Split representation of adaptively compressed polarizability operator

The polarizability operator plays a central role in density functional perturbation theory and other perturbative treatment of first principle electronic structure theories. cost computing the generally scales as $${\mathcal {O}}(N_{e}^4)$$ where $$N_e$$ is number electrons system. recently developed adaptively compressed (ACP) formulation [L. Lin, Z. Xu L. Ying, Multiscale Model. Simul. 2017] reduces such complexity to {O}}(N_{e}^3)$$ context phonon calculations with large basis set for time, demonstrates its effectiveness model problems. In this paper, we improve performance ACP by splitting into near singular component that statically compressed, smooth compressed. new split representation maintains {O}}(N_e^3)$$ complexity, accelerates nearly all components formulation, including Chebyshev interpolation energy levels, iterative solution Sternheimer equations, convergence Dyson equations. For simulation real materials, discuss how incorporate nonlocal pseudopotentials finite temperature effects. We demonstrate our method using one-dimensional problem insulating metallic regimes, well accuracy molecules solids.