Title: Rigorous Formalization of Orbital Functionals: Addressing the Noninteracting $v$-Representability Problem

Abstract: Functionals that explicitly depend on occupied, unoccupied, or fractionally-occupied orbitals are rigorously formalized using Clifford algebras, and a variational principle is established that facilitates orbital (and occupation) optimization as a formal implementation method. Theoretically, these methodologies circumvent the limitations encountered in the original Kohn-Sham and related methods, particularly when the interacting system's electron density does not match that of any noninteracting reference system. This work redefines orbital (and occupation) functionals from a novel perspective, positioning them not merely as extensions of traditional density functionals, but as superior, rigorous alternatives.