Correlation-Energy Extrapolation within Shell-Nested Virtual Spaces

We show that power-law extrapolation of the correlation energy can be carried from a sequence of orbital bases into a shell hierarchy within one molecular virtual space. In shell-nested virtual spaces (SNVS), atomic natural orbitals guide the radial and angular content of successive layers, and the correlating-orbital count beyond the valence-virtual space replaces the cardinal number in a two-point formula. The construction uses one Hartree–Fock reference in one parent basis. We calibrate and cross-validate the formula for 200 small molecular states of H and B–F at the MP2, CCSD and CCSD(T) levels, then test MP2 and CCSD transfer without refitting on 40 larger systems. In the standard ANO family, the full quadruple-zeta layer pair gives an MP2 RMS error of 0.79 mEh for the larger set, compared with 2.16 mEh for conventional extrapolation. Diffuse augmentation reduces the anion bias, and placing the excess functions in the valence-virtual space preserves the counting variable. The CCSD and reaction-energy tests show how the benefits depend on correlation level and error cancellation. Measured CCSD(T) workflow costs are comparable to those of the conventional pair. These results demonstrate the usefulness of atomic shell information for molecular extrapolation and motivate basis sets designed for the successive SNVS layers.