  X-ray photoelectron spectroscopy (XPS) combined with first principles
modeling is a powerful tool for determining the chemical composition and
electronic structure of novel materials. Of these, graphene is an especially
important model system for understanding the properties of other carbon
nanomaterials. Here, we calculate the carbon 1$\textit{s}$ core level binding
energy of pristine graphene using two methods based on density functional
theory total energy differences: a calculation with an explicit core-hole
($\Delta$KS), and a novel all-electron extension of the delta self-consistent
field ($\Delta$SCF) method. We study systematically their convergence and
computational workload, and the dependence of the energies on the chosen
exchange-correlation functional. The $\mathrm{\Delta}$SCF method is
computationally more expensive, but gives consistently higher C 1$\textit{s}$
binding energies. Although there is a significant functional dependence, the
binding energy calculated using the PBE functional is found to be remarkably
close to what has been measured for graphite.
