Antisymmetrized geminal products

B. Perturbation Theory on SOAGP Antisymmetrized Geminal Products Formal GFT... 

The earliest such attempts go back to 1953, when strongly orthogonal antisymmetrized geminal products (SOAGP) were employed [1, 2]. A strongly orthogonal geminal is such that Jgi(l,2)g2(2,3)d2 = 0, while the weaker... 

An alternative approach to improving upon HF wave functions is to include correlations between pairs of electrons more directly by means of two particle geminal functions, G xi Xj). The antisymmetrized geminal power (AGP), Pfaffian and perfect pairing wave functions are all examples of pairing wave functions each can be written as an antisymmetrized product of geminals,... 

Geminal functional theory is a very promising research area. The different varieties of antisymmetrized products are very flexible and inherently handle difficult problems, like multideterminantal molecules. The computational effort is low compared to the quality of the solutions. The perturbation theoretical approach to SSG should essentially be possible for AGP and UAGP as well. The formal definition of GFT is a flexible framework that opens up many new opportunities for exploring the nature of solutions to the Schrodinger equation. 

In this section, we consider a family of semiempirical implementations of the antisymmetrized product of the strictly local geminals (SLG). Quite naturally, this approach applies only to compounds (largely organic) with well localized two-center two-electron bonds. It had been originally developed for an old-fashioned MINDO/3 type of parametrization of the molecular Hamiltonian and then extended to the more contemporary NDDO family of parametrizations. First, the description of the wave function is given in detail and then the energy functional is described and analyzed. Its variation provides the equilibrium values of the electronic structure variables (ESVs) relevant for this method. 

The wave function of electrons in the molecule is then taken as the antisymmetrized product of the geminals given by eqs. (2.60), (2.61) ... 

In this section we have considered a family of semiempirical methods of analysis of the electronic structure of molecules, using the trial wave function in the form of the antisymmetrized product of strictly local geminals. The studies performed on these methods allow us to conclude that ... 

A possible approximation to be used for the cls function can be chosen considering two ideas. In contrast to the directionality and saturability characteristic for organic covalent bonds, those formed by metal ions do not possess these properties. Thus there is no need to invoke the HO formation on the metal ion. At the infinite separation limit, the cls wave function must flow to the antisymmetrized product of the lone pair geminals of eq. (2.61). The latter is in fact a single determinant function with all lone pair HOs doubly filled. With these arguments, we arrive at the conclusion that the single determinant (HFR) wave function is an appropriate form... 

Dmitriev Yu, Peinel G (1981) Coupled perturbation theory within the antisymmetrized product of separated geminals (APSG) framework. Int J Quantum Chem 19 763-769... 

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. 

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. 

Finally, one other non-linear wavefunction expansion will be described. If the geminals of the different electron pairs are further restricted to be identical for each electron pair, then the result is called the antisymmetrized product of identical geminals (APIG) wavefunctionThere are only (n—1) parameters in the APIG wavefunction which spans a subspace of the GP wavefunction space. Because of the severely restrictive form of this wavefunction, it has not been used extensively for MCSCF calculations but it has been used as a reference function for propagator calculationsfor which this wavefunction form has formal appeal. 

The electron-pair bond plays a central role in the qualitative understanding of molecular structure. The pair-function or geminal approach attempts to put this electron-pair concept into a more quantitative form. In this approach the total wavefunction is assumed to be of the form of an antisymmetrized product of pair-functions ... 

In the late 1970s it was found [5,6,8-10] that an antisymmetrized geminai power (AGP) [11] state was the most general 2M-electron state that was annihilated by a maximal set of one-electron operators, (for odd number of electrons generalized AGP (GAGP) states [12] are needed but their properties can be derived from AGP states). The 2M-electron AGP Ig ) state is an antisymmetrized product of M two-particle geminals... 

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. 

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... 

McWeeny proposed a generalization of the usual antisymmetrized product of one-electron wave functions in terms of an antisymmetrized product of many-elec-tron group functions. The extreme elegance of his formalism lies in the fact that it is able to encompass in a natural way the usual molecular orbital theory, the method of geminals, on the one hand, and on the other hand, it opened the way for different methods where chemically identified electron groups are treated separately. 

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]. 

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

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