  The Kondo effect is a many-body phenomenon allowing insight into the
electronic and atomistic structure of spin-polarized adsorbates on metal
surfaces. Its chemical control is intriguing because it deepens such insight,
but the underlying mechanisms are only partly understood. We study the effect
of increasing the number of CO ligands attached to a cobalt adatom on
copper(001), which correlates with an increase in the Kondo temperature $T_K$
experimentally (P. Wahl et al, Phy. Rev. Lett. 95, 166601 (2005)), by solving
an Anderson impurity model parametrized by density functional theory (DFT++).
Our results suggest that the orbital responsible for the Kondo effect is
$d_{x^2-y^2}$ for the tetracarbonyl, and its combination with $d_{z^2}$ for the
dicarbonyl. The molecular structures depend considerably on the approximate
exchange--correlation functional, which may be related to the known difficulty
of describing CO binding to metal surfaces. These structural variations
strongly affect the Kondo properties, which is not only a concern for
predictive studies, but also of interest for detecting mechanical deformations
and for understanding the effect of tip--adsorbate interactions in the scanning
tunneling microscope. Still, by constraining the tetracarbonyl to $C_{4v}$
symmetry, as suggested by experimental data, we find structures compatible with
the experimental trend for $T_K$ (employing BLYP-D3+U). This is not possible
for the tricarbonyl despite the range of computational parameters scanned. For
the tetra- and dicarbonyl, the increased $T_K$ correlates with a larger
hybridization function at the Fermi level, which we trace back to an increased
interaction of the Co $3d$ orbitals with the ligands.
