Accurate and precise ab initio anharmonic free-energy calculations for metallic crystals: Application to hcp Fe at high temperature and pressure

Abstract

A framework for computing the anharmonic free energy (FE) of metallic crystals using BornOppenheimer ab initio molecular dynamics (AIMD) simulation, with unprecedented efficiency, is introduced and demonstrated for the hcp phase of iron at extreme conditions (up to ≈ 290 GPa and 5000 K). The advances underlying this work are: (1) A recently introduced harmonically-mapped averaging temperature integration (HMA-TI) method reduces the computational cost by order(s) of magnitude compared to the conventional TI approach. The TI path starts from zero Kelvin, where it assumes the behavior is given exactly by a harmonic treatment; this feature restricts application to systems that have no imaginary phonons in this limit. (2) A Langevin thermostat with the HMA-TI method allows use of a relatively large MD time step (4 fs, which is about eight times larger than the size needed for the Andersen thermostat) without loss of accuracy. (3) AIMD sampling is accelerated by using density functional theory (DFT) with a low-level parameter set, then the measured quantities of selected configurations are robustly reweighted to a higher level of DFT. This introduces a speedup of about 20-30× compared to directly simulating the accurate system. (4a) The temperature (T ) dependence of the hcp equilibrium shape (i.e. c/a axial ratio) is determined (including anharmonicity), with uncertainty less than ±0.001. (4b) Electronic excitation is included through Mermin's finite-temperature extension of the T = 0 K DFT. A simple FE perturbation method is introduced to handle the difficulty associated with applying the TI method with a Tdependent geometry and (due to electronic excitation) potential-energy surface. (5) The FE in the thermodynamic limit is attained through extrapolation of only the (computationally inexpensive) quasiharmonic FE, because the anharmonic FE contribution has negligible finite-size effects. All methods introduced here do not affect the AIMD sampling -results are obtained through postprocessing -so established AIMD codes can be employed without modification. Analytical formulas fitted to the results for the variation of the equilibrium c/a ratio and FE components with T are provided. Notably, effects of magnetic excitations are not included, and may yet prove important to the overall FE; if so, it is plausible that such contributions can be added perturbatively to the FE values reported here. Notwithstanding these considerations, FE values are obtained with an estimated accuracy and precision of 2 meV/atom, suggesting that the capability to compute the phase diagram of iron at Earth's inner core conditions is within reach.

Fast anharmonic free energy method with an application to vacancies in ZrC

“…There are a number of approaches in the literature to calculate the anharmonic vibrational properties of crystals, [1][2][3][4][12][13][14][15][16][17][18][19][20][21][22] including thermodynamic integration 23 (TI), which is the method used in this work. In TI the anharmonic part of the full Hamiltonian, E − E qh , is switched on with the parameter λ ∈ [0, 1], in this instance linearly as E mix (λ) = E qh + λ(E − E qh ).…”

Equations of State and Anisotropy of Fe‐Ni‐Si Alloys

“…The uncertainty contributions due to the isothermal EOS parameters V 0 , K T 0 , and for hcp‐Fe, hcp‐Fe 0.91 Ni 0.09 , and hcp‐Fe 0.8 Ni 0.1 Si 0.1 are found to be of similar magnitude to the uncertainty due to γ 0 and q , which highlights the importance of accurately constraining the parameters V 0 , K T 0 , and to obtain reliable thermal equations of state. One of the greatest sources of uncertainty is due to the electronic and anharmonic contributions to the thermal pressure, which are not well constrained (e.g., Alfè et al, ; Fei et al, ; Martorell, Vočadlo, et al, ; Martorell, Brodholt, et al, , Martorell et al, ; Moustafa et al, ). To account for this, we enlarge our error bars on density, adiabatic bulk modulus, and bulk sound speed.…”

Melting properties from ab initio free energy calculations: Iron at the Earth's inner-core boundary

“…To avoid such complications, simpler and more general references are needed. For the solid phase, a natural choice is the harmonic force field [12,19,20], which can be readily determined via density functional perturbation theory (DFPT) [21] or finite differences [22]. For the liquid phase, a preferable reference is less apparent.…”

“…Accounting for this correction significantly improves the agreement between melting properties predicted by MD and those of experiments [17,33]. Alternatively, one may separate the free energy of the solid phase into harmonic and anharmonic parts [12,19,20]. The dominant harmonic part is strongly size dependent, thus it is computed on a dense q-mesh with Fourier interpolated phonon frequencies.…”