  Recent studies of core-level X-ray photoelectron spectroscopy (XPS) spectra
of silicene on ZrB$_2$(0001) were found to be inconsistent with the density of
states (DOS) of a planar-like structure that has been proposed as the ground
state by density functional theory (DFT). To resolve the discrepancy, a
reexamination of the XPS spectra and direct theoretical access of accurate
single-particle excitation energies are desired. By analyzing the XPS data
using symmetric Voigt functions, different binding energies and its sequence of
Si $2p$ orbitals can be assigned from previously reported ones where asymmetric
pseudo-Voigt functions are adopted. Theoretically, we have adopted an approach
developed very recently, which follows the sophisticated $\Delta$
self-consistent field ($\Delta$SCF) methods, to study the single-particle
excitation of core states. In the calculations, each single-particle energy and
the renormalized core-hole charge density are calculated straightforwardly via
two SCF calculations. By comparing the results, the theoretical core-level
absolute binding energies including the splitting due to spin-orbit coupling
are in good agreement with the observed high-resolution XPS spectra. The good
agreement not only resolves the puzzling discrepancy between experiment and
theory (DOS) but also advocates the success of DFT in describing many-body
interactions of electrons at the surface.
