Straightening the Hierarchical Staircase for Basis Set Extrapolations: A Low-Cost Approach to High-Accuracy Computational Chemistry

Abstract

Because the one-electron basis set limit is difficult to reach in correlated post-Hartree-Fock ab initio calculations, the low-cost route of using methods that extrapolate to the estimated basis set limit attracts immediate interest. The situation is somewhat more satisfactory at the Hartree-Fock level because numerical calculation of the energy is often affordable at nearly converged basis set levels. Still, extrapolation schemes for the Hartree-Fock energy are addressed here, although the focus is on the more slowly convergent and computationally demanding correlation energy. Because they are frequently based on the gold-standard coupled-cluster theory with single, double, and perturbative triple excitations [CCSD(T)], correlated calculations are often affordable only with the smallest basis sets, and hence single-level extrapolations from one raw energy could attain maximum usefulness. This possibility is examined. Whenever possible, this review uses raw data from second-order Møller-Plesset perturbation theory, as well as CCSD, CCSD(T), and multireference configuration interaction methods. Inescapably, the emphasis is on work done by the author's research group. Certain issues in need of further research or review are pinpointed.

Optimal diffuse augmented atomic basis sets for extrapolation of the correlation energy

“…Instead, for triplet states, the leading contribution is , while for unnatural parity singlet states, it is . From this, Equation can be readily justified . For a basis set saturated up to angular momentum L , the leading terms are therefore X −3 (for a singlet coupled electron pair), with the next being X −5 (for a triplet coupled pair).…”

“…Varandas suggested instead a uniform singlet‐ and triplet‐pair extrapolation (USTE) scheme, which avoids such a decomposition. In fact, the decomposition (a) is unnecessary for accurate results, (b) cannot be implemented for open‐shell CCSD calculations as the wave function is not a spin eigenfunction in practical implementations of CCSD theory, and (c) is extraneous to multireference calculations.…”

Validating additive correction schemes against gradient‐based extrapolations

“…The most common, rigorous, and systematic way to extrapolate results to both the complete basis set and correlated limits is by combining the energies (or energy‐related parameters such as gradients or Hessians, where applicable) of the components of an extrapolation “recipe.” The foundation of this process is an accurate reference Hartree‐Fock (HF) energy, which converges toward the basis set limit rather quickly when compared to correlated methods: a convergence within 1 mEh of the basis set limit can be expected from an extrapolation of 3‐ and 4‐ ζ results, or from a single 5‐ ζ calculation . The bulk of the correlation is typically determined from an MP2 extrapolation over two large basis sets, and can be further corrected by differences with the CCSD(T) results with a smaller basis set (at least 3‐ ζ ).…”

“…Similarly, effects of diffuse functions, relativistic effects, or even multireference character can be accounted for by further expansion of the total energy term above. For a recent review of the concepts used to design such extrapolation “recipes,” see Varandas for details. However, the ease of carrying out such extrapolations is often hampered by the inflexibility of the quantum chemical packages: if the summation of the energy derivatives of the components of the extrapolation is not easily possible, one has to resort to calculations from finite differences at fixed geometries and compute the gradients numerically .…”

Assessment of a composite method based on selected density functional theory methods and complete basis set extrapolation formulas

“…OAN(C) and OAN(P) were calibrated considering a training set of 223 molecules with energies estimated from the G3X and G2 calculations. USTE, GUSTE, USPE, and OAN methodologies are possibly the state of art of the extrapolation methods …”