EOMCCSD

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. 

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. 

CC/EOMCC equations corresponding to the approximate method A, as defined by Eq. (48). In particular, if we want to recover the full Cl energies E from the CCSD/EOMCCSD energies (the niA = 2 case), we... 

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... 

TABLE 1. Explicit algebraic expressions for the matrix elements elements of (designated by h) and other intermediates (designated by I or i9) used to construct the triply excited moments of the CCSD/EOMCCSD equations, Eq. (60). 

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... 

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] ... 

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] ... 

CCSD/EOMCCSD equations defined by Eqs. (58) and (59). These moments are easy to calculate. As implied by Eqs. (58) and (59), their determination requires the explicit consideration of the triples-reference, triples-singles, and triples-doubles blocks of the matrix representing the CCSD/EOMCCSD similarity-transformed Hamiltonian Eq. (13). 

Cl expansion of the CCSD/EOMCCSD wave function which can be easily determined using the CCSD/EOMCCSD cluster and excitation amplitudes fo(At), C(r)) md r (At)- As in the case of... 

TABLE 2. A comparison of the MMCC(2,3)/CI, MMCC(2,3)/PT, CR-EOMCCSD(T), and CR-EOMCCSD(T) jj, vertical excitation energies of the CH+ ion, as described by the [5s3pld/3slp] basis set of Olsen et al. [103], at the equilibrium geometry, with the exact, full Cl, data and other CC results. ... 

State Full Cf EOMCCSDP CC. f EOMCCSDT MMCC(2,3)/CP f MMCC(2,3)/PTf S CR-EOMCCSD(Tf CR-EOMCCSD(T)sf ... 

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. 

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