The connubium between crystallography and quantum mechanics

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.

Applicability of transferable multipole pseudo-atoms for restoring inner-crystal electronic force density fields. Chemical bonding and binding features in the crystal and dimer of 1,3-bis(2-hydroxyethyl)-6-methyluracil

We considered it timely to test the applicability of transferable multipole pseudo-atoms for restoring inner-crystal electronic force density fields. The procedure was carried out on the crystal of 1,3-bis(2-hydroxyethyl)-6-methyluracil, and some derived properties of the scalar potential and vector force fields were compared with those obtained from the experimental multipole model and from the aspherical pseudo-atom model with parameters fitted to the calculated structure factors. The procedure was shown to accurately replicate the general vector-field behavior, the peculiarities of the quantum potentials and the characteristics of the force-field pseudoatoms, such as charge, shape and volume, as well as to reproduce the relative arrangement of atomic and pseudoatomic zero-flux surfaces along internuclear regions. It was found that, in addition to the quantum-topological atoms, the force-field pseudoatoms are spatially reproduced within a single structural fragment and similar environment. In addition, the classical and nonclassical hydrogen bonds in the uracil derivative crystal, as well as the H...O, N...O and N...C interactions in the free π-stacked dimer of the uracil derivative molecules, were studied using the potential and force fields within the concepts of interatomic charge transfer and electron lone pair donation-acceptance. Remarkably, the nitrogen atoms in the N...O and N...C interactions behave rather like a Lewis base and an electron contributor. At the same time, the hydrogen atom in the H...O interaction, being a Lewis acid, also participates in the interatomic electron transfer by acting as a contributor. Thus, it has been argued that, when describing polar interatomic interactions within orbital-free considerations, it makes more physical sense to identify electronegative (electron occupier) and electropositive (electron contributor) atoms or subatomic fragments rather than nucleophilic and electrophilic sites.

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.

Multipolar Atom Types from Theory and Statistical Clustering (MATTS) Data Bank: Impact of Surrounding Atoms on Electron Density from Cluster Analysis

The multipole model (MM) uses an aspherical approach to describe electron density and can be used to interpret data from X-ray diffraction in a more accurate manner than using the spherical approximation. The MATTS (multipolar atom types from theory and statistical clustering) data bank gathers MM parameters specific for atom types in proteins, nucleic acids, and organic molecules. However, it was not fully understood how the electron density of particular atoms responds to their surroundings and which factors describe the electron density in molecules within the MM. In this work, by applying clustering using descriptors available in the MATTS data bank, that is, topology and multipole parameters, we found the topology features with the biggest impact on the multipole parameters: the element of the central atom, the number of first neighbors, and planarity of the group. The similarities in the spatial distribution of electron density between and within atom type classes revealed distinct and unique atom types. The quality of existing types can be improved by adding better parametrization, definitions, and local coordinate systems. Future development of the MATTS data bank should lead to a wider range of atom types necessary to construct the electron density of any molecule.

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.

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.

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.

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.