  Traditional finite-temperature Kohn-Sham density functional theory (KSDFT)
has an unfavorable scaling with respect to the electron number or at high
temperatures. The evaluation of the ground-state density in KSDFT can be
replaced by the Chebyshev trace (CT) method. In addition, the use of stochastic
orbitals within the CT method leads to the stochastic density functional theory
[Phys. Rev. Lett. 111, 106402 (2013)] (SDFT) and its improved theory, mixed
stochastic-deterministic density functional theory [Phys. Rev. Lett. 125,
055002 (2020)] (MDFT). We have implemented the above four methods within the
first-principles package ABACUS. All of the four methods are based on the
plane-wave basis set with the use of norm-conserving pseudopotentials and the
periodic boundary conditions with the use of $k$-point sampling in the
Brillouin zone. By using the KSDFT calculation results as benchmarks, we
systematically evaluate the accuracy and efficiency of the CT, SDFT, and MDFT
methods via examining a series of physical properties, which include the
electron density, the free energy, the atomic forces, stress, and density of
states for a few condensed phase systems. The results suggest that our
implementations of CT, SDFT, and MDFT not only reproduce the KSDFT results with
a high accuracy, but also exhibit several advantages over the tradition KSDFT
method. We expect these methods can be of great help in studying
high-temperature and large-size extended systems such as warm dense matter and
dense plasma.
