MMCC approximations

EOMCCSD energies, is very encouraging. However, we have to perform a larger number of calculations to see if CR-EOMCCSD(T) offers the same level of consistency in applications involving singly and doubly excited states as other MMCC approximations. The results of our findings will be reported elsewhere [79]. 

CC calculations, referred to as method A, recovers the full Cl ground-state energy q. The purpose of the approximate MMCC calculations is to estimate... 

The remaining issue is what do we do with the wave function F(,) in eqs (16) or (22), which in the exact MMCC theory represents the full Cl ground state. In the approximate MMCC methods considered in our earlier work, the wave functions Fo) were evaluated either by using the low-order MBPT... 

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-... 

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. 

Equation (50) (or its CCSD/EOMCCSD-based analog, Eq. (53)) defines the exact MMCC formalism for ground and excited states. This equation allows us to improve the CC/EOMCC (e.g. CCSD/EOMCCSD) results, in a state-selective manner, by adding the noniterative corrections (in practice, one of the approximate forms of or obtained using the... 

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... 

Approximate MMCC methods for excited states and their performance... 

The MMCC(2,3), CR-EOMCCSD(T), and other MMCC(mA,mij) methods are obtained by assuming that the Cl expansions of the ground- and excited-state wave functions T ) entering Eq. (50) do not contain higher-than-m -tuply excited components relative to the reference T), where niA < rriB < N. In all MMCC mA,mB) approximations, we calculate the ground- and excited-state energies as follows [47-52,61-63,72] ... 

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... 

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. 

In the specific case of the MMCC(2,3)/PT approximation, we go one step further and, after rewriting each I l p), Eq. (89), in the form of the Cl expansion relative to the reference determinant ) used in the CCSD and EOMCCSD calculations whose results we want to improve. 

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]. 

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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