Accuracy of the recursion method

In this paper, the insensitivity of the projected density of states (PDOS) is investigated. Rounding errors are treated as perturbations in the recursion process when calculating the PDOS. A generalized error theory for the PDOS is developed, which includes Paige’s theorem and the effects of truncation of the continued fraction. An analytic expression for the PDOS which isolates the error term is derived. This error term is shown to be exponentially insensitive to perturbations which are distant from the starting state; in contrast to eigenvalues and/or eigenfunctions which are very sensitive to small perturbations. This is what makes calculating PDOS and its integrated quantities a more stable approach compared to computing eigenvalues. This insensitivity is equivalent to a black-body theorem for the PDOS. This result is useful in practical computations because it enables an infinite system to be approximated by a particular finite one, and gives a bound on the error in the computed PDOS.