The CCPT Lagrangian

The CCPT energies are calculated from the energy expression (14.3.13) subject to the constraints (14.3.14). Introducing one undetermined Lagrange multiplier for each constraint (i.e. one multiplier for each amplitude r ), we arrive at the CCPT Lagrangian... 

Let us set up the CCPT Lagrangian for a closed-shell state parametrized by the cluster operators (14.4.4)-( 14.4.6). We shall follow the approach of Section 14.3, indicating only the necessary modifications. First, we derive the equivalent of (14.3.7) for the singlet Fock operator (14.4.2) and the singlet cluster operators (14.4.4)-(14.4.6). We readily obtain the following relationships... 

To arrive at the CCPT energies, we now expand the Lagrangian in orders of the perturbation ... 

In our discussion of size-extensivity in Section 14.3.5, we found that the CCPT energies that comply with the n -f- 1 rule are size-extensive term by term as well as order by order. Since the Lagrange multipliers may not separate for noninteracting systems, we must likewise make sure that the energies derived from the variational Lagrangian (14.3.44) are size-extensive in the same manner. 

Together with the first-order expressions for the amplitudes (14.4.41) and (14.4.42), these equations determine the closed-shell spin-restricted CCPT Lagrangian. Whereas the evaluation of the first-order amplitudes scales as O V, the evaluation of the second-order amplitudes scales as Cp V for the singles, for the doubles and for the triples. 

To calculate the third-order CCPT energy, we collect all terms to third order in the Lagrangian (14.3.44) ... 

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