MMCC approach

We are now equipped with all of the basic concepts of the CC/EOMCC theory which are necessary to explain the noniterative MMCC approaches to ground and excited electronic states. In this section, we focus on the exact MMCC theory. The approximate MMCC schemes for excited electronic states, including the externally corrected MMCC approaches and the CR-EOMCCSD(T) theory, and their most recent analog based on the left eigenstates of the similarity-transformed Hamiltonian, are discussed in Section 3. 

In our view, the MMCC theory represents an interesting development in the area of new CC methods for molecular PESs. The MMCC-bas renormalized CCSD(T), CCSD(TQ), and CCSDT(Q) methods and the noniterative MMCC approaches to excited states provide highly accurate results for ground and excited-state PESs, while preserving the simplicity and the black-box character of the noniterative perturbative CC schemes. In this chapter, we review the MMCC theory and new CC i pnndmations that result firom it and show the examples of the MMCC and renormalized CC calculations for ground and excited state PESs of several benchmark molecules, including HF, F2, N2, and CH" ". The review of the previously published numerical results (7,16-20) is combined with the presentation of new results for the C2, N2, and H2O molecules. 

The method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

Of the methods listed above, only the noniterative CC approaches based on the partitioning of the similarity-transformed Hamiltonian (24-28) and the (C)R-CC approaches of refs 9,13-18,20,21, which employ the MMCC formalism (P. 13, 14, 18, 19, 21, 45, 106, 107), retain the simplicity and the black-box character of the standard CCSD(T) or CCSD(TQf) methods. One of the two goals of the present work is the development of a new class of the MMCC-based black-box methods for multiple bond breaking. 

The main idea of the MMCC formalism and of the related R-CC and CR-CC approaches (9, 13-21, 45, 106, 107) is that ofthe simple, noniterative energy... 

Our recent numerical experiments with the Cl-corrected MMCC methods indicate that in looking for the extensions of the CR-CCSD[T], CRCCSD(T), and CR-CCSIXfQ) methods that would provide an accurate description of triple bond breaking one may have to consider the approximations that use the pentuply and hextuply excited moments of the CCSD equations, M (2), k = 5 and 6, respectively (21). The CR-CCSD[T] and CR-CCSD(T) methods use only the triexcited CCSD moments M (2), whereas the CR-CCSD(TQ) approaches use the tri- and tetraexcited moments, (2) and (2),... 

The purpose of the present paper is to review the most essential elements of the excited-state MMCC theory and various approximate methods that result from it, including the aforementioned CR-EOMCCSD(T) [49,51,52,59] and externally corrected MMCC ]47-50, 52] approaches. In the discussion of approximate methods, we focus on the MMCC corrections to EOMCCSD energies due to triple excitations, since these corrections lead to the most practical computational schemes. Although some of the excited-state MMCC methods have already been described in our earlier reviews [49, 50, 52], it is important that we update our earlier work by the highly promising new developments that have not been mentioned before. For example, since the last review ]52], we have successfully extended the CR-EOMCCSD(T) methods to excited states of radicals and other open-shell systems ]59]. We have also developed a new type of the externally cor-... 

Eqs. (50) and (53) are the full Cl states. Thus, we must approximate wave functions T ) in some way. A few different methods of approximating T ) in Eq. (53), leading to the aforementioned externally corrected MMCC(2,3) approaches and CR-EOMCCSD(T) schemes, and their analogs exploiting the left eigenstates of, and the performance of all of these methods... 

EXTERNALLY CORRECTED MMCC(2,3) SCHEMES AND THE CR-EOMCCSD(T) APPROACH... 

In consequence, the most expensive steps of the ground- and excited-state calculations using methods based on the MMCC(2,3) approximation are essentially identical to the n nf noniterative steps of the ground-state CCSD(T) calculations uo and are the numbers of occupied and unoccupied correlated orbitals, respectively). Similar remarks apply to the memory and disk-space requirements. Clearly, these are great simplifications in the computer effort, compared to the higher-level EOMCC approaches, such as EOMCCSDT [43,44,55,56], particularly if we realize that we only have to use the Ti and T2 clusters, obtained in the CCSD calculations, to construct matrix elements of that enter 9Jt (2), Eqs. (58) and (59). In... 

In the specific case of the Cl-corrected MMCC(2,3) approach, very good results are obtained when the wave function in Eq. (67) is replaced by the wave function obtained in the active-space CISDt calculations, which... 

As shown in Table 2, the inexpensive MMCC(2,3)/CI approach is capable of providing the results of full EOMCCSDT quality. Indeed, the errors in the vertical excitation energies for the 2 S+, 1 A, 2 A, and 2 states of CH+ that have large double excitation components, obtained with the noniterative MMCC(2,3)/CI approximation, are 0.006-0.105 eV. This should be compared to the 0.327-0.924 eV errors in the EOMCCSD results, the 0.219-0.318 eV errors obtained with the CC3 method, and the 0.504-0.882 eV errors obtained with the CISDt approach used to construct wave functions T ) for the MMCC(2,3)/CI calculations [47,48]. For the remaining states shown in Table 2 (the third and fourth states and the lowest-energy state), the errors in the CISDt-corrected MMCC(2,3) results, relative to full Cl, are 0.000-0.015 eV. Again, the only standard EOMCC method that can compete with the MMCC(2,3)/CI approach is the expensive full EOMCCSDT approximation. 

Eq. (95). Again, as in the MMCC(2,3)/CI case, one can easily extend the above MMCC(2,3)/PT approximation to higher-order MRMBPT-corrected MMCC(mA,ms) schemes, such as the MMCC(2,4)/PT approach which describes the combined effect of selected triple and quadruple excitations introduced by the MRMBPT wave functions [78]. 

As shown in Table 2, the inexpensive MMCC(2,3)/PT approach is capable of providing the results which are practically as good as the excellent MMCC(2,3)/CI results. In the case of the 2 S+ and 1 states, which have a strong double excitation character, causing the EOMCCSD approach to fail, the MMCC(2,3)/PT corrections to CCSD/EOMCCSD energies produce the results of the EOMCCSDT quality, reducing the 0.560 and 0.924 eV errors in the EOMCCSD results to 0.102 and 0.090 eV, respectively. For these two states, the errors relative to full Cl obtained with the noniterative MMCC(2,3)/PT approach are 2-3 times smaller than the errors obtained with the much more expensive and iterative CC3 method. For states such as 2 n, which have a partially biexcited character, and for states dominated by single excitations (3 1 11), the MMCC(2,3)/PT results are as... 

The CR-EOMCCSD(T) approach The black-box MMCC method for molecular excited states... 

An interesting alternative to the externally corrected MMCC methods, discussed in Section 3.1.1, is offered by the CR-EOMCCSD(T) approach [49, 51,52,59]. The CR-EOMCCSD(T) method can be viewed as an extension of the ground-state CR-CCSD(T) approach of Refs. [61,62], which overcomes the failures of the standard CCSD(T) approximations when diradicals [76,104,105] and potential energy surfaces involving single bond breaking and single bond insertion [49,50,52,60-62,65,67,69,70,72,73] are examined, to excited states. 

The CR-EOMCCSD(T) approach is a purely single-reference, blackbox method based on the MMCC(2,3) approximation, in which the wave function jH/ ) entering Eq. (67) is designed by using the singly and doubly excited cluster amplitudes tl and defining Ti and T2, respectively, obtained in the CCSD calculations, and the zero-, one- and two-body amplitudes ro p), rl pL) and r pi), defining R, 1, and respectively. 

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