Highly accurate and constrained density functional obtained with differentiable programming

Using an end-to-end differentiable implementation of the Kohn-Sham self-consistent field equations, we obtain a highly accurate neural network--based exchange and correlation (XC) functional of the electronic density. The functional is optimized using information on both energy and density while exact constraints are enforced through an appropriate neural network architecture. We evaluate our model against different families of XC approximations and show that at the meta-GGA level our functional exhibits unprecedented accuracy for both energy and density predictions. For nonempirical functionals, there is a strong linear correlation between energy and density errors. We use this correlation to define an XC functional quality metric that includes both energy and density errors, leading to an improved way to rank different approximations.