EOMCC

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 ground-state CC theory has a natural extension to excited electronic states I l ) via the EOMCC or linear response CC method, in which we write... 

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. 

As described in Section 2.1, the standard CC and EOMCC equations are obtained by projecting and on the excited determinants with n = that correspond to the particle-... 

It can be demonstrated that once the cluster and excitation operators, and R l, respectively, and the ground- and excited-state energies are determined by solving the relevant CC/EOMCC equations, Eqs. (9), (23), and (26), we can obtain the exact, full Cl, energies by adding the... 

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... 

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. 

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