Ground-state MMCC

In the ground-state MMCC theory, we focus on the noniterative energy correction... 

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 main purpose of all ground-state MMCC calculations, including the CR-CC methods considered in this work, is to estimate correction SqA Eq. (1), such that the resulting energy, defined as... 

Equation (7) is the basic equation of the ground-state MMCC formalism. The main element of eq. (7) are the SRCC equations, in which T = projected on the y-tuply excited configurations with j > niA, i.e.,... 

The Ground-State MMCC(m, ma) Methods and the Renormalized CCSD(T), CCSD(TQ), and CCSDT(Q) Schemes... 

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... 

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. 

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. 

Generalized moments of CC equations, MMCC functional, and MMCC expansions for the exact ground-state energy... 

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],... 

It has recently been demonstrated that the applicability of the ground-state SRCC approaches, including the popular noniterative approximations, such as CCSD(T), can be extended to bond breaking and quasidegenerate states, if we switch to a new type of the SRCC theory, termed the method of moments of CC equations (MMCC) (7,16-18). It has further been demon-... 

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