Theoretical Study of 1s, 2s, and 2p Core Electron Binding Energies of Third-Period Elements Calculated by the ΔSCF Method, Koopmans’ Theorem, and Slater’s Transition State Theory within the Framework of Hartree–Fock and Kohn–Sham Theory

The binding energies (BEs) of the 1s, 2s, and 2p core electrons of third-period elements were calculated using the ΔSCF method, Koopmans' theorem, and Slater's transition state theory (STS), within the frameworks of Hartree-Fock (HF) and Kohn-Sham (KS) theory, employing B3LYP and LC2gau-core-BOP (LC2gau) functionals. The total self-consistent field (SCF) energy difference (ΔSCF) between neutral and cationic states was obtained by solving the SCF equations with fractional occupation numbers. Energy functionals were plotted as a function of continuously varying occupation numbers, and the validity of Koopmans' theorem and STS theory was assessed by analyzing the derivative of the energy functional with respect to occupation number. The HF ΔSCF method reproduces the experimental core electron binding energies (CEBEs) of third-period elements with reasonable accuracy. STS theory within HF closely matches the ΔSCF values, though the errors relative to experimental results are nearly identical to those of the ΔSCF approach. In contrast, the B3LYP ΔSCF method slightly underestimates the experimental BEs of 1s and 2p electrons and significantly underestimates the 2s BEs, primarily due to self-interaction errors (SIEs) inherent in the functional. The LC2gau ΔSCF method effectively eliminates the SIEs for 2s electrons and provides BE values with improved accuracy. Koopmans' theorem holds well in the LC2gau framework, reproducing ΔSCF results with reasonable accuracy. Interestingly, the error between ΔSCF and experimental values is approximately equal in magnitude─but opposite in sign─to the error between Koopmans' theorem and ΔSCF, particularly for 1s and 2s BEs. As a result of this error cancellation, Koopmans' approach with LC2gau predicts CEBEs closer to the experimental values, achieving a high level of accuracy.