Coupled Perturbed Approach to Dual Basis Sets for Molecules and Solids. II: Energy and Band Corrections for Periodic Systems

When trying to reach convergence of quantum chemical calculations toward the complete basis set limit, crystalline solids generally prove to be more challenging than molecules. This is due both to the closer packing of atoms─hence, to linear dependencies─and to the problematic behavior of Ewald techniques used for dealing with the infinite character of Coulomb sums. Thus, a dual basis set approach is even more desirable for periodic systems than for molecules. In such an approach, the self-consistent procedure is implemented in a small basis set, and the effect of the enlargement of the basis set is estimated a posteriori. In this paper, we extend to crystalline solids our previous coupled perturbed dual basis set approach [J. Chem. Theory Comput. 2020, 16, 1, 340-353] in which the basis set enlargement is treated as a perturbation. Among the notable features of this approach are (i) the possibility of obtaining not only a correction to the energy but also to energy bands and electron density; (ii) the absence of a diagonalization step for the full Fock matrix in the large basis set; and (iii) the possibility of extrapolating low order perturbation energy corrections to infinite order. We also present here the first periodic implementation of the dual basis set method of Liang and Head-Gordon [J. Phys. Chem. A 2004, 108, 3206-3210]. The effectiveness of both approaches is, then, compared on a small, but representative, set of solids.

Anisotropic thermo-mechanical response of layered hexagonal boron nitride and black phosphorus: application as a simultaneous pressure and temperature sensor

Hexagonal boron nitride (hBN) and black phosphorus (bP) are crystalline materials that can be seen as ordered stackings of two-dimensional layers, which lead to outstanding anisotropic physical properties. Knowledge of the thermal equations of state of hBN and bP is of great interest in the field of 2D materials for a better understanding of their anisotropic thermo-mechanical properties and exfoliation mechanism towards the preparation of important single-layer materials like hexagonal boron nitride nanosheets and phosphorene. Despite several theoretical and experimental studies, important uncertainties remain in the determination of the thermoelastic parameters of hBN and bP. Here, we report accurate thermal expansion and compressibility measurements along the individual crystallographic axes, using in situ high-temperature and high-pressure high-resolution synchrotron X-ray diffraction. In particular, we have quantitatively determined the subtle variations of the in-plane and volumetric thermal expansion coefficients and compressibility parameters by subjecting these materials to hydrostatic conditions and by collecting a large number of data points in small pressure and temperature increments. In addition, based on the anisotropic behavior of bP, we propose the use of this material as a sensor for the simultaneous determination of pressure and temperature in the range of 0-5 GPa and 298-1700 K, respectively.

Structure and Bonding in Amorphous Red Phosphorus

Amorphous red phosphorus (a-P) is one of the remaining puzzling cases in the structural chemistry of the elements. Here, we elucidate the structure, stability, and chemical bonding in a-P from first principles, combining machine-learning and density-functional theory (DFT) methods. We show that a-P structures exist with a range of energies slightly higher than those of phosphorus nanorods, to which they are closely related, and that the stability of a-P is linked to the degree of structural relaxation and medium-range order. We thus complete the stability range of phosphorus allotropes [Angew. Chem. Int. Ed. 2014, 53, 11629] by now including the previously poorly understood amorphous phase, and we quantify the covalent and van der Waals interactions in all main phases of phosphorus. We also study the electronic densities of states, including those of hydrogenated a-P. Beyond the present study, our structural models are expected to enable wider-ranging first-principles investigations-for example, of a-P-based battery materials.

Probing interlayer shear thermal deformation in atomically-thin van der Waals layered materials

Atomically-thin van der Waals layered materials, with both high in-plane stiffness and bending flexibility, offer a unique platform for thermomechanical engineering. However, the lack of effective characterization techniques hinders the development of this research topic. Here, we develop a direct experimental method and effective theoretical model to study the mechanical, thermal, and interlayer properties of van der Waals materials. This is accomplished by using a carefully designed WSe₂-based heterostructure, where monolayer WSe₂ serves as an in-situ strain meter. Combining experimental results and theoretical modelling, we are able to resolve the shear deformation and interlayer shear thermal deformation of each individual layer quantitatively in van der Waals materials. Our approach also provides important interlayer coupling information as well as key thermal parameters. The model can be applied to van der Waals materials with different layer numbers and various boundary conditions for both thermally-induced and mechanically-induced deformations.

Van der Waals materials exhibit unique thermomechanical properties, but interlayer deformations are usually challenging to measure. Here, the authors exploit the strain-dependent optical properties of monolayer WSe₂ to quantitatively probe the interlayer shear thermal deformations and interlayer coupling in phosphorene and hexagonal boron nitride.

Cluster Fragments in Amorphous Phosphorus and their Evolution under Pressure

Amorphous phosphorus (a-P) has long attracted interest because of its complex atomic structure, and more recently as an anode material for batteries. However, accurately describing and understanding a-P at the atomistic level remains a challenge. Here, it is shown that large-scale molecular-dynamics simulations, enabled by a machine-learning (ML)-based interatomic potential for phosphorus, can give new insights into the atomic structure of a-P and how this structure changes under pressure. The structural model so obtained contains abundant five-membered rings, as well as more complex seven- and eight-atom clusters. Changes in the simulated first sharp diffraction peak during compression and decompression indicate a hysteresis in the recovery of medium-range order. An analysis of cluster fragments, large rings, and voids suggests that moderate pressure (up to about 5 GPa) does not break the connectivity of clusters, but higher pressure does. The work provides a starting point for further computational studies of the structure and properties of a-P, and more generally it exemplifies how ML-driven modeling can accelerate the understanding of disordered functional materials.

Gaussian Process Regression for Materials and Molecules

We provide an introduction to Gaussian process regression (GPR) machine-learning methods in computational materials science and chemistry. The focus of the present review is on the regression of atomistic properties: in particular, on the construction of interatomic potentials, or force fields, in the Gaussian Approximation Potential (GAP) framework; beyond this, we also discuss the fitting of arbitrary scalar, vectorial, and tensorial quantities. Methodological aspects of reference data generation, representation, and regression, as well as the question of how a data-driven model may be validated, are reviewed and critically discussed. A survey of applications to a variety of research questions in chemistry and materials science illustrates the rapid growth in the field. A vision is outlined for the development of the methodology in the years to come.

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.

A general-purpose machine-learning force field for bulk and nanostructured phosphorus

Elemental phosphorus is attracting growing interest across fundamental and applied fields of research. However, atomistic simulations of phosphorus have remained an outstanding challenge. Here, we show that a universally applicable force field for phosphorus can be created by machine learning (ML) from a suitably chosen ensemble of quantum-mechanical results. Our model is fitted to density-functional theory plus many-body dispersion (DFT + MBD) data; its accuracy is demonstrated for the exfoliation of black and violet phosphorus (yielding monolayers of "phosphorene" and "hittorfene"); its transferability is shown for the transition between the molecular and network liquid phases. An application to a phosphorene nanoribbon on an experimentally relevant length scale exemplifies the power of accurate and flexible ML-driven force fields for next-generation materials modelling. The methodology promises new insights into phosphorus as well as other structurally complex, e.g., layered solids that are relevant in diverse areas of chemistry, physics, and materials science.

From Anomalous to Normal: Temperature Dependence of the Band Gap in Two-Dimensional Black Phosphorus

The temperature dependence of the band gap is crucial to a semiconductor. Bulk black phosphorus is known to exhibit an anomalous behavior. Through optical spectroscopy, here we show that the temperature effect on black phosphorus band gap gradually evolves with decreasing layer number, eventually turns into a normal one in the monolayer limit, rendering a crossover from the anomalous to the normal. Meanwhile, the temperature-induced shift in optical resonance also differs with different transition indices for the same thickness sample. A comprehensive analysis reveals that the temperature-tunable interlayer coupling is responsible for the observed diverse scenario. Our study provides a key to the apprehension of the anomalous temperature behavior in certain layered semiconductors.

Thermodynamic stability and vibrational anharmonicity of black phosphorene-beyond quasi-harmonic analysis

Thermodynamic stability and vibrational anharmonicity of single layer black phosphorene (SLBP) are studied using a spectral energy density (SED) method. At finite temperatures, SLBP sheet undergoes structural deformation due to the formation of thermally excited ripples. Thermal stability of deformed SLBP sheet is analyzed by computing finite temperature phonon dispersion, which shows that SLBP sheet is thermodynamically stable and survives the crumpling transition. To analyze the vibrational anharmonicity, temperature evolution of all zone center optic phonon modes are extracted, including experimentally forbidden IR and Raman active modes. Mode resolved phonon spectra exhibits red-shift in mode frequencies with temperature. The strong anharmonic phonon-phonon coupling is the predominant reason for the observed red-shift of phonon modes, the contribution of thermal expansion is marginal. Further, temperature sensitivity of all optic modes are analyzed by computing their first order temperature co-efficient (χ), and it can be expressed asB_2g>Ag2>B3g1>B3g2>B_1g>Ag1&B_2u>B_1ufor Raman and IR active modes, respectively. The quasi-harmonicχvalues are much smaller than the SED and experimental values; which substantiate that quasi-harmonic methods are inadequate, and a full anharmonic analysis is essential to explain structure and dynamics of SLBP at finite temperatures.

Anisotropic thermal expansion of black phosphorus from nanoscale dynamics of phosphorene layers

Black phosphorus (bP) is a crystalline material which can be seen as an ordered stacking of two-dimensional layers, referred to as phosphorene. The knowledge of the linear thermal expansion coefficients (LTECs) of bP is of great interest in the field of 2D materials for a better understanding of the anisotropic thermal properties and exfoliation mechanism of this material. Despite several theoretical and experimental studies, important uncertainties remain in the determination of the LTECs of bP. Here, we report accurate thermal expansion measurements along the three crystallographic axes using in situ high temperature X-ray diffraction. From the progressive reduction of the diffracted intensities with temperature, we monitored the loss of the crystal structure of bP across the investigated temperature range, evidencing two thermal expansion regimes at temperature below and above 706 K. Below 706 K, a strong out-of-plane anisotropy can be observed, while at temperatures above 706 K a larger thermal expansion occurs along the a crystallographic direction. From our data and by taking advantage of ab initio optimization, we propose a detailed anisotropic thermal expansion mechanism of bP, which leads to an inter- and intra-layer destabilization. An interpretation of it, based on the high T perturbation of the stabilizing sp orbital mixing effect, is provided, consistent with the high pressure data.

Extending and assessing composite electronic structure methods to the solid state

A hierarchy of simplified Hartree-Fock (HF), density functional theory (DFT) methods, and their combinations has been recently proposed for the fast electronic structure computation of large systems. The covered methods are a minimal basis set Hartree-Fock (HF-3c), a small basis set global hybrid functional (PBEh-3c), and its screened exchange variant (HSE-3c), all augmented with semiclassical correction potentials. Here, we extend their applicability to inorganic covalent and ionic solids as well as layered materials. The new methods have been dubbed HFsol-3c, PBEsol0-3c, and HSEsol-3c, respectively, to indicate their parent functional as well as the correction potentials. They have been implemented in the CRYSTAL code to enable routine application for molecular as well as solid materials. We validate the new methods on diverse sets of solid state benchmarks that cover more than 90 solids ranging from covalent, ionic, semi-ionic, layered, and molecular crystals. While we focus on structural and energetic properties, we also test bandgaps, vibrational frequencies, elastic constants, and dielectric and piezoelectric tensors. HSEsol-3c appears to be most promising with mean absolute error for cohesive energies and unit cell volumes of molecular crystals of 1.5 kcal/mol and 2.8%, respectively. Lattice parameters of inorganic solids deviate by 3% from the references, and vibrational frequencies of α-quartz have standard deviations of 10 cm^-1. Overall, this shows an accuracy competitive to converged basis set dispersion corrected DFT with a substantial increase in computational efficiency.