The ab initio thermodynamic properties of simple cubic polonium (α-Po) were studied within the quasi-harmonic approximation (QHA), where both lattice vibrations (phonons) and electronic excitation s contributions are included in the Helmholtz free energy. We investigate the influence of spin-orbit coupling (SOC) by comparing the scalar relativistic (SR) and fully relativistic (FR) pseudopotentials on the thermodynamic properties of polonium and evaluate the performance of three popular exchange-correlation functionals, GGA (PBE and PBEsol), and LDA (PZ). Temperature and pressure-dependent thermodynamic properties were compared with the available experimental and theoretical studies. We found that the effect of electronic excitations is negligible for all the thermodynamic properties. LDA + SOC provides better agreement with the experimental volume, while the thermal expansion coefficients from LDA + SOC and PBEsol + SOC closely match experimental values. SOC effects appear insignificant for the isobaric heat capacity but substantially contribute to the adiabatic bulk modulus. The phonon dispersions and mode-Grüneisen parameters (γqη) were interpolated at the lattice constant corresponding to 301 K. The SR dispersions exhibit several anomalies in all directions, which were suppressed significantly by the inclusion of SOC. The effects of SOC on the elastic constant-coefficient (Cij) and elastic anisotropy factor at 0 K are also studied. The Pugh ratio confirmed that simple cubic polonium is ductile, and we observed that the elastic anisotropy factor and Pugh ratio decreased with increasing pressure. The effect of SOC on the Cauchy pressure is also determined.

We generalize a previously established ab initio approach-originally developed for hexagonal close-packed (hcp) metals-to accommodate solids with both internal and external degrees of freedom. This extension enables the thermodynamic and thermoelastic characterization of insulators, including those with non-vanishing piezoelectric and pyroelectric tensors. Utilizing Density Functional Theory (DFT) and Density Functional Perturbation Theory (DFPT) within the quasi-harmonic approximation, we derive the pressure and temperature dependence of these properties. Specifically, we investigate internal degrees of freedom using two distinct frameworks: the Zero Static Internal Stress Approximation (ZSISA) and Full Free Energy Minimization (FFEM). We then compare these approximations by computing internal and external thermal expansions, as well as temperature-dependent piezoelectric and pyroelectric tensors. Finally, we demonstrate the generalized formalism by calculating the thermodynamic properties of wurtzite ZnO across a broad range of pressures and temperatures.

This study investigates the thermodynamic properties of face-centered cubic lead (fcc-Pb) using ab-initio methods within the quasi-harmonic approximation (QHA), examining the influence of spin-orbit coupling (SOC) and the exchange-correlation functionals. Two types of ultrasoft pseudopotential (US-PP) are considered: one that excludes (scalar relativistic PP) and one that includes the SOC effects (fully relativistic PP). Further, for each PP, we test the performance of three popular exchange-correlation functionals: Perdew-Burke-Ernzerhof generalized gradient approximation (PBE) (Perdew et al. Phys. Rev. Lett. 77, 3865 (1996)), PBE modified for dense solids (PBEsol) (Perdew et al. Phys. Rev. Lett. 100, 136,406 (2008)), and local density approximation (LDA) (Perdew et al. Phys. Rev. B 23, 5048 (1981)). We calculate the Helmholtz free energy, incorporating lattice vibrations (phonons) and electronic excitation contributions. The estimated equation of state (at 4 K and 301 K), phonon dispersions (at 100 K and 300 K), mode-Gr & uuml;neisen parameters (gamma q eta) (at 100 K), volume thermal expansion coefficient (beta), isobaric heat capacity (CP), bulk modulus (BS), and thermodynamic average Gr & uuml;neisen parameter (gamma) are compared with the available experimental and theoretical studies. Moreover, the 0 K pressure- dependent elastic constant-coefficient (Cij) of fcc lead and Pugh ratio, Debye temperature, and longitudinal and transverse sound velocities for polycrystalline lead are presented. The contributions of electronic excitations in all the thermodynamic properties are found to be negligible. With increasing pressure, the role of spin-orbit effects decreases but does not vanish. Our findings demonstrate that SOC leads to results distinct from the SR approach, but agreement with the experiment is not consistently improved by including SOC.

We present a systematic ab initio study of the thermoelastic properties of hcp osmium as functions of temperature and pressure within the quasiharmonic approximation (QHA). The precision of the zero static internal stress approximation (ZSISA) and of the volume-constrained ZSISA (V-ZSISA) is rigorously assessed. For osmium, we find negligible deviations between ZSISA and a full free energy minimization approach. Also, the V-ZSISA approximation influences the results very little, as we found already in beryllium, despite the markedly different behavior of the c/a ratio with temperature in the two metals. Our QHA-derived elastic constants (ECs) show excellent agreement with available experimental data in the temperature range of 5-301 K, outperforming the results obtained from the quasistatic approximation (QSA). Additionally, we report the pressure-dependent QHA ECs at 5, 301, and 1000 K, spanning pressures from 0 to 150 kbar.

We present an alternative GPU acceleration for plane waves pseudopotentials electronic structure codes designed for systems that have small unit cells but require a large number of k points to sample the Brillouin zone as happens, for instance, in metals. We discuss the diagonalization of the Kohn and Sham equations and the solution of the linear system derived in density functional perturbation theory. Both problems take advantage from a rewriting of the routine that applies the Hamiltonian to the Bloch wave-functions to work simultaneously (in parallel on the GPU threads) on the wave-functions with different wave-vectors k, as many as allowed by the GPU memory. Our implementation is written in CUDA Fortran and makes extensive use of kernel routines that run on the GPU (GLOBAL routines) or can be called from inside the GPU threads (DEVICE routines). We compare our method with the CPUs only calculation and with the approach currently implemented in Quantum ESPRESSO that uses GPU accelerated libraries for the FFT and for the linear algebra tasks such as the matrix-matrix multiplications as well as OpenACC directives for loop parallelization. We show in a realistic example that our method can give a significant improvement in the cases for which it has been designed.

We present the ab initio thermoelastic properties of body-centered cubic molybdenum under extreme conditions obtained within the quasi-harmonic approximation including both the vibrational and electronic thermal excitation contributions to the free energy. The quasi-harmonic temperature-dependent elastic constants are calculated and compared with existing experiments and with the quasi-static approximation. We find that the quasi-harmonic approximation allows for a much better interpretation of the experimental data, confirming the trend found previously in other metals. Using the Voigt–Reuss–Hill average, we predict the compressional and shear sound velocities of polycrystalline molybdenum as a function of pressure for several temperatures, which might be accessible in experiments.

We present a systematic ab initio study of the temperature- and pressure-dependent thermoelastic properties of hcp beryllium within the quasiharmonic approximation (QHA). The accuracies of the zero static internal stress approximation (ZSISA) and of the volume-constrained ZSISA that are widely applied in ab initio thermodynamic calculations are quantified. Particularly, the effect of ZSISA on the calculation of C11 and C12 is compared with a numerical approach which minimizes the free energy with respect to the atomic positions at each strain. In beryllium, minor deviations are found within ZSISA, which gives elastic constants (ECs) in good agreement with the full free energy minimization. A substantial difference is found between the QHA and the quasistatic approximation (QSA), with the former closer to experiments. Within the QSA, we compare the ECs obtained by interpolating from a set of geometries along the "stress-pressure" isotherm at 0 K (within the constant-volume ZSISA) with a more general interpolation on a two-dimensional grid of crystal parameters, which allows the calculation of the ECs along the 0 kbar isobar. This paper provides a practical approach for the investigation of the thermoelastic properties of hcp metals at extreme conditions.

We present the ab-initio temperature and pressure dependent thermoelastic properties of body-centered cubic tungsten. The temperature dependent quasi-harmonic elastic constants (ECs) are computed at several reference volumes including both the phonon and the electronic excitations contribution to the free energy and interpolated at different temperatures and pressures. Good agreement with the experimental ECs on a single crystal at ambient pressure is found. The pressure and temperature dependence of the shear sound velocity measured on polycrystalline tungsten by Qi et al is also in agreement with theory. Some discrepancies are found instead for the compressional velocity at high temperature and this is attributed to the temperature derivative of the bulk modulus, higher in theory than in experiment. These conclusions are reached both by PBE and by PBEsol functionals. The two give elastic properties with a similar pressure and temperature dependence although the latter is closer to experiment at 0 K.

The high-pressure and high-temperature thermodynamic properties of iridium are studied using density functional theory in combination with the quasi-harmonic approximation, where both the contributions to the free energy of phonons and of electronic excitations are considered. The reliability of different exchange and correlational functionals [Perdew-Burke-Ernzerhof generalized gradient approximation (PBE) (Perdew et al. Phys. Rev. Lett. 77, 3865 (1996)), PBE modified for dense solids (PBEsol) (Perdew et al. Phys. Rev. B 100, 136406 (2008)) and local density approximation (LDA) (Perdew et al. Phys. Rev. B 23, 5048 (1981))], for studying the equation of state (EOS), the phonon dispersions, the mode-Gr\"uneisen parameter, and different thermodynamic properties like thermal pressure, volume thermal expansivity, isobaric heat capacity, bulk modulus, and the average Gr\"uneisen parameter are tested. Elastic constants are studied at T=0 K as a function of pressure. The predicted results are compared with the available experiments and previous theoretical data. We find generally a good agreement with experiments with at least one functional, but none of the three outperforms the others in all the investigated thermodynamic properties. The electronic excitations contribution is minimal in bulk modulus, but it is significant for other thermodynamic properties.

The high-pressure and high-temperature thermodynamic properties of rhodium (up to 2000 K and 300 GPa) are presented using the first principle approach within the quasi-harmonic approximation. The thermal Helmholtz free energy includes the contribution of both phonon vibrations and electronic excitations. The performance of three popular exchange-correlation functionals—local density approximation [Perdew et al., Phys. Rev. B 23, 5048 (1981)], Perdew–Burke–Ernzerhof generalized gradient approximation (PBE) [Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)], PBE modified for dense solids [Perdew et al., Phys. Rev. Lett. 100, 136406 (2008)] are shown. The simulated thermal expansion coefficient, isobaric heat capacity, mode-Grüneisen parameter, thermodynamic average Grüneisen parameter, and bulk modulus are compared with the available experimental and theoretical reports. The contribution of thermal electronic excitations to the obtained thermodynamic parameters is significant at low pressure and high temperatures, except in bulk modulus, where it is small. The pressure-dependent elastic constant coefficient (Cij) and the Debye temperature are computed at 0 K. The Pugh ratio calculated from Cij indicates that rhodium undergoes brittle to ductile transitions at an average pressure of 7.45 GPa.

The effect of atomic relaxations on the temperature-dependent elastic constants (TDECs) is usually taken into account at zero temperature by the minimization of the total energy at each strain. In this paper, we investigate the order of magnitude of this approximation on a paradigmatic example: the C44 elastic constant of diamond and zincblende materials. We estimate the effect of finite-temperature atomic relaxations within the quasi-harmonic approximation by computing ab initio the internal strain tensor from the second derivatives of the Helmholtz free-energy with respect to strain and atomic displacements. We apply our approach to Si and BAs and find a visible difference between the softening of the TDECs computed with the zero-temperature and finite-temperature atomic relaxations. In Si, the softening of C44 passes from 8.6% to 4.5%, between T = 0 K and T = 1200 K. In BAs, it passes from 8% to 7%, in the same range of temperatures. Finally, from the computed elastic constant corrections, we derive the temperature-dependent Kleinman parameter, which is usually measured in experiments.

We calculate the temperature-dependent elastic constants (ECs) of palladium, platinum, copper and gold within the quasi-harmonic approximation using a first-principles approach and evaluating numerically the second derivatives of the Helmholtz free-energy with respect to strain at the minimum of the free-energy itself. We find an overall good agreement with the experimental data although the anomalies of palladium and platinum reported at room temperature are not reproduced. The contribution of electronic excitations is also investigated: we find that it is non-negligible for theC44ECs of palladium and platinum while it is irrelevant in the other cases. Its effect is not sufficient to explain the details of the anomalies found by experiments, not even when, in the case of platinum, we take into account the electron-phonon interaction. Lastly, the effect of the exchange and correlation functional is addressed and it is found that it is important atT= 0 K, while all functionals give similar temperature dependencies.

We present ab-initio calculations of the quasi-harmonic temperature dependent elastic constants. The isothermal elastic constants are calculated at each temperature as second derivatives of the Helmholtz free energy with respect to strain and corrected for finite pressure effects. This calculation is repeated for a grid of geometries and the results interpolated at the minimum of the Helmholtz free energy. The results are compared with the quasi-static elastic constants. Thermodynamic relationships are used to derive the adiabatic elastic constants that are compared with the experimental measurements. These approaches are implemented for cubic solids in the thermo_pw code and are validated by applications to silicon, aluminum, and silver.

Using a first-principles fully relativistic scheme based on ultrasoft pseudopotentials and density functional perturbation theory, we study the magnetocrystalline anisotropy free energy of the ferromagnetic binary compound MnBi. We find that differences in the phonon dispersions due to the different orientations of the magnetization (in-plane and perpendicular to the plane) give a difference between the vibrational free energies of the high-temperature and low-temperature phases. This vibrational contribution to the magnetocrystalline anisotropy energy (MAE) constant, $K_u$, is non-negligible. When the energy contribution to the MAE is calculated by the PBEsol exchange and correlation functional, the addition of the phonon contribution allows to get a $T = 0$ K $K_u$ and a spin-reorientation transition temperature in reasonable agreement with experiments.

The implementation of ultrasoft pseudopotentials into time-dependent density-functional perturbation theory is detailed for both the Sternheimer approach and the Liouville-Lanczos (LL) method, and equations are presented in the scalar relativistic approximation for periodic solids with finite momentum transfer q. The LL method is applied to calculations of the electron energy loss (EEL) spectrum of face-centered cubic bulk Au both at vanishing and finite q. Our study reveals the richness of the physics underlying the various contributions to the density fluctuation in gold. In particular, our calculations suggest the existence in gold of two quasi-separate 5d and 6s electron gasses, each one oscillating with its own frequency at resp. 5.1 eV and 10.2 eV. We find that the contribution near 2.2 eV comes from 5d to 6s interband transitions modified by the intraband contribution to the real part of the dielectric function, which we call a mixed excitation.

We present an ab-initio study of the temperature dependent elastic constants of BAs, a semiconductor that exhibits ultra-high thermal conductivity and is under investigation for thermal management in electronics. We test the consistency of our predictions by computing the temperature dependent sound velocity of the longitudinal acoustic mode along the $[111]$ direction and comparing with experiment. Furthermore, as a byproduct, we present the room temperature phonon dispersions, and the temperature dependent thermal expansion, isobaric heat capacity, and average Grüneisen parameter comparing with the most updated experiments and previous calculations when available. Finally, we present the theoretical estimate of the temperature dependent mean square atomic displacements.

We study the electronic structure of the Re(0001) surface by means of ab-initio techniques based on the Fully Relativistic (FR) Density Functional Theory (DFT) and the Projector Augmented-Wave (PAW) method. We identify the main surface states and resonances and study in detail their energy dispersion along the main symmetry lines of the SBZ. Moreover, we discuss the effect of spin-orbit coupling on the energy splittings and the spin-polarization of the main surface states and resonances. Whenever possible, we compare the results with previously studied heavy metals surfaces. We find empty resonances, located below a gap similar to the L-gap of the (111) fcc surfaces, that have a downward dispersion and cross the Fermi level, similarly to the recently studied Os(0001) surface. Their spin polarization at the Fermi level is similar to that predicted by the Rashba model, but the usual level crossing at (Gamma) over bar is not found with our slab thickness. Moreover, for selected states, we follow the spin polarization along the high symmetry lines, discussing its behavior with respect to k(parallel to), the wave vector parallel to the surface.