  We propose here a single Pfaffian correlated variational ansatz, that
dramatically improves the accuracy with respect to the single determinant one,
while remaining at a similar computational cost. A much larger correlation
energy is indeed determined by the most general two electron pairing function,
including both singlet and triplet channels, combined with a many-body Jastrow
factor, including all possible spin-spin spin-density and density-density
terms. The main technical ingredient to exploit this accuracy is the use of the
Pfaffian for antisymmetrizing an highly correlated pairing function, thus
recovering the Fermi statistics for electrons with an affordable computational
cost. Moreover the application of the Diffusion Monte Carlo, within the fixed
node approximation, allows us to obtain very accurate binding energies for the
first preliminary calculations reported in this study: C$_2$, N$_2$ and O$_2$
and the benzene molecule. This is promising and remarkable, considering that
they represent extremely difficult molecules even for computationally demanding
multi-determinant approaches, and opens therefore the way for realistic and
accurate electronic simulations with an algorithm scaling at most as the fourth
power of the number of electrons.
