Plane-wave-based stochastic-deterministic density functional theory for extended systems

Traditional finite-temperature Kohn-Sham density functional theory (KSDFT) is one of the most popular quantum-mechanics-based methods in modeling materials since it balances the accuracy and efficiency well. However, traditional KSDFT based on the diagonalization method (DG) needs to solve all occupied bands, which increases fast as the temperature rises according to the Fermi-Dirac function. Thus, it is inefficient and also takes up a lot of memory at high temperatures, which limits its further applications. The evaluation of the ground-state density in KSDFT can be replaced by the Chebyshev trace (CT) method. Recently, 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) are developed based on the CT method and stochastic orbitals, which makes it possible to simulate high-temperature systems more efficiently. We have implemented the four methods based on the plane-wave basis set within the first-principles package ABACUS. In addition, all methods are adapted to the norm-conserving pseudopotentials and periodic boundary conditions with the use of k-point sampling in the Brillouin zone. By testing the Si and C systems from the DG method as benchmarks, we systematically evaluate the accuracy and efficiency of the SDFT, and MDFT methods by examining a series of physical properties, which include the electron density, free energy, atomic forces, stress, and density of states. We conclude that they not only reproduce the DG results with a sufficient accuracy but also exhibit several advantages over the DG method. We expect these methods can be of great help in studying high-temperature systems such as warm dense matter and dense plasmas.