Fermi energy determination for advanced smearing techniques

Smearing techniques are widely used in first-principles calculations of metallic and magnetic systems, where they improve the accuracy of Brillouin zone sampling and lessen the impact of level-crossing instabilities. Smearing works by introducing a fictitious electronic temperature that smooths the discontinuities of the integrands; consequently, a fictitious entropic term needs to be added to get the correct total free energy functional. Methfessel-Paxton and cold smearing are two approaches that are constructed to make the system's total free energy temperature-independent at least up to the third order. In doing so, these end up with non-monotonic occupation functions (and, for Methfessel-Paxton, not positive definite), which can result in the chemical potential not being uniquely defined. We explore this shortcoming in detail, and propose a protocol combining different root-finding methods to implement a data-driven solution to determine the material's correct Fermi energy. We validate the method by calculating the Fermi energy of thousands of materials and comparing them with the results of previous approaches.