pgm: A Python package for free energy calculation*

The quasiharmonic approximation (QHA) is a powerful method for computing the free energy and thermodynamic properties of materials at high pressures (P) and temperatures (T). However, anharmonicity, electronic excitations in metals, or both, introduce an intrinsic T-dependence on the phonon frequencies, making the QHA inadequate. Here we present a Python package, pgm, for free energy and thermodynamic property calculations. It is based on the concept of phonon quasiparticles and the phonon gas model (PGM). The free energy is obtained by integrating the entropy, which can be readily calculated for a system of phonon quasiparticles. This method is useful for computing the free energy in anharmonic insulators and harmonic or anharmonic metals. The current implementation offers properties in a continuum P,T range. The necessary inputs are ab initio T_el-dependent static energies and T-dependent phonon quasiparticle frequencies at several discrete volumes and the user-specified P,T range. We employ techniques like just-in-time (JIT) compiling and parallel computing to accelerate the numerical computation. We demonstrate successful applications of pgm to hcp-iron (ε-Fe) at extreme conditions [1] and cubic CaSiO3-perovskite [2], a strongly anharmonic system. Anharmonic free energies computed using thermodynamic integration (TI) agree with the results produced by this approach using an equivalent q-point mesh [3]. Therefore, a dense q-point mesh should produce anharmonic free energies equivalent to those obtained on TI simulations on 10⁴ atoms and beyond.

[1] Zhuang, J., Wang, H., Zhang, Q., & Wentzcovitch, R. M. (2021). Thermodynamic properties ε-Fe with thermal electronic excitation effects on vibrational spectra. Phys. Rev. B, 103(14), 144102.

[2] Zhang, Z., & Wentzcovitch, R. M. (2021). Ab initio anharmonic thermodynamic properties of cubic CaSiO3 perovskite. Phys. Rev. B, 103(10), 104108.

[3] Zhang, D. B., Sun, T., & Wentzcovitch, R. M. (2014). Phonon quasiparticles and anharmonic free energy in complex systems. Physical review letters, 112(5), 058501.