where C = X C ′ . The left-hand side of Eq. ( ) is in practice obtained as eigenvalues of Eq. (32). Two transformations, Eqs. ( ) and (34), are associated with the non-invariance of CASPT2-IPEA. As the above algorithm is a standard procedure for solving generalized eigenvalue equations, the equation can be numerically solved. However, the basis of nonquadratic forms cannot be linearly transformed by the above matrix multiplications, so the solution (ε and C) is not obtained with the basis defined rigorously.