Coupled cluster CCSDT

Coupled cluster calculations are similar to conhguration interaction calculations in that the wave function is a linear combination of many determinants. However, the means for choosing the determinants in a coupled cluster calculation is more complex than the choice of determinants in a Cl. Like Cl, there are various orders of the CC expansion, called CCSD, CCSDT, and so on. A calculation denoted CCSD(T) is one in which the triple excitations are included perturbatively rather than exactly. 

In Table 1.2, we have listed the valence cc-pVDZ electronic energies and AEs of N2 and HF at different levels of coupled-cluster theory. The energies are given as deviations from the FCI values. Comparing the different levels of theory, we note that the error is reduced by one order of magnitude at each level. In particular, at the CCSDT level, there is a residual error of the order of a few kJ/mol in the calculated energies and AEs, suggesting that the CCSDTQ model is usually needed to reproduce experimental measurements to within the quoted errors bars (often less than 1 kJ/mol). 

Although the calculations reported here have been carried out in a small basis, there is no reason to believe that our conclusions regarding the convergence of the coupled-cluster hierarchy would be different had the calculations been carried out in larger basis. In particular, we conclude that the CCSDT model is incapable of predicting AEs to within 1 kJ/mol. 

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

The CCS, CC2, CCSD, CC3 hierarchy has been designed specially for the calculation of frequency-dependent properties. In this hierarchy, a systematic improvement in the description of the dynamic electron correlation is obtained at each level. For example, comparing CCS, CC2, CCSD, CC3 with FCI singlet and triplet excitation energies showed that the errors decreased by about a factor 3 at each level in the coupled cluster hierarchy [18]. The CC3 error was as small as 0.016 eV and the accuracy of the CC3 excitation energies was comparable to the one of the CCSDT model [18]. 

CCSDT Coupled cluster with single, double, and triple substitution operators... 

Instead of the very demanding CCSDT calculations one often performs CCSD (T) (note the parentheses), in which the contribution of triple excitations is represented in an approximate way (not refined iteratively) this could be called coupled cluster approximate (or perturbative) triples. The quadratic configuration method (QCI) is very similar to the CC method. The most accurate implementation of this in common use is QCISD(T) (quadratic Cl singles, doubles, triples, with triple excitations treated in an approximate, non-iterative way). The CC method, which is usually only moderately slower than QCI (Table 5.6), is apparently better [102]. CCSD(T) calculations are, generally speaking, the current benchmark for practical molecular calculations on molecules of up to moderate size. 

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. 

Watts JD, Bartlett RJ (1996) Iterative and non-iterative triple excitation corrections in coupled-cluster methods for excited electronic states The EOM-CCSDT-3 and EOM-CCSD(r) methods. Chem Phys Lett 258 581-588. 

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, ... 

The exponential ansatz given in Eq. [31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, T, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If T includes contributions from all possible orbital groupings for the N-electron system (that is, T, T2, . , T ), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, T , are frequently referred to as excitation operators, since the determinants they produce from fl>o resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substi-tution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., T = Ti + f 2 CCSD T T + T2 + —> CCSDT, etc., where S, D, and... 

This equation is not restricted to the CCSD approximation, however. Since higher excitation cluster operators such as T3 and T4 cannot produce fully contracted terms with the Hamiltonian, their contribution to the coupled cluster energy expression is zero. Therefore, Eq. [134] also holds for more complicated methods such as CCSDT and CCSDTQ. Higher excitation cluster operators can contribute to the energy indirectly, however, through the equations used to determine the amplitudes, and t-h, which are needed in the energy equation above. 

Every term in the coupled cluster amplitude equations that is nonlinear in T may be factored into linear components. As a result, each step of the iterative solution of the CCSD equations scales at worst as ca. 0(X ) (where X is the number of molecular orbitals). The full CCSDT method in which all Tycon-taining terms are included requires an iterative 0(X ) algorithm, whereas the CCSD(T) method, which is designed to approximate CCSDT, requires a noniterative O(X ) algorithm. The inclusion of all T4 clusters in the CCSDTQ method scales as... 

J. D. Watts and R. J. Bartlett, Chem. Phys. Lett., 258, 581 (1996). Iterative and Noniterative Triple Excitation Corrections in Coupled-Cluster Methods for Excited Electronic States— The EOM-CCSDT-3 and EOM-CCSD(T) Methods. 

The algebraic expressions obtained from a diagrammatic evaluation of the coupled-cluster equations for a CCSDT model are resolvable into products of unmodified cluster coefficients (or trivially modified in the case of t2) and modified one- and two-electron integrals. At no stage of the calculation are tensors of rank greater than 2 required, except for the initial contraction and final expansion of the rank 3 triples cluster coefficients. 

Fig. 3. Diagrammatic representation of the 7j coupled-cluster equation [Eq. (34b)] from CCSDT with directed lines. Henceforth referred to as the D equation.

We may conclude that the recommended CC approximation would be CCSDT, which accounts for 80% of the fifth-order terms, but the basis set dependence is still ns. The inclusion of T4 in the coupled-cluster scheme would appear to cause a significant increase in computational time for what is normally considered to be a fairly small correction since T4< T for most nonmetallic cases. Of course, if a reference function is sufficiently poor, T4 and even higher clusters could be important. However, as was shown in the preceding section, the missing, i.e., beyond CCSDT, fifth-order MBPT energy terms that arise from T may be calculated by supplementing the CCSDT code with a few additional diagrams, the basis set dependence of which is n6 or less, to introduce most of the correction due to T4. 

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