SR CCSD

We have carried out several applications showing the promise of this procedure [63,64], as well as addressed the question of the size-consistency and size-extensivity [65-67], to which we wish to turn our attention again in this paper. Finally, we have also extended the idea of externally corrected (ec) SR CCSD methods [68-70] (see also Refs. [21,24]) to the MR case, introducing the (N, M)-CCSB method [71], which exploits an Preference (A R) CISD wave functions as a source of higher-than-pair clusters in an M-reference SU CCSD method. Both the CMS and (N, M)-CCSD allow us to avoid the undesirable intruder states, while providing very encouraging results. 

The essential role of the size-consistency in molecular applications is strikingly conspicuous already in the SR case. Indeed, the SR CCSD method is manifestly size-extensive, yet it fails when breaking genuine chemical bonds, as the well-known examples illustrate [82,83]. This breakdown is of course most prominent when multiple bonds are involved, as the example of the CCSD PEC for N2, shown in Fig. 1, clearly illustrates [83]. Note that even when we employ the UHF reference, we will not generate a smooth PEC in view of the presence of the triplet instability (see, e.g., [84, 85] and references therein), whose onset occurs at an intermediately stretched geometry [86]. 

It turns out that MR CISD represents again the most suitable source of the required higher-order clusters. Carefully chosen small reference space MR CISD involves a very small, yet representative, subset of such cluster amplitudes. Moreover, in this way we can also overcome the eventual intruder state problems by including such states in MR CISD, while excluding them from CMS SU CCSD. In other words, while we may have to exclude some references from Ado in order to avoid intruders, we can safely include them in the MR CISD model space Adi. In fact, we can even choose the CMS for Adi. Thus, designating the dimensions of Ado and Adi spaces by M and N, respectively, we refer to the ec SU CCSD method employing an NR-CISD as the external source by the acronym N, M)-CCSD. Thus, with this notation, we have that (N, 1)-CCSD = NR-RMR CCSD and (0, M)-CCSD = MR SU CCSD. Also, (0,1)-CCSD = SR CCSD. For details of this procedure and its applications we refer the reader to Refs. [63,64,71]. 

Turning, next, our attention to the CC methods, we first consider the SR approaches. These results are summarized in Table III. We first note a much better performance of SR CCSD over the SR CISD (2nd column in Tables III and II, respectively) The CCSD deviations from FCI are much smaller than the CISD ones, as are the corresponding NPEs. This seems to be a general feature of the SR CCSD method, which often performs amazingly well even in severely quasidegenerate situations. 

Several possible choices of an external source have been tested so far. The basic requirement is that such a source must provide a reasonable approximation of the most important three- and four-body clusters that are missing in the SR CCSD approach. At the very least, we require it to describe the essential nondynamic correlation effects. The practical aspects require that it be easily accessible. The first attempts in this direction exploited the unrestricted Har-tree Fock (UHF) wave function [of different orbitals for different spins (DODS) type]. Its implicit exploitation lead to the so-called ACPQ (approximate coupled pairs with quadruples) method [26, 27]. Recently, its explicit version was also developed and implemented [31]. Although in many cases this source enables one to reach the correct dissociation channel, its main shortcoming is the fact that for the CS systems it can only provide T4 clusters, since the 7) contribution vanishes due to the spin symmetry of the DODS wave function. Nonetheless, the ACPQ method enabled an effective handling of extended linear systems (at the semi-empirical level), which are very demanding, since the standard CCSD method completely breaks down in this case [27]. 

Relying on the above discussed complementarity of the SR CCSD and MR CISD ansatze, it seems particularly attractive to employ the latter as an external source of 7) and T4 corrections. In order to explicitly illustrate this complementarity and the scope of the formalism involved, let us consider a minimal 2-reference case, i.e. let us assume that a given SCF reference becomes quasidegenerate with another configuration. For a CS system this case arises when the one-electron active-space involves only two MOs, each belonging to a different... 

Table 1. Comparison of the SR CCSD, RMR CCSD, and ASTQ CCSD energies with the exact FCI result for the X2If state of OH at three internudear separations R, R = Re = 1.832 bohr, R = l.5Re> and R = 2Re. Except for the SCF and FCI total energies, which are reported as — (E + 75) (in hartree), the energy differences (in millihartree) relative to the FCI result are given in all cases. The nonparallelism error (NPE) for the interval R e [Re, 2Re] (in millihartree) is also given for easier comparison (see the text for details) ...

The CAS FCI or SOCI corrected CCSD results represent a definite improvement over the standard SR CCSD results. Nonetheless, for a given choice of the active-space, the performance of the ASTQ CCSD approaches deteriorates with the increasing size of the basis set. This is easily understood, since the external source does not include excitations out of the active-space, which are more important for larger basis sets. We should also mention that computationally much more demanding CAS SCF wave functions are about as effective as CAS FCI ones (cf., e.g., Tables II and III of [33]). 

On the whole, the strategy of ecCCSD is already providing useful results and its further pursuit, in whichever form it may take, should enable us to substantially extend the range of applicability of the standard SR CCSD or CCSD(T) methods that serve us so well in nondegenerate situations. 

For these and other reasons, much attention was given to the so-called state-selective or state-specific (SS) MR CC approaches. These are basically of two types (i) essentially SR CCSD methods that employ MR CC Ansatz to select a subset of important higher-than-pair clusters that are then incorporated either in a standard way [163,164], or implicitly [109-117], or via the so-called externally corrected (ec) approaches of either the amplimde [214-219] or energy [220,221] type, and (ii) those actually exploiting Bloch equations, but focusing on one state at a time [222]. The energy-correcting ec CC approaches [220,221] are in fact very closely related to the renormalized CCSD(T) method of Kowalski and Piecuch mentioned earlier [146,147]. 

Abstract The singlet-triplet splittings of the di-radicals methylene, trimethylene-methane, ortha-, meta- and para-benzynes, and cyclobutane-l,2,3,4-tetrone have become test systems for the applications of various multi-reference (MR) coupled-cluster methods. We report results close to the basis set limit computed with double ionization potential (DIP) and double electron attachment (DBA) equation-of-motion coupled-cluster methods. These diradicals share the characteristics of a 2-hole 2-particle MR problem and are commonly used to assess the performance of MR methods, and yet require more careful study unto themselves as benchmarks. Here, using our CCSD(T)/6-311G(2d,2p) optimized geometries, we report DIP/DEA-CC results and single-reference (SR) CCSD, CCSD(T), ACCSD(T) and CCSDT results for comparison. 

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