Nonmonotonic Recursive Polynomial Expansions for Linear Scaling Calculation of the Density Matrix

As it stands, density matrix purification is a powerful tool for linear scaling electronic structure calculations. The convergence is rapid and depends only weakly on the band gap. However, as will be shown in this letter, there is room for improvements. The key is to allow for nonmonotonicity in the recursive polynomial expansion. On the basis of this idea, new purification schemes are proposed that require only half the number of matrix-matrix multiplications compared to previous schemes. The speedup is essentially independent of the location of the chemical potential and increases with decreasing band gap.