A connection between GCM and DFT

We initiate a preliminary formulation of this algorithm with a self-consistent field (SCF) HF calculation for the He atom. We first obtain the best HF calculation and then replace the exchange term by Slater s Xot potential [34] Vxa, the simpler expression for DFT, but sufficient for this experiment. We interpret a as the generator coordinate and we weight the exchange-correlation term for different values of the parameter a, i.e. 

With the one-electron functions (pi so generated we constmct the determinants 

We then perform a Cl calculation leading to the wave function 

On tbe other hand. Fig. 4c shows the shape of the weight function for the excited state, pointing at optimal a values close to 0.7, although in this case also there is a secondary peak for low values of a. 

To return to the discussion of Slater in 1972 [34], it seems that, as our results suggest, a = 1.0 would be best for the ground state, while a = 0.7 would be indicated for the first excited state with the same symmetry. Perhaps GCM could be an alternative manner to study the exchange-correlation term in DFT. 

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