R12 method

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. 

The trick for turning the R12 method into a viable tool is to avoid ealeulating the three- and four-electron integrals, without jeopardizing the aeeuraey. In a complete basis, a three-eleetron integral may be written in terms of produets of two-eleetron... 

Because of the success of the r12 method in the applications, one had almost universally in the literature adopted the idea of the necessity of introducing the interelectronic distances r j explicitly in the total wave function (see, e.g., Coulson 1938). It was there-fore essential for the development that Slater,39 Boys, and some other authors at about 1950 started emphasizing the fact that a wave function of any desired accuracy could be obtained by superposition of configurations, i.e., by summing a series of Slater determinants (Eq. 11.38) built up from a complete basic one-electron set. Numerical applications on atoms and molecules were started by means of the new modern electronic computers, and the results have been very encouraging. It is true that a wave function delivered by the machine may be the sum of a very large number of determinants, but the result may afterwards be mathematically simplified and physically interpreted by means of natural orbitals.22,17... 

Klopper W, Kutzelnigg W, Muller H, Noga J, Vogtner S (1999) Extremal Electron Pairs - Application to Electron Correlation, Especially the R12 Method. 203 21-42 Knochel P, see Betzemeier B (1999) 206 61-78... 

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],...

The MP2-R12 method can be regarded as an externally contracted MP2 method with (contracted) double excitations into a complete spin orbital basis. To illustrate this, we shall in the following discuss an externally contracted MP2 method that comprises contracted double excitations into a subspace (Pp-, which has been orthogonalized against the orbital basis (pp, ... 

However, the introduction of the RI approximation led to the need for large basis sets. In old R12 method, only one single basis was used for both the electronic wave function and the RI approximation. The new formulation of R12 theory presented here uses an independent basis set denoted auxiliary basis set for the RI approximation while we employ a (much) smaller basis set for the MP2 wave function (7). This auxiliary basis set makes it possible to employ standard basis sets in explicitly correlated MP2-R12 calculations. 

To expose the essence of the R12 method of Kutzelnigg [19], consider the simplest two-electron system, the helium atom in its ground state. The exact wave function in the vicinity of an electron-electron coalescence point r can be expressed [13] as... 

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 ... 

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