Density-matrix refinement for molecular crystals

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.

X-ray molecular orbital analysis. II. Application to diformohydrazide, (NHCHO)2

The molecular orbitals (MOs) of diformohydrazide have been determined from the electron density measured by X-ray diffraction. The experimental and refinement procedures are explained in detail and the validity of the obtained MOs is assessed from the crystallographic point of view. The X-ray structure factors were measured at 100 K by a four-circle diffractometer avoiding multiple diffraction, the effect of which on the structure factors is comparable to two-centre structure factors. There remained no significant peaks on the residual density map and the R factors reduced significantly. Among the 788 MO coefficients, 731 converged, of which 694 were statistically significant. The C-H and N-H bond distances are 1.032 (2) and 1.033 (3) Å, respectively. The electron densities of theoretical and experimental MOs and the differences between them are illustrated. The overall features of the electron density obtained by X-ray molecular orbital (XMO) analysis are in good agreement with the canonical orbitals calculated by the restricted Hartree Fock (RHF) method. The bonding-electron distribution around the middle of each bond is well represented and the relative phase relationships of the π orbitals are reflected clearly in the electron densities on the plane perpendicular to the molecular plane. However, differences are noticeable around the O atom on the molecular plane. The orbital energies obtained by XMO analysis are about 0.3 a.u. higher than the corresponding canonical orbitals, except for MO10 to MO14 which are about 0.7 a.u. higher. These exceptions are attributed to the N-H...O'' intermolecular hydrogen bond, which is neglected in the MO models of the present study. The hydrogen bond is supported by significant electron densities at the saddle points between the H(N) and O'' atoms in MO7, 8, 14 and 17, and by that of O''-p extended over H(N) in MO21 and 22, while no peaks were found in MO10, 11, 13 and 15. The electron density of each MO clearly exhibits its role in the molecule. Consequently, the MOs obtained by XMO analysis give a fundamental quantum mechanical insight into the real properties of molecules.

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.

Quantum Crystallography: Current Developments and Future Perspectives

Crystallography and quantum mechanics have always been tightly connected because reliable quantum mechanical models are needed to determine crystal structures. Due to this natural synergy, nowadays accurate distributions of electrons in space can be obtained from diffraction and scattering experiments. In the original definition of quantum crystallography (QCr) given by Massa, Karle and Huang, direct extraction of wavefunctions or density matrices from measured intensities of reflections or, conversely, ad hoc quantum mechanical calculations to enhance the accuracy of the crystallographic refinement are implicated. Nevertheless, many other active and emerging research areas involving quantum mechanics and scattering experiments are not covered by the original definition although they enable to observe and explain quantum phenomena as accurately and successfully as the original strategies. Therefore, we give an overview over current research that is related to a broader notion of QCr, and discuss options how QCr can evolve to become a complete and independent domain of natural sciences. The goal of this paper is to initiate discussions around QCr, but not to find a final definition of the field.

X-ray molecular orbital analysis. I. Quantum mechanical and crystallographic framework

Molecular orbitals were obtained by X-ray molecular orbital analysis (XMO). The initial molecular orbitals (MOs) of the refinement were calculated by the ab initio self-consistent field (SCF) MO method. Well tempered basis functions were selected since they do not produce cusps at the atomic positions on the residual density maps. X-ray structure factors calculated from the MOs were fitted to observed structure factors by the least-squares method, keeping the orthonormal relationship between MOs. However, the MO coefficients correlate severely with each other, since basis functions are composed of similar Gaussian-type orbitals. Therefore, a method of selecting variables which do not correlate severely with each other in the least-squares refinement was devised. MOs were refined together with the other crystallographic parameters, although the refinement with the atomic positional parameters requires a lot of calculation time. The XMO method was applied to diformohydrazide, (NHCHO)₂, without using polarization functions, and the electron-density distributions, including the maxima on the covalent bonds, were represented well. Therefore, from the viewpoint of X-ray diffraction, it is concluded that the MOs averaged by thermal vibrations of the atoms were obtained successfully by XMO analysis. The method of XMO analysis, combined with X-ray atomic orbital (AO) analysis, in principle enables one to obtain MOs or AOs without phase factors from X-ray diffraction experiments on most compounds from organic to rare earth compounds.

Multi-temperature study of potassium uridine-5'-monophosphate: electron density distribution and anharmonic motion modelling

Uridine, a nucleoside formed of a uracil fragment attached to a ribose ring via a β-N1-glycosidic bond, is one of the four basic components of ribonucleic acid. Here a new anhydrous structure and experimental charge density distribution analysis of a uridine-5'-monophosphate potassium salt, K(UMPH), is reported. The studied case constitutes the very first structure of a 5'-nucleotide potassium salt according to the Cambridge Structural Database. The excellent crystal quality allowed the collection of charge density data at various temperatures, i.e. 10, 100, 200 and 300 K on one single crystal. Crystal structure and charge density data were analysed thoroughly in the context of related literature-reported examples. Detailed analysis of the charge density distribution revealed elevated anharmonic motion of part of the uracil ring moiety relatively weakly interacting with the neighbouring species. The effect was manifested by alternate positive and negative residual density patterns observed for these atoms, which `disappear' at low temperature. It also occurred that the potassium cation, quite uniformly coordinated by seven O atoms from all molecular fragments of the UMPH^- anion, including the O atom from the ribofuranose ring, can be treated as spherical in the charge density model which was supported by theoretical calculations. Apart from the predominant electrostatic interactions, four relatively strong hydrogen bond types further support the stability of the crystal structure. This results in a compact and quite uniform structure (in all directions) of the studied crystal, as opposed to similar cases with layered architecture reported in the literature.

Quantum crystallography

Approximate wavefunctions can be improved by constraining them to reproduce observations derived from diffraction and scattering experiments. Conversely, charge density models, incorporating electron-density distributions, atomic positions and atomic motion, can be improved by supplementing diffraction experiments with quantum chemically calculated, tailor-made electron densities (form factors). In both cases quantum chemistry and diffraction/scattering experiments are combined into a single, integrated tool. The development of quantum crystallographic research is reviewed. Some results obtained by quantum crystallography illustrate the potential and limitations of this field.

Libraries of Extremely Localized Molecular Orbitals. 2. Comparison with the Pseudoatoms Transferability

Due to both technical and methodological difficulties, determining and analyzing charge densities of very large molecular systems represents a serious challenge that, in the crystallographers community, has been mainly tackled by observing that the so-called pseudoatoms of the electron density multipole expansions are reliably transferable from molecule to molecule. This has led to the construction of pseudoatoms databanks that have allowed successful refinements of crystallographic structures of macromolecules, while taking into account their corresponding reconstructed electron distributions. A recent alternative/complement to the previous approach is represented by techniques based on extremely localized molecular orbitals (ELMOs) that, due to their strict localization on small molecular fragments (e.g., atoms, bonds, and functional groups), are also in principle exportable from system to system. The ELMOs transferability has been already tested in detail, and, in this work, it has been compared to the one of the pseudoatoms. To accomplish this task, electron distributions obtained both through the transfer of pseudoatoms and through the transfer of extremely localized molecular orbitals have been analyzed, especially taking into account topological properties and similarity indexes. The obtained results indicate that all the considered reconstruction methods give completely reasonable and similar charge densities, and, consequently, the new ELMOs libraries will probably represent new useful tools not only for refining crystal structures but also for computing approximate electronic properties of very large molecules.

Molecular Orbitals Strictly Localized on Small Molecular Fragments from X-ray Diffraction Data

Nowadays, the electron density is recognized as a fundamental property that contains most of the information concerning the electronic structure of molecules, and, therefore, its determination from high-resolution X-ray diffraction data is becoming more and more important. In this context, we propose a new strategy for the charge density analysis, strategy in which the chemical interpretability of the multipole model is combined with the quantum mechanical rigor of the wave function-based approaches. In particular, this novel technique aims at extracting molecular orbitals strictly localized on small molecular fragments (e.g., atoms, bonds, or functional groups) from a set of measured structure factors amplitudes. Preliminary tests have shown that their determination is really straightforward and, given their reliable transferability, we envisage the possibility of constructing new extremely localized molecular orbital databases as an alternative to the existing pseudoatom libraries.

Novel properties from experimental charge densities: an application to the zwitterionic neurotransmitter taurine

The charge distribution of taurine (2-aminoethane-sulfonic acid) is revisited by using an orbital-based method that describes the density in a fixed molecular orbital basis with variable orbital occupation numbers. A new neutron data set is also employed to explore whether this improves the deconvolution of thermal motion and charge density. A range of molecular properties that are novel for experimentally determined charge densities are computed, including Weinhold population analysis, Mayer bond orders, and local kinetic energy densities, in addition to charge topological analysis and quantum theory of atoms-in-molecules (QTAIM) integrated properties. The ease with which a distributed multipole analysis can be performed on the fitted density matrix makes it straightforward to compute molecular moments, the lattice energy, and the electrostatic interaction energies of molecules removed from the crystal. Results are compared with high-level (QCISD) gas-phase calculations and band structure calculations employing density functional theory. Finally, the avenues available for extending the range of molecular properties that can be calculated from experimental charge densities still further using this approach are discussed.