Hierarchical Fourier Phase Projection for Local Electronic Observables

Large-scale electronic-structure calculations require efficient access to local observables without explicitly constructing all occupied orbitals. We develop hierarchical Fourier phase projection (HPP), which organizes Fourier probes into a reusable spatial hierarchy that progressively removes short-range aliasing while exploiting density-matrix locality. The method provides systematic refinement from low-cost local estimates to the projection-exact limit of the chosen numerical occupation operator, without discarding previously evaluated responses. Tests using frozen Kohn–Sham Hamiltonians for semiconducting and metallic systems demonstrate controllable convergence of electron densities and nonlocal pseudopotential forces, weak size dependence of the probing resolution required for a fixed local accuracy, and near-linear growth of the direct computational cost at fixed probing workload. Inter-level changes further provide practical information for terminating the refinement at finite accuracy. HPP connects electronic locality, observable accuracy, and computational effort within a single hierarchical framework, providing a scalable route to local quantities in large-scale electronic-structure calculations.