  Parameterless stopping criteria for recursive polynomial expansions to
construct the density matrix in electronic structure calculations are proposed.
Based on convergence order estimation the new stopping criteria automatically
and accurately detect when the calculation is dominated by numerical errors and
continued iteration does not improve the result. Difficulties in selecting a
stopping tolerance and appropriately balancing it in relation to parameters
controlling the numerical accuracy are avoided. Thus, our parameterless
stopping criteria stands in contrast to the standard approach to stop as soon
as some error measure goes below a user-defined parameter or tolerance. We
demonstrate that the stopping criteria work well both in dense and sparse
matrix calculations and in large-scale self-consistent field calculations with
the quantum chemistry program Ergo (www.ergoscf.org).
