Automatic differentiation for orbital-free density functional theory

Differentiable programming has facilitated numerous methodological advances in scientific computing. Physics engines supporting automatic differentiation have simpler code, accelerating the development process and reducing the maintenance burden. Furthermore, fully differentiable simulation tools enable direct evaluation of challenging derivatives-including those directly related to properties measurable by experiment-that are conventionally computed with finite difference methods. Here, we investigate automatic differentiation in the context of orbital-free density functional theory (OFDFT) simulations of materials, introducing PROFESS-AD. Its automatic evaluation of properties derived from first derivatives, including functional potentials, forces, and stresses, facilitates the development and testing of new density functionals, while its direct evaluation of properties requiring higher-order derivatives, such as bulk moduli, elastic constants, and force constants, offers more concise implementations than conventional finite difference methods. For these reasons, PROFESS-AD serves as an excellent prototyping tool and provides new opportunities for OFDFT.

Structural Stability, Thermodynamic and Elastic Properties of Cubic Zr0.5Nb0.5 Alloy under High Pressure and High Temperature

Structural stability, sound velocities, elasticity, and thermodynamic properties of cubic Zr0.5Nb0.5 alloy have been investigated at high pressure and high temperature by first-principles density functional calculations combined with the quasi-harmonic Debye model. A pronounced pressure-induced shear wave velocity stiffening in Zr0.5Nb0.5 alloy is observed at pressures above ~11 GPa, owing to its structural instability under high pressure, whose anomalous behavior is also observed in the end members of Zr-Nb alloys for Zr at ~13 GPa and for Nb at ~6 GPa upon compression, respectively. In addition, high-pressure elasticity and elastic-correlated properties of cubic Zr0.5Nb0.5 are reported, as compared with previous studies on Zr-Nb alloys with different compositions. A comprehensive study of the thermodynamic properties of cubic Zr0.5Nb0.5, such as heat capacity (Cv), thermal expansion coefficients (α), and Debye temperature (ΘD), are also predicted at pressures and temperatures up to 30 GPa and 1500 K using the quasi-harmonic Debye model.

Calculation of elastic constants of embedded-atom-model potentials in the NVT ensemble

A method for the calculation of elastic constants in the NVT ensamble using molecular dynamics (MD) simulation with a realistic many-body embedded-atom-model (EAM) potential is studied in detail. It is shown that, in such NVT MD simulations, the evaluation of elastic constants is robust and accurate because it gives the elastic tensor in a single simulation which converges using a small number of time steps and particles. These results highlight the applicability of this method in (i) the calculation of local elastic constants of nonhomogeneous crystalline materials and (ii) the calibration of interatomic potentials, as a fast and accurate alternative to the common method of explicit deformation, which requires a set of consistent simulations at different conditions. The method is demonstrated for the calculation of the elastic constants of copper in the temperature range of 0-1000 K, and results agree with the target values used for the potential calibration. The various contributions to the values of the elastic constants, namely, the Born, stress fluctuation, and ideal gas terms, are studied as a function of temperature.

High-pressure monoclinic-monoclinic transition in fergusonite-type HoNbO4

In this paper we perform a high-pressure (HP) study of fergusonite-type HoNbO₄. Powder x-ray diffraction experiments andab initiodensity-functional theory (DFT) simulations provide evidence of a phase transition at 18.9(1.1) GPa from the monoclinic fergusonite-type structure (space group I2/a) to another monoclinic polymorph described by space group P2₁/c. The phase transition is reversible and the HP structural behavior is different than the one previously observed in related niobates. The HP phase remains stable up to 29 GPa. The observed transition involves a change in the Nb coordination number from 4 to 6, and it is driven by mechanical instabilities. We have determined the pressure dependence of unit-cell parameters of both phases and calculated their room-temperature equation of state. For the fergusonite-phase we have also obtained the isothermal compressibility tensor. In addition to the HP studies, we report ambient-pressure Raman and infrared (IR) spectroscopy measurements. We have been able to identify all the active modes of fergusonite-type HoNbO₄, which have been assigned based upon DFT calculations. These simulations also provide the elastic constants of the different structures and the pressure dependence of the Raman and IR modes of the two phases of HoNbO₄. According toab initiocalculations, the reported phase transition is related to a mechanical instability and a phonon softening.

High-pressure structural, elastic, and thermodynamic properties of zircon-type HoPO4 and TmPO4

Zircon-type holmium phosphate (HoPO₄) and thulium phosphate (TmPO₄) have been studied by single-crystal x-ray diffraction and ab initio calculations. We report on the influence of pressure on the crystal structure, and on the elastic and thermodynamic properties. The equation of state for both compounds is accurately determined. We have also obtained information on the polyhedral compressibility which is used to explain the anisotropic axial compressibility and the bulk compressibility. Both compounds are ductile and more resistive to volume compression than to shear deformation at all pressures. Furthermore, the elastic anisotropy is enhanced upon compression. Finally, the calculations indicate that the possible causes that make the zircon structure unstable are mechanical instabilities and the softening of a silent B _1u mode.

Charge-transfer potentials for ionic crystals: Cauchy violation, LO-TO splitting, and the necessity of an ionic reference state

In this work, we study how including charge transfer into force fields affects the predicted elastic and vibrational Γ-point properties of ionic crystals, in particular those of rock salt. In both analytical and numerical calculations, we find that charge transfer generally leads to a negative contribution to the Cauchy pressure, P(C) ≡ C12 - C66, where C12 and C66 are elements of the elastic tensor. This contribution increases in magnitude with pressure for different charge-transfer approaches in agreement with results obtained with density functional theory (DFT). However, details of the charge-transfer models determine the pressure dependence of the longitudinal optical-transverse optical splitting and that for partial charges. These last two quantities increase with density as long as the chemical hardness depends at most weakly on the environment while experiments and DFT find a decrease. In order to reflect the correct trends, the charge-transfer expansion has to be made around ions and the chemical (bond) hardness has to increase roughly exponentially with inverse density or bond lengths. Finally, the adjustable force-field parameters only turn out meaningful, when the expansion is made around ions.

Classical density functional theory and the phase-field crystal method using a rational function to describe the two-body direct correlation function

We introduce a new approach to represent a two-body direct correlation function (DCF) in order to alleviate the computational demand of classical density functional theory (CDFT) and enhance the predictive capability of the phase-field crystal (PFC) method. The approach utilizes a rational function fit (RFF) to approximate the two-body DCF in Fourier space. We use the RFF to show that short-wavelength contributions of the two-body DCF play an important role in determining the thermodynamic properties of materials. We further show that using the RFF to empirically parametrize the two-body DCF allows us to obtain the thermodynamic properties of solids and liquids that agree with the results of CDFT simulations with the full two-body DCF without incurring significant computational costs. In addition, the RFF can also be used to improve the representation of the two-body DCF in the PFC method. Last, the RFF allows for a real-space reformulation of the CDFT and PFC method, which enables descriptions of nonperiodic systems and the use of nonuniform and adaptive grids.

Unifying the criteria of elastic stability of solids

The elastic stability criterion formulated by Born is based on the convexity requirement of the equilibrium free energy F of a stress-free crystal under small strain fluctuation, that demands the elastic constant tensor C to be positive definite, |C| > 0. For a crystal subject to an external stress, Hill specifies that for the crystal to be stable, the difference between its internal energy change δE and the work done to the system δW must be positive, i.e. δE - δW > 0. Polanyi, Frenkel, and Orowan proposed a different stability criterion based on stress increment for a loaded system, τ(ε + Δε) - τ(ε) > 0 until the limit is reached at dτ/dε = 0. Although known empirically, the formal connection between the different criteria has not been established rigorously. Using finite deformation theory, we show quite simply that the different formulations of the stability criteria originate from the same necessary condition for the convexity of the free energy of the system subject to external loading, f = F - W. However, in practice caution must be taken in implementation of the different criteria; they may lead to quite different results, especially when stability bifurcation occurs.

Influence of vacancies on the melting transition of hard disks in two dimensions

We present the results of molecular dynamics simulations of two-dimensional (2D) hard disk systems in the vicinity of melting. The simulations are used to calculate the elastic constants, which can be used to estimate the location of the Kosterlitz-Thouless dislocation unbinding transition. Simulations on defect-free lattices indicate that this transition is expected to occur at essentially the same density as a first-order solid-isotropic transition and so it is not possible to rule out either a one step weak first-order transition between the solid and the isotropic fluid or a two step transition via a hexatic phase. Simulations performed on systems with vacancies indicate that the elastic constants are essentially unchanged at constant density. This result implies that vacancies have little influence on the melting of 2D hard disk solids.