Optimized Attenuated Interaction: Enabling Stochastic Bethe-Salpeter Spectra for Large Systems

We develop an improved stochastic formalism for the Bethe-Salpeter equation, based on an exact separation of the effective-interaction $W$ to two parts, $W=(W-v_W)+v_W$ where the latter is formally any translationally-invariant interaction $v_W(r-r')$. When optimizing the fit of $v_W$ exchange kernel to $W$, by using a stochastic sampling of $W$, the difference $W-v_W$ becomes quite small. Then, in the main BSE routine, this small difference is stochastically sampled. The number of stochastic samples needed for an accurate spectrum is then largely independent of system size. While the method is formally cubic in scaling, the scaling prefactor is small due to the constant number of stochastic orbitals needed for sampling $W$.