Effective molecular polarizabilities and crystal refractive indices estimated 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.

Theoretical investigation on the linear and nonlinear optical properties of DAPSH crystal

The linear polarizability, first and second hyperpolarizabilities of the asymmetric unit of DAPSH crystal are studied and compared with available experimental results. The polarization effects are included using an iterative polarization procedure, which ensures the convergence of the dipole moment of DAPSH embedded within a polarization field generated by the surrounding asymmetric units whose atomic sites are considered as point charges. We estimate macroscopic susceptibilities from the results of the polarized asymmetric units in the unit cell, considering the significant contribution of the electrostatic interactions in crystal packing. The results show that the influence of the polarization effects leads to a marked decrease of the first hyperpolarizability, compared with the respective isolated counterpart, which improves the concordance with the experiment. There is a minor influence of polarization effects on the second hyperpolarizability but our estimated result for the third-order susceptibility, related to the NLO process of the intensity dependent refractive index, is significant as compared with the results for other organic crystals, such as chalcone-derivatives. In addition, supermolecule calculations are conducted for explicit dimers in presence of the electrostatic embedding to illustrate the role played by the electrostatic interactions in the hyperpolarizabilities of the DAPSH crystal.

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.

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.

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.

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.

Crystal structure, vibrational frequencies and polarizability distribution in hydrogen-bonded salts of pyromellitic acid

Structural features of moderate-to-strong O—H...O hydrogen bonds are related to the frequencies of O—H stretching vibrations and to the electric polarizability distribution in the donor and acceptor functional groups for crystals synthesized from the 1,2,4,5-benzenetetracarboxylic (pyromellitic) acid, namely: bis(3-aminopyridinium) dihydrogen pyromellitate tetrahydrate, (1); bis(3-carboxypyridinium) dihydrogen pyromellitate, (2); bis(3-carboxyphenylammonium) dihydrogen pyromellitate dihydrate, (3); and bis(4-carboxyphenylammonium) dihydrogen pyromellitate, (4). A combination of single-crystal X-ray diffraction, powder Raman spectroscopy and first-principle calculations in both crystalline and gaseous phases has shown that changes in the O—H...O hydrogen-bond geometry can be followed by changes in the corresponding spectral modes. Vibrational properties of moderate hydrogen bonds can be estimated from correlations based on statistical analysis of several compounds [Novak (1974).Struct. Bond.18, 177–216]. However, frequencies related to very short O—H...O bonds can only be predicted by relationships built from a subset of structurally similar systems. Moreover, the way in which hydrogen bonds affect the polarizability of donor and acceptor groups depends on their strength. Moderate interactions enhance the polarizability and make it more anisotropic. Shorter hydrogen bonds may decrease the polarizability of a group as a consequence of the volume restraint implied by the neighbour molecule within a hydrogen-bonded aggregate. This is significant for evaluation of the electric susceptibility in crystals and, therefore, for estimation of refractive indices and birefringence.

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.

Is it Reasonable to Obtain Information on the Polarizability and Hyperpolarizability Only from the Electron Density?

Formulae for the static electronic polarizability and hyperpolarizability are derived in terms of moments of the ground-state electron density matrix by applying the Unsöld approximation and a generalization of the Fermi-Amaldi approximation. The latter formula for the hyperpolarizability appears to be new. The formulae manifestly transform correctly under rotations, and they are observed to be essentially cumulant expressions. Consequently, they are additive over different regions. The properties of the formula are discussed in relation to others that have been proposed in order to clarify inconsistencies. The formulae are then tested against coupled-perturbed Hartree-Fock results for a set of 40 donor-π-acceptor systems. For the polarizability, the correlation is reasonable; therefore, electron density matrix moments from theory or experiment may be used to predict polarizabilities. By constrast, the results for the hyperpolarizabilities are poor, not even within one or two orders of magnitude. The formula for the two- and three-particle density matrices obtained as a side result in this work may be interesting for density functional theories.

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.

Polarization effects on the electric properties of urea and thiourea molecules in solid phase

We present theoretical results for the dipole moment, linear polarizability, and first hyperpolarizability of the urea and thiourea molecules in solid phase. The in-crystal electric properties were determined by applying a supermolecule approach in combination with an iterative electrostatic scheme, in which the surrounding molecules are represented by point charges. It is found for both urea and thiourea molecules that the influence of the polarization effects is mild for the linear polarizability, but it is marked for the dipole moment and first hyperpolarizability. The replacement of oxygen atoms by sulfur atoms increases, in general, the electric responses. Our second-order Møller-Plesset perturbation theory based iterative scheme predicts for the in-crystal dipole moment of urea and thiourea the values of 7.54 and 9.19 D which are, respectively, increased by 61% and 58%, in comparison with the corresponding isolated values. The result for urea is in agreement with the available experimental result of 6.56 D. In addition, we present an estimate of macroscopic quantities considering explicit unit cells of urea and thiourea crystals including environment polarization effects. These supermolecule calculations take into account partially the exchange and dispersion effects. The results illustrate the role played by the electrostatic interactions on the static second-order nonlinear susceptibility of the urea crystal.

Charge density analysis for crystal engineering

This review reports on the application of charge density analysis in the field of crystal engineering, which is one of the most growing and productive areas of the entire field of crystallography. While methods to calculate or measure electron density are not discussed in detail, the derived quantities and tools, useful for crystal engineering analyses, are presented and their applications in the recent literature are illustrated. Potential developments and future perspectives are also highlighted and critically discussed. Graphical abstractᅟ

Modeling electron density distributions from X-ray diffraction to derive optical properties: constrained wavefunction versus multipole refinement

The rational design of next-generation optical materials requires an understanding of the connection between molecular structure and the solid-state optical properties of a material. A fundamental challenge is to utilize the accurate structural information provided by X-ray diffraction to explain the properties of a crystal. For years, the multipole refinement has been the workhorse technique for transforming high-resolution X-ray diffraction datasets into the detailed electron density distribution of crystalline material. However, the electron density alone is not sufficient for a reliable calculation of the nonlinear optical properties of a material. Recently, the X-ray constrained wavefunction refinement has emerged as a viable alternative to the multipole refinement, offering several potential advantages, including the calculation of a wide range of physical properties and seeding the refinement process with a physically reasonable starting point. In this study, we apply both the multipole refinement and the X-ray constrained wavefunction technique to four molecules with promising nonlinear optical properties and diverse structural motifs. In general, both techniques obtain comparable figures of merit and generate largely similar electron densities, demonstrating the wide applicability of the X-ray constrained wavefunction method. However, there are some systematic differences between the electron densities generated by each technique. Importantly, we find that the electron density generated using the X-ray constrained wavefunction method is dependent on the exact location of the nuclei. The X-ray constrained wavefunction refinement makes smaller changes to the wavefunction when coordinates from the Hartree-Fock-based Hirshfeld atom refinement are employed rather than coordinates from the multipole refinement, suggesting that coordinates from the Hirshfeld atom refinement allow the X-ray constrained wavefunction method to produce more accurate wavefunctions. We then use the experimentally derived wavefunctions to calculate the molecular dipole moment, polarizability, hyperpolarizability, and refractive index and show that these are in good agreement with the values calculated using ab initio methods. Thus, this study shows that experimental wavefunctions can be reliably generated from X-ray diffraction datasets, and that optical properties can be reliably calculated from these wavefunctions. Such a concerted interplay of experiment and computation via the X-ray constrained wavefunction refinement stands to enable the molecular engineering of tailor-made next-generation optical materials.

PolaBer: a program to calculate and visualize distributed atomic polarizabilities based on electron density partitioning

This paper describes the program PolaBer, which calculates atomic polarizability tensors from electric field perturbations of a partitioned electron density distribution. Among many possible partitioning schemes, PolaBer is currently using the quantum theory of atoms in molecules and it is interfaced to programs that apply such a partitioning. The calculation of the atomic tensors follows the idea suggested by Keith [The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, (2007), edited by C. F. Matta & R. J. Boyd. Weinheim: Wiley-VCH], which enables the removal of the intrinsic origin dependence of the atomic charge contributions to the molecular dipole moment. This scheme allows the export, within chemically equivalent functional groups, of properties calculated from atomic dipoles, such as for example the atomic polarizabilities. The software permits visualization of the tensors and calculation of straightforward optical properties of a molecule (like the molar refractive index) or a crystal (assuming the molecule in a given crystal lattice).

X-ray diffraction, vibrational and quantum chemical investigations of 2-methyl-4-nitroanilinium trichloroacetate trichloroacetic acid

The structural investigations of the molecular complex of 2-methyl-4-nitroaniline with trichloroacetic acid, namely 2-methyl-4-nitroanilinium trichloroacetate trichloroacetic acid (C(11)H(10)Cl(6)N(2)O(6)) have been performed by means of single crystal and powder X-ray diffraction method. The complex was formed with accompanying proton transfer from trichloroacetic acid molecule to 2-methyl-4-nitroaniline. The studied crystal is built up of singly protonated 2-methyl-4-nitroanilinium cations, trichloroacetate anions and neutral trichloroacetic acid molecules. The crystals are monoclinic, space group P2(1)/c, with a=14.947Å, b=6.432Å, c=19.609Å and Z=4. The vibrational assignments and analysis of 2-methyl-4-nitroanilinium trichloroacetate trichloroacetic acid have also been performed by FTIR, FT-Raman and far-infrared spectral studies. More support on the experimental findings were added from the quantum chemical studies performed with DFT (B3LYP) method using 6-31G, cc-pVDZ, 6-31G and 6-31++G basis sets. The structural parameters, energies, thermodynamic parameters and the NBO charges of 2M4NATCA were also determined by the DFT methods.

Refractive indices for molecular crystals from the response of X-ray constrained Hartree-Fock wavefunctions

Refractive indices for molecular crystals are obtained from Hartree-Fock wavefunctions constrained to reproduce a set of experimental X-ray structure factors. Coupled-perturbed Hartree-Fock theory is used to calculate the in-crystal effective polarizabilities from which the refractive indices are obtained, thus eliminating the need for the calibration procedure used in earlier work by Whitten et al. [J. Chem. Phys. 2006, 125, 174505]. The results clearly demonstrate that these X-ray constrained Hartree-Fock (XCHF) wavefunctions reflect genuine effects of intermolecular interactions in crystals. Molecular dipole moments are consistently in excellent agreement with ab initio MP2 estimates that incorporate the effects of the crystal field. Consistent agreement of the XCHF refractive indices with experimental measurements at optical frequencies confirms that this approach can provide both meaningful results and considerable insight into the relative importance of molecular properties and crystal field effects in determining the detailed nature of the refractivity tensor.