The energy representations in Eq. (4.13) or Eq. (4.17) show that the appropriate energy differences corresponding to excitations of the *N*-electron system are obtained from the excitation propagator, *i.e.*, the one with the field operator basis

where and the average is taken over the appropriate reference state or with the appropriate density operator. If the reference would be the Hartree-Fock single-determinantal ground state of Eq. (4.29), the index *k* refers to unoccupied (or particle) spin orbitals and *l* to occupied (or hole) spin orbitals, making the operators so-called particle-hole operators with the compound index *i* = (*k*,*l*). It should be noted that the adjoint (hole-particle) operators are . The matrix propagator to consider has the block structure

where in the last step, the geometric approximation is invoked. Using the basic anticommutation relations of the electron field operators and taking the average with respect to the density operator ...

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