Order-$N$ orbital-free density-functional calculations with machine learning of functional derivatives for semiconductors and metals

Abstract

Orbital-free density functional theory (OFDFT) offers a challenging way of electronic-structure calculations scaled as $\mathcal{O}(N)$ computation for system size $N$. We here develop a scheme of the OFDFT calculations based on the accurate and transferrable kinetic-energy density functional (KEDF) which is created in an unprecedented way using appropriately constructed neural network (NN). We show that our OFDFT scheme reproduces the electron density obtained in the state-of-the-art DFT calculations and then provides accurate structural properties of 24 different systems, ranging from atoms, molecules, metals, semiconductors and an ionic material. The accuracy and the transferability of our KEDF is achieved by our NN training system in which the kinetic-energy functional derivative (KEFD) at each real-space grid point is used. The choice of the KEFD as a set of training data is essentially important, because first it appears directly in the Euler equation which one should solve and second, its learning assists in reproducing the physical quantity expressed as the first derivative of the total energy. More generally, the present development of KEDF $T[\rho ]$ is in the line of systematic expansion in terms of the functional derivatives ${\delta }^{{\ell }_1}T/\delta {\rho }^{{\ell }_1}$ through progressive increase of ${\ell }_1$. The present numerical success demonstrates the validity of this approach. The computational cost of the present OFDFT scheme indeed shows the $\mathcal{O}(N)$ scaling, as is evidenced by the computations of the semiconductor SiC used in power electronics.

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• Variational principle to regularize machine-learned density functionals: the non-interacting kinetic-energy functional
• Machine learning of kinetic energy densities with target and feature averaging: better results with fewer training data
• Machine learning based nonlocal kinetic energy density functional for simple metals and alloys
• Multi-channel machine learning based nonlocal kinetic energy density functional for semiconductors
• A machine-learned kinetic energy model for light weight metals and compounds of group III-V elements
• Machine learning-guided construction of an analytic kinetic energy functional for orbital free density functional theory
• On the Kinetic Energy Density Functional: The Limit of the Density Derivative Order