On the inherited “purity” of certain extrapolated density matrices

The method of Löwdin transform – inverse Löwdin transform is extended to approximate density matrix extrapolation in cases where the reference density matrix in not “pure” that is, if it fails the test of idempotency. It is shown that a matrix expression provides a scalar measure of the overall level of impurity and for large, sparse matrices of approximate macromolecular fragment-based quantum chemistry methods the one-step Diophantine density matrix purification method is recommended if a charge-preserving improvement of the extrapolated density matrix is needed.