Diagrams unlinked

We study three different approximations for removing unlinked diagrams in EOM-CC and show that these models provide second-order properties and transition probabilities that are close to those provided by CCLR in isolated molecular systems, but in a more convenient computational structure. 

Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

The inspection of the K dependence shows that the unlinked diagrams such as 96 scale as K1 while the other (linked) diagrams as K°. Equation (2-95) is shown to behave in the same fashion, i.e.,... 

A size extensive method is frequently defined esoterically as one whose energy and amplitude/coefficient equations contain no unlinked diagrams, such as... 

Fig. 20. (a) Unlinked diagrams in fifth-order MBPT (b) illustration of factorization of unlinked diagrams. 

III MBPT eliminates the large (unlinked diagram) errors in molecular calculations that beset Cl methods, while greatly reducing the level of computational complexity for a given accuracy. 

CC theory is inherently better than an equivalent level of Cl because it eliminates unlinked diagrams and as a consequence, is size-extensive [13]. It is also inherently better than an equivalent level of MBPT because it is not hmited to finite-orders, or potential difficulties encountered in the convergence of perturbation theory. It is well known, e.g. that ordinary MP perturbation theory does not converge for the electron gas, and this has also been emphasized recently for molecules [47], though resummations (including CC theory) work fine [48]. But, the operable word is equivalent level . For Cl, that meant at least single and double excitations, and frequently some more, perhaps even from a multi-reference space. MBPT had been done with single excitations in fourth-order SDQ-MBPT(4) in the above two papers [13,46]. 

The next critical element in the development of CC theory was to incorporate the connected triple excitations, Tj,. Since even CCD puts in the dominant quadruple excitation effects, and CCSD some of the disconnected triple excitations effects, the only term left in fourth-order MBPT comes from T, and the triples will be much more important to CC theory than to Cl, since CIs unlinked diagrams have a very large role that can only be alleviated by putting in quadruple excitations (see Fig. 42.1). Triples had been explored in the ECPMET discussed above. Kvasnicka et al., Pople et al., Guest and Wilson, Urban et al., and ourselves had included triples in fourth-order MBPT = MP4 [59-64], but no attempt had been made to introduce them into general purpose CC methods. In 1984 we wrote a paper detailing the triple excitation equations in CC theory and reported results for CCSDT-1 [65], which meant the lead contribution of triples was included on top of CCSD. This also made it possible to treat triple excitations on-the-fly in the sense that we never required storage of the n N amplitudes. 

The most general diagram is unlinked and consists of a certain number of linked parts. A simple combinatorial calculation shows that the complete series of linked and unlinked diagrams is equal to the exponential of the series of linked diagrams only. This is essential as we are in fact interested in the logarithm of S(a,j8). One is then lead to the expression ... 

The reason for including only the linked diagrams in the expression for is further clarified by examining the value of an unlinked diagram, e.g., the one given in Fig. 3.3A ... 

Because, for two noninteracting subsystems, both of the disjoint sums occurring in Eq. (3.60) are size consistent (i.e., proportional to the size of the system), the product would not be size consistent. Hence, unlinked diagrams correspond directly to non-size-consistent factors, which should not be included. 

Properties computed using Eq. (11) are not size-extensive, however, because of the appearance of unlinked diagrams arising from disconnected terms implicit in Eq. (3). Such terms naturally cancel if the T and operators are not truncated, implying that the EOM-CC method is formally exact in the full-CC limit. 

One important property of Eq. (23), which manifests itself very easily in the diagrammatic evaluation of the perturbation expansions, is the linked-diagram theorem (LOT). According to this theorem, only a selected class of terms survives in the perturbation expansion of the wave operator, while all unlinked diagrams cancel in each order of perturbation theory in addition, this theorem ensures the size consistency of MBPT and coupled-cluster theory. Based on LDT, Eq. (23) simplify to... 

It is well-known, for example, that in a perturbation theory analysis of the method of configuration interaction when restricted to single- and double-excitations with respect to a single determinant reference function includes many terms, which correspond to unlinked diagrams, which are exactly canceled by terms involving higher order excitations. 

The second rule forces us to draw only diagrams that are completely connected or linked i.e., they are all in one piece. Unlinked diagrams (n = 4) like... 

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