From 4D to 2D: Efficient computation of the electron correlation energies of atoms in molecular systems

The accurate modeling of (bio)molecular systems remains a major challenge in computational chemistry. In recent years, machine learning potentials (MLPs) have emerged as a promising approach, combining the efficiency of classical force fields with near quantum-mechanical accuracy. Among these, the physics-informed MLP FFLUX employs the Interacting Quantum Atoms (IQA) energy partitioning scheme to decompose molecular energies into atomic contributions derived from quantum mechanics for machine learning. However, incorporating two-electron correlation energies into IQA, and consequently into FFLUX, is currently unfeasible for larger systems en masse, due to the steep computational scaling associated with the necessary storage and evaluation of the two-particle density matrix (2PDM). In this work, we introduce a proof-of-concept workflow that employs density fitting (DF) to compress the 2PDM into a two-index representation, thereby accelerating the IQA partitioning of two-electron correlation energies. By expressing traditionally four-center electron repulsion integrals in terms of auxiliary basis functions, the DF approximation enables a size reduction of the 2PDM from O(N4) to O(NDF2), where N is the number of basis functions and NDF is the size of the auxiliary basis set. The method proposed here is demonstrated using coupled cluster singles and doubles densities. The resulting reduction in computational cost and memory requirements enables correlated IQA analyses for significantly larger systems than previously accessible. Thus, this development provides an important step toward incorporating explicit electron correlation and hence dispersion directly into FFLUX atomic ML models, thereby improving both their physical robustness and transferability. This advance will eliminate the need to artificially add empirical potentials, or a posteriori dispersion correction.