Shell-Nested Atomic Orbitals: Restoring the Correlation-Consistent Hierarchy inside the Molecular Virtual Space

Correlation-consistent basis sets enlarge the one-electron space atom by atom and shell by shell, but the canonical molecular virtual space retains no comparable atom-resolved hierarchy. Extending the frozen natural-minimal-basis (NMB) construction, we define shell-nested atomic orbitals (SNAOs): each cycle selects the highest-occupation residual radial function from every eligible atomic angular-momentum channel, assembles the selected functions into a complete layer, and freezes that layer. The levels L1 ⊂ L2 ⊂···share one parent-basis restricted Hartree-Fock reference; only the virtual degrees of freedom available to dynamic correlation are increased. The level index therefore provides a common coordinate for examining radial and angular saturation. Benchmarks comprise five hydrides and Ne through cc-pV7Z, benzene, pyrrole, and furan, and 32 closed-shell first-row molecules in cc-pV5Z. Rank-matched frozen natural orbitals (FNOs) recover more correlation energy at each finite dimension, as expected for an occupation-optimized compressor. The SNAO sequence is less compact but more regular: for the 32-molecule set, the three-point L2-L4 limit differs from the two-point cc-pV(Q,5)Z estimate by 3.2 mEh on average, compared with 13.1 mEh for FNO. SNAO thus complements FNO by exchanging some fixed-rank recovery for a transferable, shell-resolved sequence suited to the analysis and extrapolation of correlation freedom.