Relaxing Constrained Amplitudes: Improved F12 Treatments of Orbital Optimization and Core–Valence Correlation Energies

We present a Lagrangian correction for the energy change upon releasing constraints imposed on coupled cluster amplitudes. We demonstrate that our correction (i) eliminates the systematic basis set error of the posthoc F12 treatment in orbital optimized methods and (ii) can be used to relax the F12 amplitudes and significantly reduce the sensitivity of the core-valence correlation energies to the exponent of the correlation factor, while retaining the low cost of the fixed amplitude approach.