Based on density functional perturbation theory, we have studied the vibrational properties of three crystalline phases of tellurium dioxide: paratellurite alpha-TeO2, tellurite beta-TeO2, and the new phase gamma-TeO2, recently identified experimentally. Calculated Raman and IR spectra are in good agreement with available experimental data. The vibrational spectra of alpha- and beta-TeO2 can be interpreted in terms of vibrations of TeO2 molecular units.
Based on Density Functional Theory and Density Functional Perturbation Theory we have studied the thermodynamical and vibrational properties of Mg(AlH$_4$)$_2$.
A density-functional-theory based approach to efficiently compute numerically exact vibrational free energies - including anharmonicity - for chemically complex multicomponent alloys is developed. It is based on a combination of thermodynamic integration and a machine-learning potential. We demonstrate the performance of the approach by computing the anharmonic free energy of the prototypical five-component VNbMoTaW refractory high entropy alloy.
We have studied the normal modes of hydrogenated and oxidized silicon nanocrystals, namely SiH4 (silan), H2SiO (silanon), Si10H16 and Si10H14O. The small clusters (SiH4 and H2SiO) have been used for convergence tests and their bondlengths and frequencies have been compared with experimental and theoretical reference data. For the large clusters (Si10H16 and Si10H14O) we have investigated the vibrational density of states where we have identified the oxygen-related spectral features. The vibrational modes have been also analyzed with respect to the displacement patterns. The calculations have b…
We present a theoretical first-principles method to calculate the free energy of a magnetic system in its high-temperature paramagnetic phase, including vibrational, electronic, and magnetic contributions. The method for calculating free energies is based on ab-initio molecular dynamics and combines a treatment of disordered magnetism using disordered local moments molecular dynamics (DLM-MD) with the temperature dependent effective potential (TDEP) method to obtain the vibrational contribution to the free energy. We illustrate the applicability of the method by obtaining the anharmonic free e…
We present an approach for the efficient calculation of vibrational Raman intensities in periodic systems within density functional theory. The Raman intensities are computed from the second order derivative of the electronic density matrix with respect to a uniform electric field. In contrast to previous approaches, the computational effort required by our method for the evaluation of the intensities is negligible compared to that required for the calculation of vibrational frequencies. As a first application, we study the signature of 3- and 4-membered rings in the the Raman spectra of sever…
In the present work we have performed an ab initio calculation of vibrational properties of CuTe2O5 by means of density functional theory method. One has compared calculated values with known experimental data on Raman and infrared spectroscopy in order to verify the obtained results. Lattice contribution to the heat capacity, obtained from the ab initio simulations was added to magnetic contribution calculated from the simple spin hamiltonian model in order to obtain total heat capacity. Obtained result are in good agreement to the experimental data. Thus, the DFT methods could complement the…
A study based on ab initio calculations is presented on the estructural, elastic, and vibrational properties of single-wall carbon nanotubes with different radii and chiralities. We use SIESTA, an implementation of pseudopotential-density-functional theory which allows calculations on systems with a large number of atoms per cell. Different quantities like bond distances, Young moduli, Poisson ratio and the frequencies of different phonon branches are monitored versus tube radius. The validity of expectations based on graphite is explored down to small radii, where some deviations appear relat…
The different contributions to the entropy change linked to the austenite-martensitic transition in a Ni-Mn-Sn metamagnetic shape memory alloy have been determined by combining different experimental techniques. The vibrational contribution has been inferred from the vibrational density of states of both the martensitic and austenite phases. This has been accomplished by combining time-of-flight neutron scattering measurements and ab-initio calculations. Further, the electronic part of the entropy change has also been calculated. Since the martensitic transformation takes place between two par…
The PEO3:LiCF3SO3 polymer electrolyte has attracted significant research due to its enhanced stability at the lithium/polymer interface of high conductivity polymer batteries. Experimental studies have shown that, depending on the preparation conditions, both the PEO3:LiCF3SO3 crystalline complex and the PEO3:LiCF3SO3 amorphous phase can be formed. However, previous theoretical investigations focused on the short chain amorphous PEO3:LiCF3SO3 system. We report ab initio density-functional-theory calculations of crystalline PEO3:LiCF3SO3. The calculated results about the bonding configuration, …
The structural, vibrational, thermodynamical and optical properties of technologically important, weakly coupled MAX compound, Sc2AlC are calculated using density functional theory (DFT). The structural properties of Sc2AlC are compared with results reported earlier. The vibrational, thermodynamical, and optical properties are theoretically estimated for the first time. The phonon dispersion curve has been calculated and the dynamical stability of this compound has been investigated. The optical and acoustic modes are observed clearly. We have calculated the Helmholtz free energy (F), internal…
In this work, we present our theoretical results for the equation of state and the phonon dispersions of MnAs, as well as the Mn concentration dependence of both the lattice parameter and the phonon frequencies of the cubic GaMnAs alloys. The results are in good agreement with the experimental results whenever this comparison is possible. Based on the obtained results, the lattice constants and the phonon frequencies of the alloys do not obey the Vegard rule.
Plane-wave pseudopotential methods and density functional perturbation theory are used to calculate the phonon density of states and thermodynamic functions of h-P4O10. The calculated vibrational spectrum is in good agreement with the measured spectrum, but the calculations indicate some modifications in the interpretation of the spectrum, mainly suggesting changes in the number of components used to fit a few of the observed peaks. The calculated low-temperature heat capacity is in good agreement with the measured heat capacity, with a systematic offset of approximately -5 J mol-1 K-1, indepe…
Structural stability and vibrational properties of KN$_3$ have been studied under pressure using the first principles calculations based on the density functional theory (DFT) as well as with semi empirical dispersion correction to the DFT to treat van der Waals interactions. The structural properties calculated through the dispersion correction are in good agreement with the experimental results when compared with the standard density functional theory. The lattice mode and the azide ion bending mode frequencies are found to be well reproduced by our calculations. As pressure increases, the E…
Boron nitride (BN) is a structurally versatile insulator since it can be found in several crystalline structures with interesting mechanical and electrical properties, making this material attractive for technological applications. Seeking to improve its features, experimental and simulational studies for the amorphous phase (a-BN) have been carried out by some groups, focusing on the electrical and structural properties, pressure-induced phase transformations, and hydrogenated a-BN. In this work two amorphous structures are computationally generated and studied using ab initio Molecular Dynam…
In this paper we present an approach aimed at performing many-body calculations of Born-effective charges of crystalline insulators, by including the electron-correlation effects. The scheme is implemented entirely in the real space, using Wannier-functions as single-particle orbitals. Correlation effects are computed by including virtual excitations from the Hartree-Fock mean field, and the excitations are organized as per a Bethe-Goldstone-like many-body hierarchy. The results of our calculations suggest that the approach presented here is promising.
Despite vibrational properties being critical for the ab initio prediction of the finite temperature stability and transport properties of solids, their inclusion in ab initio materials repositories has been hindered by expensive computational requirements. Here we tackle the challenge, by showing that a good estimation of force constants and vibrational properties can be quickly achieved from the knowledge of atomic equilibrium positions using machine learning. A random-forest algorithm trained on only 121 metastable structures of KZnF$_3$ reaches a maximum absolute error of 0.17 eV/$\textrmÅ…
The effect of pressure on the structural and vibrational properties of the layered molecular crystal 1,1-diamino-2,2-dinitroethelene (FOX-7) are explored by first principles calculations. We observe significant changes in the calculated structural properties with different corrections for treating van der Waals interactions to Density Functional Theory (DFT), as compared with standard DFT functionals. In particular, the calculated ground state lattice parameters, volume and bulk modulus obtained with Grimme's scheme are found to agree well with experiments. The calculated vibrational frequenci…
We report the structural, elastic and vibrational properties of five ionic-molecular solid oxidizers MNO$_3$ (M = Li, Na, K) and MClO$_3$ (M = Na, K). By treating long range electron-correlation effects, dispersion corrected method leads to more accurate predictions of structural properties and phase stability of KNO$_3$ polymorphs. The obtained elastic moduli show soft nature of these materials and are consistent with Ultrasonic Pulse Echo measurements. We made a complete assignment of vibrational modes which are in good accord with available experimental results. From calculated IR and Raman…
An ab initio equation of motion method is introduced to calculate the temperature-dependent mean square vibrational amplitudes which appear in the Debye-Waller factors in x-ray absorption, x-ray scattering, and related spectra. The approach avoids explicit calculations of phonon-modes, and is based instead on calculations of the displacement-displacement time correlation function from ab initio density functional theory molecular dynamics simulations. The method also yields the vibrational density of states and thermal quantities such as the lattice free energy. Illustrations of the method are…
The (Cd, Be)Te semiconductor alloy that exhibits a dramatic mismatch in bond covalency and stiffness clarifying its vibrational and mechanical properties is used as a benchmark to test the limits of the percolation model (PM) worked out to explain the complex Raman spectra of the related but less contrasted (Zn, Be) chalcogenides. The test is done by way of experiment (x smaller than 0.11) combining Raman scattering with X ray diffraction at high pressure and ab initio calculations (x around 0, 0.5 and 1). The (macroscopic) bulk modulus B drops below the CdTe value on minor Be incorporation, a…
We present a novel approach for parameter-free modeling of the structural, dynamical and electronic properties of non-crystalline materials based on ab-initio Molecular Dynamics, improved signal processing technique and computer visualization. The method have been extensively tested by investigating hydrogen and silicon dynamics in hydrogenated amorphous silicon (a-Si:H). By comparing the theoretical and experimental vibrational spectra we demonstrate how to relate vibrational properties to the structural stability, bonding and hydrogen diffusion. We extracted microscopic characteristics that …
Thermal vibrations alter the external potential. Allen (Phys. Rev. B 18 (1978) 5217) proved that at finite temperatures the pseudopotential form factors are corrected by a Debye-Waller Factor (DWF). We generalize this result to the crystal potential. (The generalization to the all-electron case of the nuclear potential fails due to the breakdown of the rigid-atom approximation.) This finite temperature formalism only gives thermal-averaged properties and no dynamical information can be obtained. Hence, it is labeled the Quasi Ab Initio formalism. Analogous to the use of experimental lattice pa…
We calculate the electronic and optical excitations of polythiophene using the GW approximation for the electronic self-energy, and include excitonic effects by solving the electron-hole Bethe-Salpeter equation. Two different situations are studied: excitations on isolated chains and excitations on chains in crystalline polythiophene. The dielectric tensor for the crystalline situation is obtained by modeling the polymer chains as polarizable line objects, with a long-wavelength polarizability tensor obtained from the ab-initio polarizability function of the isolated chain. With this model die…
In this paper we have used our recently developed ab initio Wannier-function-based methodology to perform extensive Hartree-Fock and correlated calculations on Li$_{2}$O and LiCl to compute their Born effective charges. Results thus obtained are in very good agreement with the experiments. In particular, for the case of Li$_{2}$O, we resolve a controversy originating in the experiment of Osaka and Shindo {[}Solid State Commun. 51 (1984) 421] who had predicted the effective charge of Li ions to be in the range 0.58--0.61, a value much smaller compared to its nominal value of unity, thereby, sug…
Predicting which crystalline modifications can be present in a chemical system requires the global exploration of its energy landscape. Due to the large computational effort involved, in the past this search for sufficiently stable minima has been performed employing a variety of empirical potentials and cost functions followed by a local optimization on the ab initio level. However, this entails the risk of overlooking important modifications that are not modeled accurately using empirical potentials. In order to overcome this critical limitation, we develop an approach to employ ab initio en…
We present the investigations on atomic dynamics and Li+ diffusion in crystalline and amorphous Li2Si2O5 using quasielastic (QENS) and inelastic neutron scattering (INS) studies supplemented by ab-initio molecular dynamics simulations (AIMD). The QENS measurements in the amorphous phase of Li2Si2O5 show a narrow temperature window (700 < T < 775 K), exhibiting significant quasielastic broadening corresponding to the fast Li+ diffusion and relaxation of SiO4 units to the crystalline phase. Our INS measurements clearly show the presence of large phonon density of states (PDOS) at low energy (low…
We investigated the structural and dynamical properties of a tetrahedrally coordinated crystalline ice from first principles based on density functional theory within the generalized gradient approximation with the projected augmented wave method. First, we report the structural behaviour of ice at finite temperatures based on the analysis of radial distribution functions obtained by molecular dynamics simulations. The results show how the ordering of the hydrogen bonding breaks down in the tetrahedral network of ice with entropy increase in agreement with the neutron diffraction data. We also…
Molecular dynamics (MD) simulations of crystalline poly(ethylene oxide) (PEO) have been carried out in order to study its vibrational properties. The vibrational density of states has been calculated using a normal mode analysis (NMA) and also through the velocity autocorrelation function of the atoms. Results agree well with experimental spectroscopic data. System size effects in the crystalline state, studied through a comparison between results for 16 unit cells and that for one unit cell has shown important differences in the features below 100 cm^-1. Effects of interchain interactions are…
In this paper we infer the structure of Pd40Ni40P20 from experimental diffraction data and ab initio interactions using Force Enhanced Atomic Refinement (FEAR). Our model accurately reproduces known experimental signatures of the system and is more efficient than conventional melt-quench schemes. We critically evaluate the local order, carry out detailed comparisons to extended X-ray absorption fine structure (EXAFS) experiments and also discuss the electronic structure. We thoroughly explore the lattice dynamics of the system, and describe a vibrational localized-to-extended transition and di…
In this paper we present an approach aimed at calculating the optical dielectric constant of crystalline insulators both at the Hartree-Fock, and correlated levels. Our scheme employs a real-space methodology, employing Wannier functions as the basic building blocks. The scheme has been applied to compute the optical dielectric constants of LiF, LiCl, MgO, and Li2O. In all the cases, results of correlated calculations, based upon systematic many-body corrections beyond Hartree-Fock, exhibit excellent agreement with the experiments. To the best of our knowledge, the calculations presented here …
Crystalline phases formed in stoichiometric Zr$_9$Ni$_{11}$ and Hf$_9$Ni$_{11}$ have been studied by perturbed angular correlation (PAC) spectroscopy, XRD and TEM/SAED measurements. In Zr$_9$Ni$_{11}$, the phases Zr$_9$Ni$_{11}$ ($\sim$89\%) and Zr$_8$Ni$_{21}$ ($\sim$11\%) have been found at room temperature from PAC measurements. At 773 K, Zr$_9$Ni$_{11}$ partially decomposes to Zr$_7$Ni$_{10}$ and at 973 K, it is completely decomposed to ZrNi and Zr$_7$Ni$_{10}$. In Hf$_9$Ni$_{11}$, a predominant phase ($\sim$81\%) due to HfNi is found at room temperature while the phase Hf$_9$Ni$_{11}$ is …
Bismuth displays puzzling superconducting properties. In its crystalline equilibrium phase, it does not seem to superconduct at accessible low temperatures. However, in the amorphous phase it displays superconductivity at ~ 6 K. Under pressure bismuth has been found to superconduct at Tcs that go from 3.9 K to 8.5 K depending on the phase obtained. So the question is: what electronic or vibrational changes occur that explains this radical transformation in the conducting behavior of this material? In a recent publication we argue that changes in the density of electronic and vibrational states…
The vibrational entropy of a solid at finite temperature is investigated from the perspective of information theory. Ab initio molecular dynamics (AIMD) simulations generate ensembles of atomic configurations at finite temperature from which we obtain the $N$-body distribution of atomic displacements, $ρ_N$. We calculate the information-theoretic entropy from the expectation value of $\ln{ρ_N}$. At a first level of approximation, treating individual atomic displacements independently, our method may be applied using Debye-Waller B-factors, allowing diffraction experiments to obtain an upper bo…
The vibrational properties of graphene fluoride and graphane are studied using ab initio calculations. We find that both sp3 bonded derivatives of graphene have different phonon dispersion relations and phonon density of states as expected from the different masses associated with the attached atoms fluorine and hydrogen, respectively. These differences manifest themselves in the predicted temperature behavior of the constant-volume specific heat of both compounds.
The correct calculation of formation enthalpy is one of the enablers of ab-initio computational materials design. For several classes of systems (e.g. oxides) standard density functional theory produces incorrect values. Here we propose the "Coordination Corrected Enthalpies" method (CCE), based on the number of nearest neighbor cation-anion bonds, and also capable of correcting relative stability of polymorphs. CCE uses calculations employing the Perdew, Burke and Ernzerhof (PBE), Local Density Approximation (LDA) and Strongly Constrained and Appropriately Normed (SCAN) exchange correlation f…
We develop a shell model (SM) for potassium dihydrogen phosphate (KDP) which is fitted to ab initio (AI) results that include nonlocal van der Waals corrections. The SM is comprehensively tested by comparing results of structural, vibrational and ferroelectric properties with AI and experimental data. The relaxed structural parameters are in very good agreement with the AI results and the available experimental data. The $Γ$-point phonons and the total phonon densities of states (DOSs) in the ferroelectric and paraelectric phases calculated with the developed SM are in good overall agreement w…
Molecular adsorbates on metal surfaces exchange energy with substrate phonons and low-lying electron-hole pair excitations. In the limit of weak coupling, electron-hole pair excitations can be seen as exerting frictional forces on adsorbates that enhance energy transfer and facilitate vibrational relaxation or hot-electron mediated chemistry. We have recently reported on the relevance of tensorial properties of electronic friction [Phys. Rev. Lett. 116, 217601 (2016)] in dynamics at surfaces. Here we present the underlying implementation of tensorial electronic friction based on Kohn-Sham Dens…
Given the unique optical properties of LiF, it is often used as an observation window in high-temperature and pressure experiments; and, hence, estimates of its transmission properties are necessary to interpret observations. Since direct measurements of the thermal conductivity of LiF at the appropriate conditions are difficult, we resort to molecular simulation methods. Using the Belonoshko et al. (2000) empirical potential validated against ab initio phonon density of states, we estimate the thermal conductivity of LiF at high temperatures (1000-4000K) and pressures (100-400 GPa) with the G…
The atomistic resolution recently achieved by ultrafast spectroscopies demands corresponding theoretical advances. Real-time time-dependent density-functional theory (RT-TDDFT) with Ehrenfest dynamics offers an optimal trade-off between accuracy and computational costs to study electronic and vibrational dynamics of laser-excited materials in the sub-picosecond regime. However, this approach is unable to account for thermal effects or zero-point energies which are crucial in the physics involved. Herein, we adopt a quantum-semiclassical method based on RT-TDDFT+Ehrenfest to simulate laser-indu…
From density-functional-theory (DFT) based methods we calculate the vibrational spectrum of the Mn_{12}O_{12}(COOH)_{16}(H_2 O)_4 molecular magnet. Calculated infrared intensities are in accord with experimental studies. There have been no ab initio attempts at determining which interactions account for the fourth-order anisotropy. We show that vibrationally induced distortions of the molecule contribute to the fourth-order anisotropy Hamiltonian and that the magnitude and sign of the effect (-6.2 K) is in good agreement with all experiments. Vibrationally induced tunnel splittings in isotopic…
We report a phonon density of states measurement of $α$-ZnCl$_{2}$ using the coherent inelastic neutron scattering technique and a lattice dynamical calculation in four crystalline phases of ZnCl$_{2}$ using a transferable interatomic potential. The model calculations agree reasonably well with the available experimental data on the structures, specific heat, Raman frequencies and their pressure variation in various crystalline phases. The calculated results have been able to provide a fair description of the vibrational as well as the thermodynamic properties of ZnCl$_{2}$ in all its four pha…
Hydrogen arranges at dislocations in palladium to form nanoscale hydrides, changing the vibrational spectra. An ab initio hydrogen potential energy model versus Pd neighbor distances allows us to predict the vibrational excitations for H from absolute zero up to room temperature adjacent to a partial dislocation and with strain. Using the equilibrium distribution of hydrogen with temperature, we predict excitation spectra to explain new incoherent inelastic neutron-scattering measurements. At 0K, dislocation cores trap H to form nanometer-sized hydrides, while increased temperature dissolves t…
Talc is an insulating layered material that is stable at ambient conditions and has high-quality basal cleavage, which is a major advantage for its use in van der Waals heterostructures. Here, we use near-field synchrotron infrared nanospectroscopy, Raman spectroscopy, and first-principles calculations to investigate the structural and vibrational properties of talc crystals, ranging from monolayer to bulk, in the 300-750 cm-1 and <60 cm-1 spectral windows. We observe a symmetry crossover from mono to bilayer talc samples, attributed to the stacking of adjacent layers. The in-plane lattice par…
Molecular dynamics calculations were preformed on organic crystals 4-N,N-dimethylamino-4'-N'-methyl-stilbazolium tosylate (DAST) and 4-N,N-dimethylamino-4'-N'-methylstilbazolium 2,4,6-trimethylbenzenesulfonate (DSTMS). Vibrational modes of the structures were investigated to examine the single unit cell phononic contribution of the organic crystals to their terahertz generating capabilities. Linear optical properties were also calculated from snapshots of the molecular dynamics structures through Green-Kubo relations, and compared with experimental transmission.
The adsorption of oxygen on the Ag(001) is investigated by means of density functional techniques. Starting from a characterization of the clean silver surfaces oxygen adsorption in several modifications (molecularly, on-surface, sub-surface, Ag$_2$O) for varying coverage was studied. Besides structural parameters and adsorption energies also work-function changes, vibrational frequencies and core level energies were calculated for a better characterization of the adsorption structures and an easier comparison to the rich experimental data.
High Entropy Alloys (HEAs) are designed by mixing multiple metallic species in nearly the same amount to obtain crystalline or amorphous materials with exceptional mechanical properties. Here we use molecular dynamics simulations to investigate the role of positional and compositional disorder in determining the low-frequency vibrational properties of CrMnFeCoNi HEAs. Our results show that the expected dependence of the density of states on the frequency as $D(ω)\simω^4$ is recovered for amorphous HEAs and is also observed for partially crystallized alloys with deviations that depend on the de…
Silicon-vacancy (SiV) center in diamond is a photoluminescence (PL) center with a characteristic zero-phonon line energy at 1.681 eV that acts as a solid-state single photon source and, potentially, as a quantum bit. The majority of the luminescence intensity appears in the zero-phonon line; nevertheless, about 30\% of the intensity manifests in the phonon sideband. Since phonons play an essential role in the operation of this system, it is of importance to understand the vibrational properties of the SiV center in detail. To this end, we carry out density functional theory calculations of dil…
In an effort to extend the reach of current ab initio calculations to simulations requiring millions of configurations for complex systems such as heterostructures, we have parameterized the third-generation Charge Optimized Many-Body (COMB3) potential using solely ab initio total energies, forces, and stress tensors as input. The quality and the predictive power of the new forcefield is assessed by computing properties including the cohesive energy and density of SiO$_2$ polymorphs, surface energies of alpha-quartz, and phonon densities of states of crystalline and amorphous phases of SiO$_2$…
The ordered structure of MgO-ZnO alloy system is a versatile tunable optical material promising for diverse optoelectronic applications. However, isovalent and isostructural alloy compositions of MgO-ZnO are generally unstable at ambient conditions. Using state-of-the-art \textit{ab initio} evolutionary simulations, we predict and study the properties of stable phases of MgO-ZnO. We establish the dynamical stability of the predicted crystal structures through the phonon and Raman spectroscopy. Detailed analyses of two of the most stable structures reveal highly tunable properties that could be…
In this work, we have calculated the vibrational modes and frequencies of the crystalline PPP (in both the Pbam and Pnnm symmetries) and PPV (in the P21/c symmetry). Our results are in good agreement with the available experimental data. Also, we have calculated the temperature dependence of their specific heats at constant volume, and of their vibrational entropies. Based on our results, at high temperatures, the PPP is more stable in the Pnnm structure than in the Pbam one.
The atomic structure of icosahedral B$_4$C boron carbide is determined by comparing existing infra-red absorption and Raman diffusion measurements with the predictions of accurate {\it ab initio} lattice-dynamical calculations performed for different structural models. This allows us to unambiguously determine the location of the carbon atom within the boron icosahedron, a task presently beyond X-ray and neutron diffraction ability. By examining the inter- and intra-icosahedral contributions to the stiffness we show that, contrary to recent conjectures, intra-icosahedral bonds are harder.
We develop an ab initio method to simulate the infrared vibrational response of metallic systems in the framework of time-dependent density functional perturbation theory. By introducing a generalized frequency-dependent Born effective charge tensor, we show that phonon peaks in the reflectivity of metals can be always described by a Fano function, whose shape is determined by the complex nature of the frequency-dependent effective charges and electronic dielectric tensor. The IR vibrational properties of graphite, chosen as a representative test case to benchmark our method, are found to be a…
It is a long-time pursuit of computations with \emph{ab initio} precision of thermal contributions to phase behaviors of condensed matters under extreme conditions. In this work, the pressure induced structural phase transitions of crystalline aluminum up to $600$ GPa at room temperature are investigated based on the criterion of Gibbs free energy derived directly from the partition function that formulated in the ensemble theory with the interatomic interactions characterized by density functional theory computations. The transition pressures of the FCC$\rightarrow$HCP$\rightarrow$BCC phase t…
