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We study energetic, magnetic, and electronic properties of dilutedsubstitutional Mn-pairs on the reconstructed $(001)$ GaAs surfaces. The studiesare based on first-principles calculations in the framework of the densityfunctional theory. We demonstrate that the stability of the systems stronglydepends on the position, orientation, and the distance between the Mn-atomsconstituting the pair. Independently of the considered surface reconstructionpattern, the Mn-pairs with Mn-atoms being the nearest neighbors (NN) oncationic sublattice turn out to be energetically more favorable than the pairswith the larger distance between the Mn-atoms. However, the preferentialbuild-up orientation of the Mn-NN-pair depends on the surface reconstructionand is parallel either to $[110]$ or $[1\bar{1}0]$ crystallographic direction.We reveal also the mechanisms of the magnetic ordering of Mn-NN-pairs. TheMn-NN-pairs along the $[110]$ crystallographic direction exhibit alwaysferromagnetic alignment of Mn spins, whereas the spins in the Mn-NN-pairs along$[1\bar{1}0]$ direction are mostly anti-ferromagnetically aligned. In theelectronic structure of the systems containing Mn-pairs with ferromagneticallyaligned spins, we observe the valence band hole states in the neighborhood ofFermi energy. This indicates that the surface ferromagnetism in this prototypeof dilute magnetic semiconductors can be explained in terms of the $p$-$d$Zener model.

Energetic, electronic and magnetic properties of Mn-dimers on reconstructed (001) GaAs surfaces