content_image/366084.jpg content_image/366085.jpg content_image/366086.jpg content_image/366087.jpg content_image/366088.jpg content_image/366089.jpg content_image/366090.jpg content_image/366091.jpg content_image/366092.jpg
we have the time-ordered formula 

<img src='content_image/366084.jpg'>

The non-interacting density matrix is the standard sum over the occupied states 

<img src='content_image/366085.jpg'>

We separate a potential U$_{0}$ from H$_{0}$ via 

<img src='content_image/366086.jpg'>

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: 

<img src='content_image/366087.jpg'>

The exact, interacting, one-particle Green’s function obeys the Dyson equation 

<img src='content_image/366088.jpg'>

where φ$_{H}$ is the Hartree potential determined by the electron density n ( x ) 

and the bare Coulomb operator is 

<img src='content_image/366089.jpg'>

<img src='content_image/366090.jpg'>

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 

<img src='content_image/366091.jpg'>

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 

<img src='content_image/366092.jpg'>

