Density-functional theory formulated in terms of functional integrals

In a previous study, the author formulated the density functional theory in terms of functional integrals. It was valid at zero and finite temperature. It was possible to derive the Hohenberg and Kohn formulation at zero temperature and the Mermin formulation at finite temperature of the density functional theory, which states that the energy or the grand potential are functionals of the true density of the system considered. In particular, the Kohn and Sham equations are proven to appear naturally by performing a saddle-point evaluation of a specific functional integral. This result is valid at zero or finite temperature. Unfortunately, the expression of the grand potential given in our previous work differs from the usual expression found in the literature. In this short paper, we derive the common expression of the grand potential in the framework of the density functional theory by starting from the expression given in this previous work. This completes the formulation of the density functional theory using functional integrals. This work could be of interest to people working in the field of quantum Monte Carlo methods at finite temperature.