In the present study, we investigate the use of the ΔSCF method and Slater's transition state (STS) theory to calculate the binding energies of the 2s and 2p electrons of third-period elements (P, S, and Cl). Both the Hartree-Fock (HF) and Kohn-Sham (KS) approximations are examined. The STS approximation performs well in reproducing the ΔSCF values. However, for the ΔSCF method itself, while the binding energy of the 2p electrons is accurately predicted, the results for 2s are fairly sensitive to the functional, exhibiting significant variations due to self-interaction errors (SIE). Nonetheless, the variations in chemical shifts between different species remain relatively small, and the values agree with experiments due to the cancellation of SIE. A notable observation is that the chemical shifts of the 2s and 2p electrons are similar, indicating a perturbation caused by the valence electrons. The error in the absolute binding energy of KS ΔSCF against the experiment is nearly constant for the same element in different molecules, and it depends largely on the functional owing to SIE. A shifting scheme previously developed can be employed to reproduce the experimental 2s and 2p binding energies, with dependence on the functional and atom but not on the molecule even for 2s KS ΔSCF binding energies. Upon obtaining the corrected binding energies, we find that the gap between 2s and 2p binding energy is nearly independent of chemical environment for a given element: 57.5, 63.9, and 70.9 eV for the elements P, S, and Cl, respectively.