Purification of the first-order density matrix using steepest descent and Newton-Raphson methods

We propose a powerful approach to purification of the first-order density matrix based on minimizing the trace of a fourth-order polynomial, representing a deviation from idempotency. Two variants of this strategy are discussed. The first, based on a steepest descent minimization is robust and efficient, especially when the trial density matrix is far from idempotency. The second, using a Newton-Raphson technique, is quadratically convergent if the trial matrix is nearly idempotent. A steepest descent method with a switch to McWeeny's purification method is found to have a lower computational cost and wider range of convergence than McWeeny's scheme alone.