Extracting molecular properties from a wave function can be performed through the linear response (LR) formalism or, equivalently, the equation of motion (EOM) formalism. For a simple model system, He in a 6-31G basis, it is shown here that calculated excitation energies depend on the specifically chosen orbitals, even when the ground-state is the FCI solution, if the LR is truncated to a singles expansion. This holds for naïve, projected, self-consistent, and state-transfer parameterizations of the LR operators. With a focus on the state-transfer parameterization, this problem is shown to also hold for more complicated systems and is also present when the LR is truncated to singles and doubles. This problem can be alleviated by performing a ground-state constrained trace optimization of the Hessian matrix before performing the LR calculation. It is finally shown that spectra can be further improved for small LR expansions by targeting only a few states in the constrained trace optimization using constrained state-averaged UCC.