  Proxima b is a rocky exoplanet in the habitable zone of the nearest star
system and a key test case in the search for extraterrestrial life. Here, we
investigate the characterization of a potential Earth-like atmosphere around
Proxima b in reflected light via molecule mapping, combining high resolution
spectroscopy (HRS) and high contrast imaging, using the first-generation
integral field spectrograph HARMONI on the $39$-m Extremely Large Telescope. We
simulate comprehensive observations of Proxima b at an assumed $45^{\circ}$
inclination using HARMONI's High Contrast Adaptive Optics mode, with spatial
resolution $\sim 8$mas ($3.88$mas/spaxel) and spectral resolving power
$R\simeq17,000$ between $1.538$--$1.678 \mu m$, containing the spectral
features of water, carbon dioxide and methane. Tellurics, stellar features, and
additional noise sources are included, and removed using established molecule
mapping techniques. We find that HARMONI's current focal plane mask (FPM) is
too large and obscures the orbit of Proxima b and thus explore smaller and
offset FPMs to yield a detection. A $\rm{S/N}=5$ detection of Proxima b's
reflected light, suitable for atmospheric characterisation, is possible with
such modifications, requiring a minimum of $20$ hours, but ideally at least
$30$ hours of integration time. We highlight that such detections do not scale
with the photon noise, hence suitably detailed simulations of future
instruments for the ELTs are needed to fully understand their ability to
perform HRS observations of exoplanet atmospheres. Alterations to the HARMONI
FPM design are feasible at this stage, but must be considered in context of
other science cases.
