Local Spectral Formulation of the One-Determines-All (ODA) Principle for Multistate Density Functionals

Multistate density functional theory (MSDFT) generalizes Kohn-Sham density functional theory to a finite subspace of interacting states through a Hamiltonian matrix functional of the matrix density D(r). A central challenge is to construct matrix functionals that preserve subspace unitary invariance while encoding nontrivial state coupling. Here we show that any local, unitary-covariant matrix functional of D(r) must be codiagonalizable with D(r), and is therefore completely specified by a scalar generator acting on its eigenvalues. This establishes a one-to-all mapping from a scalar generator to the full matrix functional. As a consequence, the construction of N² matrix elements is reduced to a single scalar mapping evaluated on the eigenvalue spectrum at each spatial grid point. The formalism is illustrated using a four-state Hubbard model, where exact reconstruction is achieved with a known scalar function, and deviations from this mapping can be systematically corrected within the spectral framework. The results provide a rigorous foundation for constructing local matrix exchange-correlation functionals with computational complexity comparable to Kohn-Sham DFT, and offer a practical route toward scalable MSDFT approximations.