Generalized Kohn-Sham iteration on Banach spaces

Area of Science:

• Quantum Chemistry
• Mathematical Physics
• Computational Science

Background:

• The Kohn-Sham (KS) algorithm is central to density-functional theory (DFT) for electronic structure calculations.
• Existing formulations often lack rigorous mathematical underpinnings, particularly in generalized settings.
• A need exists for a robust mathematical framework to ensure the reliability of KS-based methods.

Purpose of the Study:

• To provide a rigorous mathematical formulation of the Kohn-Sham (KS) algorithm.
• To develop a generalized KS iteration scheme applicable to various DFT models.
• To establish the convergence properties of the proposed generalized KS method.

Main Methods:

• Utilizing convex analysis on Banach spaces for rigorous mathematical formulation.
• Employing a Levy-Lieb-type functional and its convex, lower semi-continuous extension.
• Introducing regularization to achieve differentiability and define a novel KS iteration scheme.
• Proving convergence in a weak sense.

Main Results:

• A rigorously defined Kohn-Sham (KS) iteration scheme is established.
• The generalized formulation overcomes limitations of common unregularized approaches.
• Weak convergence of the proposed KS iteration scheme is mathematically proven.
• The framework is shown to be applicable beyond standard DFT.

Conclusions:

• The generalized Kohn-Sham (KS) algorithm provides a robust mathematical foundation for density-functional theories.
• This rigorous approach enhances the reliability and applicability of KS-based computational methods.
• The formulation may extend to models outside of traditional quantum mechanics, broadening its impact.