Reconstruction of Local Potentials from Matrix Representations in Arbitrary Basis Sets

Multiplicative (local) potential operators are commonly represented as matrices in finite basis sets. Recovering the corresponding real-space potential from such matrices is generally regarded as ill-posed because of the inevitable linear dependencies among products of basis functions. We show that this problem admits a robust solution when conventional truncation and regularization strategies are replaced by importance-based selection and linear dependencies are exploited rather than avoided. Our method iteratively identifies a subset of important, linearly independent products for which the reconstruction is well-conditioned and the corresponding matrix elements are reproduced exactly. The remaining elements are then recovered through the underlying linear dependencies. The approach is independent of the particular representation of the target potential, including auxiliary-basis expansions, and applies to arbitrarily large basis sets. In the complete-basis-set limit, it converges to the exact real-space potential.