CCSDT approach

On the other hand, it is possible to calculate these terms without employing the T4 coefficient, but in order to do that we have to write some additional code beyond the CCSDT approach. Diagrams EfT and E can be calculated, because of the above-mentioned factorization theorem, with the help of only T2 and T3 coefficients. 

As a last approach we have investigated the performance of the hybrid method - referred to as CCSDT - which uses for the ground state the standard CCSDT-3 method and - at the EOM level - it includes the 7 3 operator in a rigorous manner. This approach is justified for the IP and EA calculations, since in the full EOM-CCSDT approach the computational bottleneck occurs at the ground state n ) step. So approximating the ground state with the CCSDT-3 method we make the scaling for the two steps comparable. 

For a more accurate treatment of electron correlation, coupled-cluster (CC) approaches [23] enter into play. While the full CC singles, doubles, triples (CCSDT) model [24] and augmented CCSDT approaches that feature corrections for quadruple and even higher excitations [25, 26] are currently too expensive, CC singles and doubles (CCSD) approximation [27] and CCSD augmented by a perturbative treatment of triple excitations, CCSD(T) [28], are more feasible and rather widely adopted. 

Among the various approximate methods for including the connected triple excitations, the CCSD(T) method is the most popular [19]. In this approach, the CCSD calculation is followed by the calculation of a perturbational estimate of the triple excitations. In addition to reducing the overall scaling with respect to the number of atoms K from K8 in CCSDT [see Eq. (2.5)] to K7 in CCSD(T), the CCSD(T) method avoids completely the storage of the triples amplitudes. 

It has been well known for some time (e.g. [36]) that the next component in importance is that of connected triple excitations. By far the most cost-effective way of estimating them has been the quasiper-turbative approach known as CCSD(T) introduced by Raghavachari et al. [37], in which the fourth-order and fifth-order perturbation theory expressions for the most important terms are used with the converged CCSD amplitudes for the first-order wavefunction. This account for substantial fractions of the higher-order contributions a very recent detailed analysis by Cremer and He [38] suggests that 87, 80, and 72 %, respectively, of the sixth-, seventh-, and eighth-order terms appearing in the much more expensive CCSDT-la method are included implicitly in CCSD(T). 

The most challenging and therefore the most telling example for excitation theories is C2, whose ground state has a severe multi-determinant wave function. It is known that, to obtain quantitative results (errors < 0.1 eV), one must resort to EOM-CCSDTQ [134], Figure 2-11 compares EOM-CCSD, CCSDT, and various perturbation corrections to EOM-CCSD with FCI for three excited states of C2 [126], EOM-CCSD, which is usually highly accurate, is inadequate for the two states A and B with errors approaching 2 eV. All variants of the perturbation corrections are... 

CCSD(T) method. The question then naturally arises as to how these methods can be extended to excited states. For the iterative methods, the extension is straightforward by analyzing the correspondence between terms in the CC equations and in H, one can define an H matrix for these methods, even though it is not exactly of the form of a similarity-transformed Hamiltonian. If one follows the linear-response approach, one arrives at the same matrix in the linear response theory, one starts from the CC equations, rather than the CC wave function, and no CC wave function is assumed. This matrix also arises in the equations for derivatives of CC amplitudes. In linear response theory, this matrix is sometimes called the Jacobian [19]. The upshot is that excited states for methods such as CCSDT-1, CCSDT-2, CCSDT-3, and CC3 can be obtained by solving eigenvalue equations in a manner similar to those for methods such as CCSD and CCSDT. 

Piecuch P, Kucharski SA, Bartlett RJ (1999) Coupled-cluster methods with internal and semiinternal triply and quadruply excited clusters CCSDt and CCSDtq approaches. J Chem Phys 110 6103-6122. 

Kowalski, K. Piecuch, P The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches, ... 

As pointed out in the preceding section, the CCSD energy contains contributions identical to those of the MBPT(2) and MBPT(3) energy, but lacks triple-excitation contributions necessary for MBPT(4). Thus, a natural approach to the triples problem is to correct the CCSD energy for the missing MBPT(4) terms,using the CCSDT similarity-transformed Hamiltonian,... 

In summary, the rules for the construction and subsequent evaluation of diagrams corresponding to matrix elements in the CCSDT model have been given. The adaptation of certain features from time-dependent diagrams, not usually found in time-independent approaches, have been seen to clarify and/or expedite time-independent diagrams. 

Following the pair correlation approach of Sinanoglu,4 Cizek2 introduced the coupled-pair many-electron theory (CPMET) method, which takes into account only double excitation clusters T2. In the present paper we prefer our more systematic nomenclature for different versions of the CC method, consistently used in our previous papers. Thus the original Cizek approach will be denoted CCD (CC doubles), while its linearized version will be denoted as LCCD. By including in the CC equations single excitation clusters T, we arrive at the CCSD method, exploited in a series of papers by Bartlett and co-workers.9-11 We will also consider the linearized version of the CCSD method, LCCSD,15 the full CCSDT model that includes effects of triple excitation clusters T3, and the CCSDTQ model that also considers T. ... 

The question arises whether we need T4 amplitudes in order to generate all fifth-order diagrams. The answer depends on the approach. If we solve the CCSDT equations and obtain converged results, then we have the full fourth-order energy, but not the fifth-order energy, since we miss 168 diagrams, denoted here as Ef1 and EfQ. We do include some other diagrams that are formally of quadruple excitation type that are termed as... 

The analysis of the several different CC approaches in terms of the fifth-order energy contributions points out that within an n6 dependent scheme, i.e., LCCD to CCSD in Table I, the CCSD is much preferred since it accounts for nearly one-third of all terms and avoids potential singularities in LCCD.42 It may also be observed that it pays off to include, even partially, the triple contribution, as was done in the CCSDT-1 method.10 In this model the number of terms is nearly doubled as compared to CCSD, and, of course, this method is correct through the fourth-order energy and the second-order wave function. Also, the connected T contributions are numerically important.11-34... 

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