Learning local and semi-local density functionals from exact exchange-correlation potentials and energies

TL;DR

A data-driven pathway to learn the XC functional by using the exact density, XC energy, and XC potential, which underscores the promise of using the XC potential in modeling XC functionals and can pave the way for systematic learning of increasingly accurate XC functionals.

Abstract

Finding accurate exchange-correlation (XC) functionals remains the defining challenge in density functional theory (DFT). Despite 40 years of active development, the desired chemical accuracy is still elusive with existing functionals. We present a data-driven pathway to learn the XC functionals by utilizing the exact density, XC energy, and XC potential. While the exact densities are obtained from accurate configuration interaction (CI), the exact XC energies and XC potentials are obtained via inverse DFT calculations on the CI densities. We demonstrate how simple neural network (NN) based local density approximation (LDA) and generalized gradient approximation (GGA), trained on just five atoms and two molecules, provide remarkable improvement in total energies, densities, atomization energies, and barrier heights for hundreds of molecules outside the training set. Particularly, the NN-based GGA functional attains similar accuracy as the higher rung SCAN meta-GGA, highlighting the promise of using the XC potential in modeling XC functionals. We expect this approach to pave the way for systematic learning of increasingly accurate and sophisticated XC functionals.

Figures (9)

• Figure 1: Schematic of the ML based XC modeling. The exact spin-densities $\{\rho^{\uparrow}(\boldsymbol{\textbf{r}}),\rho^{\downarrow}(\boldsymbol{\textbf{r}})\}$ and energies ($E$) are obtained from configuration interaction (CI) calculations. The inverse DFT with finite-element basis solves a PDE-constrained optimization to find the corresponding exact XC potentials $\{v_{\rm{xc}}^{\uparrow}(\boldsymbol{\textbf{r}}),v_{\rm{xc}}^{\downarrow}(\boldsymbol{\textbf{r}})\}$ and the XC energy ($E_{\rm{xc}}$). The neural network (NN) uses the exact XC densities, potentials, and energies to model the XC functional.
• Figure 2: Comparison of enhancement factor of NNLDA (solid) with PW92 (dashed).
• Figure 3: Comparison of enhancement factor of NNGGA (solid) with PBE (dashed).
• Figure 4: Comparison of enhancement factor of NNGGA-UEG (solid) with PBE (dashed).
• Figure 5: Comparison of error in total energy ($E_{\text{tot}}$) per atom for molecules in the G2 dataset containing up to second-row elements (97 molecules). (a) Error in individual molecules. Molecules with a parenthesis indicate different isomers or spin-states. For clarity, we have shown the data for some of the molecules and have also omitted the comparison with PW92. See the SI for the remaining molecules, the comparison with PW92, and the details of individual systems. (b) Mean absolute error (MAE) for the 97 molecules in G2 containing up to second-row elements.