Rigorous Ab Initio Quantum Embedding for Quantum Chemistry Using Green's Function Theory: Screened Interaction, Nonlocal Self-Energy Relaxation, Orbital Basis, and Chemical Accuracy.

We present a detailed discussion of the self-energy embedding theory (SEET), which is a quantum embedding scheme allowing us to describe a chosen subsystem very accurately while keeping the description of the environment at a lower level. We apply SEET to molecular examples where our chosen subsystem is made out of a set of strongly correlated orbitals while the weakly correlated orbitals constitute an environment. Consequently, a highly accurate method is used to calculate the self-energy for the system, while a lower-level method is employed to find the self-energy for the environment. Such a self-energy separation is very general, and to make the SEET procedure applicable to multiple systems, a detailed and practical procedure for the evaluation of the system and environment self-energy is necessary. We list all of the intricacies for one of the possible procedures while focusing our discussion on many practical implementation aspects such as the choice of best orbital basis, impurity solver, and many steps necessary to reach chemical accuracy. Finally, on a set of carefully chosen molecular examples, we demonstrate that SEET, which is a controlled, systematically improvable Green's function method, can be as accurate as established wave function quantum chemistry methods.