Natural Atomic Orbitals at the Complete-Basis-Set Limit: A Frozen Natural-Minimal-Basis Construction

The standard natural atomic-orbital (NAO) construction is a mature and successful density-adaptive route to atom-centered orbitals. Here we examine it near the complete-basis-set limit, where the natural minimal basis is orthogonalized together with a large, nearly linearly dependent Rydberg space. Complete cc-pV $n$ Z ( $n=\text{D}$ -7) ladders for four first-row rings, supplemented by correlated-density tests from borole to a cobaltacycle, show that valence NAOs remain localized while their AO expansions acquire large, mutually canceling contributions from neighboring centers and high-angular-momentum functions. The sensitivity appears first in the orbital coefficients and, at cc-pV7Z, in NPA charges. We introduce a frozen-NMB two-cycle construction that defines and freezes the natural minimal basis in a reduced AO subspace before restoring the full Rydberg complement. The resulting NAOs reproduce standard populations and qualitative orbital shapes in compact bases, preserve the full density, remain symmetric and smoothly convergent, and provide more compact AO coefficient expansions.