When doing DFT calculations, some integrations are commonly done numerically on grids. [In fact, more than a single grid may be used at the same time for different integrals, e.g. approximations such as the resolution of identity (RIJCOSX, RIJK, etc., see e.g. J. Chem. Phys. 118, 9136 (2003)) use grid schemes too (I believe some programs call this approximation density fitting).]
There's the downside that those grid schemes introduce a source of error. In fact, I find that only by increasing the quality of the grid I can remove some imaginary frequencies. Can I rigorously compare energy values among calculations that used slightly different grid schemes?
I think yes and I reasoned as follows. Since I'm interested in energy differences, let's imagine two structures with true energies $E^*_1$ and $E^*_2$. Both energies are approximated by the calculated energies $E_i = E^*_i + \epsilon_i$, where $\epsilon_i$ is the error introduced by calculations (including grids, etc.). Now the energy difference $\Delta E^* = E^*_2 - E^*_1$ is approximated by $\Delta E = E_2 - E_1 = \Delta E^* + \Delta \epsilon$, where $\Delta \epsilon = \epsilon_2 - \epsilon_1$. Since grid schemes introduce errors that are smaller than other sources of error (e.g. implicit solvation), everything should be fine as long as $\Delta \epsilon$ is acceptably small for the particular application at hand.
Does this seem reasonable? If not, what's wrong with the above?