We report here on the two lowest, rigorous-accurate diabatic potential energy surfaces (PES), for the F + H-2 system, as calculated by including the two dominant topological effects of this system at the low energy region, namely, the Jahn Teller effect and the Renner Teller effect. Both effects were treated in the most rigorous way as demanded by the Born Oppenheimer approach. No approximations were made, and in those cases where convergence was required, it was satisfied. In other words, convergence was attained in all situations. The numerical part that includes the calculation of the two lowest ab initio adiabatic PESs and the corresponding nonadiabatic coupling terms (NACTs) was carried out using the MOLPRO program. The required diabatic potentials are calculated by employing these ab initio adiabatic PESs and the corresponding adiabatic-to-diabatic angles as obtained employing the above-mentioned ab initio NACTs. The relevance of these Renner-Teller/Jahn-Teller diabatic potentials is studied by comparing the dressed-lowest ab initio adiabatic PES and the one formed by diagonalizing the dressed-diabatic 2 X 2 potential matrix. The dressed-potentials are calculated employing the vib-rotational manifold derived for each of the three surfaces, namely, the lowest adiabatic potential and the two diabatic ones. This kind of study was recently recommended by Lipoff and Herschbach (Mol. Phys. 2010, 108, 1133) as a "blessed-practice" for the relevance of any PES. In the present case significant differences were revealed between the two types of dressed-adiabatic PESs, eventually, indicating that the lowest, ab initio PES (due to the Born Oppenheimer approximation) is not adequate for low energy processes.