pANO-F12: An atomic natural orbital-inspired route to more compact basis sets for F12 explicitly correlated methods

Abstract: Explicitly correlated methods such as MP2-F12 and CCSD(F12*) exhibit much faster basis set convergence (asymptotically $\propto L<sup>{-7}$, with L the highest angular momentum) than orbital-only approaches. Yet it has been pointed out that cc-pVnZ-F12 basis sets themselves are substantially larger than the corresponding cc-pVnZ, and specifically that cc-pVDZ-F12 is the size of cc-pVTZ. One way to generate compact basis sets in an orbital-only context are Atomic Natural Orbital (ANO) basis sets [J. Almlöf and P. R. Taylor, JCP 86, 4070 (1987)]. However, obtaining the required first-order reduced density matrix while properly accounting for the F12 geminal is problematic. In this work, we show that an energy minimization-based contraction process under linear independence constraints yields `pseudo-ANO' (pANO) basis sets that are functionally equivalent in quality. Subsequently, we apply this recipe to obtain pANO-F12 basis sets from the same elements, then validate them for several thermochemical benchmarks and for the hypersensitive out-of-plane vibrations of benzene. We show that, unlike cc-pVnZ-F12, pANO-F12 exhibits the familiar shell structure seen in cc-pVnZ and ANO basis sets, and that pANO-F12 offers a route to more compact F12 basis sets more amenable to medium-sized systems, especially in conjunction with localized pair natural orbital approaches. Overall, the pANO approach is most beneficial for the smaller double-and triple-zeta basis sets, offering either superior performance to cc-pVnZ-F12 at same cost, or similar performance at lower cost.