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The real-space density-functional perturbation theory (DFPT) for thecomputations of the response properties with respect to the atomic displacementand homogeneous electric field perturbation has been recently developed andimplemented into the all-electron, numeric atom-centered orbitals electronicstructure package FHI-aims. It is found that the bottleneck for large scaleapplications is the computation of the response density matrix, which scales as$O(N^3)$. Here for the response properties with respect to the homogeneouselectric field, we present an efficient parallel linear scaling algorithm forthe response density matrix calculation. Our scheme is based on thesecond-order trace-correcting purification and the parallel sparsematrix-matrix multiplication algorithms. The new scheme reduces the formalscaling from $O(N^3)$ to $O(N)$, and shows good parallel scalability over tensof thousands of cores. As demonstrated by extensive validation, we achieve arapid computation of accurate polarizabilities using DFPT. Finally, thecomputational efficiency of this scheme has been illustrated by making thescaling tests and scalability tests on massively parallel computer systems.

Efficient Parallel Linear Scaling Method to get the Response Density Matrix in All-Electron Real-Space Density-Functional Perturbation Theory