I’m not sure if I get the question. I’ll try to answer what I think is asked to us.
So in the slides we see both energy diagrams for the toy model, as well as the real quantum problem. I first of all see that the toy model has axial symmetry, and hence we need to have eta = 0 in the quantum version in order to get the same results for both. As also mentioned in the video, we need to flip the sign of V_zz, since only then do the two models agree with each other. Finally, the energy splitting must be the same in both cases, i.e. the seperation between the three energy levels (unperturbed, vs two perturbed ones), but for this, we need to compare the equations predicting the energy shifts, and finetune the parameters to get the same separations.