Simulating anharmonic crystals: Lights and shadows of first-principles approaches

Understanding and simulating the thermodynamic and dynamical properties of materials affected by strong ionic anharmonicity is a central challenge in material science. Two powerful methodologies have emerged as frontrunners in this exploration: the Self-Consistent Harmonic Approximation (SCHA) with its counterpart, Self-Consistent Phonons (SCP), and the Temperature-Dependent Effective Potential (TDEP). Despite their widespread use, the limitations of these methods are often overlooked, and the absence of a formal derivation for the TDEP has led to significant missteps that hamper its predictive accuracy. Here, the theoretical foundations of both SCHA and TDEP are revised, unveiling their profound interconnection. This work introduces formal derivation for TDEP, dispelling misconceptions about how to improve its accuracy and refining best practices. The perturbative limit of these methods uncovers that both SCHA and TDEP fall short in reproducing the lowest-order perturbative free energy. While TDEP excels in providing an exact treatment of the static susceptibility, its dynamical extension violates the perturbative regime by overcounting anharmonicity. Conversely, SCHA delivers an approximate static susceptibility but maintains correct static and dynamic response functions within the perturbative limit. This work introduces a corrective strategy for TDEP's dynamical extension to address these shortcomings. All the claims provided here are benchmarked against the exact (numerical) solution on a simple one-dimensional anharmonic potential.