Shell-nested virtual spaces from atomic natural orbitals

Selecting a compact molecular virtual space requires deciding which radial and angular freedom can be removed while retaining a useful description of electron correlation. A correlated molecular density can guide this choice, but obtaining it requires a preliminary correlation calculation. Here we examine whether tabulated atomic natural orbitals (ANOs) can provide radial guidance within the previously introduced shell-nested framework. Complete multicenter overlaps select directions available in the molecular parent basis, while adaptation uses the restricted Hartree–Fock reference and earlier layers remain fixed. The resulting shell-nested virtual spaces, SNVS(ANO), improve the average energy recovery over density-guided shell spaces in the tested second-order Møller–Plesset (MP2) and coupled-cluster singles and doubles (CCSD) panels. ANO guidance narrows the mean raw-energy differences from equally sized MP2-based frozen natural orbital spaces, although the latter retain smaller parent truncation errors. The prescribed shell content also makes the successive additions of spatial flexibility explicit. These results show that correlated atomic radial functions can guide molecular virtual-space selection before a correlated molecular density is available. The molecular reference controls their adaptation, and the correction and extrapolation tests show that retention of finite-parent correlation must still be distinguished from accuracy against an independent explicitly correlated reference.