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 can be reduced to nearly optimal as
with respect to the number of samples in the Brillouin zone
for the first
time. In addition, we show that the cost for applying the Bethe–Salpeter Hamiltonian to a
vector scales as
.
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
and
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
, it
takes
hours to assemble the Bethe–Salpeter Hamiltonian and
hours to fully diagonalize
the Hamiltonian using
cores when the brute force approach is used. The new method takes less than
minutes to set up
the Hamiltonian and
minutes to compute the absorption spectrum on a single core.
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