Explicitly correlated geminals

Valeev, E.F., Crawford, T.D. Simple coupled-cluster singles and doubles method with perturbative inclusion of triples and explicitly correlated geminals the CCSD(T)ri2 model. J. Chem. Phys. 2008, 128, 244113. 

The main idea of explicitly correlated CCSD theory is to extend the conventional space of excitations such that pairs of occupied orbitals are replaced by an explicitly correlated geminal function... 

Let us imagine the CH4 molecule. Let us look at it from the viewpoint of localized orbitals. With the method of explicitly correlated geminal functions for bonds we would succeed in making the electrons avoid each other within the same bond. And what should... 

Most involved is the calculation of By, the matrix elements of the Fock operator over explicitly-correlated geminals. Here, we only give a quick overview of the strategy that in the literature has been termed... 

Unfortunately, extending Hylleraas s approach to systems containing three or more electrons leads to very cumbersome mathematics. More practical approaches, known as explicitly correlated methods, are classified into two categories. The first group of approaches uses Boys Gaussian-type geminal (GTG) functions with the explicit dependence on the interelectronic coordinate built into the exponent [95]... 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

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. 

Ten-no, S. Initiation of explicitly correlated Slater-type geminal theory. Chem. Phys. Lett. 2004,398, 56-61. 

A scientific research group, concentrated mainly at Gainesville, Florida, carried out a detailed study on various quantum chemical methods, incorporating explicitly correlated Gaussian geminals into the basis set [75, 76, 77, 78, 79], The research in this direction is still in progress [80, 81, 82, 83, 84]. 

These equations are expressed in the spin-orbital formalism and the products of orbitals are assumed to be antisymmetrized. The coefficients are the explicitly correlated analogues of the conventional amplitudes. The xy indices refer to the space of geminal replacements which is usually spanned by the occupied orbitals. The operator Q12 in Eq. (21) is the strong orthogonality projector and /12 is the correlation factor. In Eq. (18) the /12 correlation factor was chosen as linear ri2 term. It is not necessary to use it in such form. Recent advances in R12 theory have shown that Slater-type correlation factors, referred here as /12, are advantageous. Depending on the choice of the Ansatz of the wave function, the formula for the projector varies, but the detailed discussion of these issues is postponed until Subsection 4.2. The minimization of the Hylleraas functional... 

The ionization potentials and electron affinities of the atoms H, C, N, O and F have been computed by means of coupled-cluster methods using doubly augmented correlation-consistent one-electron basis sets in conjunction with explicitly correlated Slater-type geminals. Excitations up to the level of connected quintuples have been accounted for, and all orbitals in the core and valence shells have been correlated. Relativistic effects (spin-orbit as well as scalar) and diagonal Born-Oppenheimer corrections have been included. 

Valeev EF (2006) Combining explicitly correlated R12 and Gaussian geminal electronic stmcture theories. J Chem Phys 125 244106-1/10... 

Furthermore, it was realized that the even-tempered basis of Gaussian-type geminals could also be applied in an uncontracted form. Some initial calculations were performed on He, and it was observed that the variationally determined linear parameters for the primitive Gaussian geminals were quite similar to the coefficients obtained from the fitting procedure. Results for Ne and H2O obtained from this (uncontracted) ansatz are presented in Table 9. The results show that augmentation of standard basis sets with a few explicitly correlated... 

Until very recently, the use of rii-dependent wave functions has been confined to two-, three-, and four-electron systems. For those systems, it is possible to perform very accurate variational calculations with explicitly correlated wave functions. See, for example, the computations on two-electron systems discussed in Section 2 or the variational calculations in a basis of Gaussian geminals presented in Section 5.2,1. 

Coupled-cluster calculations using explicitly correlated (131) Gaussian geminals were performed by Wenzel et al. in... 

The most troublesome issue about explicitly correlated theories has been left out so far the calculation of the underlying integrals. In fact it is one of the advantages of the conventional orbital expansions that the integrals are so simple Only one- and two-electron integrals are needed. Unfortunately, the introduction of geminal basis functions with ri2 dependence destroys this feature. 

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