  We present methods for calculating the Fermi-Dirac density matrix for
electronic structure problems at finite temperature while preserving
physicality by construction. These methods model cooling a state initially at
infinite temperature down to the desired finite temperature. We consider both
the grand canonical ensemble (constant chemical potential) and highlight
subtleties involved with treating the canonical ensemble (constant number of
electrons) that have been overlooked in previous works. We hope that the
discussion and results presented in this article reinvigorates interest in
density matrix minimization methods.
