Δ-machine learning for potential energy surfaces: A PIP approach to bring a DFT-based PES to CCSD(T) level of theory

"Δ-machine learning" refers to a machine learning approach to bring a property such as a potential energy surface (PES) based on low-level (LL) density functional theory (DFT) energies and gradients close to a coupled cluster (CC) level of accuracy. Here, we present such an approach that uses the permutationally invariant polynomial (PIP) method to fit high-dimensional PESs. The approach is represented by a simple equation, in obvious notation V_LL→CC = V_LL + ΔV_CC-LL, and demonstrated for CH₄, H₃O⁺, and trans and cis-N-methyl acetamide (NMA), CH₃CONHCH₃. For these molecules, the LL PES, V_LL, is a PIP fit to DFT/B3LYP/6-31+G(d) energies and gradients and ΔV_CC-LL is a precise PIP fit obtained using a low-order PIP basis set and based on a relatively small number of CCSD(T) energies. For CH₄, these are new calculations adopting an aug-cc-pVDZ basis, for H₃O⁺, previous CCSD(T)-F12/aug-cc-pVQZ energies are used, while for NMA, new CCSD(T)-F12/aug-cc-pVDZ calculations are performed. With as few as 200 CCSD(T) energies, the new PESs are in excellent agreement with benchmark CCSD(T) results for the small molecules, and for 12-atom NMA, training is done with 4696 CCSD(T) energies.