hydrogen bonds. The π-π or CH-π and hydrogen bonding intermolecular interactions were found in both crystals, which were also analyzed using a Hirshfeld surface analysis program. To compare these results with experimental results, a density functional theory (DFT) calculation was also carried out based on the crystal structures to obtain calculated electron density using a conventional Gaussian program. These results revealed that the axial Mn-O coordination bonds of Mn were relatively weaker than the in-plane M-N or M-O coordination bonds.

Climbing Jacob's Ladder of Structural Refinement: Introduction of a Localized Molecular Orbital-Based Embedding for Accurate X-ray Determinations of Hydrogen Atom Positions

The positions of hydrogen atoms in molecules are fundamental in many aspects of chemistry. Nevertheless, most molecular structures are obtained from refinements of X-ray data exploiting the independent atom model (IAM), which uses spherical atomic densities and provides bond lengths involving hydrogen atoms that are too short compared to the neutron reference values. To overcome the IAM shortcomings, the wave function-based Hirshfeld atom refinement (HAR) method has been recently proposed, emerging as a promising strategy able to give element-hydrogen bond distances in excellent agreement with the neutron ones in terms of accuracy and precision. In this Letter, we propose a significant improvement of HAR based on the idea of describing the crystal environment explicitly in the underlying wave function calculation through a quantum mechanical embedding strategy that exploits extremely localized molecular orbitals. Test-bed refinements on a crystal structure characterized by strong intermolecular interactions are also discussed.

A Step toward the Quantification of Noncovalent Interactions in Large Biological Systems: The Independent Gradient Model-Extremely Localized Molecular Orbital Approach

The independent gradient model (IGM) is a recent electron density-based computational method that enables to detect and quantify covalent and noncovalent interactions. When applied to large systems, the original version of the technique still relies on promolecular electron densities given by the sum of spherically averaged atomic electron distributions, which leads to approximate evaluations of the inter- and intramolecular interactions occurring in systems of biological interest. To overcome this drawback and perform IGM analyses based on quantum mechanically rigorous electron densities also for macromolecular systems, we coupled the IGM approach with the recently constructed libraries of extremely localized molecular orbitals (ELMOs) that allow fast and reliable reconstructions of polypeptide and protein electron densities. The validation tests performed on small polypeptides and peptide dimers have shown that the novel IGM-ELMO strategy provides results that are systematically closer to the fully quantum mechanical ones and outperforms the IGM method based on the crude promolecular approximation, but still keeping a quite low computational cost. The results of the test calculations carried out on proteins have also confirmed the trends observed for the IGM analyses conducted on small systems. This makes us envisage the future application of the novel IGM-ELMO approach to unravel complicated noncovalent interaction networks (e.g., in protein-protein contacts) or to rationally design new drugs through molecular docking calculations and virtual high-throughput screenings.

HgH2 meets relativistic quantum crystallography. How to teach relativity to a non-relativistic wavefunction

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 HgH₂ molecule. 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.

Quantum Mechanics/Extremely Localized Molecular Orbital Embedding Strategy for Excited States: Coupling to Time-Dependent Density Functional Theory and Equation-of-Motion Coupled Cluster

The QM/ELMO (quantum mechanics/extremely localized molecular orbital) method is a recently developed embedding technique in which the most important region of the system under examination is treated at fully quantum mechanical level, while the rest is described by means of transferred and frozen extremely localized molecular orbitals. In this paper, we propose the first application of the QM/ELMO approach to the investigation of excited states, and, in particular, we present the coupling of the QM/ELMO philosophy with Time-Dependent Density Functional Theory (TDDFT) and Equation-of-Motion Coupled Cluster with single and double substitutions (EOM-CCSD). The proposed TDDFT/ELMO and EOM-CCSD/ELMO strategies underwent a series of preliminary tests that were already considered for the validation of other embedding methods for excited states. The obtained results showed that the novel techniques allow the accurate description of localized excitations in large systems by only including a relatively small number of atoms in the region treated at fully quantum chemical level. Furthermore, for TDDFT/ELMO, it was also observed that (i) the method enables to avoid the presence of artificial low-lying charge-transfer states that may affect traditional TDDFT calculations, even using functionals that do not take into account long-range corrections, and (ii) the novel approach can be also successfully exploited to investigate local electronic transitions in quite large systems (e.g., reduced model of the Green Fluorescent Protein), and the accuracy of the results can be improved by including a sufficient number of chemically crucial fragments/residues in the quantum mechanical region. Finally, concerning EOM-CCSD/ELMO, it was also seen that, despite the quite crude approximation of an embedding potential given by frozen extremely localized molecular orbitals, the new strategy is able to satisfactorily account for the effects of the environment. This work paves the way to further extensions of the QM/ELMO philosophy for the study of local excitations in extended systems, suggesting the coupling of the QM/ELMO approach with other quantum chemical strategies for excited states, from the simplest ΔSCF techniques to the most advanced and computationally expensive multireferences methods.

Accurate crystal structures and chemical properties from NoSpherA2

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.

Bonding and metastability for Group 12 dications

Electronic structure and bonding properties of the Group 12 dications M₂ ²⁺ (M = Zn, Cd, Hg) are investigated and electron density-derived quantities are used to characterize the metastability of these species. Of particular interest are the complementary descriptions afforded by the Laplacian of the electron density ∇² ρ(r) and the one-electron Bohm quantum potential (Q =  ∇ 2 ρ r / 2 ρ r ) along the bond path. Further, properties derived from the pair density including the localization-delocalization matrices (LDMs) and the interacting quantum atoms (IQA) energies are analyzed within the framework of the quantum theory of atoms in molecules (QTAIM). From the crossing points of the singlet (ground) and triplet (excited) potential energy curves, the barriers for dissociation (BFD) are estimated to be 25.2 kcal/mol (1.09 eV) for Zn₂ ²⁺ , 22.8 kcal/mol (0.99 eV) for Cd₂ ²⁺ , and 26.4 kcal/mol (1.14 eV) for Hg₂ ²⁺ . For comparison and benchmarking purposes, the case of N₂ ²⁺ is considered as a texbook example of metastability. At the equilibrium geometries, LDMs, which are used here as an electronic fingerprinting tool, discriminate and group together Group 12 M₂ ²⁺ from its isoelectronic Group 11 M₂ . While "classical" bonding indices are inconclusive in establishing regions of metastability in the bonding, it is shown that the one-electron Bohm quantum potential is promising in this regard.

On the accuracy and precision of X-ray and neutron diffraction results as a function of resolution and the electron density model

X-ray diffraction is the main source of three-dimensional structural information. In total, more than 1.5 million crystal structures have been refined and deposited in structural databanks (PDB, CSD and ICSD) to date. Almost 99.7% of them were obtained by approximating atoms as spheres within the independent atom model (IAM) introduced over a century ago. In this study, X-ray datasets for single crystals of hydrated α-oxalic acid were refined using several alternative electron density models that abandon the crude spherical approximation: the multipole model (MM), the transferable aspherical atom model (TAAM) and the Hirshfeld atom refinement (HAR) model as a function of the resolution of X-ray data. The aspherical models (MM, TAAM, HAR) give far more accurate and precise single-crystal X-ray results than IAM, sometimes identical to results obtained from neutron diffraction and at low resolution. Hence, aspherical approaches open new routes for improving existing structural information collected over the last century.

Orbital-free quantum crystallography: view on forces in crystals

Quantum theory of atoms in molecules and the orbital-free density functional theory (DFT) are combined in this work to study the spatial distribution of electrostatic and quantum electronic forces acting in stable crystals. The electron distribution is determined by electrostatic electron mutual repulsion corrected for exchange and correlation, their attraction to nuclei and by electron kinetic energy. The latter defines the spread of permissible variations in the electron momentum resulting from the de Broglie relationship and uncertainty principle, as far as the limitations of Pauli principle and the presence of atomic nuclei and other electrons allow. All forces are expressed via kinetic and DFT potentials and then defined in terms of the experimental electron density and its derivatives; hence, this approach may be considered as orbital-free quantum crystallography. The net force acting on an electron in a crystal at equilibrium is zero everywhere, presenting a balance of the kinetic F kin( r ) and potential forces F ( r ). The critical points of both potentials are analyzed and they are recognized as the points at which forces F kin( r ) and F ( r ) individually are zero (the Lagrange points). The positions of these points in a crystal are described according to Wyckoff notations, while their types depend on the considered scalar field. It was found that F ( r ) force pushes electrons to the atomic nuclei, while the kinetic force F kin( r ) draws electrons from nuclei. This favors formation of electron concentration bridges between some of the nearest atoms. However, in a crystal at equilibrium, only kinetic potential v kin( r ) and corresponding force exhibit the electronic shells and atomic-like zero-flux basins around the nuclear attractors. The force-field approach and quantum topological theory of atoms in molecules are compared and their distinctions are clarified.

Deep Learning the Hohenberg-Kohn Maps of Density Functional Theory

A striking consequence of the Hohenberg-Kohn theorem of density functional theory is the existence of a bijection between the local density and the ground-state many-body wave function. Here we study the problem of constructing approximations to the Hohenberg-Kohn map using a statistical learning approach. Using supervised deep learning with synthetic data, we show that this map can be accurately constructed for a chain of one-dimensional interacting spinless fermions in different phases of this model including the charge ordered Mott insulator and metallic phases and the critical point separating them. However, we also find that the learning is less effective across quantum phase transitions, suggesting an intrinsic difficulty in efficiently learning nonsmooth functional relations. We further study the problem of directly reconstructing complex observables from simple local density measurements, proposing a scheme amenable to statistical learning from experimental data.