Ab initio stochastic time-domain formulation of the Bethe-Salpeter equation*

We present a reduced scaling formulation of the Bethe-Salpeter equation (BSE) through a combination of a time-dependent approach and stochastic representations, which allows efficient ab initio calculations of optical absorption spectra for semiconductor nanoparticles. The linear response of the system to external fields is simulated by propagating quasiparticle orbital dynamics in real time, followed by Fourier transforming the dipole-dipole correlation function to obtain the absorption spectrum. The spatially dependent screening of the system is described within the random phase approximation (RPA) and combines several efficient stochastic techniques, including factorization with stochastic representations, and time-dependent Hartree propagation of stochastic occupied orbitals. The computational cost of the new BSE formulation is quadratic scaling with respect to system size, which is a significant improvement compared with the conventional symplectic eigenvalue representation of the BSE. We discuss preliminary results from the applications of the method to silicon and CdSe nanocrystals.