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.

翻译：针对采用原子轨道线性组合离散的含原子中心势线性薛定谔算子的源问题与特征值问题，我们建立了可保证且实际可计算的后验误差界。我们证明离散化误差的能量范数可通过残差对偶能量范数进行估计，该残差可进一步分解为表征原子局域误差的原子贡献分量。此外，我们指出原子残差对偶范数的实际计算涉及径向薛定谔算子的对角化操作，该操作在实践中易于预先计算。我们通过多个测试案例展示了此类后验分析方法的数值性能，结果表明误差界能精确估计实际误差，且局域化误差分量可为自适应基组优化提供依据。