Antisymmetrized product of strongly orthogonal geminals

The first pair theory was proposed as long ago as 1953 the antisymmetrized product of strongly orthogonal geminals (APSG) of Hurley et a  

There is clearly no extension of (1) aiming at the description of correlation effects among all possible pairs of electrons—or better (spin) orbitals—within a product ansatz for the total wavefunction. As a consequence, pair theories have developed in various directions and were not a really uniform undertaking. Their development was, of course, intimately tied to other techniques of electronic structure calculations, such as the configuration-interaction (Cl) or perturbation theory methods. 

Pair theories gener ly share the following two features  

The treatment of an n-electron system is reduced to (effective) two-electron systems, which may be coupled. (The scope of this fact is elucidated by the properties of two-electron functions—pair functions—which will be discussed below.) 

The expression for the (total, correlation) energy scales properly with the size of the system (size extensivity of the energy). 

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The antisymmetrized product of strongly orthogonal geminals is denoted by the acronym APSG. Properties of this wave function, its construction and use, will be discussed in the forthcoming sections. 

Since our wave function is most conveniently written in the form (69), i.e. as an antisymmetrized product of strongly orthogonal geminals i), the name APSG is now commonly used for this ansatz. 

The separated electron pair concept, which was first proposed by Hurley et al. [14] and which was later referred to as antisymmetrized product of strongly orthogonal geminals (APSG) [15], is also a special case of the group function concept. This kind of wave function is qualitatively correct at all internuclear distances and it can be improved either perturbationally [16, 17] or variationally [18]. 

The perturbation theory has widely been used in quantum chemistry to account for the dynamical electron correlation in single Slater [i.e., Hartree-Fock (HF)] and multideter-minantal states [1]. Suijdn has woiked on the perturbation theories for both HF and non-HF references. For non-HF reference functions, he and his cowoikers proposed a series of multiconfiguration perturbation (MCPT) theories [2-6]. Because the MCPT theories are applicable to any reference functions, they have occasionally been applied to the antisymmetric product of strongly orthogonal geminals wave functions [7-9]. 

The total wave function of a many-electron system can be constructed as an antisymmetrized product of individual geminals [see Eq. (3)]. Dealing with this product is substantially simpler if the geminals are kept orthogonal to each other in the strong sense, i.e. 

We have in fact to compare the energy of an APG function (Antisymmetrized Product of Geminals, without strong orthogonality) with the sum of individual pair energies (without factorization of the Hilbert space). The difference... 

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... 

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