The Regal project has so far produced five journal publications. In the first [ACS Phys. Chem Au 3 334, 2023] and third [ACS Phys. Chem Au 3 492, 2023] publications of the project, we reviewed the Hohenberg–Kohn theorem, which is traditionally seen as the foundation of density-functional theory (DFT) for describing electronic systems in their ground state using one-body particle density. We examined the actual role of the Hohenberg–Kohn theorem within standard DFT. In addition, we explored extensions of DFT that incorporate magnetic fields. We argued that the Hohenberg–Kohn theorem is not the fundamental basis of DFT but rather a result of a broader mathematical framework. This insight is particularly valuable for developing generalized versions of DFT. A particular emphasis was also placed on the Moreau-Yosida regularization, which is the main focus of the Regal project.
The second article of the Regal project [Electron. Struct. 5 014009, 2023] explored how computation of effective potentials for a given electron density in non-interacting systems relates to the Moreau–Yosida regularization of density functionals. In a rigorous mathematical setting, such potentials arise through a limit process of the regularization parameter and can be used to describe density-to-potential inversion in a unifying mathematical framework. The choice of space where the one-body particle density lives in becomes the crucial choice to make.
In [J. Chem. Phys. 160, 024103, 2024], we studied an approach to ground-state DFT that replaces energy functionals with exact force expressions. This method aims to improve approximations for the exchange–correlation potential and energy, which are the key unknowns in DFT. This force-based approach offers several advantages over traditional energy-based methods and is especially relevant for the Regal project since it does not assume differentiability of the universal functional.
In [J. Chem. Phys. 160, 084115, 2024], we revisited the exact constraint of exchange-only virial relation by Levy and Perdew and studied it from a mathematical perspective in the spirit of one of the main objectives of Regal - to derive exact constraints without the assumption of differentiability of the universal density functional. In this work we managed to derive the exchange energy in terms of the right derivative of the coupling constant dependent version of the universal functional. This is an interesting result in formal density-functional theory and the discovery came from using the adiabatic connection that allows a non-interacting system to be connected to the fully interacting physical system under the assumption that the one-body particle density stays the same. It is our belief that such results provide a deeper understanding, and offers a robust framework for further developments in DFT that we are currently investigating within the framework of regularized- and extended versions of standard DFT.
In the unpublished manuscript [arXiv:2409.04372] we have continued the work in [Electron. Struct. 5 014009, 2023] with a more numerical focus, but also containing new mathematical results. The work here presented the use of an exact Moreau-Yosida regularized formulation to compute the exchange-correlation potential for periodic systems. This approach uniquely integrates rigorous mathematical principles with efficient numerical implementation, representing the first application of a Moreau-Yosida-based inversion in physical systems. We developed a mathematically rigorous inversion algorithm with mathematical guarantees, including error bounds, and verified it numerically in bulk silicon. Furthermore, this work falls within the framework of an exact constraint that the effective potential must satisfy, developed within the Moreau-Yosida regularization framework.
