Lattice thermal expansion and anisotropic displacements in 𝜶-sulfur from diffraction experiments and first-principles theory

Rigid covalent bond ofα-sulfur investigated via temperature-dependent EXAFS

This study performs extended x-ray absorption fine structure (EXAFS) measurements for the S-K edge in the temperature range of 10 and 300 K in the transmission mode using a photodiode to detect the transmitted x-rays. It provides the first report of temperature variations in the structural parameters ofα-S. As the temperature increases from 10 to 300 K in the Fourier transform ofkχ(k)the first peak corresponding to the covalent bond of the eight-membered ring becomes slightly low anomalously despite thermal disturbances. However, as in normal materials, the second peak at 300 K decreases to approximately half of that at 10 K, which contains several intra- and inter-ring correlations. All structural parameters of the covalent bond obtained by nonlinear least squares fitting exhibit missing temperature variations. A value of zero for the asymmetric parameter in the EXAFS (C3) implies that the potential of the covalent bond is symmetric, and the constant value of the mean square relative displacement (MSRD) with temperature implies that the potential is extremely high. The Einstein model fitting for the temperature variation in the MSRD yields an Einstein temperature of 942 K and force constant (K) of 405 N m-1. The value ofKis the largest among those of chalcogen elements.

Temperature-Resolved Anisotropic Displacement Parameters from Theory and Experiment: A Case Study

Anisotropic displacement parameters (ADPs) for an organopalladium complex were obtained from synchrotron diffraction data between 100 and 250 K and compared to the results from first-principles calculations at the harmonic approximation. Calculations and experiments agree with respect to the orientation of displacement ellipsoids and hence the directionality of atomic movement, but the harmonic approximation underestimates the amplitudes of motion by about 20%. This systematic but modest underestimation can only be reliably detected with a high-quality experimental benchmark at hand. Our experiments comprised diffraction data at 20 K intervals from 130–250 K on the same crystal. An additional high-resolution data set was collected at 100 K on a second crystal and underlined the robustness of our approach with respect to the individual sample, resolution, and instrumentation. In the temperature range relevant for our study and for many diffraction experiments, the discrepancy between experimentally determined and calculated displacement appears as an almost constant temperature offset. The systematic underestimation of harmonic theory can be accounted for by calculating the ADPs for a temperature 20 K higher than that of the actual diffraction. This entirely empirical “+20 K rule” lacks physical relevance but may pave the way for application in larger systems where a more reliable quasi-harmonic approximation remains computationally demanding or even entirely unaffordable.

Thermal properties of energetic materials from quasi-harmonic first-principles calculations

The structure and properties at a finite temperature are critical to understand the temperature effects on energetic materials (EMs). Combining dispersion-corrected density functional theory with quasi-harmonic approximation, the thermodynamic properties for several representative EMs, including nitromethane, PETN, HMX, and TATB, are calculated. The inclusion of zero-point energy and temperature effect could significantly improve the accuracy of lattice parameters at ambient condition; the deviations of calculated cell volumes and experimental values at room temperature are within 0.62%. The calculated lattice parameters and thermal expansion coefficients with increasing temperature show strong anisotropy. In particular, the expansion rate (2.61%) of inter-layer direction of TATB is higher than intra-layer direction and other EMs. Furthermore, the calculated heat capacities could reproduce the experimental trends and enrich the thermodynamic data set at finite temperatures. The predicted isothermal and adiabatic bulk moduli could reflect the softening behavior of EMs. These results would fundamentally provide a deep understanding and serve as a reference for the experimental measurement of the thermodynamic parameters of EMs.

Displacement parameters from density-functional theory and their validation in the experimental charge density of tartaric acid

Advanced theory matches advanced experiment: anisotropic displacement parameters for tartaric acid have been calculated in the quasi-harmonic approximation and determined experimentally based on a charge density study.

Sublimation Properties of α,ω-Diamines Revisited from First-Principles Calculations

Sublimation enthalpies of alkane-α,ω-diamines exhibit an odd-even pattern within their homologous series. First-principles calculations coupled with the quasi-harmonic approximation for crystals and with the conformation mixing model for the ideal gas are used to explain this phenomenon from the theoretical point of view. Crystals of the odd and even alkane-α,ω-diamines distinctly differ in their packing motifs. However, first-principles calculations indicate that it is a delicate interplay of the cohesive forces, phonons, molecular vibrations and conformational equilibrium which governs the odd-even pattern of the sublimation enthalpies within the homologous series. High molecular flexibility of the alkane-α,ω-diamines predetermines higher sensitivity of the computational model to the quality of the optimized geometries and relative conformational energies. Performance of high-throughput computational methods, such as the density functional tight binding (DFTB, GFN2-xTB) and the explicitly correlated dispersion-corrected Møller-Plesset perturbative method (MP2C-F12), are benchmarked against the consistent state-of-the-art calculations of conformational energies and interaction energies, respectively.

Can we trust the experiment? Anisotropic displacement parameters in 1-(halomethyl)-3-nitrobenzene (halogen = Cl or Br)

1-(Chloromethyl)-3-nitrobenzene, C₇H₆NClO₂, and 1-(bromomethyl)-3-nitrobenzene, C₇H₆NBrO₂, were chosen as test compounds for benchmarking anisotropic displacement parameters (ADPs) calculated from first principles in the harmonic approximation. Crystals of these compounds are isomorphous, and theory predicted similar ADPs for both. In-house diffraction experiments with Mo Kα radiation were in apparent contradiction to this theoretical result, with experimentally observed ADPs significantly larger for the bromo derivative. In contrast, the experimental and theoretical ADPs for the lighter congener matched reasonably well. As all usual quality indicators for both sets of experimental data were satisfactory, complementary diffraction experiments were performed at a synchrotron beamline with shorter wavelength. Refinements based on these intensity data gave very similar ADPs for both compounds and were thus in agreement with the earlier in-house results for the chloro derivative and the predictions of theory. We speculate that strong absorption by the heavy halogen may be the reason for the observed discrepancy.

A new tool for validating theoretically derived anisotropic displacement parameters with experiment: directionality of prolate displacement ellipsoids

How well do anisotropic displacement parameters from theory match experiment? The orientation of prolate ellipsoids contributes to the answer!

Dispersion-Corrected Modified Embedded-Atom Method Bond Order Interatomic Potential for Sulfur

An interatomic potential for sulfur has been developed using the bond order addition to the modified embedded-atom method (MEAM-BO). In order to correctly model the interaction between molecules, dispersion forces have been included via the DFT-D3 modification. It is demonstrated that this semiempirical classical potential correctly reproduces the behavior of the S₂ dimer, various cyclic sulfur rings, the molecular solids α-, β-, and γ-sulfur, and a number of theoretical, high symmetry sulfur structures. This potential will serve as a useful tool in the atomistic modeling of sulfur and, ultimately, in the modeling of sulfur containing organic compounds using this updated MEAM-BO formalism.

Orthorhombic sulfur from Cap Garonne, Mine du Pradet

Natural orthorhombic sulfur (α-S8), grown on galena crystals from Cap Garonne, Mine du Pradet, France, were studied by single crystal X-ray diffraction at 150 K: Fddd, a=1036.75(5), b=1273.54(7), c=2437.85(13) pm, wR=0.0380, 1433 F 2 values (all data) and 37 variables. Refinements of the occupancy parameters along with EDX data indicate pure sulfur.

Lattice thermal expansion and anisotropic displacements in urea, bromomalonic aldehyde, pentachloropyridine, and naphthalene

Anisotropic displacement parameters (ADPs) are commonly used in crystallography, chemistry, and related fields to describe and quantify thermal motion of atoms. Within the very recent years, these ADPs have become predictable by lattice dynamics in combination with first-principles theory. Here, we study four very different molecular crystals, namely, urea, bromomalonic aldehyde, pentachloropyridine, and naphthalene, by first-principles theory to assess the quality of ADPs calculated in the quasi-harmonic approximation. In addition, we predict both the thermal expansion and thermal motion within the quasi-harmonic approximation and compare the predictions with the experimental data. Very reliable ADPs are calculated within the quasi-harmonic approximation for all four cases up to at least 200 K, and they turn out to be in better agreement with the experiment than those calculated within the harmonic approximation. In one particular case, ADPs can even reliably be predicted up to room temperature. Our results also hint at the importance of normal-mode anharmonicity in the calculation of ADPs.

Thermal Expansion in Layered Na x MO2

Layered oxide Na_xMO₂ (M: transition metal) is a promising cathode material for sodium-ion secondary battery. Crystal structure of O3- and P2-type Na_xMO₂ with various M against temperature (T) was systematically investigated by synchrotron x-ray diffraction mainly focusing on the T-dependences of a- and c-axis lattice constants (a and c) and z coordinate (z) of oxygen. Using a hard-sphere model with minimum Madelung energy, we confirmed that c/a and z values in O3-type Na_xMO₂ were reproduced. We further evaluated the thermal expansion coefficients (α_a and α_c) along a- and c-axis at 300 K. The anisotropy of the thermal expansion was quantitatively reproduced without adjustable parameters for O3-type Na_xMO₂. Deviations of z from the model for P2-type Na_xMO₂ are ascribed to Na vacancies characteristic to the structure.

Plane-Wave Density Functional Theory Meets Molecular Crystals: Thermal Ellipsoids and Intermolecular Interactions

Molecular compounds, organic and inorganic, crystallize in diverse and complex structures. They continue to inspire synthetic efforts and "crystal engineering", with implications ranging from fundamental questions to pharmaceutical research. The structural complexity of molecular solids is linked with diverse intermolecular interactions: hydrogen bonding with all its facets, halogen bonding, and other secondary bonding mechanisms of recent interest (and debate). Today, high-resolution diffraction experiments allow unprecedented insight into the structures of molecular crystals. Despite their usefulness, however, these experiments also face problems: hydrogen atoms are challenging to locate, and thermal effects may complicate matters. Moreover, even if the structure of a crystal is precisely known, this does not yet reveal the nature and strength of the intermolecular forces that hold it together. In this Account, we show that periodic plane-wave-based density functional theory (DFT) can be a useful, and sometimes unexpected, complement to molecular crystallography. Initially developed in the solid-state physics communities to treat inorganic solids, periodic DFT can be applied to molecular crystals just as well: theoretical structural optimizations "help out" by accurately localizing the elusive hydrogen atoms, reaching neutron-diffraction quality with much less expensive measurement equipment. In addition, phonon computations, again developed by physicists, can quantify the thermal motion of atoms and thus predict anisotropic displacement parameters and ORTEP ellipsoids "from scratch". But the synergy between experiment and theory goes much further than that. Once a structure has been accurately determined, computations give new and detailed insights into the aforementioned intermolecular interactions. For example, it has been debated whether short hydrogen bonds in solids have covalent character, and we have added a new twist to this discussion using an orbital-based theory that once more had been developed for inorganic solids. However, there is more to a crystal structure than a handful of short contacts between neighboring residues. We hence have used dimensionally resolved analyses to dissect crystalline networks in a systematic fashion, one spatial direction at a time. Initially applied to hydrogen bonding, these techniques can be seamlessly extended to halogen, chalcogen, and pnictogen bonding, quantifying bond strength and cooperativity in truly infinite networks. Finally, these methods promise to be useful for (bio)polymers, as we have recently exemplified for α-chitin. At the interface of increasingly accurate and popular DFT methods, ever-improving crystallographic expertise, and new challenging, chemical questions, we believe that combined experimental and theoretical studies of molecular crystals are just beginning to pick up speed.

Ab initio thermodynamic properties and their uncertainties for crystalline α-methanol

To investigate the performance of quasi-harmonic electronic structure methods for modeling molecular crystals at finite temperatures and pressures, thermodynamic properties are calculated for the low-temperature α polymorph of crystalline methanol and their computational uncertainties are analyzed.