Efficient Construction of 4-point Green’s Function in Real-Space Representation using Permutation Sampling Monte Carlo method

A principle challenge in constructing the 1-particle Green’s function is the steep scaling of computational cost with increasing system size. We address this problem by transforming the Green’s function into real-space representation and calculating all required integrals using the Permutation Sampling Monte Carlo method (PSMC). We started with the frequency-domain representation of the self-energy and performed Laplace transformation to obtain a 4-point representation of the self-energy in position basis. The main benefit of PSMC is that it avoids the steep scaling associated with the traditional methods of constructing the self-energy. Specifically, we avoided AO-to-MO integral transformation and explicit representation of the self-energy in particle-hole basis. In this work we demonstrated the linear scaling of computational cost with the number of molecular-orbital basis functions that was achieved by the PSMC method. Consequently, PSMC can be applied to systems that are computationally impractical using conventional methods. We used PSMC to calculate the ionization potentials of PbS quantum dots (Pb4S4-P140S140).