Linear R12 methods

This approximation inherent in the CI-RI2 method is important in view of the related methods for many-electron systems that will be discussed in Section 5.3 (these methods are denoted as linear RI2 methods). The approximation becomes exact in the limit where the corresponding one-electron problem (e.g., H2 or H3 ) is solved exactly. This is the fundamental idea on which the approximations in the linear R12 methods are based. It is much easier to converge to near-basis set completeness in one-electron than in two-electron space. It is therefore understood that atomic basis sets of near-Hartree-Fock limit quality are used as a starting point for linear R12 calculations. 

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. 

In order to achieve a high aceuraey, it would seem desirable to explicitly include terms in the wave functions which are linear in the intereleetronie distanee. This is the idea in the R12 methods developed by Kutzelnigg and co-workers. The first order correction to the HF wave funetion only involves doubly exeited determinants (eqs. (4.35) and (4.37)). In R12 methods additional terms are included which essentially are the HF determinant multiplied with faetors. 

With an appropriate /(r12) function, e.g., in the original linear form f(r-[2) — C12, the operator product r firu) is no longer singular. Such cancellation is not possible with Slater determinants alone and this is what allows explicitly correlated wave functions to achieve accurate correlation energies with relatively small basis sets. With the single explicitly correlated term, therefore, we effectively include a linear combination of an infinite set of Slater determinants, but without the need to solve an infinite set of equations to determine the corresponding amplitudes. The R12 method constructs wave functions that are more compact and computationally tractable than naive Slater-determinant-based counterparts. 

It is instructive to discuss the MP2-R12 method [37] before going into more involved CC-R12. As in MP2, the wave function of MP2-R12 (IT1)) is a linear combination of the reference HF determinant ( o)) and doubly excited determinants produced by the action of a two-electron excitation operator T ... 

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]... 

Valeev, E.F., Janssen, C.L. Second-order Moller-Plesset theory with linear R12 terms (MP2-R12) revisited auxiliary basis set method and massively parallel implementation. J. Chem. Phys. 2004, 121, 1214-27. 

Second-order Moller-Plesset theory with linear R12 terms (MP2-R12) revisited Auxiliary basis set method and massively parallel implementation101... 

For completeness, it is noted that a detailed derivation of the coupled-cluster theory with linear R12 terms has been given by Noga and Kutzelnigg, and that Gdanitz has presented a compilation of the formulas required for the implementation of multireference CI-R12 methods. The MP2-R12 method in its early form has also been investigated by Bearpark et al, ... 

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. ... 

Explicitly correlated Gaussians (ECG) methods have already been developed earlier 10,11,12) and have been used for accurate calculations on small molecules. The main difference between these ECG methods and the use of Gaussian geminals in the framework of R12 theory is that in the latter, their purpose is to dampen the linear r term more than being a correlation factor on its own. 

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