Periodic Reporting for period 1 - REGAL (Regularized Density-Functional Analysis)
Período documentado: 2022-09-01 hasta 2025-02-28
The Kohn-Sham approach of density-functional theory (DFT) stands as a cornerstone in quantum chemistry based on its practical utility. Despite its widespread use, the central component of DFT, the exact universal density functional, remains unknown and is furthermore nondifferentiable, posing significant challenges to its theoretical and practical applications. Extensive research has established exact conditions for this functional, yet obtaining universal and accurate approximations remain a top priority.
The primary motivation behind this project is to address the nondifferentiability of the universal density functional in DFT. By applying the Moreau-Yosida regularization to DFT, this project aims to achieve differentiability, thereby resolving, e.g. the problem of potential-representability and providing global solutions to the underlying variational problem. The first objective is to develop a robust mathematical foundation for a regularized DFT, analogous to the unregularized standard DFT. This involves a close interplay between different theories that utilize more than just the particle density as variables, guiding the development of the regularized theory. With a regularized formulation in place, the second objective focuses on developing new and understanding existing exact constraints for the density functional and it component. This is expected to open new avenues for method development in approximate functionals, enhancing the accuracy and reliability of DFT calculations. The third objective is to investigate the regularized Kohn-Sham iteration scheme, aiming to prove guaranteed convergence. Additionally, the project will explore how regularization can speed up convergence by, e.g. utilizing bounds on energy curvature.
The proposed approach is unconventional and has the potential to transform the development of approximate functionals and the iterative Kohn-Sham scheme. By achieving differentiability and addressing potential-representability, the project aims to significantly enhance the theoretical and, hopefully, practical aspects of DFT. This could lead to more accurate and efficient quantum chemical calculations, ultimately advancing the field of quantum chemistry.
The primary motivation behind this project is to address the nondifferentiability of the universal density functional in DFT. By applying the Moreau-Yosida regularization to DFT, this project aims to achieve differentiability, thereby resolving, e.g. the problem of potential-representability and providing global solutions to the underlying variational problem. The first objective is to develop a robust mathematical foundation for a regularized DFT, analogous to the unregularized standard DFT. This involves a close interplay between different theories that utilize more than just the particle density as variables, guiding the development of the regularized theory. With a regularized formulation in place, the second objective focuses on developing new and understanding existing exact constraints for the density functional and it component. This is expected to open new avenues for method development in approximate functionals, enhancing the accuracy and reliability of DFT calculations. The third objective is to investigate the regularized Kohn-Sham iteration scheme, aiming to prove guaranteed convergence. Additionally, the project will explore how regularization can speed up convergence by, e.g. utilizing bounds on energy curvature.
The proposed approach is unconventional and has the potential to transform the development of approximate functionals and the iterative Kohn-Sham scheme. By achieving differentiability and addressing potential-representability, the project aims to significantly enhance the theoretical and, hopefully, practical aspects of DFT. This could lead to more accurate and efficient quantum chemical calculations, ultimately advancing the field of quantum chemistry.
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.
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.
There has been partial advancement beyond state-of-the-art. The achievements in [Electron. Struct. 5 014009, 2023] and the manuscript [arXiv:2409.04372] mark significant theoretical advancements in the field of formal density-functional theory (DFT). These works have addressed the practical usefulness of the exact (lossless) Moreau-Yosida regularization in connection with density-to-potential inversion. Accurately computing effective Kohn-Sham potentials is of great fundamental value to DFT, as it forms the basis of forward Kohn-Sham computations, which are integral to many quantum chemistry simulations today. The Moreau-Yosida density-to-potential inversion scheme that we have developed within the Regal project provides a rigorous mathematical framework for determining effective potential within a Kohn-Sham setting. While it remains to convince the community and prove its efficiency and rigor over existing methods for challenging systems, it holds promise for advancing density-to-potential inversion beyond the current state-of-the-art. This advancement opens a new pathway for analyzing Kohn-Sham inversion methods and it is hoped that this will promote the development of mathematical approaches for creating better approximate functionals and algorithms for the quantum chemistry community.