In this work we study the effect of electron correlations in molecular transistors with molecular bridges based on 1,4-benzene-dithiol (BDT) and 2-nitro-1,4-benzene-dithiol (nitro-BDT) by using ab initio electron propagator calculations. We find that there is no gate field effect for the BDT based transistor in accordance with the experimental data. After verifying the computational method on the BDT molecule, we consider a transistor with a nitro-BDT molecular bridge. From the electron propagator calculations, we predict strong negative differential resistance at small positive and negative values of source-drain voltages. The explanation of the peak and the minimum in the current is given in terms of the molecular orbital picture and switch-on (-off) properties due to the voltage dependencies of the Dyson poles (ionization potentials). When the current is off, the electronic states on both electrodes are populated resulting in the vanishing tunneling probability due to the Pauli principle. Besides the minimum and the maximum in the I-V characteristics, we find a strong gate field effect in the conductance where the peak at Vsd=0.15eV and Eg=4×10−3a.u. switches to the minimum at Eg=−4×10−3a.u. A similar behavior is discovered at the negative Vsd. Such a feature can be used for fast current modulation by changing the polarity of a gate field.