Communication: Relating the pure and ensemble density matrix functional.

Area of Science:

• Quantum Chemistry
• Computational Physics
• Density Matrix Theory

Background:

• Reduced Density Matrix Functional Theory (RDMFT) is a powerful tool for electronic structure calculations.
• A key theorem posits the equivalence of pure and ensemble functionals within their shared domain.
• Understanding the relationship between pure and ensemble RDMFT is crucial for developing accurate theoretical models.

Purpose of the Study:

• To rigorously examine the relationship between universal pure and ensemble functionals in RDMFT.
• To refute the theorem suggesting their coincidence on the domain of pure N-representable one-matrices.
• To elucidate the precise mathematical connection between pure and ensemble functionals.

Main Methods:

• Detailed analysis of Levy's constrained search, a foundational method in density functional theory.
• Geometric interpretation of the functional landscape in RDMFT.
• Convex analysis to determine the relationship between pure and ensemble functionals.

Main Results:

• The theorem stating the coincidence of pure and ensemble functionals is refuted.
• The ensemble functional is demonstrated to be the lower convex envelope of the pure functional.
• The pure functional dictates the ensemble functional even beyond the pure N-representable domain.

Conclusions:

• The equivalence of pure and ensemble functionals in RDMFT is not universally valid.
• The complexity of N-representability conditions is inherent and may be transferred, not resolved, by ensemble methods.
• This work provides a deeper understanding of the mathematical structure of RDMFT functionals.