Highly accurate theoretical predictions of transition energies in the radium monofluoride molecule, 226RaF, and radium cation, 226Ra+, are reported. The considered transition X2ς1/2 → A2Π1/2 in RaF is one of the main features of this molecule and can be used to laser-cool RaF for a subsequent measurement of the electron electric dipole moment. For molecular and atomic predictions, we go beyond the Dirac-Coulomb Hamiltonian and treat high-order electron correlation effects within the coupled cluster theory with the inclusion of quadruple and ever higher amplitudes. The effects of quantum electrodynamics (QED) are included non-perturbatively using the model QED operator that is now implemented for molecules. It is shown that the inclusion of the QED effects in molecular and atomic calculations is a key ingredient in resolving the discrepancy between the theoretical values obtained within the Dirac-Coulomb-Breit Hamiltonian and the experiment. The remaining deviation from the experimental values is within a few meV. This is more than an order of magnitude better than the "chemical accuracy,"1 kcal/mol = 43 meV, that is usually considered as a guiding thread in theoretical molecular physics.