Detour Explicitly Correlated Wave Functions

The two-electron interaction operators g i,j) in the many-electron Hamiltonian are the reason why a product ansatz for the electronic wave function Yg/( r, ) that separates the coordinates of the electrons is not the proper choice if the exact function is to be obtained in a single product of one-electron functions. Instead, the electronic coordinates are coupled and the motion of electrons is correlated. It is therefore natural to assume that a suitable ansatz for the many-electron wave function requires functions that depend on two coordinates. Work along these lines within the clamped-nuclei approximation has a long history [320-328]. The problem, however, is then what functional form to choose for these functions. First attempts in molecular quantum mechanics used simply terms that are linear in the interelectronic distance Tij = ki — Tjl [329-333], while it could be shown that an exponential ansatz is more efficient [334-337]. 

It turns out that the functional form of a Gaussian distribution is very suitable when it comes to the evaluation of many-electron integrals (compare chapter 10). Therefore, a Gaussian geminal. 

The geminal ansatz still requires more effort than the standard one-electron approach of the independent particle model. It is therefore usually restricted to small molecules for feasibility reasons. As an example how the nonlinear optimization problem can be handled we refer to the stochastic variational approach [340]. However, the geminal ansatz as presented above has the useful feature that all elementary particles can be treated on the same footing. This means that we can actually use such an ansatz for total wave functions without employing the Born-Oppenheimer approximation, which exploits the fact that nuclei are much heavier than electrons. Hence, electrons and nuclei can be treated on the same footing [340-342] and even mixed approaches are possible, where protons and electrons are treated in the external field of heavier nuclei [343-346]. The integrals required for the matrix elements are hardly more complicated than those over one-electron Gaussians [338,339,347]. 

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