Fourth-order energy from coupled-cluster doubles

Whereas the singles occur to fourth order, the doubles appear only quadratically in these exjxessions. Moreover, from (13.2.39), it is easily verified that, except for the singles and the doubles, the amplitudes of the highest excitation level always occur oidy linearly in the coupled-cluster equations. These addifional simplifications in the coupled-cluster equations - beyond what is dictated by the termination of the BCH expansion (13.2.37) - occur because of the restrictions on the excitation levels in the projection space (/r,j. If instead we had calculated the energy fhrm the variation principle, the expansion of the exponentials would not terminate (except, of coinse, for the fact that we have only a finite number of electrons to excite from the occupied spin orbitals). The evaluation... 

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