Fig. 1. Implementation of NeuralXC.

Starting from the electron density in real space, obtained with a converged DFT calculation (using the baseline functional E_base), the projector maps this density to a set of descriptors c_nlm. The symmetrizer creates rotationally invariant versions of these descriptors d_nl, which, after preprocessing (not depicted here), are passed through a Behler–Parrinello type neural network architecture. By using the same network for descriptors of a given atomic species, we ensure permutation invariance. Once the energy E_ML is obtained, its derivative can be backpropagated using the chain rule to obtain the machine-learned potential V_ML. V_ML is added back to the baseline potential V_base = δE_base/δn(r), to create the full V_NXC(r), which can be used in subsequent self-consistent calculations.