  In this work we perform a first-principles study of the adsorption properties
of an In adatom deposited on 1.75 monolayers (ML) InAs, forming a wetting layer
on GaAs$(001)$ with the $\alpha_2 (2\times4)$ or $\beta_2 (2\times4)$
reconstruction. The structural properties of these reconstructions have been
studied: we determine the equilibrium geometry of the surfaces and their
stability for various growth conditions. We have then carried out a detailed
study of the potential energy surface (PES) for an In adsorbate, finding the
minima and the saddle points. The main characteristics of the PES and the
bonding configurations of the In adatom on the surface are analyzed by
comparing with analogous studies reported in the literature, trying to extract
the effects due to: (i) the compressive strain to which the InAs adlayer is
subjected, (ii) the particular surface reconstruction, and (iii) the wetting
layer composition. We found that, in general, stable adsorption sites are
located at: (i) locations besides the As in-dimers, (ii) positions bridging two
As in-dimers, (iii) between two adjacent ad-dimers (only in $\beta_2$), and
(iv) locations bridging two As ad-dimers. We find also other shallower
adsorption sites which are more reconstruction specific due to the lower
symmetry of the $\alpha_2$ reconstruction compared to the $\beta_2$
reconstruction.
