Excited-state MMCC

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-... 

The main idea of the excited-state MMCC theory (98) is that of the noniterative energy corrections... 

Equation (22), with Cn-j mA) defined by eq (8) and defined by eqs (24) and (26), defines the excited-state MMCC theory. In analogy to the groimd-state case, the main elements of eq (22) are the generalized moments of the EOMCC equations defining approximate method A, i.e., the EOMCC equations, in which T is approximated by and Rk is approximated by projected onto the excited configurations not included in method A. For example, if we want to correct the exdted-state energies obtained in EOMCCSD calculations (the mA — 2 case) and recover the full Cl energies Ek, we must calculate the EOMCCSD equations projected on triples, quadruples, etc., or... 

As in the case of the ground-state MMCC and CR-CC methods [49,50,52,61-63,65-77], the key to a successful description of excited states by the CR-EOMCCSD(T) and other MMCC methods is the very good control of accuracy that all of these methods offer by directly addressing the quantity of interest, which is the difference between the exact, full Cl, and EOMCC (e.g., EOMCCSD) energies. The MMCC formalism provides us with precise information about the many-body structure of these differences, suggesting several useful types of noniterative corrections to EOMCCSD or other EOMCC energies. 

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... 

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... 

Figure 2. Potential energy curves for the CH+ ion (energies in hartree and the C-H distance in bohr see Refs. [44,47] for the EOMCCSD and MMCC(2,3)/CI data see Refs. [45,103] for the full Cl data). The results include the ground state and the two lowest excited states of the symmetry (the full Cl curves are indicated by the dotted lines and other results are represented by ), the lowest two fl states (the full Cl curves are indicated by the dashed-dotted lines and other results are represented by 0)> the lowest state (the fuU Cl curve is indicated by the dashed hne and other results are indicated by A), (a) A comparison of the EOMCCSD and full Cl results, (b) A comparison of the CISDt and full Cl results, (c) A comparison of the MMCC(2,3)/CI and fuU Cl results. 

The Cl-corrected MMCC methods provide us with an excellent description of excited states dominated by double excitations and excited-state potentials along bond breaking coordinates, but one may think of reducing the costs of the CISDt-corrected MMCC(2,3) and similar cal-... 

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) sf method is currently under development, so that we cannot show too many examples of the actual applications yet. However, we have already tested the CR-EOMCCSD(T) approach using the electronic excitations in the CH+ ion as an example. The CR-EOMCCSD(T)j > results for the three lowest-energy excited states of the symmetry and two lowest-energy states of the H and symmetries, obtained at the equilibrium geometry Rc-h = Re = 2.13713 bohr and the same [5s3pld/3slp] basis set of Ref. [103] as used in the MMCC(2,3)/CI, MMCC(2,3)/PT, and CR-EOMCCSD(T) calculations discussed in Section 3.1, are shown in Table 2. As one can see, the CR-EOMCCSD(T) 5f approach is as effective in improving the EOMCCSD results as the CR-EOMCCSD(T) method analyzed in Section 3.1.2. This is particularly true for the 2 1 A, and 2 A states that are dominated by double exci-... 

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]. 

In a few numerical examples, we have demonstrated that all of the above MMCC and CR-EOMCC methods provide considerable improvements in the EOMCCSD results, particularly when the excited states of interest gain a significant double excitation or multi-reference character. The MMCC(2,3)/CI, MMCC(2,3)/PT, CR-EOMCCSD(T), and... 

CR-EOMCCSD(T) methods can often compete with the much more expensive EOMCCSDT approach. In fact, there are cases, such as the lowest-energy state of the C2 molecule, where the MMCC(2,3)/CI and CR-EOMCCSD(T) methods balance the ground and excited state correlation effects better than full EOMCCSDT. Even if this particular case is a result of the fortuitous cancellation of errors, it is very encouraging to see that the low-cost and easy-to-use MMCC(2,3)/CI, MMCC(2,3)/PT, CR-EOMCCSD(T), and CR-EOMCCSD(T) sf methods can be as accurate as the high-level and very expensive EOMCC methods, such as EOMCCSDT. 

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