The R12 methods

The special case of ECG functions, referring to two-electron systems, is called Gaussian-type geminal (GTG). The mathematical form of the GTG basis function can be written 

Expansions based on GTG basis functions [Eq. 17] have been used as Ansatze for the explicitly correlated MP2 and CC wave functions [15]. 

It is worth to mention that each basis function within the ECG expansion [Eq. (14)] correlates all electrons, which yields in total rib 3N + N N +1)/2) nonlinear parameters to be optimized, where ribas is the number of the basis functions in the expansion. Each ECG basis function depends on the coordinates of all electrons, but the resulting 3N-dimensional integrals can be computed in clewed form. The accuracy obtained with ECGs is unprecedented [15], but due to the time-consuming nonlinear optimization the high accuracy could be obtained only for few-electron atoms and molecules. 

The common feature of the explicitly correlated approaches discussed so far is that the whole wave function is expanded in explicitly correlated basis functions. Kutzelnigg and Klopper proposed a different approach, initially at the MP2 level of theory ]16]. The general idea of Kuztelnigg and Klopper was to supplement the conventional Cl expansion with the explicitly correlated part in the following way 

The pair function Eq. (19) contains the conventional part expanded in products of the virtual orbitals 

---

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],...

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. 

In the original formulation of the R12 method [23], which is often referred to as "the standard approximation" or SA, a large uncontracted OBS was needed as the RI basis, strongly limiting the applicability of the R12 method to relatively small systems. Klopper and Samson [26] later pioneered the use of a separate RI basis set and considerably improved the applicability and accuracy of MP2-R12. Valeev developed an alternative formulation of the RI procedure [27] known as the complementary auxiliary basis set (CABS) approach, which had smaller RI errors. Ten-no, on the other hand, proposed the use of a numerical quadrature for the same purpose [28]. Kedzuch et al. later used the CABS approach to develop a particularly elegant and practical formalism for the MP2-R12 energies [30]. 

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]. 

Kirtman B (1999) Local Space Approximation Methods for Correlated Electronic Structure Calculations in Large Delocalized Systems that are Locally Perturbed. 203 147-166 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 BetzemeierB (1999) 206 61-78 Kozhushkov SI, see de Meijere A (1999) 201 1 -42 Kozhushkov SI, see de Meijere A (2000) 207 89-147 Kozhushkov SI, see de Meijere A (2000) 207 149-227... 

---