we have the time-ordered formula The non-interacting density matrix is the standard sum over the occupied states We separate a potential U$_{0}$ from H$_{0}$ via where T is the kinetic operator and U$_{ion}$ the electron-ion interaction potential. The static and Hermitian U$_{0}$ represents approximately the effects of the Coulomb interaction (Hartree, exchange, and correlation). For example, in DFT U$_{0}$ is a local potential that is the sum of the Hartree and exchange-correlation potentials. However, in general, we allow for a non- local U$_{0}$ , i.e. U$_{0}$ ( x, x $^{′}$) = 0 for ⃗ r = ⃗ r $^{′}$. For a fixed nuclear configuration and thus U$_{ion}$ , G$_{0}$ is determined by U$_{0}$ and vice versa: The exact, interacting, one-particle Green’s function obeys the Dyson equation where φ$_{H}$ is the Hartree potential determined by the electron density n ( x ) and the bare Coulomb operator is The self-energy Σ$_{xc}$( ω ) is frequency-dependent (dynamic) and non-Hermitian and encodes the complex exchange and correlation effects of the many-body system. We can write the Dyson equation equally as This shows that to obtain the true interacting Green’s function, we replace the static, Her- mitian U$_{0}$ by the dynamic, non-Hermitian φ$_{H}$ + Σ$_{xc}$( ω ). The interacting electron density n ( x ) and density matrix ρ ( x, x $^{′}$) can be computed from the Green’s function via G$_{0}$ ( t ) = ∫ ∞ −∞ dω 2 π e − $^{iωt}$G$_{0}$ ( ω ) = − iθ ( t ) ∑ c | c 〉〈 c | e − iϵ$_{c}$t + iθ ( − t ) ∑ v | v 〉〈 v | e − iϵ$_{v}$t . (2) ρ$_{0}$ = ∑ n | n 〉 θ ( µ − ϵ$_{n}$ ) 〈 n | = ∑ v | v 〉〈 v | = − i G$_{0}$ ( − 0 $^{+}$) = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$G$_{0}$ ( ω ) . H$_{0}$ = T + U$_{ion}$ + U$_{0}$ G$_{0}$ ( ω ) − 1 = ωI − H$_{0}$ = ωI − T − U$_{ion}$ − U$_{0}$ (3) G ( ω ) − 1 = ωI − T − U$_{ion}$ − φ$_{H}$ − Σ$_{xc}$( ω ) φ$_{H}$ ( x ) = ∫ dx ′ V ( x, x $^{′}$) n ( x $^{′}$) V ( x, x $^{′}$) = δ$_{σ,σ}$ ′ | ⃗ r − ⃗ r $^{′}$| . G − $^{1}$( ω ) = G − 1 0 ( ω ) − $^{[}$φ + Σ$_{xc}$( ω ) − U$_{0}$ $^{]}$. (4) n ( x ) = ρ ( x, x ) , ρ ( x, x $^{′}$) = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$G ( x, x $^{′}$, ω ) = − i G ( x, x $^{′}$, t = − 0 $^{+}$) (5) 5