For ω in the X-ray regime, computing firstorder response wave functions within standard EOM-CC (Eq. ( )) often results in erratic behavior of solvers. 11-13 One reason for this is similar to the case of computing core-level states with standard EOM-CC. Since the response wave functions can be expressed as a linear combination of all states of the system (Eq. ( )) and the contribution of high-lying states to the response states is often dominant due to the near-resonance condition, the coupling of these response states to the valence continuum is significant, which leads to divergences. This coupling can be omitted by using the CVS Hamiltonian Ĥ, which is block diagonal in the core-excited and valence-excited determinant spaces. However, simply using Ĥ instead of Ĥ for computing response wave functions in Eq. ( ) is not sufficient for fixing the erratic behavior of the solvers. The X-ray response wave functions can also have non-negligible contributions (coming from Ĥval ) from high-lying valence resonances, which can also decay via the coupling to the valence doubly excited determinants; this coupling contributes to the erratic behavior of the valence doubly excited determinants in the course of the iterative procedure. In addition, the diagonal preconditioner associated with the doubles-doubles block of Ĥval is no longer a good approximation to it in the Xray energy regime, crippling the convergence. As noted in Refs. 11, 12, and 13, lower-level theories such as TDDFT, CIS (configuration interaction singles), CC2, and ADC(2) either do not have doubly excited determinants in the excitation manifold that are primarily responsible for the erratic behavior of solvers or include them in a perturbative fashion with a diagonal doubles-doubles block (the preconditioner is exact). Therefore, the decay channels are blocked and the problematic convergence is avoided in the response equations.