  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. [Phys. Rev. B. 66 (2002), p.
155115], which are accelerated by a scale-and-fold technique [J. Chem. Theory
Comput. 7 (2011), p. 1233]. 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 represents an
improvement of the conditioning phase, which typically gives a significant
reduction of the computational cost.
