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Recent studies of core-level X-ray photoelectron spectroscopy (XPS) spectraof silicene on ZrB$_2$(0001) were found to be inconsistent with the density ofstates (DOS) of a planar-like structure that has been proposed as the groundstate by density functional theory (DFT). To resolve the discrepancy, areexamination of the XPS spectra and direct theoretical access of accuratesingle-particle excitation energies are desired. By analyzing the XPS datausing symmetric Voigt functions, different binding energies and its sequence ofSi $2p$ orbitals can be assigned from previously reported ones where asymmetricpseudo-Voigt functions are adopted. Theoretically, we have adopted an approachdeveloped very recently, which follows the sophisticated $\Delta$self-consistent field ($\Delta$SCF) methods, to study the single-particleexcitation of core states. In the calculations, each single-particle energy andthe renormalized core-hole charge density are calculated straightforwardly viatwo SCF calculations. By comparing the results, the theoretical core-levelabsolute binding energies including the splitting due to spin-orbit couplingare in good agreement with the observed high-resolution XPS spectra. The goodagreement not only resolves the puzzling discrepancy between experiment andtheory (DOS) but also advocates the success of DFT in describing many-bodyinteractions of electrons at the surface.

Single-particle excitation of core states in epitaxial silicene