Completing density functional theory by machine learning hidden messages from molecules

Kohn-Sham density functional theory (DFT) relies on the exchange-correlation (xc) energy functional to capture electron-electron interactions beyond classical models, but a universal exact form remains unknown and practical studies depend on approximations with known issues in accuracy and transferability. This study shows that xc functionals can be systematically constructed from accurate reference density distributions and energies of molecules via machine learning. A neural-network functional trained on only a few molecules achieves accuracy comparable to standard functionals across hundreds of molecules containing first- and second-row elements. The approach relates density and energy through a flexible feed-forward neural network, enabling functional derivatives via back-propagation. Incorporating a simple nonlocal density descriptor further improves accuracy, making nonlocal effects practical. This machine-learning framework enriches DFT by exploiting rapidly advancing ML techniques.