  We report density functional theory plus exact diagonalization of the
multi-orbital Anderson impurity model calculations for the Co adatom on the top
of Cu(001) surface. For the Co atom $d$-shell occupation $n_d \approx$ 8, a
singlet many-body ground state and Kondo resonance are found, when the
spin-orbit coupling is included in the calculations. The differential
conductance is evaluated in a good agreement with the scanning tuneling
microscopy measurements. The results illustrate the essential role which the
spin-orbit coupling is playing in a formation of Kondo singlet for the
multi-orbital impurity in low dimensions.
