Positivity Preserving Density Matrix Minimization for Finite and Zero Temperatures

Knowledge of the density matrix is critical for understanding the dynamics of many materials. We present methods for calculating the Fermi-Dirac density matrix for electronic structure problems at both finite and zero temperature while preserving physicality. In either case, we consider both the grand canonical ensemble (constant chemical potential) and the canonical ensemble (constant number of electrons). The methods for calculating the finite temperature case are based around the minimization of the density matrix, while the methods for calculating the zero temperature case are based on self-consistent iterations. Our presented methods are able to calculate the density matrix with more accuracy than previous methods while still scaling linearly with the size of the system in question.