We incorporate in the Kohn-Sham self-consistent equation a trained neural-network projection from the charge density distribution to the Hartree-exchange-correlation potential n → VHxc for a possible numerical approach to the exact Kohn-Sham scheme. The potential trained through a newly developed scheme enables us to evaluate the total energy without explicitly treating the formula of the exchange-correlation energy. With a case study of a simple model, we show that the well-trained neural-network VHxc achieves accuracy for the charge density and total energy out of the model parameter range used for the training, indicating that the property of the elusive ideal functional form of VHxc can approximately be encapsulated by the machine-learning construction. We also exemplify a factor that crucially limits the transferability—the boundary in the model parameter space where the number of the one-particle bound states changes—and see that this is cured by setting the training parameter range across that boundary. The training scheme and insights from the model study apply to more general systems, opening a novel path to numerically efficient Kohn-Sham potential.
The long range part of the interaction was truncated within |x1 − x2| < 2, whereas we approximated 1/|x1 − x2| by a large value at the singularity x1 = x2.
Two sets of 1000 training data were generated for Ntot = 1 and 2.