Prediction of Core Level Binding Energies in Density Functional Theory: Rigorous Definition of Initial and Final State Contributions and Implications on the Physical Meaning of Kohn-Sham Energies

A systematic study of the N(1s) core level binding energies (BE's) in a broad series of molecules is presented employing Hartree-Fock (HF) and the B3LYP, PBE0, and LC-BPBE density functional theory (DFT) based methods with a near HF basis set. The results show that all these methods give reasonably accurate BE's with B3LYP being slightly better than HF but with both PBE0 and LCBPBE being poorer than HF. A rigorous and general decomposition of core level binding energy values into initial and final state contributions to the BE's is proposed that can be used within either HF or DFT methods. The results show that Koopmans' theorem does not hold for the Kohn-Sham eigenvalues. Consequently, Kohn-Sham orbital energies of core orbitals do not provide estimates of the initial state contribution to core level BE's; hence, they cannot be used to decompose initial and final state contributions to BE's. However, when the initial state contribution to DFT BE's is properly defined, the decompositions of initial and final state contributions given by DFT, with several different functionals, are very similar to those obtained with HF. Furthermore, it is shown that the differences of Kohn-Sham orbital energies taken with respect to a common reference do follow the trend of the properly calculated initial state contributions. These conclusions are especially important for condensed phase systems where our results validate the use of band structure calculations to determine initial state contributions to BE shifts.