Variational density functional perturbation theory for metals

Density functional perturbation theory (DFPT) is a well-established method to study responses of molecules and solids, especially responses to atomic displacements or to different perturbing fields (electric, magnetic). Like for density functional theory (DFT), the treatment of metals is delicate, due to the Fermi-Dirac (FD) statistics and electronic bands crossing the Fermi energy. At zero temperature, there is an abrupt transition from occupied states to unoccupied ones, usually addressed with smearing schemes. Also, at finite temperature, fractional occupations are present, and the occupation numbers may vary in response to the perturbation. In this paper, we establish the characteristics of DFPT stemming from the underlying variational principle, in the case of metals. After briefly reviewing variational DFT for metals, the convexity of the entropy function of the occupation number is analyzed, and at finite temperature, the benefit of resmearing the FD broadening with the Methfessel-Paxton one is highlighted. Then the variational expressions for the second-order derivative of the free energy are detailed, exposing the different possible gauge choices. The influence of the inaccuracies in the unperturbed wave functions from the prior DFT calculation is studied. The whole formalism is implemented in the ABINIT software package.