Exgem

Extended Geminal Models

The extended geminal model is a particular way of constructing a full configuration interaction (FCI) wavefunction. Contrary to conventional approaches where the "building block" is the orbital, i.e. an one-electron function, the "building block" in extended geminal models is the geminal, i.e. a two-electron function. For a closed shell 2N-electron system we have the following ansatz for the wavefunction

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In this equation $\Phi ^{APSG}$ denotes an antisymmetric product of strongly orthogonal geminals (APSG), is the APSG approximation of the (2N-2)-electron cluster generated by deleting the geminal of electron pair K in $\Phi ^{APSG}$ , $\Omega _{K}^{[2]}$ is a two-electron correction function, are normalization constants and is the antisymmetrizer. There are analogous definition of the higher order terms in the equation. In the more compact representation, $\Psi _{K}$ represents a single pair correction term ( with respect to $\Phi ^{APSG}$ ), $\Psi _{K_{1}K_{2}}$ a double pair correction term, and so on. The expansion $\Phi ^{EXG}$ is identical to a FCI expansion.

The basic approximation, or root function, $\Phi ^{APSG}$ , has the form

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where $M^{[N]}$ is a normalization constant determined by the equation

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The geminals, i.e. two-electron functions, , are products of an orbital part and a spin function which is singlet coupled, i.e.

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The spatial two-electron geminal can be expressed in terms of natural orbitals (NOs)

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If the number of NOs is greater than 1, i.e. n $_{K}>1$ , then the optimization of $\Phi ^{APSG}$ leads to unique geminals. In most cases, the geminals have a localized character. Hence, in a molecule, the geminals can be associated with familiar chemical concepts like core electrons, bond pairs and lone pairs. If n $_{K}=1$ , the optimization of the wavefunction no longer leads to uniquely defined orbitals. Hovewer, by localizing the orbitals, one can obtain a similar interpretation of the geminals.