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 a 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: C2, N2, and O2 and the benzene molecule. This is promising and remarkable, considering that they represent extremely difficult molecules even for computationally demanding multideterminant 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.