As a part of the research group at Uppsala University I am developing linear scaling methods which use various approximation techniques. Their cost depends on the number of sparse matrix multiplications.
The computation of eigenvectors of the effective Hamiltonian matrix can be facilitated by the recursive density matrix expansions which give accelerated convergence of iterative methods for eigenvalue problems. We further develop this idea and combine it with the eigenvalue estimates which can be extracted from the recursive expansion by a simple and robust procedure at a negligible computational cost. I implemented the proposed algorithms in the quantum chemistry program Ergo. The orbitals are calculated “on the fly” adding only a small overhead to the polynomial expansion.
The goal is to show that the linear scaling is achieved for systems where the homo-lumo gap and the distance between the frontier orbitals and the rest of the eigenvalue spectrum is preserved with increasing system size.