s is the identity operator of the ξ th excitation subspace and ∆ Ĥ(ξ) = Ĥ(ξ) -(ξ ε) • 1(ξ) s , and where ξ ε is the reference energy for the ξ th excitation subspace (i.e., ε for single excitation and 2ε for double excitation subspace). Since the identity operator commutes with any other operators in its respective subspace, using Eq. 41, the propagator now becomes