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Exact and model exchange-correlation potentials for open-shell systems

The conventional approaches to the inverse density functional theory problem typically assume non-degeneracy of the Kohn-Sham (KS) eigenvalues, greatly hindering their use in open-shell systems. We present a generalization of the inverse density functional theory problem that can seamlessly admit degenerate KS eigenvalues. Additionally, we allow for fractional occupancy of the Kohn-Sham orbitals to also handle non-interacting ensemble-v-representable densities, as opposed to just non-interacting pure-v-representable densities. We present the exact exchange-correlation (XC) potentials for six open-shell systems -- four atoms (Li, C, N, and O) and two molecules (CN and $\text{CH}_2$) -- using accurate ground-state densities from configuration interaction calculations. We compare these exact XC potentials with model XC potentials obtained using non-local (B3LYP, SCAN0) and local/semi-local (SCAN, PBE, PW92) XC functionals. Although the relative errors in the densities obtained from these DFT functionals are of $\mathcal{O}(10^{-3}-10^{-2})$, the relative errors in the model XC potentials remain substantially large -- $\mathcal{O}(10^{-1}-10^0)$.