# Projectability-guided AO Hamiltonian construction

Converting plane-wave DFT Hamiltonians into atomic-orbital form now uses projectability-selected eigenstates to complete the subspace instead of an empirical shift, giving 1–3 meV band errors for five sodium phases with 13 orbitals per atom where real-space reconstruction needs 23 orbitals and still errs by 12–28 meV.

Converting plane-wave DFT Hamiltonians into atomic-orbital form now uses projectability-selected eigenstates to complete the subspace instead of an empirical shift, giving 1–3 meV band errors for five sodium phases with 13 orbitals per atom where real-space reconstruction needs 23 orbitals and still errs by 12–28 meV.