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

Delta -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 + Delta VCC-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 Delta VCC-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.

Automated summary

The study presents a machine learning approach to improve the accuracy of potential energy surfaces based on low-level density functional theory energies and gradients. By using a simple equation and fitting high-dimensional PESs with the permutationally invariant polynomial method, the approach is demonstrated to be effective for multiple molecules. Results show excellent agreement with benchmark results, even with a relatively small number of CCSD(T) energies used for training.