On-the-Fly Computation of Frontal Orbitals in Density Matrix Expansions.

We propose a method for computation of frontal (homo and lumo) orbitals in recursive polynomial expansion algorithms for the density matrix. Such algorithms give a computational cost that increases only linearly with system size for sufficiently sparse systems, but a drawback compared to the traditional diagonalization approach is that molecular orbitals are not readily available. Our method is based on the idea to use the polynomial of the density matrix expansion as an eigenvalue filter giving large separation between eigenvalues around homo and lumo [ Rubensson et al. J. Chem. Phys. 2008 , 128 , 176101 ]. This filter is combined with a shift-and-square (folded spectrum) method to move the desired eigenvalue to the end of the spectrum. In this work we propose a transparent way to select recursive expansion iteration and shift for the eigenvector computation that results in a sharp eigenvalue filter. The filter is obtained as a byproduct of the density matrix expansion, and there is no significant additional cost associated either with its construction or with its application. This gives a clear-cut and efficient eigenvalue solver that can be used to compute homo and lumo orbitals with sufficient accuracy in a small fraction of the total recursive expansion time. Our algorithms make use of recent homo and lumo eigenvalue estimates that can be obtained at negligible cost [ Rubensson et al. SIAM J. Sci. Comput . 2014 , 36 , B147 ]. We illustrate our method by performing self-consistent field calculations for large scale systems.

Parameterless Stopping Criteria for Recursive Density Matrix Expansions.

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 stand 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 ) .