Energy levels and transition probabilities from the Rayleigh-Ritz variation method: C I and O III

•Energy levels and transition parameters for the electric dipole transitions in C I and O III are studied by the Rayleigh-Ritz variation method with moderate-scale multi-configurations wavefunctions.•The present calculated energy levels are in fair agreement with the NIST experimental data.•Transition parameters for 5 electric dipole transitions in C I and O III are new reported, and uncertainties for 19 electric dipole transitions in C I and O III are improved.