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.
Gao et al. (Tue,) studied this question.