Termwise size-extensivity

The additive separability of each commutator leads to a formulation of coupled-cluster theory where each term (i.e. each expectation-value expression) in the energy or in the amplitude equations is separately size-extensive. The linked equations are therefore said to he termwise size-extensive. No terms that violate the size-extensivity arise and no cancellation of such terms ever occurs. 

Termwise size-extensivity makes the linked formulation of coupled-cluster theory particularly convenient for developing approximate but rigoously size-extensive models from the energy or amplitude equations, we may omit any commutator contribution without destroying size-extensivity. We shall see examples of this technique in our discussion of quadratic Cl theory in Section 13.8.2 and in the development of perturbation theory in Section 14.6. 

It should be realized that, when multiplied by (0 0a ), each of the two terms in (14.1.92) gives rise to an energy contribution in (14.1.90) that is not size-extensive - such a cancellation of terms that are not size-extensive is a general feature of RSPT and occurs exactly to each order in the perturbation. The RSPT energy corrections are thus not termwise size-extensive. 

Returning now to the RSPT energies in the -P 1 form (14.1.41), we note that this form does not contradict the requirement of termwise size-extensivity. Indeed, if the RSPT energies are calculated... 

We shall later see that any Mpller-Plesset energy correction may be written as the Hartree-Fock expectation value of such commutators. As discussed in Section 13.3.2, these commutators are important for establishing termwise size-extensivity. 

It was suggested in Section 14.2.4 that it is advantageous to express the Mpller-Plesset energy corrections in terms of nested commutators. We shall in this subsection see that such commutator expressions are convenient since they make the energy corrections termwise size-extensive, just like the similarity-transformed formulation of coupled-cluster theory makes the coupled-cluster equations termwise size-extensive, as shown in Section 13.3.2. We first consider the perturbed wave function for two noninteracting systems and then go on to consider the separability of the Mpller-Plesset energies. 

In conclusion, it is possible to restructure the M0ller-Plesset energy and wave-function corrections such that their separability and size-extensivity become apparent. However, the separability of the corrections is not obvious in the original formulation of M0ller-Plesset theory but becomes transparent only when commutators are introduced. In Section 14.3, we shall develop coupled-cluster perturbation theory, where the connected (termwise size-extensive) commutator form arises naturally, without the need to restructure the expressions by hand. 

In this section, we present an alternative formulation of MPPT, designed to give corrections that are termwise size-extensive. In Section 13.3, termwise size-extensivity was seen to arise in the similarity-transformed representation of the coupled-cluster equations. Developing perturbation theory in the same representation, we shall And that some of the results arrived at laboriously in MPPT - for example, termwise size-extensivity and the factorization of the quadruples contribution to the second-order wave function - are now obtained effortlessly. 

The arguments for size-extensivity in CCPT follow closely those for linked coupled-cluster theory in Section 13.3. Thus, whereas the order-by-order size-extensivity of the energy follows ftom the exponential ansatz for the wave function, the termwise size-extensivity follows from the use of a similarity-transformed Hamiltonian, assuming that both the zero-order Fock operator and the first-order fluctuation operator separate for noninteracting systems. 

Clearly, only multipliers that refer to a single subsystem (A or B) contribute to the total Lagrangian, ensuring that the energies obtained from the Lagrangian are termwise size-extensive. 

As discussed in Section 4.3. a computational method is said to be size-extensive if a calculation on the compound system AB consisting of two noninteracting systems A and B yields a total energy equal to the sum of the energies obtained in separate calculations on the two subsystems. This property of the coupled-cluster model is demonstrated for the linked formulation in Section 13.3.1, leading to the concept of termwise size-extensiAdty in Section 13.3.2. In Section 13.3.3, we consider size-extensivity in the unlinked formulation of coupled-cluster theory, demonstrating how size-extensivity in this case arises from a cancellation of terms that individually violate size-extensivity. 

In conclusion, in the unlinked formulation of coupled-cluster theory, the amplitude equations yield solutions that are size-extensive but not termwise so. In general, therefore, it is not trivial to make size-extensive approximations to the unlinked coupled-cluster equations and the linked equations are to be preferred for such purposes. 

The advantage of the commutator form of the energy corrections should now be evident. The commutators ensure that the additivity of the energy follows directly from the separability of the Hartree-Fock wave function and from the additivity of the fluctuation potential and the amplitudes. The energy corrections are termwise separable - no terms that violate the size-extensivity arise and no cancellation ever occurs. We sometimes express this by stating that the commutators provide a linked form of the energy corrections. 

---