In this excitation class, one electron is excited out from the active block. This excitation class does not actually give linear dependence as long as the active space is sensible, as the overlap matrix S G,γ tu is just the one-particle RDM, and its eigenvalue is the natural occupation number. Nevertheless, the origin of the non-invariance is already clear; the third term in Eq. ( ) has a quadratic dependence on both indices t and u, so the term, specific to CASPT2-IPEA, cannot be written in a quadratic form and the basis cannot be uniquely transformed by simply multiplying a transformation matrix. CASPT2-IPEA is therefore non-invariant with respect to rotations in the basis: non-orthogonal ICB and active MOs, specifically. Consequently, first, the perturbation energy depends on the choice of the basis, and second, even if the derivative contribution of the IPEA shift term (the third term in Eq. ( )) is evaluated, derivatives computed using algorithms that rely on the invariance with respect to these rotations can have a relatively large error of up to 10 -4 hartrees/bohr even for a simple molecule; see "Analytic0" in Table . Necessary modifications to formulate analytic derivatives are discussed in Sections II C 3 and II C 4.