X‐ray Constrained Spin‐Coupled Wavefunction: a New Tool to Extract Chemical Information from X‐ray Diffraction Data

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.

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.

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.

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.

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 Crystallography in the Last Decade: Developments and Outlooks

In this review article, we report on the recent progresses in the field of quantum crystallography that has witnessed a massive increase of production coupled with a broadening of the scope in the last decade. It is shown that the early thoughts about extracting quantum mechanical information from crystallographic experiments are becoming reality, although a century after prediction. While in the past the focus was mainly on electron density and related quantities, the attention is now shifting toward determination of wavefunction from experiments, which enables an exhaustive determination of the quantum mechanical functions and properties of a system. Nonetheless, methods based on electron density modelling have evolved and are nowadays able to reconstruct tiny polarizations of core electrons, coupling charge and spin models, or determining the quantum behaviour at extreme conditions. Far from being routine, these experimental and computational results should be regarded with special attention by scientists for the wealth of information on a system that they actually contain.

Localized Molecular Orbital-Based Embedding Scheme for Correlated Methods

Embedding strategies currently provide the best compromise between accuracy and computational cost in modeling chemical properties and processes of large and complex systems. In this framework, different methods have been proposed all over the years, from the very popular QM/MM approaches to the more recent and very promising density matrix and density functional embedding techniques. Here, we present a further development of the quantum mechanics/extremely localized molecular orbital technique (QM/ELMO) method, a recently proposed multiscale embedding strategy in which the chemically active region of the investigated system is treated at a fully quantum mechanical level, while the rest is described by frozen extremely localized molecular orbitals previously transferred from proper libraries or tailor-made model molecules. In particular, in this work we discuss and assess in detail the extension of the QM/ELMO approach to density functional theory and post-Hartree-Fock techniques by evaluating its performances when it is used to describe chemical reactions, bond dissociations, and intermolecular interactions. The preliminary test calculations have shown that, in the investigated cases, the new embedding strategy enables the results of the corresponding fully quantum mechanical computations to be reproduced within chemical accuracy in almost all the cases but with a significantly reduced computational cost, especially when correlated post-Hartree-Fock strategies are used to describe the quantum mechanical subsystem. In light of the obtained results, we already envisage the future application of the new correlated QM/ELMO techniques to the investigation of more challenging problems, such as the modeling of enzyme catalysis, the study of excited states of biomolecules, and the refinement of macromolecular X-ray crystal structures.

The Origin of the σ-Hole in Halogen Atoms: a Valence Bond Perspective

A detailed Valence Bond-Spin Coupled analysis of a series of halogenated molecules is here reported, allowing to get a rigorous ab initio demonstration of the qualitative models previously proposed to explain the origin of halogen bonding. The concepts of σ-hole and negative belt observed around the halogen atoms in the electrostatic potential maps are here interpreted by analysis of the relevant Spin Coupled orbitals.

On the use of the Obara–Saika recurrence relations for the calculation of structure factors in quantum crystallography

Modern methods of quantum crystallography are techniques firmly rooted in quantum chemistry and, as in many quantum chemical strategies, electron densities are expressed as two-centre expansions that involve basis functions centred on atomic nuclei. Therefore, the computation of the necessary structure factors requires the evaluation of Fourier transform integrals of basis function products. Since these functions are usually Cartesian Gaussians, in this communication it is shown that the Fourier integrals can be efficiently calculated by exploiting an extension of the Obara–Saika recurrence formulas, which are successfully used by quantum chemists in the computation of molecular integrals. Implementation and future perspectives of the technique are also discussed.

X-ray constrained spin-coupled technique: theoretical details and further assessment of the method

One of the well-established methods of modern quantum crystallography is undoubtedly the X-ray constrained wavefunction (XCW) approach, a technique that enables the determination of wavefunctions which not only minimize the energy of the system under examination, but also reproduce experimental X-ray diffraction data within the limit of the experimental errors. Initially proposed in the framework of the Hartree–Fock method, the strategy has been gradually extended to other techniques of quantum chemistry, but always remaining limited to a single-determinant ansatz for the wavefunction to extract. This limitation has been recently overcome through the development of the novel X-ray constrained spin-coupled (XCSC) approach [Genoni et al. (2018). Chem. Eur. J. 24, 15507–15511] which merges the XCW philosophy with the traditional spin-coupled strategy of valence bond theory. The main advantage of this new technique is the possibility of extracting traditional chemical descriptors (e.g. resonance structure weights) compatible with the experimental diffraction measurements, without the need to introduce information a priori or perform analyses a posteriori. This paper provides a detailed theoretical derivation of the fundamental equations at the basis of the XCSC method and also introduces a further advancement of its original version, mainly consisting in the use of molecular orbitals resulting from XCW calculations at the Hartree–Fock level to describe the inactive electrons in the XCSC computations. Furthermore, extensive test calculations, which have been performed by exploiting high-resolution X-ray diffraction data for salicylic acid and by adopting different basis sets, are presented and discussed. The computational tests have shown that the new technique does not suffer from particular convergence problems. Moreover, all the XCSC calculations provided resonance structure weights, spin-coupled orbitals and global electron densities slightly different from those resulting from the corresponding unconstrained computations. These discrepancies can be ascribed to the capability of the novel strategy to capture the information intrinsically contained in the experimental data used as external constraints.

Libraries of Extremely Localized Molecular Orbitals. 3. Construction and Preliminary Assessment of the New Databanks

The fast and reliable determination of wave functions and electron densities of macromolecules has been one of the goals of theoretical chemistry for a long time, and in this context, several linear scaling techniques have been successfully devised over the years. Different approaches have been adopted to tackle this problem, and one of them exploits the fact that, according to the traditional chemical perception, molecules can be seen as constituted of recurring units (e.g., functional groups) with well-defined chemical features. This has led to the development of methods in which the global wave functions or electron densities of macromolecules are obtained by simply transferring density matrices or fuzzy electron densities associated with molecular fragments. In this context, we propose an alternative strategy that aims at quickly reconstructing wave functions and electron densities of proteins through the transfer of extremely localized molecular orbitals (ELMOs), which are orbitals strictly localized on small molecular units and, for this reason, easily transferable from molecule to molecule. To accomplish this task we have constructed original libraries of ELMOs that cover all the possible elementary fragments of the 20 natural amino acids in all their possible protonation states and forms. Our preliminary test calculations have shown that, compared to more traditional methods of quantum chemistry, the transfers from the novel ELMO databanks allow to obtain wave function and electron densities of large polypeptides and proteins at a significantly reduced computational cost. Furthermore, notwithstanding expected discrepancies, the obtained electron distributions and electrostatic potentials are in very good agreement with those obtained at Hartree-Fock and density functional theory (DFT) levels. Therefore, the results encourage to use the new libraries as alternatives to the popular pseudoatom-databases of crystallography in the refinement of crystallographic structures of macromolecules. In particular, in this context, we have already envisaged the coupling of the ELMO databanks with the promising Hirshfeld atom refinement technique to extend the applicability of the latter to very large systems.