Fast optical absorption spectra calculations for periodic solid state systems
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We present a method to construct an efficient approximation to the bare exchange and screened direct interaction kernels of the Bethe-Salpeter Hamiltonian for periodic solid state systems via the interpolative separable density fitting technique. We show that the cost of constructing the approximate Bethe-Salpeter Hamiltonian scales nearly optimally as O(N_k) with respect to the number of samples in the Brillouin zone N_k. In addition, we show that the cost for applying the Bethe-Salpeter Hamiltonian to a vector scales as O(N_k log N_k). Therefore the optical absorption spectrum, as well as selected excitation energies can be efficiently computed via iterative methods such as the Lanczos method. This is a significant reduction from the O(N_k^2) and O(N_k^3) scaling associated with a brute force approach for constructing the Hamiltonian and diagonalizing the Hamiltonian respectively. We demonstrate the efficiency and accuracy of this approach with both one-dimensional model problems and three-dimensional real materials (graphene and diamond). For the diamond system with N_k=2197, it takes 6 hours to assemble the Bethe-Salpeter Hamiltonian and 4 hours to fully diagonalize the Hamiltonian using 169 cores when the brute force approach is used. The new method takes less than 3 minutes to set up the Hamiltonian and 24 minutes to compute the absorption spectrum on a single core.
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