Short-Range Δ-Machine Learning: A Cost-Efficient Strategy to Transfer Chemical Accuracy to Condensed Phase Systems

DFT-based machine-learning potentials (MLPs) are now routinely trained for condensed-phase systems, but surpassing DFT accuracy remains challenging due to the cost or unavailability of periodic reference calculations. Our previous work ( Phys. Rev. Lett. 2022, 129, 226001) demonstrated that high-accuracy periodic MLPs can be trained within the CCMD framework using extended yet finite reference calculations. Here, we introduce short-range Δ-Machine Learning (srΔML), a method that starts from a baseline MLP trained on low-level periodic data and adds a Δ-MLP correction based on high-level cluster calculations at the CC level. Applied to liquid water, srΔML reduces the required cluster size from (H₂O)₆₄ to (H₂O)₁₅ and significantly lowers the number of clusters needed, resulting in a 50–200× reduction in computational cost. The resulting potential closely reproduces the target CC potential and accurately captures both two- and three-body structural descriptors.