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

Thermal properties of solid-state materials are a fundamental topic of study with important practical implications. For example, anisotropic displacement parameters (ADPs) are routinely used in physics, chemistry, and crystallography to quantify the thermal motion of atoms in crystals. ADPs are commonly derived from diffraction experiments, but recent developments have also enabled their first-principles prediction using periodic density functional theory (DFT). Here, we combine experiments and dispersion-corrected DFT to quantify lattice thermal expansion and ADPs in crystalline {\alpha}-sulfur (S8), a prototypical elemental solid that is controlled by the interplay of covalent and van der Waals interactions. We first report on single-crystal and powder X-ray diffraction (XRD) measurements that provide new and improved reference data from 10 K up to room temperature. We then use several popular dispersion-corrected DFT methods to predict vibrational and thermal properties of {\alpha}-sulfur, including the anisotropic lattice thermal expansion. Hereafter, ADPs are derived in the commonly used harmonic approximation (in the computed zero-Kelvin structure) and also in the quasi-harmonic approximation (QHA) which takes the predicted lattice thermal expansion into account. At the PBE+D3(BJ) level, the latter leads to excellent agreement with experiments. Finally, more general implications of this study for realistic materials modeling at finite temperature are discussed.