Naturally, the orthogonality between the independent and dependence vectors should be maintained. However, extra care is needed in the case with IPEA shift. The above transformation is numerically straightforward, but this term is sufficient for analytic derivatives only when the basis of B e,γ can be uniquely transformed linearly. This is not the case with IPEA shift. Considering the fact that perturbation energies are dependent on the orthonormalization procedure only when linear dependent vectors exist as mentioned in Section II B 3, it may be hypothesized that dependent vector space is mistakenly exploited because of the non-invariance of CASPT2-IPEA. Therefore, we need an additional constraint condition that makes the independent and dependent vector spaces orthogonal. This is the reason that the additional term is added to the PT2 Lagrangian (the fourth term in Eq. ( )). If there are no linearly dependent vectors, although the dependence on active MOs (discussed in Section II C 4) remains, the orthogonal ICB spans exactly the same space as the nonorthogonal one. Therefore, the derivative contribution in the orthogonal ICB can be exactly expressed in the non-orthogonal ICB, and the following modification is not necessary; after derivation, we can confirm that no contributions arise.