Real-space Hubbard-corrected density functional theory

TL;DR

The paper develops and validates a real-space DFT+U framework with explicit energy, force, and stress expressions tailored for finite-difference discretization, implemented in SPARC for large-scale parallelism. It demonstrates accuracy by benchmarking against planewave results and shows significant performance advantages, achieving near-linear to quadratic strong scaling up to thousands of cores. The authors apply the method to TiO$_2$ polymorphs, analyzing exchange–correlation consistency in local orbitals and proposing a hybrid-functional-based scheme to optimize the Hubbard parameter, yielding band gaps in good agreement with experiment. Collectively, the work provides a robust, scalable alternative to planewave approaches for DFT+U calculations, with practical insights for orbital generation and Hubbard parameter determination that enhance predictive capability for strongly correlated materials.

Abstract

We present an accurate and efficient framework for real-space Hubbard-corrected density functional theory. In particular, we obtain expressions for the energy, atomic forces, and stress tensor suitable for real-space finite-difference discretization, and develop a large-scale parallel implementation. We verify the accuracy of the formalism through comparisons with established planewave results. We demonstrate that the implementation is highly efficient and scalable, outperforming established planewave codes by more than an order of magnitude in minimum time to solution, with increasing advantages as the system size and/or number of processors is increased. We apply this framework to examine the impact of exchange-correlation inconsistency in local atomic orbital generation and introduce a scheme for optimizing the Hubbard parameter based on hybrid functionals, both while studying TiO$_2$ polymorphs.

Figures (6)

• Figure 1: Convergence of the energy, atomic forces, and stress tensor with respect to the local atomic orbital truncation threshold. The energy error is defined as the absolute difference, while the force and stress errors are defined as the maximum difference in any component. Results obtained with a truncation threshold of $10^{-4}$ bohr$^{-3/2}$ serve as the reference.
• Figure 2: Convergence of the energy, atomic forces, and stress tensor with respect to grid spacing. The energy error is defined as the absolute difference, while the force and stress errors are defined as the maximum difference in any component. Results obtained with a grid spacing of $\sim 0.10$ bohr serve as the reference.
• Figure 3: Strong scaling performance of an AIMD step in SPARC for TiO$_2$ systems with PBE+U. All parameters have been chosen so as to achieve accuracy of $10^{-3}$ ha/atom in the energy.
• Figure 4: Variation in the DFT+U total energy differences between the anatase and brookite polymorphs of TiO$_2$ relative to the rutile polymorph, for atomic orbitals generated using either r$^2$SCAN or PBE. The green region marks the experimental $\left( E_{\text{anatase}} - E_{\text{rutile}} \right)$ range of values, while the pink region denotes the values of $U$ that result in the correct polymorph stability order.
• Figure 5: Non self-consistent HSE hybrid functional total energy evaluated at the r$^2$SCAN+U ground state as a function of the Hubbard parameter $U$.