  Highly accurate theoretical predictions of transition energies in the radium
monofluoride molecule, $^{226}$RaF and radium cation, $^{226}$Ra$^+$, are
reported. The considered transition $X~^2\Sigma_{1/2} \to A~^2\Pi_{1/2}$ in RaF
is one of the main features of this molecule and can be used to laser cool RaF
for 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. Effects of quantum
electrodynamics (QED) are included non-perturbatively using the model QED
operator that is implemented now for molecules. It is shown that the inclusion
of 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.
