A posteriori error estimates for Schr{\"o}dinger operators discretized with linear combinations of atomic orbitals

TL;DR

This paper develops guaranteed, computable a posteriori error bounds for Schrödinger operators discretized with atomic orbitals, enabling localized error estimation and adaptive basis set optimization.

Contribution

It introduces a novel a posteriori error estimation method that decomposes errors into atomic contributions for Schrödinger operators with atomic potentials.

Findings

Error bounds accurately estimate discretization errors.

Atomic residuals enable localized error analysis.

Adaptive basis sets improve computational efficiency.

Abstract

We establish guaranteed and practically computable a posteriori error bounds for source problems and eigenvalue problems involving linear Schr{\"o}dinger operators with atom-centered potentials discretized with linear combinations of atomic orbitals. We show that the energy norm of the discretization error can be estimated by the dual energy norm of the residual, that further decomposes into atomic contributions, characterizing the error localized on atoms. Moreover, we show that the practical computation of the dual norms of atomic residuals involves diagonalizing radial Schr{\"o}dinger operators which can easily be precomputed in practice. We provide numerical illustrations of the performance of such a posteriori analysis on several test cases, showing that the error bounds accurately estimate the error, and that the localized error components allow for optimized adaptive basis sets.