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

Quantum chemistry relies on numerical approximations. 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. This is, to our knowledge, the first end-to-end forward uncertainty quantification demonstration for an iterative ab initio method, and the first applied to both energy and wave function.