  Pristine single-walled carbon nanotubes (SWCNTs) are rather inert to O$_2$
and N$_2$, which for low doses chemisorb only on defect sites or vacancies of
the SWCNTs at the ppm level. However, very low doping has a major effect on the
electronic properties and conductivity of the SWCNTs. Already at low O$_2$
doses (80 L), the X-ray photoelectron spectroscopy (XPS) O 1s signal becomes
saturated, indicating nearly all the SWCNT's vacancies have been oxidized. As a
result, probing vacancy oxidation on SWCNTs via XPS yields spectra with rather
low signal-to-noise ratios, even for metallicity-sorted SWCNTs. We show that,
even under these conditions, the first principles density functional theory
calculated Kohn-Sham O 1s binding energies may be used to assign the XPS O 1s
spectra for oxidized vacancies on SWCNTs into its individual components. This
allows one to determine the specific functional groups or bonding environments
measured. We find the XPS O 1s signal is mostly due to three O-containing
functional groups on SWCNT vacancies: epoxy (C$_2$$>$O), carbonyl
(C$_2$$>$C$=$O), and ketene (C$=$C$=$O), as ordered by abundance. Upon
oxidation of nearly all the SWCNT's vacancies, the central peak's intensity for
the metallic SWCNT sample is 60\% greater than for the semiconducting SWCNT
sample. This suggests a greater abundance of O-containing defect structures on
the metallic SWCNT sample. For both metallic and semiconducting SWCNTs, we find
O$_2$ does not contribute to the measured XPS O~1s spectra.
