Geminal functional theory formal

Another route to construction of the approximate 1-RDM functional involves employment of expressions for E and D afforded by some size-consistent formalism of electronic structure theory. Mazziotti [42] proposed a geminal functional theory (GET) where an antisymmetric two-particle function (geminal) serves as the fundamental parameter. The one-matrix-geminal relationship allowed him to define a D-based theory from GET [43]. He generalized Levy s constrained search to optimize the universal functionals with respect to 2-RDMs rather than wavefunctions. 

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. 

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. 

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... 

Jeziorski et al, have formulated a first-quantization form of the CCD equations where the pair functions are not expressed in terms of double replacements but as expansions in Gaussian geminals, In the original derivation of the theory, they have employed a spin-adapted formulation in terms of singlet and triplet pairs, but a spin-orbital formalism will be used in the following for the sake of a compact presentation,... 

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