Revisiting the Atomic Natural Orbital Approach for Basis Sets: Robust Systematic Basis Sets for Explicitly Correlated and Conventional Correlated <i>ab initio</i> Methods?

The performance of several families of basis sets for correlated wave function calculations on molecules is studied. The widely used correlation-consistent basis set family cc-pVXZ (n = D, T, Q, 5) is compared to a systematic series of atomic natural orbital basis sets (ano-pVXZ). These basis sets are built from the cc-pV6Z primitives in atomic multireference average coupled pair functional (MR-ACPF) calculations. Segmented basis sets optimized for self-consistent field calculations (def2-SVP, def2-TZVPP, and def2-QZVPP as well as "pc-n", n = 1, 2, 3) were also tested. Reference Hartree-Fock energies are determined with the uncontracted aug-cc-pV6Z basis set for a set of 21 small molecules built from H, B, C, N, O, and F. Reference coupled cluster CCSD(T) correlation energies were determined from extrapolation at the cc-pV5Z/cc-pV6Z level. It is found that the ano-pVXZ basis sets outperform the other basis sets. The error in the SCF energies compared to cc-pVXZ basis sets is reduced by about a factor of 3 at each cardinal number. In addition, the ano-pVXZ consistently recovers more correlation energy than their competitors at each cardinal number. The ability of the four families of basis sets to extrapolate SCF and correlation energies to the basis set limit has been investigated. A conclusion by Truhlar is confirmed that the optimum exponent for correlation energy extrapolations at the DZ/TZ level is ∼2.4. All TZ/QZ basis set pairs lead to an optimum exponent close to the expected value of 3. The SCF energy extrapolation proposed by Petersson and co-workers is found to be effective. At the DZ/TZ level, errors in total energies of less than 2 mEh are found for the test set, while at the TZ/QZ level one obtains the total energies within ∼0.3 mEh of the basis set limit. For extrapolation, the "cc" and "ano" bases are found to be similarly successful. Extrapolation results were compared to explicitly correlated calculations with dedicated basis sets (cc-pVXZ-F12) as well as the ano-pVXZ bases. It is found that the ano-pVXZ+ basis sets perform as well as the cc-pVXZ-F12 family (both are of comparable size); additional improvement should be possible by reoptimizing the ANO basis sets for explicitly correlated calculations. The error of the extrapolated energies is about 2-3 times smaller than what was found in the explicitly correlated calculations. However, the error in the explicitly correlated calculations is more systematic, and hence the same conclusion may not hold for the computation of energy differences.