The X-ray restrained/constrained wavefunction (XRW/XCW) fitting approach [1, 2] is one of the well-established methods of modern quantum crystallography [3-5], an approach that allows the determination of wavefunctions from experimental X-ray diffraction measurements by minimizing a functional (i.e., the so-called Jayatilaka functional) given by the sum of the electronic energy of the system under exam and of the statistical agreement between experimental and computed structure factor amplitudes.

Initially proposed in the framework of the restricted Hartree-Fock formalism [1, 2], over the years the method has been gradually extended to other techniques of quantum chemistry (e.g., unrestricted Hartree-Fock formalism, density functional theory, relativistic methods and extremely localized molecular orbital (ELMO) strategy), but practically remaining always limited to a single Slater determinant ansatz for the wavefunction to be determined.

In this presentation, two recent XRW fitting techniques that went beyond the previous limitation will be discussed: the X-ray restrained extremely localized molecular orbital – valence bond (XR-ELMO-VB) method [6, 7] and the X-ray restrained spin-coupled (XRSC) strategy [8, 9]. Both of them are multi-determinant XRW approaches and, above all, they are strongly rooted in Valence Bond theory, thus allowing the extraction of traditional chemical descriptors (e.g., weights of resonance structures) compatible with measured X-ray diffraction data

The XR-ELMO-VB technique, which can be considered as the first prototype multi-determinant XRW approach, exploits the use of pre-computed and frozen ELMOs to define the basis of pre-determined Slater determinants on which expanding the desired wavefunction [6]. The method is particularly useful to investigate systems characterized by a multi-reference character. The performed calculations have also clearly shown that the technique is able to reveal how the relative importance of resonance structures changes when structure factors measured at different conditions (e.g., at ambient or high pressure) are used as external restraints [7].

The XCSC method can be considered a step forward compared to the XR-ELMO-VB strategy. In fact, through the coupling of the XRW philosophy with the spin-coupled technique of Valence Bond theory, the approach enables not only the determination of resonance structure weights, but also the optimization of the so-called spin-coupled orbitals, which are quite localized and allow us to shed light on the spatial rearrangements of the electronic clouds and the hybridization of atoms [8, 9]. Preliminary tests have shown that XCSC computations can provide resonance structure weights, spin-coupled orbitals and global electron density distributions that are different from those obtained through corresponding gas-phase calculations [9]. These differences are probably due to the capability of the X-ray restrained spin-coupled approach in getting information contained in the experimental data employed in the computations (e.g., correlation and crystal filed effects).

References:

[1] Jayatilaka, D. (1998). Phys. Rev. Lett. 80, 798.

[2] Jayatilaka, D. & Grimwood, D. J. (2001). Acta Cryst. A57, 76.

[3] Genoni, A., Bučinský, L., Claiser, N., Contreras-García, J., Dittrich, B.; Dominiak, P. M., Espinosa, E., Gatti, C., Giannozzi, P., Gillet, J.-M., Jayatilaka, D., Macchi, P., Madsen, A. Ø., Massa, L. J., Matta, C. F., Merz, K. M. Jr., Nakashima, P. N. H., Ott, H., Ryde, U., Schwarz, K., Sierka, M. & Grabowsky, S. (2018). Chem. Eur. J. 24, 10881.

[4] Grabowsky, S., Genoni, A. & Bürgi, H.-B. (2017). Chem. Sci. 8, 4159.

[5] Genoni, A. & Macchi, P. (2020) Crystals 10, 473.

[6] Genoni, A. (2017). Acta Cryst. A73, 312.

[7] Casati, N., Genoni, A., Meyer, B., Krawczuk, A. & Macchi, P. (2017). Acta Cryst. B73, 584.

[8] Genoni, A., Franchini, D., Pieraccini, S. & Sironi, M. (2018). Chem. Eur. J. 24, 15507.

[9] Genoni, A., Macetti, G., Franchini, D., Pieraccini, S. & Sironi, M. (2019). Acta Cryst. A75, 778.

Acknowledgments:

The French Research Agency (ANR) is gratefully acknowledged for financial support of this work through the Young Investigator Project “QuMacroRef” (Grant No. ANR-17-CE29-0005-01).

We are developing in the CRM2 laboratory a new software: Mollynx. As MoPro or XD, Mollynx is derived from Molly (Hansen and Coppens, 1978) but allows to differentiate the electron spins. This new algorithm has been successively applied to paramagnetic coordination compounds (Deutsch et al., 2012, Deutsch et al., 2014) to organic radicals (Voufack et al., 2017) and to small unit cells inorganic crystals (Voufack et al., 2019).

A more general model based on atomic orbitals (Tanaka, 1988; Tanaka, 1993; Bytheway et al., 2001; Schweitzer, 2006), has also been coded in the Mollynx software. This model should allow calculation of properties derived from the atomic wave functions such as covalency, populations of atomic orbitals, energy, optical properties and can in principle describe bonded pair of atoms with orbitals centered on different atoms (two centres orbital products). This model, extended to spin resolved orbitals, has been applied to the YTiO₃ perovskite (figure 1, Kibalin et al., 2021) The radial extension, orientation and population of outer atomic orbitals for each atom have been modelled leading to a clear description of the bonding in this crystal .

Thus Mollynx can refine a spin resolved electron density model based on multipolar or on orbital approach. This presentation will describe some of these results and will focus on the experimental spin resolved atomic orbitals model obtained on the YTiO₃ including a comparison with a refinement based on theoretical structure factors calculated using density functional theory and the WIEN2k code (Blaha et al., 2020).

Blaha, P.; Schwarz, K; Tran, F.; Laskowski, R; Madsen G. K. H. and Marks, L. D., WIEN2k: An APW+lo program for calculating the properties of solids editors-pick, J. Chem. Phys. 152, 074101 (2020); https://doi.org/10.1063/1.5143061

Bytheway, I., Figgis, B. N. & Sobolev, A. N., 2001. Charge density in Cu(glygly)(OH2)2·H2O at 10 K and the reproducibility of atomic orbital populations. J. Chem. Soc., Dalton Trans., Volume 22, pp. 3285-3294.

Deutsch, M., Claiser, N., Pillet, S., Ciumacov, Y., Becker, P. J., Gillon, B., Gillet, J.-M., Lecomte, C. & Souhassou, M. (2012). Acta Cryst. A 68, 675-686.

Deutsch, M., Gillon, B., Claiser, N., Gillet, J.-M., Lecomte, C. & Souhassou, M. (2014). IUCR J, 194-199.

Hansen, N. K. & Coppens, P. (1978). Acta Cryst. A34, 909–921.

Kibalin I., Voufack A.B., Souhassou M., Gillon B., Gillet J.M., Claiser N., Gukasov A., Porcher F. and Lecomte C. (2021) Acta Cryst A77, 96-104

Schweizer, J. (2006). Chapter 4: Polarized Neutrons and Polarization analysis, in Neutron Scattering from magnetic materials. Ed. T. Chatterji, Elsevier

Voufack, A. B. et al., 2017. When combined X-ray and polarized neutron diffraction data challenge high-level calculations: spin-resolved electron density of an organic radical. Acta Cryst., Volume B73, pp. 544-549.

Voufack, A. B. et al., 2019. Spin resolved electron density study of YTiO3 in its ferromagnetic phase: signature of orbital ordering. IUCrJ, 6(5), pp. 884-894.

Tanaka K., (1993). Acta Cryst., B49, 1001-1010.

Tanaka, K., (1988) Acta Cryst., A44, 1002-1008.

This work retraces different methods that have been explored to account for the atomic thermal motion in the reconstruction of one-electron reduced density matrices from experimental X-ray structure factors (XSF) and directional Compton profiles (DCP).
Attention has been paid to propose the simplest possible model, which obeys the necessary N-representability conditions, while accurately reproducing all available experimental data.
The deconvolution of thermal effects makes it possible to obtain an experimental static density matrix, which can directly be compared with theoretical 1-RDM. It is found that above a 1% statistical noise, the role played by Compton scattering data becomes negligible and no accurate 1-RDM becomes reachable.
Since no thermal 1-RDM is available as a reference, the quality of an experimentally derived temperature-dependent matrix is difficult to assess. However, the accuracy of the obtained static 1-RDM is strong evidence that the Semi-Definite Programming method is robust and well-adapted to the reconstruction of an experimental dynamical 1-RDM.

Several physico-chemical properties of materials and biological molecules crucially depend on the hydrogen atom positions. Therefore, obtaining reliable three-dimensional structures of molecules and materials is a crucial step to have accurate results.

Unfortunately, the weak X-rays scattering power of hydrogen atoms makes them usually very hard to detect accurately. Besides, the choice of the refinement method has also a strong influence. The most widely used approach, the independent atom model (IAM), describes the total electron density as the sum of spherical atomic densities centred on the nuclei. This approximation evidently fails when applied to hydrogen atoms because their only electron is delocalized in forming a bond. As a result, bonds lengths involving hydrogen atoms are generally too short.

Over the years, different methods have been proposed to overcome this drawback. Within the field of quantum crystallography, the Hirshfeld atom refinement (HAR) approach is one of the most promising strategies [1]. HAR is a technique exploiting fully quantum mechanical calculations to obtain tailor-made ab initio electron densities, which are partitioned into aspherical atomic contributions to fit experimental structure factors without further approximations. HAR is able to provide X-H bond distances that are in very good agreement with those obtained from neutron diffraction experiments [1-3]. Moreover, the introduction of crystal environment effects is crucial to carry out better refinements, especially when strong intermolecular interactions are present. This is usually done by adding point charges at symmetry-related atomic positions around the selected reference crystal unit [1-3].

In this contribution, we introduce an improvement to the description of crystal field effects in HAR exploiting the recently developed quantum mechanics/extremely localized molecular orbitals (QM/ELMO) technique [4,5]. In our new modified version of HAR, the reference crystal unit is treated variationally at quantum mechanical level as in the traditional HAR, while the symmetry-related crystal units are described using pre-computed frozen extremely localized molecular orbitals [6]. The ELMOs contribution describing the crystal environment is included in the Hamiltonian of the reference crystal unit through an electrostatic embedding.

Other than discussing the theoretical framework at the basis of the new strategy, we will show test refinements performed on the xylitol crystal, a system characterized by an extended network of strong hydrogen bonds. The results show that the new ELMO-embedded HAR approach gives bond lengths involving hydrogen atoms in optimal agreement with neutron results, outperforming not only the traditional HAR but also the charge-embedded HAR technique in practically all the cases [6].

Given the promising results, we envisage the application of the new ELMO-embedded HAR technique to refine structures of crystals with strong intermolecular interactions. However, further test-bed refinements will be necessary to draw final conclusions, also considering other aspects such as basis-sets and theoretical methods dependence.

[1] Jayatilaka, D. & Dittrich, B. (2008). Acta Cryst. A64, 383.

[2] Woińska, M., Grabowsky, S., Dominiak, P. M., Woźniak, K. & Jayatilaka, D. (2016). Sci. Adv. 2, e1600192.

[3] Capelli, S., Bürgi, H.-B., Dittrich, B., Grabowsky, S. & Jayatilaka, D. (2014). IUCrJ 1, 361.

[4] Macetti, G. & Genoni, A. (2019). J. Phys. Chem. A 123, 9420.

[5] Macetti, G, Wieduwilt, E. K., Assfeld, X. & Genoni, A. (2020). J. Chem. Theory Comput. 16, 3578.

[6] Wieduwilt, E. K., Macetti, G. & Genoni, A. (2021). J. Phys. Chem. Lett. 12, 463.

Relativistic effects in chemistry manifest themselves in many ways and influence various physical and chemical properties of materials. The well-known of them is the yellow color of gold or the high voltage of the lead-acid car battery^1,2. Therefore, a description of these effects is of great importance for a better understanding of the chemistry of heavy atoms.

A perspective method is quantum crystallography that relies on the high-resolution and high-quality XRD data to describe crystal structure in unprecedented detail^3-4. Intensities of the diffracted beam are affected not only by relativistic effects but also by absorption⁵, anharmonic motion⁶, anomalous dispersion⁷, and many other effects which highly influence electron density distribution in the crystal and, in consequence, derived properties.

We collected the data sets for the chloro(triphenylphosphine)gold(I) and di(triphenylphosphine)mercury(II) nitrate using Mo X-ray source at 100K, where the data sets for (3-(4-chlorophenyl)-3-oxoprop-1-yn-1-yl)(triphenylphosphine)gold(I) and chloro(dimethylsulfide) gold(I) were collected using synchrotron radiation at 80K.

Here, we present the results of relativistic Hirshfeld atom refinements⁸ carried out as implemented in NoSpherA2⁹ for high-resolution X-ray diffraction data sets. The outcome of DFT-based refinements with the nonrelativistic and quasi-relativistic approaches will be compared, including analysis of the influence of disorder on relativistic effects, description of aurophilic interactions, and the nature of the Me–X bonds in Au and Hg crystals.

Acknowledgment:

Support of this work by the National Science Centre, Poland through grant PRELUDIUM no. UMO-2018/31/N/ST4/02141 is gratefully acknowledged.

The experiment was carried out at the Spring-8 with the approval of the Japan Synchrotron Radiation Research Institute (Proposal Number 2019A1069).

References:

1. Pyykko, P. (1988). Chemical Reviews, 88, 563–594.

2. Yatsimirskii, K. B. (1995) Theoretical and Experimental Chemistry, 31, 153–168.

3. L. J. Farrugia, C. Evans, D. Lentz and M. Roemer, Journal of the American Chemical Society, 2009, 131, 1251–1268.

4. T. S. Koritsanszky and P. Coppens, Chem. Rev., 2001, 101, 1583–1628.

5. J. Als‐Nielsen and D. McMorrow, in Elements of Modern X-ray Physics, John Wiley & Sons, Ltd, 2011, pp. 1–28.

6. R. Herbst-Irmer, J. Henn, J. J. Holstein, C. B. Hübschle, B. Dittrich, D. Stern, D. Kratzert and D. Stalke, The Journal of Physical Chemistry A, 2013, 117, 633–641.

7. S. Caticha-Ellis, Anomalous dispersion of x-rays in crystallography, University College Cardiff Press, Cardiff, Wales, 1981.

8. Bučinský, L., Jayatilaka, D., Grabowsky, S. (2016) The Journal of Physical Chemistry A, 120, 6650–6669.

9. Kleemiss, F.; Dolomanov, O. V.; Bodensteiner, M.; Peyerimhoff, N.; Midgley, L.; Bourhis, L. J.; Genoni, A.; Malaspina, L. A.; Jayatilaka, D.; Spencer, J. L.; White, F.; Grundkötter-Stock, B.; Steinhauer, S.; Lentz, D.; Puschmann, H.; Grabowsky, S. (2021) Chem. Sci., 12, 1675-1692.