CCSDT model

Not only are sueh integrals difficult to calculate, but when the MOs are expanded in a basis set consisting of M AOs, there will be on the order of thrce-electron integrals and on the order of M four-eleetron integrals. Such methods are therefore inherently more expensive than for example the full CCSDT model. 

Although the calculations reported here have been carried out in a small basis, there is no reason to believe that our conclusions regarding the convergence of the coupled-cluster hierarchy would be different had the calculations been carried out in larger basis. In particular, we conclude that the CCSDT model is incapable of predicting AEs to within 1 kJ/mol. 

The CCS, CC2, CCSD, CC3 hierarchy has been designed specially for the calculation of frequency-dependent properties. In this hierarchy, a systematic improvement in the description of the dynamic electron correlation is obtained at each level. For example, comparing CCS, CC2, CCSD, CC3 with FCI singlet and triplet excitation energies showed that the errors decreased by about a factor 3 at each level in the coupled cluster hierarchy [18]. The CC3 error was as small as 0.016 eV and the accuracy of the CC3 excitation energies was comparable to the one of the CCSDT model [18]. 

Currently the full CCSDT model is far too expensive for routine calculations. To save time, we first carry out a CCSD calculation, which is then followed by a computation of a perturbative estimate of the triple excitations. Such an approximate method is called CCSD(T). 

Urban M, Noga J, Cole SJ, Bartlett RJ (1985) Towards a full CCSDT model for electron correlation. J Chem Phys 83 4041 1046... 

Noga J, Bartlett RJ (1987) The full CCSDT model for molecular electronic structure. J Chem Phys 86 7041-7050. 

M. Urban,. Noga, S. J. Cole, and R. J. Bartlett,/. Chem. Phys., 83,4041 (1985). Towards a Full CCSDT Model for Electron Correlation. 

CCSDT Model for Electron Correlation. CCSDT-n Models. 

G. E. Scuseria and H. F. Schaefer, Chem. Phys. Lett., 152, 382 (1988). A New Implementation of the Full CCSDT Model for Molecular Electronic Structure. 

The specific equations used to determine the cluster coefficients in the CCSDT model will now be given. Projection of the Schrodinger equation [cf. Eq. (77)] onto the singly excited space gives... 

The component diagrams necessary for a CCSDT model have now... 

There are four types of nontrivial single-index set permutation operators required in a CCSDT model i.e.,... 

In summary, the rules for the construction and subsequent evaluation of diagrams corresponding to matrix elements in the CCSDT model have been given. The adaptation of certain features from time-dependent diagrams, not usually found in time-independent approaches, have been seen to clarify and/or expedite time-independent diagrams. 

The equations for the cluster coefficients and the correlated energy in a CCSDT model were given in operator form in Section III [cf. Eqs. (80)— (83)] this form is, of course, not amenable to calculations. In Section V the time-independent techniques discussed in Section IV are applied to evaluate the requisite matrix elements in terms of cluster coefficients and one- and two-electron integrals over the spin-orbital basis. 

The algebraic expressions obtained from a diagrammatic evaluation of the coupled-cluster equations for a CCSDT model are resolvable into products of unmodified cluster coefficients (or trivially modified in the case of t2) and modified one- and two-electron integrals. At no stage of the calculation are tensors of rank greater than 2 required, except for the initial contraction and final expansion of the rank 3 triples cluster coefficients. 

Following the pair correlation approach of Sinanoglu,4 Cizek2 introduced the coupled-pair many-electron theory (CPMET) method, which takes into account only double excitation clusters T2. In the present paper we prefer our more systematic nomenclature for different versions of the CC method, consistently used in our previous papers. Thus the original Cizek approach will be denoted CCD (CC doubles), while its linearized version will be denoted as LCCD. By including in the CC equations single excitation clusters T, we arrive at the CCSD method, exploited in a series of papers by Bartlett and co-workers.9-11 We will also consider the linearized version of the CCSD method, LCCSD,15 the full CCSDT model that includes effects of triple excitation clusters T3, and the CCSDTQ model that also considers T. ... 

In the single reference CC world, the full CCSDT model (23-26) has seen modest use because it requires a substantial effort over CCSD to code, and because the resource requirements—CPU time, memory, disk—are significantly greater. As a result, a number of approximations to the full CCSDT method have been proposed and have proven quite effective at incorporating important effects of triple excitations while at the same time minimizing the above concerns (27-31). In the same spirit, we examine the FSCCSDT equations with an eye to approximating or ignoring the most expensive terms to evaluate. In this sense, the simplest possible approximation would be to evaluate the triple... 

For all considered schemes (IP, EA, EE) we have one more variant, i.e., EOM-CCSDT which scales as EOM-CCSDT model. 

For a more accurate treatment of electron correlation, coupled-cluster (CC) approaches [23] enter into play. While the full CC singles, doubles, triples (CCSDT) model [24] and augmented CCSDT approaches that feature corrections for quadruple and even higher excitations [25, 26] are currently too expensive, CC singles and doubles (CCSD) approximation [27] and CCSD augmented by a perturbative treatment of triple excitations, CCSD(T) [28], are more feasible and rather widely adopted. 

Response functions have also been derived for the CC3 model (Christiansen et al., 1995a), which is an approximation to the CCSDT model (Hald et al, 2001) in the same way as CC2 is to CCSD. The immense number of triple excitations included in the CCS Jacobian makes it necessary to formulate it in a partitioned form using Eq. (10.14). However, the partitioned CC3 Jacobian then depends also on its own eigenvalues similar to the partitioned SOPPA Hessian in Eq. (10.49),... 

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