Does Your Self-Consistent Field Program Know How Wrong It Is? Introducing Quantum Chemistry Uncertainty Propagation

Establishing the correctness and limits of these approximations traditionally requires laborious benchmarking. We introduce a more direct approach, quantum chemistry uncertainty propagation (QCUP), and apply it to a full iterative self-consistent field (SCF) calculation.

QCUP-SCF estimates uncertainty from floating-point arithmetic, primitive-Gaussian screening, and incomplete SCF convergence, and propagates it to the final energy and density. Across all systems tested, predictions are typically within 1 kcal/mol compared to analytic error bars and within an order of magnitude of the true error  enough to establish significant figures and support runtime decision making. We also show how thresholded-affine forms improve on interval bounds by exploiting guaranteed error cancellation.