Fourth-order contributions to the correlation energy

When the canonical Hartree-Fock orbitals are used to construct the reference function with respect to which the perturbation expansion is made, only one non-vanishing renormalization term occurs in the fourth-order energy component. The fourth-order energy can be written as 

The importance of the linked cluster theorem in fourth-order of the energy expansion, lies in the fact that the two disconnected contributions to the direct term in eq. (3.197) are exactly cancelled by renormalization terms. This result allows the total fourth-order energy to be written as 

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