Interior Eigenvalues from Density Matrix Expansions in Quantum Mechanical Molecular Dynamics

An accelerated polynomial expansion scheme to construct the density matrix in quantum mechanical molecular dynamics simulations is proposed. The scheme is based on recursive density matrix expansions, e.g., [A. M. N. Niklasson, Phys. Rev. B, 66 (2002), 155115], which are accelerated by a scale-and-fold technique [E. H. Rubensson, J. Chem. Theory Comput., 7 (2011), pp. 1233--1236]. The acceleration scheme requires interior eigenvalue estimates, which may be expensive and cumbersome to come by. Here we show how such eigenvalue estimates can be extracted from the recursive expansion by a simple and robust procedure at a negligible computational cost. Our method is illustrated with density functional tight-binding Born--Oppenheimer molecular dynamics simulations, where the computational effort is dominated by the density matrix construction. In our analysis we identify two different phases of the recursive polynomial expansion, the conditioning and purification phases, and we show that the acceleration repres...