A one-step, diophantine solution to the density matrix purification problem

Abstract

Approximate first-order density matrices can be ‘purified', and converted into idempotent density matrices satisfying the McWeeny condition using a new, single-step process based on a Diophantine matrix approach. The new, direct, single-step method replaces the conventional many-step McWeeny iterative method for density matrix purification. Besides the obvious advantages of a single-step, direct method over the traditional iterative method of many steps that also requires convergence criteria to be tested at each step and eventually approximately satisfied, the new approach requires no convergence tests, it is exact within the numerical accuracy of the eigenvalue routine employed, and it is also conceptually simple and has some additional advantages when compared with the traditional iterative density matrix purification method based on the McWeeny condition. One application of the technique is suggested for the generation of idempotent macromolecular density matrices of the linear scaling, adjustable density matrix assembler (ADMA) method, within the framework of the additive fuzzy density fragmentation (AFDF) approach.