Hierarchical Fourier Phase Projection for Local Electronic Observables

Abstract

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.