How localized is ``local?'' Efficiency vs. accuracy of $O(N)$ domain decomposition in local orbital based all-electron electronic structure theory

Numeric atom-centered local orbitals (NAO) are efficient basis sets for all-electron electronic structure theory. The locality of NAO's can be exploited to render (in principle) all operations of the self-consistency cycle $O(N)$. This is straightforward for 3D integrals using domain decomposition into spatially close subsets of integration points, enabling critical computational savings that are effective from $\sim$tens of atoms (no significant overhead for smaller systems) and make large systems (100s of atoms) computationally feasible. Using a new all-electron NAO-based code,$^1$ we investigate the quantitative impact of exploiting this locality on two distinct classes of systems: Large light-element molecules [Alanine-based polypeptide chains (Ala)$_n$], and compact transition metal clusters. Strict NAO locality is achieved by imposing a cutoff potential with an onset radius $r_c$, and exploited by appropriately shaped integration domains (subsets of integration points). Conventional tight $r_c\le$~3{\AA} have no measurable accuracy impact in (Ala)$_n$, but introduce inaccuracies of 20-30 meV/atom in Cu$_n$. The domain shape impacts the computational effort by only 10-20~\% for reasonable $r_c$. \newline $^1$ V. Blum, R. Gehrke, P. Havu, V. Havu, M. Scheffler, \emph{The FHI Ab Initio Molecular Simulations (aims) Project}, Fritz-Haber-Institut, Berlin (2006).