Extrapolating Molecular Correlation Energies to the Complete Basis Set Limit with Symbolic Regression

Abstract

Basis set extrapolation is an important strategy for reducing basis set incompleteness error (BSIE) in correlation energy calculations. However, extrapolating correlation energies from double-ζ/triple-ζ (DZ/TZ) levels to the complete basis set (CBS) limit remains challenging. To address this challenge, we apply the sure independence screening and sparsifying operator (SISSO), a symbolic regression method, to discover formulas for the correction from TZ correlation energies to the near-CBS limit based on information from DZ and TZ calculations. In short, a subset of QM9 containing 834 small molecules with no more than 10 atoms was first constructed for subsequent training and test. After that, eight different input feature sets were constructed by combining the TZ/DZ correlation energy difference (ΔE_TD) with the basis-function increment (ΔN_Basis-TD), the effective atom count (N_eff), and the angular-momentum-resolved shell increments (ΔN_s,p,d and ΔN_f), followed by a systematic comparison of the models trained using these feature sets. Among them, the formula obtained using ΔE_TD, ΔN_Basis-TD, and N_eff as input features achieved the best balance between internal accuracy and external test set performance. These results demonstrate that symbolic regression can generate compact and interpretable formulas for complete basis set extrapolation of correlation energies, while input features with clear physical interpretation improve the accuracy and transferability of basis set extrapolation.