[Paper Review] Lieb variation principle in density-functional theory

Trygve Helgaker, Andrew M. Teale|arXiv (Cornell University)|Apr 26, 2022

Machine Learning in Materials Science4 citations

TL;DR

This paper presents Lieb's convex formulation of density-functional theory (DFT) as a unifying, pedagogical framework that emphasizes the duality between the ground-state energy and the universal density functional. It introduces the Lieb variation principle—a computational tool enabling high-accuracy calculation of the Kohn–Sham potential, exchange–correlation functional, and adiabatic connection via high-precision many-body methods, offering both theoretical insight and benchmark data for functional development.

ABSTRACT

Lieb's convex formulation of density-functional theory is presented in a pedagogical manner, emphasizing its connection to Hohenberg-Kohn theory and to Levy's constrained-search theory. The Hohenberg-Kohn and Lieb variation principles are discussed, highlighting the dual relationship between the ground-state energy and the universal density functional. Applications of the Lieb variation principle are reviewed, demonstrating how it may be utilized to calculate the Kohn-Sham potential of atoms and molecules, to study the exchange-correlation functional and the adiabatic connection by high-precision many-body methods, and to calculate the exchange-correlation hole and energy densities of atoms and molecules.

Motivation & Objective

• To provide a pedagogical and accessible introduction to Lieb’s convex formulation of DFT, which unifies Hohenberg–Kohn and Levy’s constrained-search theories.
• To highlight the underappreciated role of Lieb’s 1983 convex formulation in clarifying the foundational structure of DFT.
• To demonstrate the practical utility of the Lieb variation principle for computing the universal density functional and Kohn–Sham potential with high precision.
• To extend the applicability of Lieb’s framework to orbital-free DFT and systems under magnetic fields, unifying diverse DFT variants.
• To provide benchmark-quality data for the exchange–correlation functional and adiabatic connection using high-precision quantum-chemical methods.

Proposed method

• Utilizes Lieb’s convex formulation, where the ground-state energy is a concave functional of the external potential, enabling a variational principle over densities.
• Applies the Lieb variation principle: for a given electron density, compute the corresponding external potential and universal density functional via energy minimization.
• Employs high-precision many-body methods (e.g., coupled-cluster theory) to compute the universal density functional and exchange–correlation energy densities.
• Uses the subgradient inequality of the energy to establish the duality between the energy and the density functional, linking potentials and densities via the Hohenberg–Kohn mapping.
• Applies the formalism to Kohn–Sham DFT by inverting the density to find the effective potential and to orbital-free DFT by directly computing the universal functional.
• Extends the framework to systems in magnetic fields by generalizing the convex formulation to include vector potentials and spin degrees of freedom.

Experimental results

Research questions

• RQ1 How does Lieb’s convex formulation unify and clarify the foundations of DFT, particularly in relation to the Hohenberg–Kohn and Levy’s constrained-search theories?
• RQ2 Can the Lieb variation principle be used to compute the Kohn–Sham potential and universal density functional with high accuracy using high-precision quantum-chemical methods?
• RQ3 What insights does the Lieb formulation provide into the structure of the exchange–correlation functional and the adiabatic connection in DFT?
• RQ4 How can the Lieb variation principle be extended to orbital-free DFT and systems under external magnetic fields?
• RQ5 What benchmark data for the exchange–correlation hole and energy density can be generated using this approach?

Key findings

• The Lieb variation principle enables high-accuracy computation of the universal density functional and Kohn–Sham potential from a given electron density, providing a practical realization of the Hohenberg–Kohn mapping.
• The method yields benchmark-quality data for the exchange–correlation energy and hole densities in atoms and molecules using high-precision many-body techniques.
• The convex formulation reveals a dual relationship between the ground-state energy and the universal density functional, enhancing theoretical clarity and pedagogical accessibility.
• The framework successfully extends to orbital-free DFT and systems under magnetic fields, demonstrating its unifying power across DFT variants.
• The subgradient inequality underpins the variational structure, showing that the energy is a concave functional of the potential, which enables stable and accurate inversion procedures.
• The approach provides a systematic way to study the adiabatic connection and the exact exchange–correlation functional, offering critical benchmarks for functional development.