Gap edge eigenpairs from density matrix purification using moments of the Dirac distribution

In this work, we propose a simple method to resolve the eigenstates located at the band gap edges of an electronic eigenspectrum using only the quasi-purified one-particle density matrix as input. The theoretical framework relies on the decomposition of the occupation number variance into a particle and a hole moment. These moments, when purified using power narrowing iterations, allow us to isolate the highest occupied and lowest unoccupied single-state projectors, \colp{giving ready access} to the corresponding eigenpairs. We demonstrate that when degeneracy is encountered, power narrowing remains able to deliver relevant mixed states. From a benchmark of selected molecules, we show that the method is robust and efficient since it requires from four to sixteen matrix-matrix multiplications in the worst case. The possibility of reducing the computational cost using the Lanczos subspace approach is discussed. The algorithm simplicity of power narrowing makes it very easy to implement in electronic structure codes or libraries already featuring Fermi operator expansion or density matrix purification.