Where Does the Anion-Redox Hole Go? Reliability of Open-Shell Projection-Based Embedding for Li-Rich Cathodes, from DFT Environments to Quantum Solvers

Anion redox in Li-rich layered oxides places holes on oxygen next to an antiferromagnetic Mn(IV) d3 lattice, a strongly correlated open-shell problem for which projection-based wavefunction-in-DFT embedding offers a route to correlated and quantum solvers, but whose errors are known mainly for closed-shell molecules. We develop a spin-unrestricted projection-based embedding, exact at the DFT-in-DFT level to the microhartree scale for closed-shell, broken-symmetry and high-spin references with point charges, and apply DMRG-in-DFT to point-charge clusters of delithiated Li(2-x)MnO3, varying the environment functional (0-100% exact exchange), geometry, active region, active-space spin, virtual space, cluster size and magnetic reference. In every setting the lowest SCF state places the hole on an oxygen pair, with no O-O bond. How the hole is shared depends on the functional together with the geometry: hybrids give a full, uncoupled hole at the DFT+U geometry but a partial, weakly covalent hole at their own geometry, a change of up to 0.5 e on one oxygen. Where spin spreads across the active/environment boundary, the integer spin projection of the active space becomes a choice worth up to 0.26 e, and at most 0.09 e once the two Mn neighbors are included. An extended state search found missed lower states for two functionals, changing a pair-oxygen spin by up to 0.45 e. Across all 82 runs the embedded mean field gives the same hole (median DMRG change 0.02-0.04 e); correlation mainly adds O-O covalency. State control and geometry consistency, not the solver, dominate the uncertainty, and total energies are not converged at 30-40 orbitals. For quantum solvers, the embedded Hamiltonian must be spin-unrestricted: a shared-orbital form reverses the hole, while sample-based quantum diagonalization reproduces exact diagonalization, as accurately as, but not better than, classical selected CI.