a) Electronic mail: herbert@chemistry.ohio-state.edu Whereas the ∆SCF approach includes orbital relaxation via an independent-particle framework, many-body interactions (screening by the core hole, electron-electron scattering, etc.) are not included explicity. Many-body methods such as coupled-cluster theory do incorporate these effects and can achieve errors as small as 0.2-0.5 eV for core-level ionization, yet these methods are cost-prohibitive except for very small molecules. In addition, the use of a core-hole reference state for the description of dynamical correlation can sometimes lead to singularities because there is a strongly-bound orbital in the virtual space. A popular many-body alternative, especially for periodic solids, is the GW approach that is based on the single-particle Green's function, for which errors of 0.2-0.5 eV for CEBEs (computed as quasiparticle energies within the GW framework) are also typical. As with coupled-cluster theory, these calculations are inherently more expensive than DFT, scaling as O(N 6 ) with respect to system size. This can be reduced to O(N 4 ) with a large prefactor in some recent implementations, but the cost remains much higher than O(N 3 ) DFT calculations and is higher still for GW calculations that target core states. In addition, GW calculations are subject to arbitrary choices that include the choice of representation, leading to apparently unresolvable discrepancies of 0.1-0.2 eV between different implementations. More significantly, "GW " means not just one but a family of methods with various levels of self-consistency. If the GW calculations are not fully self-consistent (as is usually the case) then the XC functional that is used to generate the orbitals must be chosen carefully, and artially self-consistent approach may not afford continuous potential energy surfaces. (Note that analytic gradients are not available for any of these methods. ) Finally, GW calculations of core-ionized states are prone to spurious solutions, such that GW cannot be considered a black-box method in such cases. For all of these reasons, the ∆SCF approach remains the workhorse tool for low-cost calculation of CEBEs.