On combining temperature and pressure effects on structural properties of crystals with standard ab initio techniques

Temperature-dependent Raman spectroscopy and thermodynamic behaviors of energetic NTO crystal

The vibrational and thermodynamic properties of energetic materials (EMs) are critical to understand their structure responses at finite temperature. In this work, the zero-point energy and temperature effects are incorporated into dispersion-corrected density functional theory to improve the calculated accuracy for vibrational responses and thermodynamic behaviors of 3-nitro-1,2,4-triazole-5-one (NTO). Based on temperature-dependent Raman spectroscopy, the emergence and disappearance of new peaks as well as discontinuous Raman shifts indicate the distinct changes of molecular configuration and intermolecular interactions within the temperature of 250-350 K. From Hirshfeld surface and structure analysis, the subtle changes of intermolecular hydrogen bonds (HBs) related with the shrinkage of thermal expansion coefficient, are treated as an essential step of a potential structural transformation of NTO. Moreover, the calculated heat capacity, entropy and bulk moduli could reflect the softening behavior of NTO and further enrich the thermodynamic data set of EMs. These results demonstrate the evolution of NTO molecules controlled by non-covalent interactions and provide vital insights into the thermodynamic behaviors at finite temperature.

Frontiers of molecular crystal structure prediction for pharmaceuticals and functional organic materials

The reliability of organic molecular crystal structure prediction has improved tremendously in recent years. Crystal structure predictions for small, mostly rigid molecules are quickly becoming routine. Structure predictions for larger, highly flexible molecules are more challenging, but their crystal structures can also now be predicted with increasing rates of success. These advances are ushering in a new era where crystal structure prediction drives the experimental discovery of new solid forms. After briefly discussing the computational methods that enable successful crystal structure prediction, this perspective presents case studies from the literature that demonstrate how state-of-the-art crystal structure prediction can transform how scientists approach problems involving the organic solid state. Applications to pharmaceuticals, porous organic materials, photomechanical crystals, organic semi-conductors, and nuclear magnetic resonance crystallography are included. Finally, efforts to improve our understanding of which predicted crystal structures can actually be produced experimentally and other outstanding challenges are discussed.

CRYSTAL23: A Program for Computational Solid State Physics and Chemistry

The Crystal program for quantum-mechanical simulations of materials has been bridging the realm of molecular quantum chemistry to the realm of solid state physics for many years, since its first public version released back in 1988. This peculiarity stems from the use of atom-centered basis functions within a linear combination of atomic orbitals (LCAO) approach and from the corresponding efficiency in the evaluation of the exact Fock exchange series. In particular, this has led to the implementation of a rich variety of hybrid density functional approximations since 1998. Nowadays, it is acknowledged by a broad community of solid state chemists and physicists that the inclusion of a fraction of Fock exchange in the exchange-correlation potential of the density functional theory is key to a better description of many properties of materials (electronic, magnetic, mechanical, spintronic, lattice-dynamical, etc.). Here, the main developments made to the program in the last five years (i.e., since the previous release, Crystal17) are presented and some of their most noteworthy applications reviewed.

Thermodynamic Properties of Crystalline Cellulose Allomorphs Studied with Dispersion-Corrected Density Functional Methods

The phonon properties and thermodynamics of four crystalline cellulose allomorphs, Iα, Iβ, II, and III1, have been investigated using dispersion-corrected density functional theory (DFT). In line with experimental findings, the free energy differences between the studied cellulose allomorphs are small, less than 1 kJ/mol per atom. The calculated specific heat at constant volume (Cv) has been compared with the available experimental data in the temperature range 10–300 K. Quasiharmonic approximation has been employed to study thermodynamics and specific heat at constant pressure (Cp). For the studied temperature range of 10–400 K, the specific heat of all cellulose allomorphs shows very similar behavior. The calculated and experimental specific heat agree well at low temperatures below 100 K, but the deviation between theory and experiment increases with temperature. This may be due to increasing phonon anharmonicity as the temperature increases.

Anharmonic Coupling of Stretching Vibrations in Ice: A Periodic VSCF and VCI Description

The anharmonicity of O-H stretching vibrations of water ice is characterized by use of a periodic implementation of the vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) methods, which take phonon-phonon couplings explicitly into account through numerical evaluation of high-order terms of the nuclear potential. The low-temperature, proton-ordered phase of water ice (namely, ice XI) is investigated. The net effect of a coupled anharmonic treatment of stretching modes is not just a rigid blue-shift of the respective harmonic spectral frequencies but rather a complex change of their relative spectral positions, which cannot be captured by simple scaling strategies based on harmonic calculations. The adopted techniques allow for a hierarchical treatment of anharmonic terms of the nuclear potential, which is key to an effective identification of leading factors. We show that the anharmonic independent-mode approximation─only describing the "intrinsic anharmonicity" of the O-H stretches─is unable to capture the correct physics, and that couplings among O-H stretches must be described. Inspection of harmonic normal coordinates allows identification of specific features of the O-H stretching motions which most likely enable strong mode-mode couplings. Finally, by coupling O-H stretches to all other possible modes of ice XI (THz collective vibrations, molecular librations, bendings), we identify specific types of motion which significantly affect O-H stretching states: in particular, molecular librations are found to affect the stretching states more than molecular bendings.

On the use of a volume constraint to account for thermal expansion effects on the low-frequency vibrations of molecular crystals

A volume-constraint method is presented as a means to capture the influence of thermal expansion on the low-frequency vibrations in molecular crystals. In particular, the room-temperature terahertz absorption spectra of L-tartaric acid, α-lactose monohydrate, and α-para-aminobenzoic acid (PABA) have been simulated using dispersion-corrected, solid-state density functional theory (DFT-D). By comparing the normal modes obtained with a unit cell optimised without constraints to those obtained with a unit cell optimised while constrained to keep its experimental volume, wholesale improvements to the resultant spectrum is achieved when using the constrained geometry by inhibiting cell contraction. These improvements are demonstrated over a range of popular density functionals and basis sets up to triple-zeta complexity. A correlation method is then presented as a means to quantitatively compare the vibrational pattern of normal modes obtained from both unit cells. This analysis reveals that thermal expansion can effect the character and relative frequency of normal modes, with the choice of geometry ultimately affecting the assignment of the experimental absorptions. The sensibility of using the experimental volume as an approximation is then discussed, where it is speculated that large basis sets or hybrid functionals are necessary to ensure that the thermal expansion effect is not overestimated. The low-frequency absorption spectrum of PABA is then fully characterised using the PBE-D3BJ/6-311G(2d,2p) method.

QUANTAS: a Python software for the analysis of thermodynamics and elastic behavior of solids from ab initio quantum mechanical simulations and experimental data

This paper presents QUANTAS, an open-source Python-based software aimed at providing a fast, flexible and easy-to-use framework to calculate the thermodynamics and elastic properties of crystalline solids. QUANTAS could be of use for researchers involved in various fields of solid-state chemistry, physics and mineralogy.

Mineralogy, petrology and materials science are fundamental disciplines not only for the basic knowledge and classification of solid phases but also for their technological applications, which are becoming increasingly demanding and challenging. Characterization and design of materials are of utmost importance and usually need knowledge of the thermodynamics and mechanical stability of solids. Alongside well known experimental approaches, in recent years the advances in both quantum mechanical methods and computational power have placed theoretical investigations as a complementary useful and powerful tool in this kind of study. In order to aid both theoreticians and experimentalists, an open-source Python-based software, QUANTAS, has been developed. QUANTAS provides a fast, flexible, easy-to-use and extensible platform for calculating the thermodynamics and elastic behavior of crystalline solid phases, starting from both experimental and ab initio data.

Study of the variation of the optical properties of calcite with applied stress, useful for specific rock and material mechanics

Calcite (CaCO3, trigonal crystal system, space group [Formula: see text]) is a ubiquitous carbonate phase commonly found on the Earth's crust that finds many useful applications in both scientific (mineralogy, petrology, geology) and technological fields (optics, sensors, materials technology) because of its peculiar anisotropic physical properties. Among them, photoelasticity, i.e., the variation of the optical properties of the mineral (including birefringence) with the applied stress, could find usefulness in determining the stress state of a rock sample containing calcite by employing simple optical measurements. However, the photoelastic tensor is not easily available from experiments, and affected by high uncertainties. Here we present a theoretical Density Functional Theory approach to obtain both elastic and photoelastic properties of calcite, considering realistic experimental conditions (298 K, 1 atm). The results were compared with those available in literature, further extending the knowledge of the photoelasticity of calcite, and clarifying an experimental discrepancy in the sign of the p41 photoelastic tensor component measured in past investigations. The methods here described and applied to a well-known crystalline material can be used to obtain the photoelastic properties of other minerals and/or materials at desired pressure and temperature conditions.

Analysis of the relative stability of lithium halide crystal structures: Density functional theory and classical models

All lithium halides exist in the rock salt crystal structure under ambient conditions. In contrast, common lithium halide classical force fields more often predict wurtzite as the stable structure. This failure of classical models severely limits their range of application in molecular simulations of crystal nucleation and growth. Employing high accuracy density functional theory (DFT) together with classical models, we examine the relative stability of seven candidate crystal structures for lithium halides. We give a detailed examination of the influence of DFT inputs, including the exchange-correlation functional, basis set, and dispersion correction. We show that a high-accuracy basis set, along with an accurate description of dispersion, is necessary to ensure prediction of the correct rock salt structure, with lattice energies in good agreement with the experiment. We also find excellent agreement between the DFT-calculated rock salt lattice parameters and experiment when using the TMTPSS-rVV10 exchange-correlation functional and a large basis set. Detailed analysis shows that dispersion interactions play a key role in the stability of rock salt over closely competing structures. Hartree-Fock calculations, where dispersion interactions are absent, predict the rock salt structure only for LiF, while LiCl, LiBr, and LiI are more stable as wurtzite crystals, consistent with radius ratio rules. Anion-anion second shell dispersion interactions overcome the radius ratio rules to tip the structural balance to rock salt. We show that classical models can be made qualitatively correct in their structural predictions by simply scaling up the pairwise additive dispersion terms, indicating a pathway toward better lithium halide force fields.

Hydroxylapatite and Related Minerals in Bone and Dental Tissues: Structural, Spectroscopic and Mechanical Properties from a Computational Perspective

Hard tissues (e.g., bone, enamel, dentin) in vertebrates perform various and different functions, from sustaining the body to haematopoiesis. Such complex and hierarchal tissue is actually a material composite whose static and dynamic properties are controlled by the subtle physical and chemical interplay between its components, collagen (main organic part) and hydroxylapatite-like mineral. The knowledge needed to fully understand the properties of bony and dental tissues and to develop specific applicative biomaterials (e.g., fillers, prosthetics, scaffolds, implants, etc.) resides mostly at the atomic scale. Among the different methods to obtains such detailed information, atomistic computer simulations (in silico) have proven to be both corroborative and predictive tools in this subject. The authors have intensively worked on quantum mechanical simulations of bioapatite and the present work reports a detailed review addressed to the crystal-chemical, physical, spectroscopic, mechanical, and surface properties of the mineral phase of bone and dental tissues. The reviewed studies were conducted at different length and time scales, trying to understand the features of hydroxylapatite and biological apatite models alone and/or in interaction with simplified collagen-like models. The reported review shows the capability of the computational approach in dealing with complex biological physicochemical systems, providing accurate results that increase the overall knowledge of hard tissue science.

Modeling the α- and β-resorcinol phase boundary via combination of density functional theory and density functional tight-binding

The ability to predict not only what organic crystal structures might occur but also the thermodynamic conditions under which they are the most stable would be extremely useful for discovering and designing new organic materials. The present study takes a step in that direction by predicting the temperature- and pressure-dependent phase boundary between the α and β polymorphs of resorcinol using density functional theory (DFT) and the quasi-harmonic approximation. To circumvent the major computational bottleneck associated with computing a well-converged phonon density of states via the supercell approach, a recently developed approximation is employed, which combines a supercell phonon density of states from dispersion-corrected third-order density functional tight binding [DFTB3-D3(BJ)] with frequency corrections derived from a smaller B86bPBE-XDM functional DFT phonon calculation on the crystallographic unit cell. This mixed DFT/DFTB quasi-harmonic approach predicts the lattice constants and unit cell volumes to within 1%-2% at lower pressures. It predicts the thermodynamic phase boundary in almost perfect agreement with the experiment, although this excellent agreement does reflect fortuitous cancellation of errors between the enthalpy and entropy of transition.

Thermoelasticity of Flexible Organic Crystals from Quasi-harmonic Lattice Dynamics: The Case of Copper(II) Acetylacetonate

A computationally affordable approach, based on quasi-harmonic lattice dynamics, is presented for the quantum-mechanical calculation of thermoelastic moduli of flexible, stimuli-responsive, organic crystals. The methodology relies on the simultaneous description of structural changes induced by thermal expansion and strain. The complete thermoelastic response of the mechanically flexible metal-organic copper(II) acetylacetonate crystal is determined and discussed in the temperature range 0-300 K. The elastic moduli do not just shrink with temperature but they do so anisotropically. The present results clearly indicate the need for an explicit account of thermal effects in the simulation of mechanical properties of elastically flexible organic materials. Indeed, predictions from standard static calculations on this flexible metal-organic crystal are off by up to 100%.

Negative thermal expansion of Cu2O studied by quasi-harmonic approximation and cubic force-constant method

Cubic cuprous oxide, Cu₂O, is characterized by a peculiar structural response to temperature: it shows a relatively large negative thermal expansion below 250 K, then followed by a positive thermal expansion at higher temperatures. The two branches of its thermal expansion (negative and positive) are almost perfectly symmetric at low temperatures, with the minimum of its lattice parameter at about 250 K and with the lattice parameter at 500 K almost coinciding with that at 0 K. We perform lattice-dynamical quantum-mechanical calculations to investigate the thermal expansion of Cu₂O. Phonon mode-specific Grüneisen parameters are computed, which allows us to identify different spectral regions of atomic vibrations responsible for the two distinct regimes of thermal expansion. Two different computational approaches are explored, their results compared, and their numerical aspects critically assessed: a well-established method based on the quasiharmonic approximation, where harmonic frequencies are computed at different lattice volumes, and an alternative approach, where quadratic and cubic interatomic force-constants are computed at a single volume. The latter scheme has only recently become computationally feasible in the context of lattice thermal conductivity simulations. When proper numerical parameters are used (phonon sampling, tolerances, etc.), the two approaches are here shown to provide a very consistent description, yet at a rather different computational cost. All of the experimentally observed features of the complex thermal expansion of Cu₂O are correctly reproduced up to 500 K, with a slight overall underestimation of the volume contraction.

Improving Predicted Nuclear Magnetic Resonance Chemical Shifts Using the Quasi-Harmonic Approximation

Ab initio nuclear magnetic resonance chemical shift prediction plays an important role in the determination or validation of crystal structures. The ability to predict chemical shifts more accurately can translate to increased confidence in the resulting chemical shift or structural assignments. Standard electronic structure predictions for molecular crystal structures neglect thermal expansion, which can lead to an appreciable underestimation of the molar volumes. This study examines this volume error and its impact on 68 ¹³C- and 28 ¹⁵N-predicted chemical shifts taken from 20 molecular crystals. It assesses the ability to recover more realistic room-temperature crystal structures using the quasi-harmonic approximation and how refining the structures impacts the chemical shifts. Several pharmaceutical molecular crystals are also examined in more detail. On the whole, accounting for quasi-harmonic expansion changes the ¹³C and ¹⁵N chemical shifts by 0.5 and 1.0 ppm on average. This, in turn, reduces the root-mean-square errors relative to experiment by 0.3 ppm for ¹³C and 0.7 ppm for ¹⁵N. Although the statistical impacts are modest, changes in individual chemical shifts can reach multiple ppm. Accounting for thermal expansion in molecular crystal chemical shift prediction may not be needed routinely, but the systematic trend toward improved accuracy with the experiment could be useful in cases where discrimination between structural candidates is challenging, as in the pharmaceutical theophylline.

The impact of lattice vibrations on the macroscopic breathing behavior of MIL-53(Al)

The mechanism inducing the breathing in flexible metal-organic frameworks, such as MIL-53(Al), is still not fully understood. Herein, the influence of lattice vibrations on the breathing transition in MIL-53(Al) is investigated to gain insight in this phenomenon. Through solid-state density-functional theory calculations, the volume-dependent IR spectrum is computed together with the volume-frequency relations of all vibrational modes. Furthermore, important thermodynamic properties such as the Helmholtz free energy, the specific heat capacity, the bulk modulus, and the volumetric thermal expansion coefficient are derived via these volume-frequency relations using the quasi-harmonic approximation. The simulations expose a general volume-dependency of the vibrations with wavenumbers above 300 cm−1 due to their localized nature. In contrast, a diverse set of volume-frequency relations are observed for vibrations in the terahertz region (<300 cm−1) containing the vibrations exhibiting collective behavior. Some terahertz vibrations display large frequency differences over the computed volume range, induced by either repulsion or strain effects, potentially triggering the phase transformation. Finally, the impact of the lattice vibrations on the thermodynamic properties is investigated. This reveals that the closed pore to large pore phase transformation in MIL-53(Al) is mainly facilitated by terahertz vibrations inducing rotations of the organic linker, while the large pore to closed pore phase transformation relies on two framework-specific soft modes.

Thermo-Elasticity of Materials from Quasi-Harmonic Calculations

An effective algorithm for the quasi-harmonic calculation of thermo-elastic stiffness constants of materials is discussed and implemented into the Crystal program for quantum-mechanical simulations of extended systems. Two different approaches of increasing complexity and accuracy are presented. The first one is a quasi-static approximation where the thermal dependence of elastic constants is assumed to be due only to the thermal expansion of the system. The second one is fully quasi-harmonic, takes into account thermal expansion, and explicitly computes Helmholtz free energy derivatives with respect to strain. The conversion of isothermal into adiabatic thermo-elastic constants is also addressed. The algorithm is formally presented and applied to the description of the thermo-elastic response of the forsterite mineral.

Thermal Gradient Approach for the Quasi-harmonic Approximation and Its Application to Improved Treatment of Anisotropic Expansion

We present a novel approach to efficiently implement thermal expansion in the quasi-harmonic approximation (QHA) for both isotropic and more importantly, anisotropic expansion. In this approach, we rapidly determine a crystal's equilibrium volume and shape at a given temperature by integrating along the gradient of expansion from 0 Kelvin up to the desired temperature. We compare our approach to previous isotropic methods that rely on a brute-force grid search to determine the free energy minimum, which is infeasible to carry out for anisotropic expansion, as well as quasi-anisotropic approaches that take into account the contributions to anisotropic expansion from the lattice energy. We compare these methods for experimentally known polymorphs of piracetam and resorcinol and show that both isotropic methods agree to within error up to 300 K. Using the Grüneisen parameter causes up to 0.04 kcal/mol deviation in the Gibbs free energy, but for polymorph free energy differences there is a cancellation in error with all isotropic methods within 0.025 kcal/mol at 300 K. Anisotropic expansion allows the crystals to relax into lattice geometries 0.01-0.23 kcal/mol lower in energy at 300 K relative to isotropic expansion. For polymorph free energy differences all QHA methods produced results within 0.02 kcal/mol of each other for resorcinol and 0.12 kcal/mol for piracetam, the two molecules tested here, demonstrating a cancellation of error for isotropic methods. We also find that with expansion in more than a single volume variable, there is a non-negligible rate of failure of the basic approximations of QHA. Specifically, while expanding into new harmonic modes as the box vectors are increased, the system often falls into alternate, structurally distinct harmonic modes unrelated by continuous deformation from the original harmonic mode.

Efficient Calculation of the Negative Thermal Expansion in ZrW2O8

We present a study of the origin of the negative thermal expansion (NTE) on ZrW₂O₈ by combining an efficient approach for computing the dynamical matrix with the Lanczos algorithm for generating the phonon density of states in the quasi-harmonic approximation. The simulations show that the NTE arises primarily from the motion of the O-sublattice, and in particular, from the transverse motion of the O atoms in the W–O and W–O–Zr bonds. In the low frequency range these combine to keep the WO₄ tetrahedra rigid and induce internal distortions in the ZrO₆ octahedra. The force constants associated with these distortions become stronger with expansion, resulting in negative Grüneisen parameters and NTE from the low frequency modes that dominate the positive contributions from the high frequency modes. This leads us to propose an anharmonic, two-frequency Einstein model that quantitatively captures the NTE behavior.

Thermal Expansion of Carbamazepine: Systematic Crystallographic Measurements Challenge Quantum Chemical Calculations

We report systematic temperature-dependent X-ray measurements on the most stable carbamazepine polymorph. This active pharmaceutical ingredient is used to demonstrate how the thermal expansion can probe certain intermolecular interactions resulting in anisotropic expansion behavior. We show that most structural features can be captured by electronic structure calculations at the quasi-harmonic approximation (QHA) provided a dispersion-corrected density functional based method is employed. The impact of thermal expansion on the phonon modes and hence free energy contributions is large enough to impact the relative stability of different polymorphs.

Intermolecular anharmonicity in molecular crystals: interplay between experimental low-frequency dynamics and quantum quasi-harmonic simulations of solid purine

A combined experimental spectroscopic and theoretical quasi-harmonic investigation of intermolecular anharmonicity in crystalline purine.

Thermal properties of molecular crystals through dispersion-corrected quasi-harmonic ab initio calculations: the case of urea

An ab initio quantum-mechanical theoretical framework is presented to compute the thermal properties of molecular crystals. The present strategy combines dispersion-corrected density-functional-theory (DFT-D), harmonic phonon dispersion, quasi-harmonic approximation to the lattice dynamics for thermal expansion and thermodynamic functions, and quasi-static approximation for anisotropic thermo-elasticity. The proposed scheme is shown to reliably describe thermal properties of the urea molecular crystal by a thorough comparison with experimental data.

Raman spectroscopic features of the neutral vacancy in diamond from ab initio quantum-mechanical calculations

Ab initio calculations of the Raman spectrum of the neutral vacancy in diamond reproduce observed spectral features of irradiated diamond up to the first-order peak.

Anharmonic thermal oscillations of the electron momentum distribution in lithium fluoride

Anharmonic thermal effects on the electron momentum distribution of a lithium fluoride single crystal are experimentally measured through high-resolution Compton scattering and theoretically modeled with ab initio simulations, beyond the harmonic approximation to the lattice potential, explicitly accounting for thermal expansion. Directional Compton profiles are measured at two different temperatures, 10 and 300 K, with a high momentum space resolution (0.10 a.u. in full width at half maximum), using synchrotron radiation. The effect of temperature on measured directional Compton profiles is clearly revealed by oscillations extending almost up to |p|=4  a.u., which perfectly match those predicted from quantum-mechanical simulations. The wave-function-based Hartree-Fock method and three classes of the Kohn-Sham density functional theory (local-density, generalized-gradient, and hybrid approximations) are adopted. The lattice thermal expansion, as described with the quasiharmonic approach, is found to entirely account for the effect of temperature on the electron momentum density within the experimental accuracy.

Assessing thermochemical properties of materials through ab initio quantum-mechanical methods: the case of α-Al2O3

Accurate ab initio calculations of thermodynamic and structural thermal properties of corundum demonstrate its quasi-harmonic nature up to the melting temperature.

Katoite under pressure: an ab initio investigation of its structural, elastic and vibrational properties sheds light on the phase transition

Structural, elastic and vibrational properties of the katoite hydrogarnet are investigated as a function of pressure, up to 65 GPa, with an ab initio simulation. A group-subgroup phase transition is found to occur in the 5–15 GPa range of pressure.