Stochastic Real-Time Second-Order Green's Function Theory for Neutral Excitations in Molecules and Nanostructures

We present a real-time second-order Green's function method (TD-GF2) for computing neutral excitations in molecules and nanostructures. The framework is combined with the stochastic resolution of the identity to decouple the 4-index electron repulsion integrals (ERI) in the system Hamiltonian. This leads to the reduction of the computational cost to $O(N^3)$ with system size. The stochastic implementation recovers deterministic results for the electronic dynamics and excitation energies, and reproduces benchmark results from the analogous linear-response implementation in frequency. This approach is further combined with the Dynamic Mode Decomposition (DMD) technique to predict the nonlinear long-time dynamics of the density matrix. The statistical error due to the incorporation of the stochastic resolution of the identity and DMD extrapolation is analyzed in terms of the number of stochastic orbitals, system size, and propagation time. Overall, this approach offers an efficient route to investigate excited states in finite systems containing hundreds of electrons.