Resolvents and Greens functions

The resolvent in eq. (1.208) is called the one-electron Green s function and the notation for it reads G (z). The integration contour may be set in such a way that it encloses all the poles of the resolvent corresponding to the occupied MOs giving by this the required total projection operator. In the spin-orbital occupation number and the second quantization representations related to each other, one can write the operator projecting to the occupied (spin)-MO as an operator of the number of particles in it. Indeed, the expression 

Let us assume from now on that by definition the one-electron Green s function in the energy domain is  

By going to the time domain and back (see for details [30]) one obtains the relation for the exact ground state energy of the system which reads  

---