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At least 487 records · Page 27

High-Performance Computing Optimization for Aladyn – Adaptive Neural Network Molecular Dynamics Mini-Application

This report provides a description and performance evaluation of the optimization techniques for high performance computing (HPC) implementation of the open source Computational Materials mini-application Aladyn (https://github.com/nasa/aladyn). Aladyn is a basic molecular dynamics code written in FORTRAN 2003, which is designed to demonstrate the use of adaptive neural networks (ANNs) in atomistic simulations. The role of ANNs is to efficiently reproduce the very complex energy landscape resulting from the atomic interactions in materials with the accuracy of the more expensive quantum mechanics-based calculations. The ANN is trained on a large set of atomic structures calculated using the density functional theory (DFT) method. While achieving orders of magnitude faster computational performance than DFT, the ANN-based approach was still very computationally demanding compared to the conventional approach of using empirically fitted energy functions. After its initial development, Aladyn was evaluated and optimized by experts at the NASA Advanced Supercomputing (NAS) division to exploit modern supercomputer architectures. The code has been optimized for execution on multicore central processing units (CPUs), including Intel® Skylake microarchitecture, and on graphic accelerators, such as Nvidia® V100 graphic processing units (GPUs), using Open Multi-Processing (OpenMP) and Open Accelerators (OpenACC) programming interfaces. The optimization achieved a speedup of 4.7 times the baseline version on CPU performance and an additional 2.4 times on CPU+GPU performance. Atomistic computer simulations are a fundamental tool in materials research to model material properties form physics-based first principles. Atomic interaction, governed by Quantum Mechanics (QM) require sophisticated and highly computationally demanding mathematical models to calculate [1]. Classical methods use approximate functional forms, empirically fitted through a set of variable parameters to emulate atomic energies as direct functions of atomic coordinates [2]. While empirical potentials are computationally much simpler, allowing simulations of large-scale systems of up to a trillion (1012) atoms [3], they are substantially less accurate compared to quantum calculations and applicable only to very specific atomic configurations or predefined crystallographic phases. A recently suggested approach is to use heuristic machine learning methods [4], such as those based on Adaptive Neural Networks (ANNs) to predict atomic energies, after being trained on a sufficiently large database of QM-calculated structures [5,6]. This approach reduces significantly the computational complexity, allowing for simulations of orders of magnitude larger systems compared to QM-based methods without compromising accuracy. Still, compared to classical methods using empirical energy functions, ANN methods remain two- to three orders of magnitude more computationally demanding. Hence, the computational cost of simulations, together with the need for extensive training of ANNs, still makes the practical implementation of ANN-based methods quite challenging. The purpose of the Aladyn mini-application software [7], available as open source at https://github.com/nasa/aladyn, is to be a testbed for exploring possible optimization strategies to develop highly scalable parallel algorithms for ANN-based atomistic simulations. Aladyn is aimed at utilizing the architecture of the high-end modern highperformance computing (HPC) hardware based on multicore central processing units (CPUs) equipped with graphic processing unit (GPU) accelerators. Specifically, the goal is to optimize the performance on a single HPC compute node, before implementing scaling to multi-node parallelization using message passing interface (MPI). At the same time, the open source code of Aladyn can serve as a training model for students and professors in academia.

Yamakov, Vesselin I.↗

Activation, Dehydrogenation, and Carbon–Carbon Coupling of Methane by Iridium Cations Studied by Infrared Multiple Photon Dissociation Spectroscopy and Density Functional Theory

Products resulting from the sequential activation of one, two, three, and four methane molecules by atomic iridium cations were characterized by gas-phase infrared multiple photon dissociation spectroscopy and density functional theory (DFT) calculations. Iridium cations were generated using a laser ablation source and reacted with methane in a linear radiofrequency ion trap before mass analysis and spectroscopic interrogation in a Fourier transform ion cyclotron resonance mass spectrometer coupled to the free-electron laser for intracavity experiments (FELICE) beamline. Product ions were irradiated using infrared light over the 250–1500 cm –1 range. Comparisons between the experimental and DFT-calculated spectra enabled structural determination of the products formed. The observed products include HIrCH + , Ir(CH 2 ) 2 + , H s Ir(C 3 H 5 ) + , and Ir(CH 3 ) s (C 3 H 5 ) + , where the subscript s denotes a syn orientation of the two ligands. Furthermore, formation of the latter two products provides evidence for efficient C–H bond activation and subsequent C–C coupling on the atomic iridium cation.

Chemical reactions↗

Custom-trained Machine-learning Interatomic Potentials: ZnCl2 Aqueous Solution

This dataset was generated using an iterative active-learning strategy implemented in the ArcaNN software package (https://github.com/arcann-chem/arcann_training) to train machine-learning interatomic potentials for aqueous ZnCl2 solutions. Each active-learning cycle consisted of three stages: training, exploration, and labeling. The initial training set combined configurations generated in this work from enhanced-sampling ab initio molecular dynamics simulations with configurations from a previously reported neural-network-potential study of aqueous ZnCl2. The enhanced-sampling ab initio molecular dynamics simulations involved Zn–Cl separation and the chloride coordination number around Zn²? as collective variables. These configurations served as the seed dataset. Subsequent active-learning cycles expanded the training set by identifying and labeling configurations that were poorly represented by the current models, thereby improving coverage of ion-association states and changes in local coordination and charge-state environments relevant to the solution free-energy landscape. For all selected configurations, single-point calculations of the total energies and atomic forces were performed within density functional theory using the CP2K Quickstep module. Reference calculations employed the revPBE-D3 and r2SCAN exchange-correlation functionals. Motivated by recent work on aqueous Zn²?, the main revPBE calculations omitted D3 dispersion contributions involving Zn²?, while retaining the D3 correction for water and chloride. For comparison, fully dispersion-corrected revPBE-D3 reference calculations were also performed, with D3 applied to all species, including Zn²?. Valence electrons were treated explicitly, while core electrons were represented using norm-conserving Goedecker–Teter–Hutter pseudopotentials. The wave functions were expanded using the mixed Gaussian-and-plane-wave scheme with TZV2P-MOLOPT basis sets for all elements and a 600 Ry auxiliary plane-wave cutoff for the electron density. Self-consistent-field convergence was accelerated using the orbital-transformation and Direct Inversion in the Iterative Subspace algorithms, with a convergence threshold of 10?6. All single-point calculations were performed in periodic orthorhombic cells. The CELL_REF keyword in CP2K was used to define a fixed reference cell with a box length of 25 Å. This treatment ensured a consistent reference for configurations extracted from NpT trajectories with fluctuating cell dimensions. The resulting DFT energies and atomic forces constitute the ground-truth labels used to train the MLIPs. The resulting MLIP was trained for aqueous ZnCl2 solutions spanning concentrations from 0 to 30 molal and a broad pH range, from strongly acidic to strongly basic conditions. Representative examples of configurations included in the MLIP training dataset are provided below. These include 1) Representative configurations from the dataset labeled at the revPBE-D3 level, with D3 dispersion interactions involving Zn2+ excluded (revPBE-wo-D3). 2) Representative configurations from the dataset labeled at the fully dispersion-corrected revPBE-D3 level, with D3 interactions applied to all species, including Zn2+ (revPBE-D3). 3) Representative configurations from the dataset labeled at the r2SCAN level of theory (r2SCAN).

Dinpajooh, Mohammadhasan [Pacific Northwest Nation↗

Interactions involved in the adsorption of ethylene glycol and 2-hydroxyethoxide on the Au(111) surface: a Density Functional Theory study

Abstract Context:The monolayers of ethylene glycol and 2-hydroxyethoxide on gold surfaces have been used in hybrid materials as biosensors. In this article, the adsorption of ethylene glycol and 2-hydroxyethoxide on the Au(111) surface was analyzed. For the first system, ethylene glycol on Au(111), there are Au$$\cdot \cdot \cdot $$ · · · O and Au$$\cdot \cdot \cdot $$ · · · H interactions. To the best of our knowledge, the Au$$\cdot \cdot \cdot $$ · · · H interaction has been overlooked until now. However, in this work, there is strong evidence that this interaction is important to stabilize the system. For the second system, the atomic interactions mentioned previously are also predicted, although there is an additional interaction between 2-hydroxyethoxide molecules. Such an interaction induces the link -O-H-O-, with high values of the electron density at the critical points of the corresponding bond path of the O-H interaction. These links suggest the forming of ethylene glycol chains.Methods:The calculations were performed using two exchange-correlation functionals: BEEF-vdW and C09$$_{x}$$ x -vdW; both functionals incorporate dispersion effects within the Kohn-Sham approach in Density Functional Theory as implemented in GPAW code and ASE computational packages. The contacts between the molecules considered in this article and the Au(111) surface were analyzed through the Quantum Theory of Atoms in Molecules implemented in GPUAM code.

Biochemistry & Molecular Biology↗

Single vs double atom catalyst for N 2 activation in nitrogen reduction reaction: A DFT perspective

Abstract Ammonia synthesis through electrochemical reduction of nitrogen molecules is a promising strategy to significantly reduce the energy consumption in traditional industrial process. Detailed mechanism study of multistep complex nitrogen reduction reaction is prerequisite for the design of highly efficient catalyst. Stable atomically dispersed catalyst with unique geometric and electronic structure is suitable for the mechanism clarification of such a complex reaction. In this study, d ‐block transition‐metal (TM) anchored C 2 N single layer catalyst is investigated by the density functional theory (DFT) calculation. Both single TM‐anchored single atom catalyst (SAC) and double TM‐anchored double atom catalyst (DAC) exhibit good thermodynamic stability in atomically dispersed catalyst. In the case of SACs, IVB metals (Ti, Zr, Hf) exhibit the highest reactivity and lowest overpotential. While in the case of DACs, Cr─Cr system leads to the NH 3 formation, but V─V system leads to the N 2 H 4 formation. The SACs show much lower overpotential and stronger activation of N 2 molecule than the DACs due to the different activation mechanisms: traditional σ ‐donation/ π ‐backdonation N 2 activation mechanism is found in SACs, while a new π ‐donation/π‐backdonation N 2 activation mechanism is found in the DACs. The present work demonstrates that the different catalytic effect for NRR between SAC and DAC and their corresponding electronic structure origin, which gives more insight into the single atom catalyst. image

Qian, Yumin↗

Recent advances in coupled cluster computations of open-shell atomic nuclei

In this contribution, we report on recent progress in coupled-cluster simulations of open-shell atomic nuclei using interactions consistently derived from chiral effective field theory. In particular, we compare different coupled-cluster approaches by computing binding energies and electric dipole polarizabilities in medium-mass calcium isotopes.

Marino, Francesco [Johannes Gutenberg Univ., Mainz↗

An electronic origin to the oscillatory segregation behavior in Ni-Cr and other BCC defects in FCC metals

Ni-Cr alloys exhibit oscillatory segregation behaviors near low index surfaces, in which the preferred segregation species changes from Ni in the first layer to Cr in the second layer. In many dilute-alloy systems, this oscillatory pattern is attributed to the elastic release of stresses in the local lattice around the segregating solute or impurity atom. These stresses are mostly thought to originate from mismatches in the atomic size of the solute and host atoms. In Ni-Cr alloys, however, an appreciable mismatch in atomic size is not present, leading to questions about the origins of the oscillatory behavior in this alloy. Therefore, using density functional theory, we have modeled the segregation of a single Cr atom in the and surfaces of FCC Ni, an alloy which exhibits this oscillatory behavior. Using Bader charge analysis, we show that the negative energy correlates directly with the amount of charge on the Cr atom. As Ni atoms strip valence charge from the Cr, the Cr contracts slightly in size. The greatest contraction and highest positive charge for the Cr occurs when it is in the second layer of the surface where the system exhibits the oscillating negative segregation energy. We then find that this behavior persists in other alloy systems (Ag-Nb, Cu-Cr, Pt-Nb, and Pt-V), which exhibit similar atomic radii and electronegativity differences between host and solute to Ni-Cr. These represent alloys in which the host metal exhibits an FCC ground-state structure while the solute metal exhibits a BCC ground-state structure.

36 MATERIALS SCIENCE↗

High-throughput dataset of impurity adsorption on common catalysts in biomass upgrading applications

Abstract An extensive dataset consisting of adsorption energies of pernicious impurities present in biomass upgrading processes on common catalysts and support materials has been generated. This work aims to inform catalyst and process development for the conversion of biomass-derived feedstocks to fuels and chemicals. A high-throughput workflow was developed to execute density functional theory calculations for a diverse set of atomic (Al, B, Ca, Cl, Fe, K, Mg, Mn, N, Na, P, S, Si, Zn) and molecular (COS, H 2 S, HCl, HCN, K 2 O, KCl, NH 3 ) species on 35 unique surfaces for transition-metal (Ag, Au, Co, Cu, Fe, Ir, Ni, Pd, Pt, Re, Rh, Ru) and metal-oxide (Al 2 O 3 , MgO, anatase-TiO 2 , rutile-TiO 2 , ZnO, ZrO 2 ) catalysts and supports. Approximately 3,000 unique adsorption geometries and corresponding adsorption energies were obtained.

09 BIOMASS FUELS↗

DFT-based mesoscopic interactions of N adatoms on Cu(100)

The first-principles density functional theory (DFT) method is used to directly calculate the interactions of N atoms of various configurations on Cu(100) surfaces. This allows us to derive a mesoscopic substratemediated interaction model. We find that, for these Cu(100)-supported two-dimensional N clusters, oscillating substrate-mediated interactions obtained from an elastic eigenvector approach including a mesoscopic interacting range (called the mesoscopic model) have reasonable agreement with the DFT results. The DFT periodic boundary conditions are properly mapped into the mesoscopic model. The model is used to analyze the energies of c(2×2) square islands independent of their sizes. Previous experimental studies have shown that, in the low coverage region, N atoms agglomerate to isolated square islands of about 5 nm × 5 nm in size with a c(2×2) structure. With increasing coverage, those islands form regular patterns but do not touch below about 0.4 monolayers. From previous experiments and calculations, the square island pattern shows significant elastic effects which are now confirmed.

74 ATOMIC AND MOLECULAR PHYSICS↗

Datasets for Custom-trained Machine-learning Interatomic Potentials: Nitric Acid Aqueous Solution

This dataset was generated using an iterative active learning strategy with the ArcaNN software package (https://github.com/arcann-chem/arcann_training) to train machine-learning interatomic potentials (MLIPs) for aqueous nitric acid. Each active-learning cycle consisted of three stages: (1) training, (2) exploration, and (3) labeling. The initial training set comprised approximately 800 randomly selected configurations from a previous study by Lewis et al. (https://doi.org/10.1021/jp205510q), which investigated nitric acid solutions at 2, 3, 4, and 5 mol/L. For all configurations, single-point calculations of atomic forces and total energies were performed at the quantum density functional theory BLYP-D2 and PBE-D3 levels of theory using the CP2K Quickstep module. Valence electrons were treated explicitly, while core electrons on all atoms were represented by norm-conserving Goedecker–Teter–Hutter (GTH) pseudopotentials. Long-range dispersion interactions were accounted for using Grimme dispersion corrections. Wave functions were expanded in a mixed Gaussian-and-plane-wave scheme using TZV2P-MOLOPT basis sets for all elements and an 800 Ry auxiliary plane-wave cutoff for the electron density. Self-consistent field convergence was accelerated using orbital transformation and Direct Inversion in the Iterative Subspace, with a convergence threshold of 10^{-6}. All single-point calculations were carried out in periodic orthorhombic cells whose dimensions match those of the molecular configurations sampled from earlier trajectories. The CELL_REF keyword in CP2K was used to define a fixed reference cell, ensuring consistency in the reference data used for MLIP training, particularly when cell fluctuations are present in NpT simulations. The resulting high-fidelity energies and forces constitute the ground-truth labels used to train the MLIPs contained in this dataset.

Dinpajooh, Mohammadhasan [Pacific Northwest Nation↗

Hydrogen irradiation-driven computational surface chemistry of lithium oxide and hydroxide

We have investigated, using molecular dynamics, the surface chemistry of hydrogen incident on the amorphous and crystalline lithium oxide and lithium hydroxide surfaces upon being slowed down by a collision cascade and retained in the amorphous surface of either Li 2 O or LiOH. We looked for the bonding of H to the resident structures in the surface to understand a possible chain of chemical reactions that can lead to surface transformation upon H atom impact. Our findings, using Density-Functional Theory (DFT) trained ReaxFF force field/electronegativity equalization method potentials, stress the importance of inclusion of polarization in the dynamics of a Li–O–H system, which is also illustrated by DFT energy minimization and quantum–classical molecular dynamics using tight binding DFT. The resulting polar-covalent chemistry of the studied systems is complex and very sensitive to the instantaneous positions of all atoms as well as the ratio of concentrations of various resident atoms in the surface.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Combined density functional theory/kinetic Monte Carlo investigation of surface morphology during cycling of Li-Cu electrodes

In this work, we employed a combined density functional theory and kinetic Monte Carlo approach in order to investigate the nanoscale behavior of lithium stripping and plating on lithium-copper nanoalloys. Density functional theory calculations were used to develop an atom-by-atom understanding of the nucleation behavior of lithium on copper surfaces with different surface morphologies. We found that nanometric pits resulted in significantly more favorable lithium deposition when compared to pristine or lightly-defective Cu surfaces, which is in agreement with previous experimental studies. We then used density functional theory and constrained ab initio molecular dynamics to develop interaction and reaction parameters for a larger-scale kinetic Monte Carlo model in order to provide insight into the stripping and plating behavior of nanostructured Li-Cu nanoalloy anodes. Substantial losses in capacity and strong indications of morphological change were observed for a wide range of Cu:Li ratios over 100 cycles. We observed a complex relationship between the nanoparticle composition and the properties of the formed solid-electrolyte interphase.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Material Effects on Electron-Capture Decay in Cryogenic Sensors

Several current searches for physics beyond the standard model are based on measuring the electron-capture (EC) decay of radionuclides implanted into cryogenic high-resolution sensors. The sensitivity of these experiments has already reached the level where systematic effects related to atomic state energy changes from the host material are a limiting factor. One example is a neutrino mass study based on the nuclear EC decay of 7 Be to 7 Li inside cryogenic Ta-based sensors. Here, to understand the material effects at the required level, we use density-functional theory to model the electronic structure of lithium atoms in different atomic environments of the polycrystalline Ta absorber film. The calculations reveal that the Li 1s binding energies can vary by more than 2 eV due to insertion at different lattice sites, at grain boundaries, in disordered Ta, and in the vicinity of various impurities. However, the total range of Li 1s shifts does not exceed 4 eV, even for extreme amorphous disorder. Furthermore, when investigating the effects on the Li 2s levels, we find broadening of more than 5 eV due to hybridization with the Ta band structure. Material effects are shown to contribute significantly to peak broadening in Ta-based sensors that are used to search for physics beyond the standard model in the EC decay of 7 Be, but they do not explain the full extent of observed broadening. Understanding these in-medium effects will be required for current- and future-generation experiments that observe low-energy radiation from the EC decay of implanted isotopes to evaluate potential limitations on the measurement sensitivity.

36 MATERIALS SCIENCE↗

Double-Atom Catalysts Featuring Inverse Sandwich Structure for CO 2 Reduction Reaction: A Synergetic First-Principles and Machine Learning Investigation

Electrocatalytic CO 2 reduction reactions (CO 2 RR) based on scalable and highly efficient catalysis provide an attractive strategy for reducing CO 2 emissions. Here in this work, we combined first-principles density functional theory (DFT) and machine learning (ML) to comprehensively explore the potential of double-atom catalysts (DACs) featuring an inverse sandwich structure anchored on defective graphene (gra) to catalyze CO 2 RR to generate C 1 products. We started with five homonuclear M 2 ⊥gra (M = Co, Ni, Rh, Ir, and Pt), followed by 127 heteronuclear MM'⊥gra (M = Co, Ni, Rh, Ir, and Pt, M' = Sc–Au). Stable DACs were screened by evaluating their binding energy, formation energy, and dissolution potential of metal atoms, as well as conducting first-principles molecular dynamics simulations with and without solvent water molecules. Based on DFT calculations, Rh 2 ⊥gra DAC was found to outperform the other four homonuclear DACs and the Rh-based single- and double-atom catalysts of noninverse sandwich structures. Out of the 127 heteronuclear DACs, 14 were found to be stable and have good catalytic performance. An ML approach was adopted to correlate key factors with the activity and stability of the DACs, including the sum of radii of metal and ligand atoms (d M–M' , d M–C , and d M'–C ), the sum and difference of electronegativity of two metal atoms (P M + P M' , P M – P M '), the sum and difference of first ionization energy of two metal atoms (I M + I M' , I M – I M '), the sum and difference of electron affinity of two metal atoms (A M + A M' , A M – A M '), and the number of d-electrons of the two metal atoms (Nd). The obtained ML models were further used to predict 154 potential electrocatalysts out of 784 possible DACs featuring the same inverse sandwich configuration. Overall, this work not only identified promising CO 2 RR DACs featuring the reported inverse sandwich structure but also provided insights into key atomic characteristics associated with high CO 2 RR activity.

30 DIRECT ENERGY CONVERSION↗

Why Is the Range of Timescale So Wide in Glass-Forming Liquid?

The viscosity and the relaxation time of a glass-forming liquid vary over 15 orders of magnitude before the liquid freezes into a glass. The rate of the change with temperature is characterized by liquid fragility. The mechanism of such a spectacular behavior and the origin of fragility have long been discussed, but it remains unresolved because of the difficulty of carrying out experiments and constructing theories that bridge over a wide timescale from atomic (ps) to bulk (minutes). Through the x-ray diffraction measurement and molecular dynamics simulation for metallic liquids we suggest that large changes in viscosity can be caused by relatively small changes in the structural coherence which characterizes the medium-range order. Here the structural coherence does not imply that of atomic-scale structure, but it relates to the coarse-grained density fluctuations represented by the peaks in the pair-distribution function (PDF) beyond the nearest neighbors. The coherence length is related to fragility and increases with decreasing temperature, and it diverges only at a negative temperature. This analysis is compared with several current theories which predict a phase transition near the glass transition temperature.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Unique Features of Polarization in Ferroelectric Ionic Conductors

Abstract Ferroelectrics that are also ionic conductors offer possibilities for novel applications with high tunability, especially if the same atomic species causes both phenomena. In particular, at temperatures just below the Curie temperature, polarized states may be sustainable as the mobile species is driven in a controlled way over the energy barrier that governs ionic conduction, resulting in unique control of the polarization. This possibility is recently demonstrated in CuInP 2 S 6 , a layered ferroelectric ionic conductor in which Cu ions cause both ferroelectricity and ionic conduction. Here, it is shown that the commonly used approach to calculate the polarization of evolving atomic configurations in ferroelectrics using the modern theory of polarization, namely concerted (synchronous) migration of the displacing ions, is not well suited to describe the polarization evolution as the Cu ions cross the van der Waals gaps. An asynchronous Cu‐migration scheme is introduced, which reflects the physical process by which Cu ions migrate, resolves the difficulties, and describes the polarization evolution both for normal ferroelectric switching and for transitions across the van der Waals gaps, providing a single framework to discuss ferroelectric ionic conductors.

O'Hara, Andrew↗

Strengthening of tungsten by coherent rhenium precipitates formed during low fluence irradiation

Experimental data show that the accumulation of rhenium and osmium from transmutation reactions severely affect the microstructural evolution and property degradation of tungsten-based materials under neutron irradiation. Theory and modeling have confirmed that Re atom transport in W is by irradiation-produced migrating self-interstitial atoms. With this diffusion mode in operation, a specific microstructure evolution is realized when at relatively low neutron fluence the Re-rich precipitates are formed, while the void and interstitial loop population development is suppressed, affecting the mechanical properties. Here, this research shows the effect of small coherent Re-rich precipitates on the dislocation glide under stress, investigated using the molecular dynamics approach with empirical interatomic potentials. The results are compared with an earlier simulation of void hardening in W. It is demonstrated that small coherent Re-rich precipitates of less than 6 nm diameter represent relatively weak obstacles for moving edge dislocations. The implication of these results on the interpretation of experimental results is discussed.

36 MATERIALS SCIENCE↗

DFTTK: Density Functional Theory ToolKit for high-throughput lattice dynamics calculations

In this work, we present a software package in Python for high-throughput first-principles calculations of thermodynamic properties at finite temperatures, which we refer to as DFTTK (Density Functional Theory ToolKit). DFTTK is based on the atomate package and integrates our experiences in the last decades on the development of theoretical methods and computational softwares. It includes task submissions on all major operating systems and task executions on high-performance computing environments. Furthermore, the distribution of the DFTTK package comes with examples of calculations of phonon density of states, heat capacity, entropy, enthalpy, and free energy under the quasi-harmonic phonon scheme for the stoichiometric phases of Al, Ni, Al 3 Ni, AlNi, AlNi 3 , Al 3 Ni 4 , and Al 3 Ni 5 , and the fcc solution phases treated using the special quasirandom structures at the compositions of Al 3 Ni, AlNi, and AlNi 3 .

97 MATHEMATICS AND COMPUTING↗