On the optimal symmetric purification scheme of the one-particle density matrix

Density matrix purification is an efficient way of avoiding the expensive cubic scaling diagonalization in self-consistent field calculations. Although there are a number of different algorithms suggested to reduce the number of matrix multiplications for purification, there is no rigorous mathematical proof which scheme is optimal. In this Letter, we show analytically that the repeated application of the fifth-order Holas polynomial throughout all iterations is an optimal scheme that reduces the error symmetrically for both occupied and virtual occupations, and either the use of lower/higher-order polynomials throughout or mixed use of polynomials of different degree at different iteration results in higher cost.