MMCC schemes

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. 

One of the main advantages of the MRMBPT-corrected MMCC schemes, such as MMCC(2,3)/PT, is their low cost, compared to the already rather... 

THE MMCC SCHEMES EXPLOITING THE LEFT EIGENSTATES OF THE SIMILARITY-TRANSFORMED HAMILTONIAN THE CR-EOMCCSD(T) APPROACH... 

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... 

We limit our discussion to the low-order MMCC(myi, ms) schemes with ruA = 2 and niB =, which can be used to correct the results of the CCSD/EOMCCSD calculations for the effects of triple excitations (for the description of the MMCC(2,4) and other higher-order MMCC mA,mB) methods, see Refs. [48-50,52,61-63,72]). The MMCC(2,3) energy expression is as follows [47-52,61-63, 72] ... 

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 in the case of the MMCC(2,3) schemes, discussed in Section 2, we can use Eq. (128) to design the analog of the CR-EOMCCSD(T) method employing the left eigenstates of. In order to accomplish this and for-... 

There are two issues that have to be addressed before one can use Eqs. (25) or (28) in practical calculations. First of all, the exact MMCC corrections SgA) and < qCCSD, Eqs. (25) and (28), respectively, have the form of long many-body expansions involving all n-tuply excited configurations with n == i/ia + I, ., /V, where N is the number of correlated electrons in a system. Thus, in order to propose the computationally inexpensive MMCC methods, we have to truncate the many-body expansions for SgA> or excitation level This leads to the so-called MMCC( i, mB) schemes [11-15,24,33,34,39,48,120,121], The CR-CCSD(T) and CR-CCSD(TQ) methods [11-14,24,33,34], reviewed and tested in this work, are the MMCC( u, mB) schemes with mA = 2 and mB = 3 (the CR-CCSD(T) case) or 4 (the CR-CCSD(TQ) case). Second of all, the wave function % that enters the exact Eqs. (25) or (28) is a full Cl ground state, which we usually do not know (if we knew the exact ko> state, we would not have to perform any calculations ). Thus, in order to propose the computationally tractable approaches based on the MMCC theory defined by Eqs. (25) and (28), we must approximate fi o) in some way as well. The CR-CCSD(T) and CR-CCSD(TQ) methods employ the low-order MBPT-like expressions to define fi o) [11-14,24,33,34],... 

Similar remarks apply to practical computational schemes, such as CR-CC(2,3), based on the biorthogonal MMCC theory, as defined by Eqs. (30) and (39). In this case, to avoid the calculation of the entire set of moments of CC equations for a given... 

In this article, we focus on the MMCC(/ a- tng) schemes with mA = 2, which can be used to correct the CCSD energy. Two truncation schemes of this type are particularly useful, namely, MMCC(2,3), which results in the CR-CCSD(T) method, and MMCC(2,4), which leads to the CR-CCSD(TQ),x (x = a, b) approximations. In the MMCC(2,3) and MMCC(2,4) approaches, we add the relevant energy corrections, So(2, 3) and So(2, 4), respectively, to the CCSD energy E0ccsl)l to obtain the following total energies [11-15,24,33,34,39,48,120,121] ... 

The idea of renormalizing the CCSD(T) method via the MMCC formalism, as described above, can be easily extended to the CCSD(TQ) case. The resulting CR-CCSD(TQ) approaches are examples of the MMCC(2,4) scheme, defined by Eq. (44), in which we improve the results of the CCSD calculations by adding the non-iterative corrections Sq(2, 4), defined in terms of moments 9Jt fc(2) and 0Jll lcd(2), to the CCSD energies. Two variants of the CR-CCSD(TQ) method, labeled by the extra letters a and b , are particularly useful. The CR-CCSD(TQ),a and CR-CCSD(TQ),b energies are calculated in the following manner [11-14,24,33,34] ... 

An example of the MMCC(mA,tnB) / method is the MMCC(2,3) > approximation, in which, in analogy to the MMCC(2,3) truncation scheme defined by Eq. (43), we correct the results of the CCSD calculations by adding the triples correction... 

In analogy to the CR-CCSD(T) and CR-CCSD(TQ) methods, in order to propose practical computational methods, based on the MMCC(ota, m b ) truncation schemes such as MMCC(2,3), we have to come up with reasonably accurate... 

This leads to the so-called MMCC(my, ttib) schemes. The renormalized and completely renormalized CCSD(T) and CCSD(TQ) methods discussed in... 

The completely renormalized CCSD(T) method (the CR-CCSD(T) approach) is an MMCC(2,3) scheme, in which the wave function o) is replaced by the very simple, MBPT(2)[SDT]-like, expression. 

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. 

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