Lower Semicontinuity of the Universal Functional in Paramagnetic Current-Density Functional Theory

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• Current-density functional theory (CDFT) is an extension of density functional theory (DFT).
• The paramagnetic formulation of CDFT is crucial for describing systems with magnetic fields.
• Establishing the mathematical rigor of CDFT is essential for its reliable application.

Purpose of the Study:

• To rigorously prove fundamental mathematical properties of the CDFT constrained-search functional.
• To establish the lower semicontinuity and expectation-valuedness of the CDFT functional.
• To solidify the mathematical framework of CDFT, aligning it with standard DFT.

Main Methods:

• Mathematical proof of lower semicontinuity.
• Mathematical proof of expectation-valuedness.
• Analysis of the CDFT constrained-search universal density functional.

Main Results:

• The lower semicontinuity of the CDFT constrained-search functional is proven.
• The expectation-valuedness of the CDFT constrained-search functional is proven.
• Existence of a minimizing density matrix for the CDFT functional is demonstrated.

Conclusions:

• The mathematical framework of CDFT is rigorously established.
• CDFT's mathematical foundation is now on par with standard DFT.
• These proofs support the continued development and application of CDFT.