CCSD -R12 method

Owing to its complexity, the CC-R12 method was initially realized in various approximate forms. The first implementation of the CCSD-R12 method including noniterative connected triples [CCSD(T)-R12] was reported by Noga et al. [31,32,57-60] within the SA. The use of the same basis set for the orbital expansion and the RI in the SA rendered many diagrammatic terms to vanish and, thereby, drastically simplified the CCSD-R12 amplitude equations, easing its implementation effort. However, the simplified equations also meant that large basis sets (such as uncontracted quintuple- basis set) were needed to obtain reliable results and, therefore, the SA CCSD-R12 method was useful only in limited circumstances. 

The initial development of the full CCSD-R12 method and its higher-ranked analogues, CCSDT-R12 and CCSDTQ-R12, reported recently by us [33-35], was... 

The initial benchmark results obtained with the full CCSD-R12 method [34] testified that the various simplified CCSD-R12 methods reported earlier were highly accurate approximations to the full CCSD-R12 method unless the basis set was too small. The assumptions about the relative importance of diagrammatic terms made in these simplified methods were proven to be valid. However, these neglected terms do not increase the computational cost scaling of CCSD-R12 and there appears no need to eliminate them from full CCSD-R12, once they are implemented. In other words, it is important to distinguish whether a certain approximation is motivated by a compromise between accuracy and the computational cost or by that between accuracy and the development cost. The latter has become increasingly unjustifiable with the advent of computerized derivation and implementation. 

Table 1.8 CCSD(T) total energies calculated using the R12 method as well as the extrapolation formula (5.14) with the basis sets cc-pCVXZ, compared with the corresponding experimental total energies (Eh). The last row contains the mean absolute deviations from the experimental energies. All calculations have been carried out at the optimized all-electron CCSD(T) /cc-pCVQZ geometries [25],...

Subsequently, Klopper and coworkers developed the CCSD(R12) and CCSD(T)(R12) methods [61-63] in which the use of the SA was avoided, while maintaining the simplicity of the equations. The "(R12)" approximation retains the terms that are at most linear in ff and thus simplifies the amplitude equations considerably. Equations (20)—(22) are, therefore, replaced by [61]... 

This approximation can be justified from a perturbation theory viewpoint that assumes the smallness of ff and is analogous to the treatment of connected triples in CCSDT-1 [64]. The simplification in the equations allowed the CCSD(R12) and CCSD(T)(R12) methods to be implemented by a modest extension of the computational elements developed in the MP2-R12 implementations. Since they do not rely on the SA, they need an auxiliary basis set for the RI, but the rapid basis-set convergence can be obtained. 

An even more radical yet effective approximation to the R12 method was proposed by Ten-no [28,43], in which the coefficients multiplying the correlation function were held fixed at the values implied by the first-order cusp condition and hence were not to be determined iteratively or noniteratively. Several variants of the CCSD(T)-R12 methods were developed on the basis of this promising approximation by Adler et al. [68], Tew et al. [69], Bokhan et al. [70], and Torheyden et al. [66]. 

The algebraic equations and efficient computational sequences were derived by smith and reported by us [33] for CCSD-, CCSDT-, and CCSDTQ-R12, their excited-state analogues via the equation-of-motion (EOM) formalisms (EOM-CC-R12 up to EOM-CCSDTQ-R12), and the so-called A equations for the analytical gradients and response properties, again up to A-CCSDTQ-R12. The full CCSD-, CCSDT-, and CCSDTQ-R12 methods [34,35] were implemented by smith into efficient computer codes that took advantage of spin, spatial, and index-permutation symmetries. 

A remarkable progress in the CC-R12 methods has been made in the last few years. A variety of approximate, but accurate CCSD-R12 and CCSD(T)-R12 methods as well as full CC-R12 methods through and up to CCSDTQ-R12 have... 

The concept of extremal electron pairs is discussed in the context of coupled-cluster theory and the MP2-R12 method. Using extremal pairs the numerical stability of R12-methods is considerably improved, which is demonstrated for CCSD(T)-R12 calculations of the molecules F2, N2, and Be2-... 

The only case documented here, where the orbital-invariant approach diverges is that for the all-electron calculations of N2 at CCSD-R12 and CCSD(T)-R12 levels, for all three basis sets. Even for this example the MP2-R12/A and MP2-R12/B results are numerically stable, and in valence-only calculations even CCSD-R12 and CCSD(T)-R12 converge. Our general experience is that the approaches based on extremal pairs don t suffer from numerical instabilities, at variance with the orbital-invariant method. To avoid divergencies we use standard pairs as default option. 

Truncate this operator to third order with CCSD(T) and it still reproduces an estimated 97% of the correlation description [9]. (It is worth noting that methods exist which explicitly include the interelectronic potential. Recent calculations on the helium atom using Hylleraas-type r12 methods were able to match the exact non-relativistic energy to an astounding 10 12 kcal/mol [10].)... 

A very accurate determination of the interaction-induced polarizability of He2 at the experimental internuclear separation of 5.6 ao was reported by Jaszunski et al The authors used a very large Ils8p6d5f4g3h basis set for He and high-precision explicitly correlated R12 methods. Their most accurate results for the mean and the anisotropy polarizability were calculated at the CCSD(T)-R12 level of theory and are are aint= 0.00104 and Aa = 0.06179 e ao Eh These values represent reference estimates of the interaction-induced dipole polarizability of two helium atoms. 

In the following, the theory of Kutzelnigg s linear R12 functions shall be presented and analyzed in the framework of the coupled-cluster doubles (CCD) method, To illustrate the ideas and approximations employed in the linear R12 methods, it is sufficient to consider the CCD model, as the corresponding CCD-R12 theory exhibits all properties of the R12 theories. It is a relatively simple matter to include singles (CCSD) or even triples (for example in the CCSD(T) method), and CI-R12-type wave functions or MPn-R12 energies require essentially the same computational procedures as the CCD-R12 approach. 

Calculations on selected benchmark atoms and molecules (Be, Ne, LiH, HF, H2O) have revealed the great potential of the coupled-cluster R12 method. The CCSD(T)-R12 calculations on the 10-electron systems represent the most accurate calculations at this level to date. At the second-order Moller-Plesset level, explicitly correlated wave functions have provided accurate and reliable potential energy surfaces for systems such as the HF dimer and the H2O trimer. 

Historically the first approach was the CCSD(F12) method. Originally, the approach was developed for linear R12 theory and did not make use of the SP ansatz (which does not work well for R12 theory). The method was then extended to F12 theory and also combined with fixed amplitudes. It was noted somewhat later, that basically all computationally expensive terms give negligible contributions and a modified method, CCSD(F12 ) was established. ... 

The development of these explicit-rjj methods has yielded a database of benchmark results for small polyatomic molecules. These calculations are listed as MP2-R12 and CCSD(T)-R12 in our tables. We have selected the version called MP2-R12/A as a benchmark reference for our study of the convergence to the MP2 limit. This is the version that Klopper et al. found to agree best with our interference effect. The close agreement with extrapolations of one-electron basis set expansions justifies this choice. 

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